Lévy driven finance theory (equity, interest rate theory, credit, reserve capital)

Introduction
Lévy processes
Lévy Driven Financial Models
Financial
modeling
Valuation
Risk Managem.
Ernst Eberlein
Interest rate
Department of Mathematical Stochastics
and
Freiburg Center for Data Analysis and Modeling
University of Freiburg
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Méthodes Statistiques et Applications en Actuariat et Finance
Université Cadi Ayyad, Marrakech
April 8–13, 2013
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 1
Deutsche Bank
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Introduction
Lévy processes
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Financial
modeling
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Valuation
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Calibration
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Interest rate
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Lévy LIBOR
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Risk Managem.
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Cross-currency
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Range options
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Duality theory
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Capital
Requirements
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Spread Option
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Quantiles of Standard Normal
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 2
QQ plots for Deutsche Bank
Introduction
Lévy processes
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Financial
modeling
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Valuation
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Deutsche Bank
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Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
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Net Asset Value
-2
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2
4
Quantiles of Standard Normal
Reserve Capital
References
c Eberlein, Uni Freiburg, 3
zero-bond log-returns (1985-95), 5 years to maturity
empirical densities calculated from zero-yield data for Germany
empirical
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normal
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Introduction
Lévy processes
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300
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density( x )
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Financial
modeling
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Valuation
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Risk Managem.
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Interest rate
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Calibration
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Lévy LIBOR
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Cross-currency
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0.0
log return x of zero-bond
0.005
0.010
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 4
6
4
2
log( density( x ) )
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0
Introduction
empirical
normal
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Lévy processes
Financial
modeling
Valuation
Risk Managem.
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Calibration
Lévy LIBOR
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Cross-currency
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Interest rate
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Range options
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Credit risk
-6
Duality theory
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Capital
Requirements
Spread Option
-0.010
-0.005
0.0
log return x of zero-bond
0.005
0.010
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 5
6
4
log( density( x ) )
0
2
Introduction
empirical
NIG
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Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
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-4
•
-0.010
-0.005
0.0
log return x of zero-bond
0.005
Range options
•
Duality theory
•
Capital
Requirements
•
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Cross-currency
Credit risk
•
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Lévy LIBOR
•
Spread Option
0.010
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 6
Lévy processes
Introduction
Lévy processes
L = (Lt )t≥0
process with stationary and independent increments
Financial
modeling
Valuation
on a probability space (Ω, F, (Ft )t≥0 , P)
Risk Managem.
càdlàg paths: right-continuous with left limits
Interest rate
Calibration
canonical representation
Lt = bt +
Lévy LIBOR
√
cWt + Zt +
X
Cross-currency
∆Ls 1{|∆Ls |>1}
s≤t
b and c ≥ 0 real numbers,
(Zt )t≥0
Range options
Credit risk
(Wt )t≥0 standard Brownian motion
purely discontinuous martingale independent of (Wt )t≥0
Duality theory
Capital
Requirements
Spread Option
∆Ls = Ls − Ls−
jump at time s > 0
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 7
Semimartingale representation
Introduction
Lévy processes
(Xt )t≥0 semimartingale with X0 = 0
Xt −
X
∆Xs 1{|∆Xs |>1}
Financial
modeling
process with bounded jumps
s≤t
Valuation
Risk Managem.
→ special semimartingale: unique decomposition into a local
martingale and a predictable process with finite variation
X
Xt = Vt + Mt +
∆Xs 1{|∆Xs |>1}
s≤t
Mt = Mtc + Mtd
M c continuous, M d purely discontinuous
For Lévy processes:
(Zt )t≥0 = (Mtd )t≥0
√
Vt = bt and M c = cWt
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
the purely discontinuous local martingale
(of a Lévy process)
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 8
The purely discontinuous martingale
X
Introduction
∆Xs 1{|∆Xs |≤1}
does not converge in general
X
Z
→ compensating
Lévy processes
s≤t
P − lim
ε↓0
∆Xs 1{ε≤|∆Xs |≤1} − t
x1{ε≤|x|≤1} F (dx)
Financial
modeling
Valuation
s≤t
Risk Managem.
What is F ?
Interest rate
µX (ω; dt, dx) =
X
1{∆Xs (ω)6=0} E(s,∆Xs (ω)) (dt, dx)
Calibration
s>0
Lévy LIBOR
random measure of jumps of X = (Xt )t≥0
µX (ω; [0, t] × A)
Cross-currency
counts how many jumps of size within A occur for
path ω from 0 to t
Range options
Credit risk
Duality theory
E µX (·; [0, t] × A) = tF (A),
Mtd =
Z tZ
0
F intensity measure or Lévy measure
Capital
Requirements
Spread Option
x1{|x|≤1} µX (ds, dx) − dsF (dx)
Net Asset Value
R
Reserve Capital
cannot be separated in general
References
c Eberlein, Uni Freiburg, 9
Local characteristics of a Lévy process (1)
Introduction
Lévy processes
Financial
modeling
Triplet of local characteristics of (Lt )t≥0 : (b, c, F )
Valuation
√
Lt = bt + cWt +
Z tZ
0
Z tZ
+
0
x1{|x|≤1} µL (ds, dx) − dsF (dx)
Risk Managem.
Interest rate
R
Calibration
x1{|x|>1} µL (ds, dx)
Lévy LIBOR
R
Cross-currency
Range options
ν = L(L1 ) is infinitely divisible
Credit risk
Duality theory
L(L1 ) = L(L1/n ) ∗ · · · ∗ L(L1/n )
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 10
Local characteristics of a Lévy process (2)
Introduction
Lévy processes
Fourier transform in Lévy –Khintchine form
Z 1
E[exp(iuL1 )] = exp iub − u 2 c +
eiux − 1 − iux1{|x|≤1} F (dx)
2
R
Financial
modeling
Valuation
Risk Managem.
Interest rate
= exp(ψ(u))
Calibration
Z
Lévy LIBOR
min(1, x 2 )F (dx) < ∞
F Lévy measure:
Cross-currency
R
Range options
Credit risk
pricing of derivatives: E[f (LT )]
uses
Duality theory
E[exp(iuLT )] = exp(ψ(u))T
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 11
Examples of Lévy measures
Introduction
Lévy processes
Financial
modeling
Valuation
5
5
4
4
3
3
Calibration
2
2
Lévy LIBOR
1
1
Risk Managem.
Interest rate
Cross-currency
Range options
Credit risk
The density of the Lévy measure of the normal inverse Gaussian (left)
and the α-stable process.
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 12
Integrability properties of the
Lévy measure
Introduction
Lévy processes
Financial
modeling
Finiteness of the moments of the process depends on the frequency of
the large jumps
Valuation
Risk Managem.
Interest rate
Calibration
Proposition
Lévy LIBOR
Let L be a Lévy process with triplet (b, c, F ).
Z
1 E[|Lt |p ] < ∞ for p ∈ R+ if and only if
Cross-currency
p
|x| F (dx) < ∞.
{|x|>1}
Range options
Credit risk
Duality theory
2 E[exp(pLt )] < ∞ for p ∈ R if and only if
Capital
Requirements
Z
exp(px)F (dx) < ∞.
Spread Option
{|x|>1}
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 13
Consequences for the representation
Introduction
If L(L1 ) has a finite expectation then
→ add −
R
R
xF (dx) < ∞
{|x|>1}
iux1{|x|>1} F (dx) to the characteristic exponent
Lévy processes
Financial
modeling
Valuation
Z
1
E[exp(iuL1 )] = exp iub − u 2 c +
eiux − 1 − iux F (dx)
2
R
(different b)
Interest rate
Calibration
Lévy LIBOR
Z tZ
x1{|x|>1} µL (ds, dx) − dsF (dx) can be
In the same way
0
Risk Managem.
R
Cross-currency
Range options
added to (Mtd )t≥0 , consequently
Credit risk
Lt = bt +
√
Z tZ
x µL (ds, dx) − dsF (dx)
cWt +
0
Duality theory
Capital
Requirements
R
Spread Option
Martingale if b = E[X1 ] = 0
Net Asset Value
Submartingale if b > 0, supermartingale if b < 0
Reserve Capital
References
c Eberlein, Uni Freiburg, 14
Example (Poisson process)
Lévy measure
F = λE1 ,
No Gaussian component:
Introduction
λ intensity parameter
Lévy processes
Financial
modeling
c=0
Valuation
→ jumps of size 1 occur with average rate λ per unit time
Risk Managem.
Interest rate
Fourier transform
Calibration
E[exp(iuLt )] = exp[λt(eiu − 1)]
Lévy LIBOR
Cross-currency
Range options
Canonical representation
Credit risk
Lt = λt + (Lt − λt)
X
= λt +
1{Tn ≤t} − λt
Duality theory
Capital
Requirements
n≥1
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 15
Generalized hyperbolic distributions
(O.E. Barndorff-Nielsen (1977))
Introduction
Lévy processes
Density:
2
dGH (x) = a(λ, α, β, δ) δ + (x − µ)
2
(λ−1/2)/2
p
×Kλ−1/2 a δ 2 + (x − µ)2 exp(β(x − µ))
Financial
modeling
Valuation
Risk Managem.
Interest rate
λ/2
α2 − β 2
p
a(λ, α, β, δ) = √
2παλ−1/2 δ λ Kλ δ α2 − β 2
Calibration
Lévy LIBOR
Cross-currency
Kλ modified Bessel function of the third kind with index λ
Range options
Credit risk
Parameters:
Duality theory
λ∈R
Class parameter
µ∈R
Location
Capital
Requirements
α>0
Shape
δ>0
Scale parameter
Spread Option
β with 0 ≤ |β| < α
Skewness
Net Asset Value
(Volatility)
Reserve Capital
References
c Eberlein, Uni Freiburg, 16
0.5
0.6
Introduction
Normal
NIG (0.02,0,0.5,0)
NIG (6,-5.5,1,1)
NIG (7,6,1,-1)
Lévy processes
Financial
modeling
Valuation
0.4
Risk Managem.
Interest rate
y
0.3
Calibration
Lévy LIBOR
0.2
Cross-currency
Range options
0.1
Credit risk
Duality theory
0.0
Capital
Requirements
Spread Option
-4
-2
0
x
2
4
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 17
0
Introduction
Lévy processes
-2
Financial
modeling
-4
Valuation
Risk Managem.
-6
Interest rate
y
Calibration
-8
Lévy LIBOR
Cross-currency
-10
Range options
Credit risk
-12
Duality theory
Capital
Requirements
Spread Option
-4
-2
0
x
2
4
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 18
Special cases
Introduction
Lévy processes
Hyperbolic
Financial
modeling
Normal inverse Gaussian (NIG)
Valuation
Risk Managem.
Normal reciprocal inverse Gaussian (NRIG)
Interest rate
Variance gamma
Calibration
Lévy LIBOR
Student t (limiting case)
Cross-currency
Range options
Cauchy (limiting case)
Credit risk
Skewed Laplace
Duality theory
Normal (limiting case)
Capital
Requirements
Spread Option
Generalized inverse Gaussian (limiting case)
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 19
Hyperbolic distribution (λ = 1)
Introduction
p
p
α2 − β 2
exp −α δ 2 + (x − µ)2 + β(x − µ)
p
dH (x) =
2αδK1 δ α2 − β 2
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Eb., Keller (1995); Eb., Keller, Prause (1998)
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Normal inverse Gaussian (NIG) (λ = −1/2)
Range options
p
K (αg (x − µ))
αδ
δ
1
dNIG (x) =
exp δ α2 − β 2 + β(x − µ)
π
gδ (x − µ)
√
where gδ (x) = δ 2 + x 2
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
O.E. Barndorff-Nielsen (1998)
Reserve Capital
References
c Eberlein, Uni Freiburg, 20
0.6
Lévy processes
Financial
modeling
Valuation
Risk Managem.
0.4
0.5
Introduction
Normal
NIG (0.02,0,0.5,0)
NIG (6,-5.5,1,1)
NIG (7,6,1,-1)
Interest rate
y
0.3
Calibration
Lévy LIBOR
0.2
Cross-currency
Range options
0.1
Credit risk
Duality theory
0.0
Capital
Requirements
-4
-2
0
x
2
4
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 21
0
Introduction
-2
Lévy processes
Financial
modeling
-4
Valuation
Risk Managem.
-6
Interest rate
y
Calibration
-8
Lévy LIBOR
-10
Range options
-12
Cross-currency
Duality theory
Credit risk
Capital
Requirements
-4
-2
0
x
2
4
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 22
Introduction
Credit profit and loss distribution
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
0.0
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 23
Activity and variation
Introduction
Proposition
Lévy processes
Let L be a Lévy process with triplet (b, c, F ).
Financial
modeling
1 If F (R) < ∞ then almost all paths of L have a finite number of
jumps on every compact interval. In that case, the Lévy process
has finite activity.
2 If F (R) = ∞ then almost all paths of L have an infinite number of
jumps on every compact interval. In that case, the Lévy process
has infinite activity.
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Proposition
Duality theory
Let L be a Lévy process with triplet (b, c, F ).
R
1 If c = 0 and |x|≤1 |x|F (dx) < ∞ then almost all paths of L have
finite variation.
R
2 If c 6= 0 or |x|≤1 |x|F (dx) = ∞ then almost all paths of L have
infinite variation.
c Eberlein, Uni Freiburg, 24
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
GH Levy process with marginal densities
100.6
Introduction
Lévy processes
100.4
Valuation
Risk Managem.
Interest rate
100.2
Calibration
Lévy LIBOR
Cross-currency
100.0
Range options
Credit risk
Duality theory
Capital
Requirements
99.8
values of GH (-0.5,100,0,1,0.1) Levy process
Financial
modeling
Spread Option
0.0
0.5
1.0
1.5
2.0
Net Asset Value
t
Reserve Capital
References
c Eberlein, Uni Freiburg, 25
Fine structure of the paths
Introduction
Lévy processes
Z
1 F (R) < ∞ ⇐⇒
F (dx) < ∞
⇒ finite activity
F (dx) = ∞
⇒ infinite activity
{|x|≤1}
2 F (R) = ∞ ⇐⇒
{|x|≤1}
|x|F (dx) < ∞ (and c = 0)
3
⇒ finite variation
{|x|≤1}
the sum of the small jumps converges and
L
R
Lévy LIBOR
Cross-currency
Range options
Credit risk
Z tZ
L
x µ (ds, dx) − dsF (dx) =
0
Interest rate
Calibration
Z
Z tZ
Valuation
Risk Managem.
Z
→
Financial
modeling
Z
xµ (ds, dx) − t
0
R
xF (dx)
R
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 26
Financial modeling
Introduction
Lévy processes
Stock prices and indices: geometric Brownian motion (Samuelson 1965)
Financial
modeling
Valuation
dSt = µSt dt + σSt dWt
Risk Managem.
Interest rate
solution
Calibration
St = S0 exp σWt + µ −
2
σ
t
2
Lévy LIBOR
Cross-currency
Range options
Log returns:
log St+1 − log St ∼ N µ −
σ2
, σ2
2
Credit risk
Duality theory
Correct return distributions: key ingredient
Capital
Requirements
Consistency of the model along different time grids
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 27
Deutsche Bank
0.6
Introduction
Lévy processes
••
Financial
modeling
•
0.5
•
•
Valuation
•
Calibration
•
0.4
•
0.3
Interest rate
•
•
Lévy LIBOR
•
•
0.2
Risk Managem.
•
•
Cross-currency
•
•
•
Range options
•
0.1
•
•••••••••••••
-4
•••••••••••••
••
-2
••••
Credit risk
•••
•
0.0
densities
•
•
•
Duality theory
•
••
•
••••
••
••••••••
Capital
Requirements
••••••••••••••••••••••••
Spread Option
0
2
4
Quantiles of Standard Normal
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 28
zero-bond log-returns (1985-95), 5 years to maturity
empirical densities calculated from zero-yield data for Germany
empirical
••
• •
normal
• •
NIG
300
•
•
density( x )
200
100
Lévy processes
•
Financial
modeling
•
•
•
0
Introduction
•
•
•
•
•
•
•
•
•
•
•
•
•
••
••
•
•
•
•
••
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•
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••••••••••••••••••••••••••••••••••••••••••••
-0.010
-0.005
Valuation
•
•
•
•
Risk Managem.
•
•
•
•
•
•
•
•
•
Interest rate
•
Calibration
•
•
Lévy LIBOR
•
•
Cross-currency
•
•
•
•
•
•
•
••
••
••
••
•
•
•
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•••••••••••••••••••••
0.0
log return x of zero-bond
0.005
0.010
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 29
6
4
2
log( density( x ) )
-2
0
Introduction
empirical
normal
••••••••••
•••
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•
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•
•
•
•
•
•
•
•
Lévy processes
Financial
modeling
Valuation
Risk Managem.
•
••
••
•
•
•
•
•
•
Calibration
Lévy LIBOR
•
•
Cross-currency
•
•
•
-4
Interest rate
•
•
Range options
•
Credit risk
-6
Duality theory
-8
Capital
Requirements
Spread Option
-0.010
-0.005
0.0
log return x of zero-bond
0.005
0.010
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 30
6
4
log( density( x ) )
0
2
Introduction
empirical
NIG
•••••••••
••
•••
••
•••
••
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•
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•
••
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
••
••
•
•
•
•
•
•
•
-2
•
-4
•
-0.010
-0.005
0.0
log return x of zero-bond
0.005
Range options
•
Duality theory
•
Capital
Requirements
•
•
Cross-currency
Credit risk
•
•
Lévy LIBOR
•
Spread Option
0.010
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 31
Introduction
Lévy processes
6
Financial
modeling
Valuation
Risk Managem.
Interest rate
y
4
Calibration
Lévy LIBOR
Cross-currency
2
Range options
Credit risk
Duality theory
0
Capital
Requirements
Spread Option
-1
0
1
2
x
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 32
Exponential Lévy model
Introduction
Lévy processes
Financial
modeling
St = S0 exp(Lt )
Valuation
L = (Lt )t≥0
Lévy process with L(L1 ) = ν
Risk Managem.
Interest rate
Along a time grid with span 1: exact log returns
Calibration
Alternative description by a stochastic differential equation
Z
c
dSt = St− dLt + dt + (ex − 1 − x)µL (dt, dx)
2
R
Lévy LIBOR
can be written as
Duality theory
dSt = St− dLet
Cross-currency
Range options
Credit risk
Capital
Requirements
Spread Option
where (e
Lt )t≥0 is a Lévy process with jumps > −1
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 33
GH Levy process with marginal densities
100.6
Introduction
Lévy processes
100.4
Valuation
Risk Managem.
Interest rate
100.2
Calibration
Lévy LIBOR
Cross-currency
100.0
Range options
Credit risk
Duality theory
Capital
Requirements
99.8
values of GH (-0.5,100,0,1,0.1) Levy process
Financial
modeling
Spread Option
0.0
0.5
1.0
1.5
2.0
Net Asset Value
t
Reserve Capital
References
c Eberlein, Uni Freiburg, 34
100
Introduction
Lévy processes
98
Financial
modeling
•
•
• ••••• •
• •
•• •• ••
•
•
•
• • •
•• •
•
• •
•
stock price [DM]
94
96
•
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Valuation
•
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Risk Managem.
•
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•
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•
• • •• •
• •
• •
•• ••
••
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
92
Credit risk
Duality theory
90
Capital
Requirements
Spread Option
10
12
14
16
time [hours]
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 35
Consistency along different time grids
Introduction
Lévy processes
Models are typically fitted (calibrated) on the basis of daily data (e.g.
daily closing prices)
Does this model describe the price movements for an intraday or weekly
horizon?
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Classical Gaussian model:
Log-returns are always normally distributed (selfsimilarity of Brownian
motion)
Cross-currency
Range options
Credit risk
Duality theory
GH model:
Empirical investigation shows that the model provides rather good
Capital
Requirements
Spread Option
distributions along other time grids as well
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 36
Daily vs. one hour returns of Bayer data (Jan. 1992 - Aug. 1994)
Introduction
120
•
100
empirical densities
40
60
80
20
0
Lévy processes
one hour returns
• • and fitted hyperbolic density
Financial
modeling
Valuation
•
Risk Managem.
•
Interest rate
•
Calibration
•
Lévy LIBOR
•
Cross-currency
•
••
•
••
•
••
• ••
•
• • daily returns
• •
• ••
•
• • ••
••
••
••• ••• • • • • •
•••
•
••• ••••••••• • • • • • • • • • • • • • •
• • • • • • • • • • • • ••••••••••
-0.04
-0.02
0.0
0.02
0.04
0.06
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 37
Daily vs. one hour returns of Bayer data (Jan. 1992 - Aug. 1994)
Introduction
120
•
••
100
empirical densities
40
60
80
20
0
Lévy processes
one hour returns
Financial
modeling
Valuation
•
Risk Managem.
•
Interest rate
•
Calibration
•
Lévy LIBOR
•
Cross-currency
•
••
•
••
•
••
• ••
•
• • daily returns
• •
• ••
•
• • ••
••
••
••• ••• • • • • •
•••
••• ••••••••• • • • • • • • • • • • • • •
• • • • • • • • • • • • • ••••••••••
-0.04
-0.02
0.0
0.02
0.04
0.06
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 38
Martingality of exponential Lévy models
Introduction
Lévy processes
St = S0 exp(Lt )
Pricing of derivatives:
Financial
modeling
Valuation
martingale model
Risk Managem.
Necessary assumption:
E[St ] = S0 E[exp(Lt )] < ∞
Interest rate
Calibration
This excludes a priori the class of stable processes in general
Lévy LIBOR
E[exp(Lt )] < ∞
⇒
E[Lt ] < ∞
consequently
√
Lt = bt + cWt +
Cross-currency
Range options
Z tZ
x µ (ds, dx) − dsF (dx)
0
St = S0 exp(Lt ) is a martingale if
L
Mt =
Credit risk
Duality theory
R
c
b=− −
2
Use either Itô’s formula or verify that
Z
ex − 1 − x F (dx)
R
Capital
Requirements
Spread Option
exp(Lt )
is a martingale
E[exp(Lt )]
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 39
Equivalent martingale measures (EMMs)
Introduction
e−rt St )t≥0 has to be a martingale
Lévy processes
Financial
modeling
In general large set of EMMs: market is incomplete
Valuation
Characterization of the set of all EMMs
(Eberlein and Jacod (1997))
Characterization of those EMMs under which L is again a Lévy process
The range of call option prices under all EMMs spans the whole
no-arbitrage interval (Eberlein and Jacod (1997))
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
(S0 − Ke−rT )+ , S0
Credit risk
Duality theory
Criteria to choose an EMM: Esscher transform, minimal distance MM,
minimal entropy MM, utility functions, . . .
Capital
Requirements
Spread Option
Net Asset Value
→ whole industry
Reserve Capital
References
c Eberlein, Uni Freiburg, 40
Pricing of derivatives
Introduction
Lévy processes
f (ST ) payoff of the option at maturity T
f (x) = (x − K )+
European call option
f (x) = (K − x)+
European put option
Financial
modeling
Valuation
Risk Managem.
Interest rate
Similarly: digitals, quantos, asset-or-nothing, power options, . . .
Given a specific martingale measure (calibration to market data)
Calibration
Lévy LIBOR
Cross-currency
V =E e
−rT
f (ST )
Range options
Credit risk
Explicit formula for European call
Z ∞
Z
∗T
V = S0
dGH
(x; θ+1) dx − e−rT K
γ
∞
Duality theory
∗T
dGH
(x; θ) dx
γ
Capital
Requirements
Spread Option
∗T
where γ = ln(K /S0 ) and dGH
GH-density under risk-neutral measure
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 41
Raible’s method
Introduction
Lévy processes
Numerical evaluation based on bilateral Laplace transforms (Raible
(2000)).
We want to price a European call option;
Z
V = e−rT IE[(ST − K )+ ] = e−rT (ST − K )+ dP
Ω
Z
Z
−rT
x
+
=e
(S0 e − K ) dPLT (x) = e−rT (S0 ex − K )+ ρ(x)dx
R
R
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
if PLT is absolutely continuous with respect to the Lebesgue measure λ\
with density ρ. Define g(x) = (e−x − K )+ and ζ = − log S0 , then
Z
V = e−rT
g(ζ − x)ρ(x)dx = e−rT (g ∗ ρ)(ζ)
R
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 42
Raible’s method (cont.)
Introduction
Lévy processes
is a convolution at point ζ. Passing to bilateral Laplace transforms
Lh (z), z ∈ C
Z
LV (z) = e−rT
e−zx (g ∗ ρ)(x)dx
R
Z
Z
= e−rT
e−zx g(x)dx
e−zx ρ(x)dx = e−rT Lg (z)Lρ (z).
R
R
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Lg can be calculated explicitly; Lρ can be expressed in terms of the
characteristic function ϕLT (Lévy–Khintchine formula!).
Cross-currency
Range options
By numerically inverting the Laplace transform, we recover the option
price.
Credit risk
The method applies to any European – hence path-independent –
payoff, such as call, put, digital, self-quanto and power options.
Capital
Requirements
Duality theory
Spread Option
The Lévy motions we are interested in, e.g. generalized hyperbolic,
have a (known) Lebesgue density.
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 43
Supremum and infimum processes
Introduction
Lévy processes
Let X = (Xt )0≤t≤T be a stochastic process. Denote by
X t = sup Xu
0≤u≤t
and
X t = inf Xu
Valuation
0≤u≤t
Risk Managem.
the supremum and infimum process of X respectively. Since the
exponential function is monotone and increasing
S T = sup St = sup S0 eLt = S0 esup0≤t≤T Lt = S0 eLT .
0≤t≤T
Financial
modeling
Interest rate
Calibration
Lévy LIBOR
(1)
Cross-currency
0≤t≤T
Range options
Similarly
Credit risk
S T = S0 eLT .
(2)
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 44
Valuation formulas – payoff functional
We want to price an option with payoff Φ(St , 0 ≤ t ≤ T ), where Φ is a
measurable, non-negative functional.
Introduction
Lévy processes
Financial
modeling
Separation of payoff function from the underlying process:
Valuation
Risk Managem.
Example
Interest rate
Calibration
Fixed strike lookback option
Lévy LIBOR
+
LT
(S T − K ) = (S0 e
+
LT +log S0
− K) = e
−K
+
Cross-currency
Range options
Credit risk
1 The payoff function is an arbitrary function f : R → R+ ; for
example f (x) = (ex − K )+
or
f (x) = 1{ex >B} ,
for K , B ∈ R+ .
2 The underlying process denoted by X , can be the log-asset price
process or the supremum/infimum or an average of the log-asset
price process (e.g. X = L or X = L).
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 45
Valuation formulas
Consider the option price as a function of S0 or better of s = − log S0
Lévy processes
Financial
modeling
X driving process (X = L, L, L, etc.)
Φ(S0 eLt , 0 ≤ t ≤ T ) = f (XT − s)
⇒
Introduction
Valuation
Risk Managem.
Interest rate
Time-0 price of the option (assuming r ≡ 0)
Calibration
Vf (X ; s) = E Φ(St , 0 ≤ t ≤ T ) = E[f (XT − s)]
Lévy LIBOR
Cross-currency
Range options
Valuation formulas based on Fourier and Laplace transforms
Carr and Madan (1999)
Raible (2000)
Duality theory
plain vanilla options
Capital
Requirements
general payoffs, Lebesgue densities
Borovkov and Novikov (2002)
Credit risk
plain vanilla and lookback options
Spread Option
Net Asset Value
In these approaches: Some sort of continuity assumption (payoff or
random variable)
c Eberlein, Uni Freiburg, 46
Reserve Capital
References
Valuation formulas – assumptions
Introduction
Lévy processes
MXT
−Rx
g(x) = e
L1bc (R)
Financial
modeling
moment generating function of XT
f (x)
(for some R ∈ R)
dampened payoff function
bounded, continuous functions in L1 (R)
Valuation
Risk Managem.
Interest rate
Calibration
Assumptions
(C1)
g ∈ L1bc (R)
(C2)
MXT (R) exists
(C3)
Lévy LIBOR
Cross-currency
Range options
Credit risk
1
b ∈ L (R)
g
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 47
Valuation formulas
Introduction
Lévy processes
Theorem
Assume that (C1)–(C3) are in force. Then, the price Vf (X ; s) of an
option on S = (St )0≤t≤T with payoff f (XT ) is given by
Z
e−Rs
Vf (X ; s) =
eius ϕXT (−u − iR) bf (u + iR)du,
2π R
Financial
modeling
Valuation
Risk Managem.
(3)
Interest rate
Calibration
Lévy LIBOR
where ϕXT denotes the extended characteristic function of XT and bf
denotes the Fourier transform of f .
Cross-currency
Range options
Proof
Credit risk
Duality theory
Z
f (XT − s)dP = e−Rs
Vf (X ; s) =
Ω
Z
eRx g(x − s)PXT (dx).
(4)
R
cont. next page
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 48
Proof (cont.)
Introduction
b is well-defined. With (C3)
Under assumption (C1), g ∈ L1 (R) and g
b ∈ L1bc (R).
g
Z
1
b(u)du.
g(x) =
e−ixu g
2π R
Lévy processes
Financial
modeling
(5)
Valuation
Risk Managem.
Interest rate
Returning to the valuation problem (4) we get
Calibration
Vf (X ; s) = e−Rs
Z
eRx
R
e−Rs
=
2π
Z
e−Rs
=
2π
Z
ius
1
2π
Z
e
R
Z
!
Lévy LIBOR
b(u)du PXT (dx)
e−i(x−s)u g
Cross-currency
R
Range options
!
i(−u−iR)x
e
b(u)du
PXT (dx) g
Credit risk
R
ius
Duality theory
e ϕXT (−u − iR) bf (u + iR)du.
(6)
R
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 49
Discussion of assumptions
Introduction
Alternative choice:
Lévy processes
(C1’) g ∈ L1 (R) ∩ L∞ (R)
Financial
modeling
1
R. P
(C3’) e\
XT ∈ L (R)
Valuation
(C3’) =⇒ eR. PXT has a cont. bounded Lebesgue density
Risk Managem.
Interest rate
Calibration
Recall:
Lévy LIBOR
b ∈ L1 (R)
(C3) g
Cross-currency
Sobolov space
Range options
H 1 (R) = g ∈ L2 (R) | ∂g exists and ∂g ∈ L2 (R)
Credit risk
Duality theory
Capital
Requirements
Lemma
Spread Option
b ∈ L1 (R)
g ∈ H 1 (R) =⇒ g
Net Asset Value
Similar for the Sobolev–Slobodeckij space H s (R) (s > 12 )
Reserve Capital
References
c Eberlein, Uni Freiburg, 50
Examples of payoff functions
Introduction
Lévy processes
Example (Call and put option)
Financial
modeling
Valuation
Call payoff f (x) = (ex − K )+ , K ∈ R+ ,
Risk Managem.
Interest rate
1+iu−R
bf (u + iR) =
K
,
(iu − R)(1 + iu − R)
R ∈ I1 = (1, ∞).
(7)
Calibration
Lévy LIBOR
Cross-currency
x +
Range options
Similarly, if f (x) = (K − e ) , K ∈ R+ ,
Credit risk
K 1+iu−R
bf (u + iR) =
,
(iu − R)(1 + iu − R)
R ∈ I1 = (−∞, 0).
(8)
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 51
Example (Digital option)
Introduction
Call payoff
1{ex >B} , B ∈ R+ .
bf (u + iR) = −B iu−R 1 ,
iu − R
Lévy processes
R ∈ I1 = (0, ∞).
(9)
Valuation
Similarly, for the payoff f (x) = 1{ex <B} , B ∈ R+ ,
bf (u + iR) = B iu−R
1
,
iu − R
Financial
modeling
Risk Managem.
Interest rate
R ∈ I1 = (−∞, 0).
(10)
Calibration
Lévy LIBOR
Cross-currency
Range options
Example (Double digital option)
The payoff of a double digital call option is
1 iu−R
bf (u + iR) =
B
− B iu−R ,
iu − R
Credit risk
Duality theory
1{B<ex <B} , B, B ∈ R+ .
R ∈ I1 = R\{0}.
Capital
Requirements
(11)
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 52
Example (Asset-or-nothing digital)
Call payoff
Put payoff
Introduction
f (x) = ex 1{ex >B}
1+iu−R
bf (u + iR) = − B
,
1 + iu − R
f (x) = ex 1{ex <B}
bf (u + iR) =
B 1+iu−R
,
1 + iu − R
Lévy processes
R ∈ I1 = (1, ∞)
Financial
modeling
Valuation
Risk Managem.
Interest rate
R ∈ I1 = (−∞, 1)
Calibration
Lévy LIBOR
Cross-currency
Range options
Example (Self-quanto option)
Call payoff
Credit risk
Duality theory
f (x) = ex (ex − K )+
bf (u + iR) =
K 2+iu−R
,
(1 + iu − R)(2 + iu − R)
Capital
Requirements
R ∈ I1 = (2, ∞)
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 53
Non-path-dependent options
European option on an asset with price process St = eLt
XT ≡ L T ,
Lévy processes
Financial
modeling
Examples: call, put, digitals, asset-or-nothing, double
digitals, self-quanto options
−→
Introduction
Valuation
Risk Managem.
i.e. we need ϕLT
Interest rate
Calibration
Generalized hyperbolic model (GH model):
Eberlein, Keller (1995),
Eberlein, Keller, Prause (1998),
Eberlein (2001)
p
λ/2 Kλ δ α2 − (β + iu)2 α2 − β 2
p
ϕL1 (u) = eiuµ 2
α − (β + iu)2
Kλ δ α 2 − β 2
I2 = (−α − β, α − β)
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
ϕLT (u) = (ϕL1 (u))T
Net Asset Value
Reserve Capital
similar: NIG, CGMY, Meixner
References
c Eberlein, Uni Freiburg, 54
Computation of ϕL1
Introduction
Lévy processes
Let (Lt )t≥0 be a gamma process, then
Z
c γ γ−1 −cx
E[euL1 ] =
eux
x
e dx
Γ(γ)
Z
c γ γ−1 −(c−u)x
=
x
e
dx
Γ(γ)
Z
(c − u)γ γ−1 −(c−u)x
cγ
=
x
e
dx
γ
(c − u)
Γ(γ)
γ
c
for u < c
=
c−u
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
ϕL1 (u) = E[e
iuL1
]=
c
c − iu
γ
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 55
Non-path-dependent options II
Introduction
Stochastic volatility Lévy models:
Z
Stochastic clock
Yt =
Carr, Geman, Madan, Yor (2003)
Eberlein, Kallsen, Kristen (2003)
Lévy processes
Financial
modeling
Valuation
t
ys ds
(ys > 0)
Risk Managem.
0
Interest rate
e.g. CIR process
Calibration
1/2
dyt = K (η − yt ) dt + λyt
dWt
Lévy LIBOR
Cross-currency
Range options
Define for a pure jump Lévy process X = (Xt )t≥0
Credit risk
Ht = XYt
Then
(0 ≤ t ≤ T )
Duality theory
Capital
Requirements
ϕYt (−iϕXt (u))
ϕHt (u) =
(ϕYt (−iuϕXt (−i)))iu
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 56
Classification of option types
Introduction
Lévy model
St = S0 eLt
Lévy processes
Financial
modeling
payoff
payoff function
+
(ST − K )
call
x
f (x) = (e − K )
distributional properties
+
PLT usually has a density
Valuation
Risk Managem.
Interest rate
Calibration
00
f (x) = 1{ex >B}
1{ST >B}
Lévy LIBOR
– –
Cross-currency
digital
Range options
ST − K
lookback
+
1{S T >B}
x
f (x) = (e − K )
+
density of PLT ?
Credit risk
Duality theory
f (x) = 1{ex >B}
Capital
Requirements
–00 –
Spread Option
digital barrier
= one touch
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 57
Valuation formula for the last case
Introduction
Payoff function f maybe discontinuous
PXT does not necessarily possess a Lebesgue density
Lévy processes
Assumption
Valuation
(D1)
g ∈ L1 (R) ∩ L∞ (R)
(D2)
MXT (R) exists
Financial
modeling
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Theorem
Cross-currency
Range options
Assume (D1)–(D2) then
Credit risk
−Rs
Vf (X ; s) = lim
A→∞
e
2π
Z
A
−A
e−ius ϕXT (u − iR)bf (iR − u) du
(12)
if Vf (X ; ·) is of bounded variation in a neighborhood of s and Vf (X ; ·) is
continuous at s.
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 58
Lookback options
Introduction
Lévy processes
Fixed strike lookback call: (S T − K )+
(analogous for lookback put).
We get
1
CT (S; K ) =
2π
Financial
modeling
Valuation
Z
K 1+iu−R
S0R−iu ϕLT (−u − iR)
du
(iu − R)(1 + iu − R)
R
(13)
Risk Managem.
Interest rate
Calibration
where
Lévy LIBOR
1
ϕLT (−u − iR) = lim
A→∞ 2π
Z
A
−A
eT (Y +iv ) κ(Y + iv , 0)
dv
Y + iv κ(Y + iv , iu − R)
(14)
Cross-currency
Range options
Credit risk
for R ∈ (1, M) and Y > α∗ (M).
Duality theory
• The floating strike lookback option, ( S T − ST )+ , is treated by a duality
formula (Eb., Papapantoleon (2005)).
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 59
One-touch options
Introduction
Lévy processes
One-touch call option:
1{S T >B} .
Driving Lévy process L is assumed to have infinite variation or has
infinite activity and is regular upwards. L satisfies assumption (EM),
then
Z A
1
B −R−iu
DCT (S; B) = lim
S0R+iu ϕLT (u − iR)
du
(15)
A→∞ 2π −A
R + iu
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
= P(LT > log(B/S0 ))
Range options
∗
for R ∈ (0, M), ϕ = α (M) and
Z N T (Y +iv )
κ(Y + iv , 0)
e
1
dv .
ϕLT (u − iR) = lim
N→∞ 2π −N Y + iv κ(Y + iv , −R − iu)
Credit risk
Duality theory
(16)
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 60
Equity default swap (EDS)
Introduction
Lévy processes
• Fixed premium exchanged for payment at “default”
Financial
modeling
• default: drop of stock price by 30 % or 50 % of S0 → first passage
time
Valuation
• fixed leg pays premium K at times T1 , . . . , TN , if Ti ≤ τB
Interest rate
Risk Managem.
Calibration
• if τB ≤ T : protection payment C, paid at time τB
Lévy LIBOR
• premium of the EDS chosen such that initial value equals 0; hence
CE e−r τB 1{τB ≤T }
K = PN
(17)
.
−rTi 1
{τB >Ti }
i=1 E e
• Calculations similar to touch options, since 1{τB ≤T } = 1{S T ≤B} .
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 61
Options on multiple assets
Introduction
Basket options
Lévy processes
Options on the minimum:
(ST1 ∧ · · · ∧ STd − K )+
Financial
modeling
Multiple functionals of one asset
Valuation
(ST − K )+ 1{S T >B}
Barrier options:
Risk Managem.
(ST − K )
Slide-in or corridor options:
N
X
+
Interest rate
1{L<ST <H}
i
Calibration
i=1
Lévy LIBOR
Modelling:
Sti
=
S0i
exp(Lit )
Cross-currency
(1 ≤ i ≤ d)
Range options
f : Rd −→ R+
g(x) = e−hR,xi f (x)
Assumptions:
Credit risk
(x ∈ Rd )
Duality theory
(A1) g ∈ L (R ) ∩ L (R )
Capital
Requirements
(A2) MXT (R) exists
Spread Option
(A3) %b ∈ L1 (Rd ) where %(dx) = ehR,xi PXT (dx)
Net Asset Value
1
d
∞
d
Reserve Capital
References
c Eberlein, Uni Freiburg, 62
Sensitivities – Greeks
Introduction
Lévy processes
1
Vf (X ; S0 ) =
2π
Z
S0R−iu MXT (R
− iu)bf (u + iR) du
Financial
modeling
R
Valuation
Risk Managem.
Interest rate
Delta of an option
Calibration
∂V(X ; S0 )
1
=
∆f (X ; S0 ) =
∂S0
2π
Z
S0R−1−iu MXT (R
R
bf (u + iR)
du
− iu)
(R − iu)−1
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Gamma of an option
∂ 2 Vf (X ; S0 )
1
Γf (X ; S0 ) =
=
∂ 2 S0
2π
Z
S0R−2−iu
R
MXT (R − iu)bf (u + iR)
du
(R − 1 − iu)−1 (R − iu)−1
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 63
Numerical examples
Introduction
Lévy processes
12
9
8
Financial
modeling
10
7
6
8
5
4
3
6
Valuation
4
Risk Managem.
2
2
1
Interest rate
1
0
85
90
95
100
105
110
115
1
0
85
0.5
90
95
100
0
Option prices in the 2d Black-Scholes model with
negative correlation.
Calibration
0.5
105
110
0
115
Option prices in the 2d stochastic volatility
model.
Lévy LIBOR
Cross-currency
Range options
Credit risk
10
10
9
Duality theory
9
8
8
7
6
5
5
4
4
3
3
2
2
Spread Option
Net Asset Value
1
1
0
85
Capital
Requirements
7
6
1
90
95
0.5
100
105
110
115
0
0
85
1
90
95
0.5
100
105
110
115
0
Option prices in the 2d GH model with positive (left) and negative (right)
correlation.
c Eberlein, Uni Freiburg, 64
Reserve Capital
References
Consequences for risk management
Introduction
Lévy processes
More precise quantification of market risk
Financial
modeling
The stochastic uncertainty of a book or portfolio corresponding
to a specified time horizon is given by its
Valuation
Risk Managem.
Interest rate
P&L-distribution
Calibration
Lévy LIBOR
Risk measures (e.g. VaR, volatility, shortfall measure)
Cross-currency
Range options
simple functions of the P&L-distribution
Credit risk
Duality theory
Capital
Requirements
Chance measures (e.g. expected return)
Spread Option
also functions of the P&L-distribution
Net Asset Value
−→ portfolio management
Reserve Capital
References
c Eberlein, Uni Freiburg, 65
P&L-distribution
Introduction
Probability
Lévy processes
1
Financial
modeling
0,9
Valuation
0,8
Risk Managem.
Confidence Interval
0,7
Interest rate
Calibration
0,6
Lévy LIBOR
0,5
Cross-currency
0,4
Range options
Credit risk
0,3
Duality theory
0,2
Capital
Requirements
0,1
Spread Option
0
-500
Loss
-400
-300
-200
-100
0
100
Value at Risk (in $ 1000)
200
300
400
500
Profit
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 66
Standard risk measure: Value at Risk
Introduction
Lévy processes
P[Xt < uα ] = α
Financial
modeling
Valuation
α-quantile of return distribution
Risk Managem.
Interest rate
VaR(α) = S0 − S0 exp(uα )
Calibration
Lévy LIBOR
Cross-currency
Functional value at risk
Range options
Credit risk
α −→ VaR(α)
Duality theory
Improvement: Shortfall measure
Capital
Requirements
Spread Option
Shortfall(α, t) = E[S0 − S0 exp(Xt ) | Xt < uα ]
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 67
Profit-and-Loss Distribution
Introduction
Lévy processes
Financial
modeling
Valuation
Density
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
1%-Quantile
Spread Option
Net Asset Value
Value at
Risk
Reserve Capital
References
0
c Eberlein, Uni Freiburg, 68
Value at Risk
Introduction
0.08
empirical
Lévy processes
normal
Financial
modeling
hyperbolic
Valuation
Value at Risk
0.04
0.06
NIG
Risk Managem.
GH
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
0.02
Credit risk
Duality theory
0.0
Capital
Requirements
Spread Option
0.0
0.01
0.02
0.03
level of probability
0.04
0.05
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 69
Stochastic volatility
Introduction
Basic model
Lévy processes
St = S0 exp(Xt ) where Xt = µt + σLt
(Lt )t≥0 standardized Lévy process:
E[L1 ] = 0 and Var(L1 ) = 1
Financial
modeling
Valuation
σ −→ (σt )t≥0
volatility process
Dynamic version:
dXt = σt dLt
Discrete version:
∆Xt = σt ∆Lt
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Various models for (σt )t≥0 : historic volatility
Credit risk
Ornstein–Uhlenbeck process
d(log σt2 ) = −a(log σt2 − c) dt + bdBt
Duality theory
Capital
Requirements
GARCH model
Spread Option
implicit volatility
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 70
implied volatility
|
||
| |
||
Lévy processes
|
500
|
Introduction
times of excessive losses
|
|
|
|
Financial
modeling
400
Valuation
VaR and actual losses
200
300
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
100
Range options
Credit risk
Duality theory
0
Capital
Requirements
600
800
1000
1200
trading day
1400
1600
1800
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 71
Interest rate models
Introduction
Lévy processes
Financial
modeling
Valuation
Interest rate models should be able to reproduce
Risk Managem.
• the observable term structures of interest rates,
• market prices of interest rate derivatives (caps, floors, swaptions)
Interest rate
Calibration
Lévy LIBOR
but they should also be
Cross-currency
• analytically tractable.
Range options
Credit risk
Idea: Use an HJM-type model driven by a (possibly non-homogeneous)
Lévy process
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 72
Real and estimated interest rates of the USA
Svensson parameters: b0 = 0.053 b1 = −0.042 b2 = −0.041 b3 = −0.009 tau1 = 1.479 tau2 = 0.329
5
real
estimated
Introduction
Lévy processes
4
Financial
modeling
3
Risk Managem.
Interest rate
Calibration
2
Lévy LIBOR
Cross-currency
Range options
1
Credit risk
Duality theory
Capital
Requirements
0
interest rates in percent
Valuation
Spread Option
0
2
4
6
8
10
Net Asset Value
time to maturity
Reserve Capital
References
Termstructure, February 17, 2004
c Eberlein, Uni Freiburg, 73
Comparison of estimated interest rates (least squares Svensson)
5
Euroland
Japan
Switzerland
USA
Introduction
Lévy processes
4
Financial
modeling
3
Risk Managem.
Interest rate
Calibration
2
Lévy LIBOR
Cross-currency
Range options
1
Credit risk
Duality theory
Capital
Requirements
0
interest rate in percent
Valuation
Spread Option
0
2
4
6
8
10
Net Asset Value
time to maturity
Reserve Capital
References
Termstructure, February 17, 2004
c Eberlein, Uni Freiburg, 74
Nelson–Siegel (1987) curves m = maturity, parameters β0 , β1 , β2 , τ1
−1
m
m
s(m) = β0 + β1 1 − exp −
τ1
τ1
−1
!
m
m
m
+ β2
1 − exp −
− exp −
τ1
τ1
τ1
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Improved curves: Svensson (1994)
parameters β0 , β1 , β2 , β3 , τ1 , τ2
−1
m
m
s(m) = β0 + β1 1 − exp −
τ1
τ1
−1
!
m
m
m
+ β2
1 − exp −
− exp −
τ1
τ1
τ1
+ β3
−1
!
m
m
m
1 − exp −
− exp −
τ2
τ2
τ2
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 75
15
Introduction
Lévy processes
Financial
modeling
1-month rate in %
5
10
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
0
Capital
Requirements
1970
1975
1980
1985
1990
Spread Option
1995
time
Net Asset Value
One month rate at the German market, March 01, 1967 – March 31, 1997
Reserve Capital
References
!"#$#%&'()!"#*+,!.- "#/01
23
45-6!879+;
:8 Freiburg,
c Eberlein,
Uni
76
Short rate dynamics
Introduction
Merton (1970)
drt = θ dt + σ dBt
Vasiček (1977)
drt = k (θ − rt ) dt + σ dBt
Dothan (1978)
drt = art dt + σrt dBt
Brennan-Schwartz
(1979)
drt = (θ(t) − art ) dt + σrt dBt
Financial
modeling
Valuation
Risk Managem.
Interest rate
3/2
Constantinides-Ingersoll drt = σrt
(1984)
Cox-Ingersoll-Ross
(1985)
Lévy processes
dBt
Calibration
Lévy LIBOR
√
drt = k (θ − rt ) dt + σ rt dBt
Cross-currency
Range options
Credit risk
Ho-Lee (1986)
drt = θ(t) dt + σ dBt
Black-Derman-Toy
(1990)
drt = rt θ(t) − a ln rt + 12 σ 2 (t) dt + σ(t)rt dBt
Hull-White (1990)
(extended CIR)
√
drt = k (θ(t) − rt ) dt + σ(t) rt dBt
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Sandmann-Sondermann drt = (1− e−rt ) θ(t)− 21 (1− e−rt )σ 2 dt +σ dBt
(1993)
c Eberlein, Uni Freiburg, 77
Reserve Capital
References
Classical modeling of the
dynamics of term structures
B(t, T ) price at time t ∈ [0, T ] of a default-free zero coupon bond with
maturity T ∈ [0, T ∗ ] (B(T , T ) = 1)
Z T
f (t, T ) instantaneous forward rate: B(t, T ) = exp −
f (t, u) du
t
Heath, Jarrow, Morton (HJM) framework
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
df (t, T ) = α(t, T ) dt + σ(t, T )> dWt
(Wt )t≥0
σ(t, T )
Introduction
d-dimensional Brownian motion
volatility structure (e.g. Vasiček)
Lévy LIBOR
Cross-currency
Range options
Credit risk
Under the risk-neutral measure
Z t
Z
Z t
1 t ∗
2
|σ (s, T )| ds + σ ∗ (s, T )> dWs
B(t, T ) = B(0, T ) exp
r (s) ds −
2 0
0
0
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
where r (t) = f (t, t) short rate
Reserve Capital
References
c Eberlein, Uni Freiburg, 78
zero-bond log-returns (1985-95), 10 years to maturity
empirical densities calculated from zero-yield data for Germany
empirical
••••
• •
normal
•
•
100
•
•
•
•
•
•
0
20
40
density( x )
60
80
•
-0.02
-0.01
Introduction
Lévy processes
Financial
modeling
•
•
•
Valuation
•
•
•
•
Risk Managem.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
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•
•
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•
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•
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••••••••••••••••••••••••••••••
•
•
•
0.0
0.01
log return x of zero-bond
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
0.02
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 79
zero-bond log-returns (1985-95), 5 years to maturity
empirical densities calculated from zero-yield data for Germany
empirical
••
• •
normal
• •
NIG
300
•
•
density( x )
200
100
Lévy processes
•
Financial
modeling
•
•
•
0
Introduction
•
•
•
•
•
•
•
•
•
•
•
•
•
••
••
•
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••••••••••••••••••••••••••••••••••••••••••••
-0.010
-0.005
Valuation
•
•
•
•
Risk Managem.
•
•
•
•
•
•
•
•
•
Interest rate
•
Calibration
•
•
Lévy LIBOR
•
•
Cross-currency
•
•
•
•
•
•
•
••
••
••
••
•
•
•
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•••••••••••••••••••••
0.0
log return x of zero-bond
0.005
0.010
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 80
The driving process
Introduction
1
d
L = (L , . . . , L ) is a d-dimensional time-inhomogeneous Lévy
process, i.e. L has independent increments and the law of Lt is given by
the characteristic function
Z t
θs (iu) ds
with
IE[exp(ihu, Lt i)] = exp
θs (z) = hz, bs i +
1
hz, cs zi +
2
Financial
modeling
Valuation
Risk Managem.
Interest rate
0
Z
Lévy processes
ehz,xi − 1 − hz, xi Fs (dx)
Rd
where bt ∈ Rd , ct is a symmetric nonnegative-definite d × d-matrix
and Ft is a Lévy measure
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
T∗
Z
Integrability:
Z
|bs | + |cs | +
x 2 Fs (dx) ds < ∞
{|x|≤1}
0
T
Z
∗
Z
exp(ux)Fs (dx) ds < ∞
0
for |u| ≤ M
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
{|x|>1}
Reserve Capital
References
c Eberlein, Uni Freiburg, 81
Description in terms of modern
stochastic analysis
Introduction
Lévy processes
Financial
modeling
L = (Lt ) is a special semimartingale with canonical representation
Z t
Z t
Z tZ
1/2
Lt =
bs ds +
cs dWs +
x(µL − ν)(ds, dx)
0
and characteristics
Z t
At =
bs ds,
0
0
0
Rd
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
t
Z
Ct =
cs ds,
ν(ds, dx) = Fs (dx) ds
Cross-currency
Range options
0
W = (Wt ) is a standard d-dimensional Brownian motion,
Credit risk
Duality theory
µL the random measure of jumps of L and ν is the compensator of µL
L is also called a process with independent increments and absolutely
continuous characteristics (PIIAC)
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 82
Simulation of a GH Lévy motion
Introduction
NIG Levy process with marginal densities
105.4
Lévy processes
Financial
modeling
Valuation
Interest rate
Calibration
105.0
Lévy LIBOR
Cross-currency
Range options
Credit risk
104.8
Duality theory
Capital
Requirements
Spread Option
104.6
values of NIG(100,0,1,0) Levy process
105.2
Risk Managem.
Net Asset Value
Reserve Capital
0.0
0.5
1.0
1.5
2.0
References
t
c Eberlein, Uni Freiburg, 83
Simulation of a Lévy process
NIG(10,0,0.100,0) on [0,1]
NIG(10,0,0.025,0) on [1,3]
Introduction
Lévy processes
Financial
modeling
Valuation
0.2
Risk Managem.
Interest rate
0.1
Calibration
Lévy LIBOR
Cross-currency
0.0
Range options
Credit risk
−0.1
Duality theory
−0.2
Capital
Requirements
Spread Option
Net Asset Value
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Reserve Capital
References
t
c Eberlein, Uni Freiburg, 84
Lévy forward rate approach
Eberlein, Raible (1999), Eberlein, Özkan (2003),
Eberlein, Jacod, Raible (2005), Eberlein, Kluge (2006)
Introduction
Lévy processes
Financial
modeling
Valuation
df (t, T ) = ∂2 A(t, T ) dt − ∂2 Σ(t, T ) dLt
∗
(0 ≤ t ≤ T ≤ T )
Risk Managem.
Interest rate
Calibration
Σ and A are deterministic functions with values in Rd and R respectively
whose paths are continuously differentiable in the second variable.
Lévy LIBOR
Cross-currency
Range options
The volatility structure is bounded 0 ≤ Σi (t, T ) ≤ M
(i ∈ {1, . . . , d}).
Credit risk
Duality theory
Furthermore, Σ(t, T ) 6= 0 for t < T and Σ(T , T ) = 0 for T ∈ [0, T ∗ ].
Capital
Requirements
Spread Option
The drift condition A(t, T ) = θs (Σ(t, T )) holds.
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 85
Implications
Introduction
Lévy processes
Financial
modeling
Savings account and default-free zero coupon bond prices are given by
Z t
Z t
1
Bt =
exp
θs (Σ(s, t)) ds −
Σ(s, t) dLs
and
B(0, t)
0
0
Valuation
Risk Managem.
Interest rate
Calibration
Z t
Z t
B(t, T ) = B(0, T )Bt exp − θs (Σ(s, T )) ds +
Σ(s, T ) dLs .
0
0
Bond prices, once discounted by the savings account, are martingales.
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
In case d = 1, the martingale measure is unique (see Eberlein, Jacod,
and Raible (2004)).
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 86
Key tool
Introduction
L = (L1 , . . . , Ld )
d-dimensional time-inhomogeneous Lévy process
Z
Lévy processes
Financial
modeling
t
θs (iu) ds
where
IE[exp(ihu, Lt i)] = exp
0
Z 1
ehz,xi − 1 − hz, xi Fs (dx)
θs (z) = hz, bs i + hz, cs zi +
2
Rd
in case L is a (time-homogeneous) Lévy process, θs = θ is the cumulant
(log-moment generating function) of L1 .
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Proposition
Eberlein, Raible (1999)
Range options
d
i
Suppose f : R+ → C is a continuous function such that |R(f (x))| ≤ M
for all i ∈ {1, . . . , d} and x ∈ R+ , then
Z t
Z t
θs (f (s))ds
f (s)dLs
= exp
IE exp
0
0
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Take
Reserve Capital
P
f (s) = (s, T ) for some T ∈ [0, T ∗ ]
References
c Eberlein, Uni Freiburg, 87
Pricing of European options
Introduction
Lévy processes
Z
B(t, T ) = B(0, T ) exp
Z t
t
(r (s) + θs (Σ(s, T ))) ds + Σ(s, T )dLs
0
0
Financial
modeling
Valuation
Risk Managem.
where r (t) = f (t, t) short rate
Interest rate
V (0, t, T , w) time-0-price of a European option with maturity t and
payoff w(B(t, T ), K )
V (0, t, T , w) = IEP∗ [Bt−1 w(B(t, T ), K )]
Calibration
Volatility structures
Range options
Lévy LIBOR
Cross-currency
Credit risk
σ
b
(1 − exp(−a(T − t)))
Σ(t, T ) =
a
(Vasiček)
Duality theory
Σ(t, T ) = σ
b(T − t)
(Ho–Lee)
Capital
Requirements
Spread Option
Fast algorithms for Caps, Floors, Swaptions, Digitals, Range options
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 88
Forward measure associated
with data T ≤ T ∗
Introduction
Lévy processes
Financial
modeling
Density
1
dPT
=
dP∗
BT B(0, T )
or
IEP∗
B(t, T )
dPT
|
F
=
t
dP∗
Bt B(0, T )
Valuation
Risk Managem.
Interest rate
Calibration
For the case of the Lévy term structure model this equals
Z t
Z t
exp
Σ(s, T ) dLs −
θs (Σ(s, T )) ds
0
Lévy LIBOR
Cross-currency
Range options
0
Credit risk
ν T (dt, dx) = ehΣ(t,T ),xi ν(dt, dx)
Z t
1/2
Standard Brownian motion under PT : WtT = Wt −
cs Σ(s, T ) ds
Compensator of µL under PT :
0
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 89
Pricing formula for caps
(Eberlein, Kluge (2006))
Introduction
Lévy processes
w(B(t, T ), K ) = (B(t, T ) − K )
+
Financial
modeling
Valuation
Call with strike K and maturity t on a bond that matures at T
C(0, t, T , K ) =
IEP∗ [Bt−1 (B(t, T )
+
− K) ]
= B(0, t)IEPt [(B(t, T ) − K )+ ]
t
Z
Interest rate
Calibration
Lévy LIBOR
Cross-currency
(Σ(s, T ) − Σ(s, t))dLs , then
Write X =
Risk Managem.
Range options
0
Credit risk
1
KB(0, t) exp(Rξ)
C(0, t, T , K ) =
2π Z
∞
×
Duality theory
eiuξ (R + iu)−1 (R + 1 + iu)−1 MtX (−R − iu) du
Capital
Requirements
Spread Option
−∞
Net Asset Value
where ξ is a constant and R < −1.
Reserve Capital
Analogous for the corresponding put and for swaptions
c Eberlein, Uni Freiburg, 90
References
Calibration to market data
Eberlein–Kluge (2006)
Introduction
Lévy processes
Financial
modeling
Valuation
Calibration performed for a driving homogeneous as well as for a
time-inhomogeneous Lévy process
Time-inhomogeneous case: piecewise Lévy process (maturities up to 1
year, 1 to 5 years, greater than 5 years)
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Minimize the sum of
Range options
model price − market price
ATM market price for the respective maturity
2
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 91
Caplet market data
Introduction
Lévy processes
Financial
modeling
0.3
Valuation
0.28
Risk Managem.
0.26
Interest rate
0.24
Calibration
0.22
Lévy LIBOR
0.2
Cross-currency
0.18
0.16
Range options
0.14
Credit risk
0.12
Duality theory
0.1
0.02
0.04
0.06
0.08
Strike rates
0.1
10
8
6
4
2
0
Capital
Requirements
Spread Option
Maturity (years)
Euro caplet implied volatility surface on February 19, 2002
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 92
Calibration results
Introduction
Lévy processes
Financial
modeling
Valuation
0.09
Risk Managem.
0.08
0.07
Interest rate
0.06
Calibration
0.05
Lévy LIBOR
0.04
Cross-currency
0.03
0.02
Range options
0.01
Credit risk
0
Duality theory
0.03
1
0.04
0.05
Strike rates
5
0.06
0.08
0.10
10
Maturity (years)
Absolute differences between implied volatility of model and market
price
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 93
Implied volatility curve for 2 years
Introduction
Lévy processes
32
Market price
Model price (homogeneous)
Model price (non−homogeneous)
30
Implied volatility (in %)
Financial
modeling
Valuation
Risk Managem.
28
Interest rate
Calibration
26
Lévy LIBOR
24
Cross-currency
Range options
22
Credit risk
20
Duality theory
Capital
Requirements
18
Spread Option
16
Net Asset Value
3
4
5
6
7
Strike rate (in %)
8
9
10
Reserve Capital
References
c Eberlein, Uni Freiburg, 94
Implied volatility curve for 5 years
Introduction
Lévy processes
26
Market price
Model price (homogeneous)
Model price (non−homogeneous)
24
Implied volatility (in %)
Financial
modeling
Valuation
Risk Managem.
Interest rate
22
Calibration
Lévy LIBOR
20
Cross-currency
18
Range options
Credit risk
16
Duality theory
14
Capital
Requirements
Spread Option
12
Net Asset Value
3
4
5
6
7
Strike rate (in %)
8
9
10
Reserve Capital
References
c Eberlein, Uni Freiburg, 95
Implied volatility curve for 10 years
Introduction
Lévy processes
24
Market price
Model price (homogeneous)
Model price (non−homogeneous)
22
Implied volatility (in %)
Financial
modeling
Valuation
Risk Managem.
Interest rate
20
Calibration
Lévy LIBOR
18
Cross-currency
16
Range options
Credit risk
14
Duality theory
12
Capital
Requirements
Spread Option
10
Net Asset Value
3
4
5
6
7
Strike rate (in %)
8
9
10
Reserve Capital
References
c Eberlein, Uni Freiburg, 96
Basic interest rates
Introduction
Lévy processes
B(t,T ): price at time t ∈ [0, T ] of a default-free zero coupon bond
Financial
modeling
Valuation
Risk Managem.
f (t,T ): instantaneous forward rate
Interest rate
R
T
B(t,T ) = exp − t f (t,u) du
Calibration
Lévy LIBOR
L(t,T ): default-free forward Libor rate for the interval T to T + δ
B(t,T )
L(t,T ) := δ1 B(t,T
−1
+δ)
Cross-currency
Range options
Credit risk
FB (t,T ,U): forward price process for the two maturities T and U
FB (t,T ,U) :=
=⇒
1 + δL(t,T ) =
B(t,T )
B(t,U)
Duality theory
Capital
Requirements
B(t,T )
= FB (t,T,T + δ)
B(t,T + δ)
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 97
LIBOR market model
Introduction
Lévy processes
T0
T1
T2
T3
TM+1=T *
Financial
modeling
Valuation
with δ = Tn+1 − Tn (fixed accrual period)
Risk Managem.
L(t, T ) forward LIBOR rate for the interval T to T + δ as of time t ≤ T
B(t, T )
1
δ-forward LIBOR rate
L(t, T ) =
−1
δ B(t, T + δ)
Interest rate
Calibration
Lévy LIBOR
Cross-currency
For two maturities T , U define the forward process
FB (t, T , U) =
Range options
Credit risk
B(t, T )
B(t, U)
Duality theory
Capital
Requirements
=⇒ 1 + δL(t, T ) = FB (t, T , T + δ)
Spread Option
Sandmann, Sondermann, Miltersen (1995); Miltersen, Sandmann,
Sondermann (1997); Brace, Gatarek, Musiela (1997); Jamshidian
(1997)
c Eberlein, Uni Freiburg, 98
Net Asset Value
Reserve Capital
References
The Lévy Libor model
Introduction
(Eberlein, Özkan (2005))
Lévy processes
∗
Tenor structure T0 < T1 < · · · < TM < TM+1 = T
with Ti+1 − Ti = δ, set Ti∗ = T ∗−iδ for i = 1, . . . , M
TM*
*
TM−1
T2 *
Financial
modeling
T1 *
Valuation
Risk Managem.
T0
T1
T2
T3
TM−1
TM
T*
Interest rate
Calibration
Assumptions
Lévy LIBOR
Cross-currency
(LR.1): For any maturity Ti there is a bounded deterministic function
λ(·, Ti ), which represents the volatility of the forward Libor
rate process L(·, Ti ).
(LR.2): We assume a strictly decreasing and strictly positive initial term
structure B(0, T ) (T ∈]0, T ∗ ]). Consequently the initial term
structure of forward Libor rates is given by
B(0, T )
1
−1
L(0, T ) =
δ B(0, T + δ)
c Eberlein, Uni Freiburg, 99
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Backward Induction (1)
Introduction
Lévy processes
Given a stochastic basis (Ω, F
T0
TM*
*
TM−1
T1
T2
T∗
,P
T∗
, (Ft )
0≤t≤T ∗
T2 *
TM−1
T3
)
Financial
modeling
T1 *
TM
Valuation
Risk Managem.
T*
Interest rate
Calibration
We postulate that under PT ∗
Z
where
∗
LTt
t
∗
Lévy LIBOR
L(t, T1∗ ) = L(0, T1∗ ) exp
λ(s, T1∗ )dLTs
0
Z t
Z t
Z tZ
∗
1/2
T∗
=
bs ds +
cs dWs +
x(µL − ν T ,L )(ds, dx)
0
0
0
Cross-currency
Range options
Credit risk
R
Duality theory
L
is a non-homogeneous Lévy process with random measure of jumps µ
∗
and PT ∗ -compensator ν T ,L (ds, dx) = Fs (dx) ds, Fs ({0}) = 0, where Fs
satisfies some integrability conditions
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 100
Backward Induction (2)
Introduction
Lévy processes
In order to make L(t, T1∗ ) a PT ∗ -martingale, choose the drift
characteristic (bs ) s.t.
Z
Z t
1 t
cs λ2 (s, T1∗ ) ds
λ(s, T1∗ )bs ds= −
2 0
0
Z tZ ∗
∗
−
eλ(s,T1 )x − 1 − λ(s, T1∗ )x ν T ,L (ds, dx)
0
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
R
Cross-currency
Transform L(t, T1∗ ) in a stochastic exponential
L(t, T1∗ )
=
Range options
L(0, T1∗ )E(H(t, T1∗ ))
Credit risk
Duality theory
where
H(t, T1∗ ) =
t
Z
0
Z tZ ∗
1/2
λ(s, T1∗ )cs dWsT +
0
Capital
Requirements
e
λ(s,T1∗ )x
∗
− 1 (µL −ν T ,L )(ds, dx)
R
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 101
Backward Induction (3)
Introduction
Lévy processes
Financial
modeling
Valuation
Equivalently
Risk Managem.
dL(t, T1∗ )
=
L(t−, T1∗ )
+
Z ∗
1/2
λ(t, T1∗ )ct dWtT
e
λ(t,T1∗ )x
Interest rate
Calibration
∗
− 1 (µL − ν T ,L )(dt, dx)
Lévy LIBOR
Cross-currency
R
Range options
with initial condition
L(0, T1∗ ) =
1
δ
Credit risk
B(0, T1∗ )
−
1
B(0, T ∗ )
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 102
Backward Induction (4)
Introduction
Recall
FB (t, T1∗ , T ∗ )
=1+
δL(t, T1∗ ),
Lévy processes
therefore,
Financial
modeling
dFB (t, T1∗ , T ∗ ) = δdL(t, T1∗ )
∗
δL(t−, T1∗ )
1/2
∗
∗
λ(t, T1∗ ) ct dWtT
= FB (t−, T1 , T )
1 + δL(t−, T1∗ )
|
{z
}
Valuation
Risk Managem.
Interest rate
= α(t,T1∗ ,T ∗ )
δL(t−, T1∗ )
Z
+
R
1 + δL(t−, T1∗ )
|
{z
∗
eλ(t,T1 )x
= β(t,x,T1∗ ,T ∗ )−1
dPT ∗
Mt1
t
Z
= ET1∗ (M 1 )
∗
1/2
α(s, T1∗ , T ∗ )cs dWsT +
=
0
Lévy LIBOR
Cross-currency
Range options
Credit risk
Define the forward martingale measure associated with
dPT1∗
Calibration
∗
− 1 (µL − ν T ,L )(dt, dx)
}
T1∗
Duality theory
Capital
Requirements
where
Spread Option
Z tZ
0
(β(s, x, T1∗ , T ∗ )
R
L
− 1) (µ −ν
T ∗,L
)(ds, dx) Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 103
Backward Induction (5)
T1∗
∗
= WtT −
Then Wt
t
Z
Introduction
1/2
α(s, T1∗ , T ∗ )cs
ds
Lévy processes
0
Financial
modeling
is the forward Brownian motion for date T1∗ and
∗
ν T1 ,L (dt, dx) = β(t, x, T1∗ , T ∗ )ν T
∗
,L
Valuation
(dt, dx) is the PT1∗ -compensator for µL .
Risk Managem.
Interest rate
Second step
T1
T0
T2
T2 *
T1 *
TM−1
TM
Calibration
Lévy LIBOR
T*
Cross-currency
Range options
Credit risk
We postulate that under PT1∗
L(t, T2∗ ) = L(0, T2∗ ) exp
t
Z
T∗
λ(s, T2∗ ) dLs 1
Duality theory
where
0
Capital
Requirements
Spread Option
T∗
Z
Lt 1 =
t
T∗
Z
bs 1 ds +
0
t
1/2
cs
0
T∗
Z tZ
dWs 1 +
0
∗
x(µL − ν T1 ,L )(ds, dx)
R
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 104
Backward Induction (6)
Introduction
Lévy processes
Financial
modeling
Valuation
L(t, T2∗ )
t
Z
0
is a PT1∗ -martingale if
T∗
T∗
(bs 1 )
Risk Managem.
is chosen s.t.
Interest rate
1 t
cs λ2 (s, T2∗ ) ds
2 0
Z tZ ∗
∗
−
eλ(s,T2 )x − 1 − λ(s, T2∗ )x ν T1 ,L (ds, dx)
λ(s, T2∗ )bs 1 ds = −
Z
0
R
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 105
Backward Induction (7)
Introduction
Lévy processes
Second measure change
dPT2∗
dPT1∗
Financial
modeling
2
= ET2∗ (M )
Valuation
Risk Managem.
where
Interest rate
Mt2
t
Z
=
1/2
α(s, T2∗ , T1∗ )cs
T∗
dWs 1
Calibration
0
Lévy LIBOR
Z tZ
+
0
β(s, x, T2∗ , T1∗ )
∗
− 1 (µL − ν T1 ,L )(ds, dx)
Cross-currency
Range options
R
Credit risk
Tj∗
This way we get for each time point
in the tenor structure a Libor rate
∗
process which is under the forward martingale measure PTj−1
of the
form
Z
t
L(t, Tj∗ ) = L(0, Tj∗ ) exp
T∗
λ(s, Tj∗ ) dLs j−1
0
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 106
Comparison with Lévy forward
rate approach
Introduction
Lévy processes
Financial
modeling
Valuation
FB (t, T1∗ , T ) =
Risk Managem.
B(t, T1∗ )
B(t, T ∗ )
Interest rate
Calibration
Lévy LIBOR
∗
∗
dPT1∗ dPT1∗ dP ∗ =
= B(t, T1 ) Bt B(0, T )
∗
∗
∗
F
F
dPT ∗ t
dP dPT ∗ t
Bt B(0, T1 ) B(t, T )
=
B(0, T ∗ )
FB (t, T1∗ , T ∗ )
B(0, T1∗ )
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 107
Forward process model (1)
Introduction
Lévy processes
Financial
modeling
Postulate
1 + δL(t, T1∗ ) = (1 + δL(0, T1∗ )) exp
t
Z
λ(s, T1∗ ) dLsT
∗
Valuation
Risk Managem.
0
Interest rate
equivalently
Calibration
FB (t, T1∗, T ∗ ) = FB (0, T1∗ , T ∗ ) exp
t
Z
λ(s, T1∗ ) dLTs
∗
0
Cross-currency
Range options
In differential form
dFB (t, T1∗ , T ∗ )
Lévy LIBOR
Credit risk
=
∗
1/2
λ(t, T1∗ )ct dWtT
Z ∗
∗
+
eλ(t,T1 )x − 1 (µL − ν T ,L )(dt, dx)
FB (t−, T1∗ , T ∗ )
Duality theory
Capital
Requirements
Spread Option
R
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 108
Forward process model (2)
Introduction
Lévy processes
Financial
modeling
Valuation
Define the forward martingale measure associated with
dPT1∗
dPT ∗
T1∗
Interest rate
= ET1∗ (M̃ 1 )
Calibration
where
Z t
Z tZ ∗
∗
∗
1/2
M̃t1 =
λ(s, T1∗ )cs dWsT +
eλ(s,T1 )x − 1 (µL − ν T ,L )(ds, dx).
0
0
Risk Managem.
Lévy LIBOR
Cross-currency
Range options
Credit risk
R
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 109
Forward process model (3)
Introduction
Then
T∗
Wt 1
=
∗
WtT
t
Z
−
motion for date T1∗ and
1/2
λ(s, T1∗ )cs
ds is the forward Brownian
0
Financial
modeling
∗
∗
ν T1 ,L (dt, dx) = exp(λ(t, T1∗ )x) ν T ,L (dt, dx) is the PT1∗ -compensator of µL .
Continuing this way we get for each time point Tj∗ in the tenor structure
∗
a Libor rate process under PTj−1
in the form
1+
δL(t, Tj∗ )
= 1+
δL(0, Tj∗ ) exp
Z
t
∗
Tj−1
∗
λ(s, Tj ) dLs
.
0
ν Tj ,L (dt, dx) = exp
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
with successive compensators
∗
Lévy processes
Credit risk
X
j
λ(t, Ti∗ )x Ft (dx) dt.
i=1
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Consequence of this alternative approach:
negative Libor rates can occur
Reserve Capital
References
c Eberlein, Uni Freiburg, 110
Pricing of caps and floors (1)
Introduction
Lévy processes
Time-Tj -payoff of a cap settled in arrears
Financial
modeling
Nδ(L(Tj−1 , Tj−1 ) − K )+
N
notional amount (set N = 1)
Valuation
K
strike rate
Risk Managem.
Interest rate
Calibration
Time-t value
Ct =
n
X
j=1
=
n
X
"
IEP∗
Bt
δ(L(Tj−1 , Tj−1 ) − K )+ | Ft
BTj
#
B(t, Tj )IEPTj δ(L(Tj−1 , Tj−1 ) − K )+ | Ft
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
j=1
Capital
Requirements
Analogous for floor
Spread Option
+
Net Asset Value
Nδ(K − L(Tj−1 , Tj−1 ))
Reserve Capital
References
c Eberlein, Uni Freiburg, 111
Pricing of caps and floors (2)
Introduction
Lévy processes
Dynamics of L(t, Tj−1 ) under PTj (purely discontinuous case)
dL(t, Tj−1 ) = L(t−, Tj−1 )
Z eλ(t,Tj−1 )x
− 1 µL −ν Tj ,L (dt, dx)
Valuation
Risk Managem.
R
Solution
Interest rate
Z
t
T
λ(s, Tj−1 ) dLs j
L(t, Tj−1 ) = L(0, Tj−1 ) exp
Z
= L(0, Tj−1 ) exp
Cross-currency
t
T
bs j λ(s, Tj−1 ) ds
0
Z tZ
(xλ(s, Tj−1 )) µL −ν Tj ,L (ds, dx)
+
0
Rt
0
Calibration
Lévy LIBOR
0
Write Xt =
Financial
modeling
R
T
λ(s, Tj−1 ) dLs j then L(t, Tj−1 ) = L(0, Tj−1 ) exp(Xt ) is
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
a martingale with respect to PTj
Reserve Capital
References
c Eberlein, Uni Freiburg, 112
Numerical evaluation
Introduction
Lévy processes
Denote ζj = − ln(L(0, Tj−1 )) and vK (x) = (e
−x
− K)
Z
Bilateral Laplace transform of vK :
+∞
L[vK ](z) =
+
Financial
modeling
Valuation
e−zx vK (x) dx
Risk Managem.
−∞
Interest rate
Characteristic function of XTj−1 :
χ(u) = IEPTj [exp(iuXTj−1 ]
Assume mgf(−R) < ∞, then the time-0 price of the j-th caplet is given
by
Z
eζj R +∞ iuζj
Vj (ζj , K ) = δB(0, Tj )
e L[vK ](R +iu)χ(iR −u) du
2π −∞
whenever the right-hand side exists
χ(u) easy to compute for generalized hyperbolic Lévy motion
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 113
Representation as convolution
Introduction
Lévy processes
e ζj R
Vj (ζj , K ) = δB(0, Tj )
2π
Z
+∞
e
iuζj
L[vK ](R +iu)χ(iR −u) du
−∞
Financial
modeling
Valuation
Risk Managem.
Vj (ζj , K ) = δB(0, Tj )IEPTj [(L(Tj−1 , Tj−1 ) − K )+ ]
Interest rate
Calibration
= δB(0, Tj )IEPTj
h
i
vK (ζj − XTj−1 )
Z
= δB(0, Tj )
R
XT
vK (ζj − x) PTj
j−1
Lévy LIBOR
Cross-currency
Range options
(dx)
Credit risk
Duality theory
Z
vK (ζj − x)ρ(x) dx
= δB(0, Tj )
Capital
Requirements
R
Spread Option
= δB(0, Tj ) (vK ∗ ρ) (ζj ).
Net Asset Value
And
L[Vj ](R + iu) = δB(0, Tj )L[vK ](R + iu)L[ρ](R + iu)
for u ∈ R.
Reserve Capital
References
c Eberlein, Uni Freiburg, 114
Extensions of the basic Lévy
market model
Multi-currency setting
(Eb–Koval (2006))
Lévy market model
(Eb–Özkan (2005))
−→
−−
−
−
−−
−−
−
−→
−−
−−−−
−
−
−
−
−−−−
−−−−
−−
−→
−−
−−
−−
−−
−−
→
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Credit risk model
(Eb–Kluge–Schönbucher (2006))
Calibration
Lévy LIBOR
Cross-currency
Range options
Swap rate model
(Eb–Liinev (2006))
Credit risk
Duality theory
Capital
Requirements
Duality principle
(Eb–Kluge–Papapantoleon
(2006))
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 115
Cross-currency Lévy market model
Introduction
Foreign Market
Domestic Market
∗
P0,T -forward measure
F i (·, T ∗ )
X
∗
Pi,T -forward measure
-
?
?
P0,TN -forward measure
Pi,TN -forward measure
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
?
P
0,TN−1
?
-forward measure
P
i,TN−1
Calibration
-forward measure
Lévy LIBOR
?
P
0,Tj+1
?
-forward measure
P
i,Tj+1
Range options
F i (·, Tj , Tj+1 )
B
FB (·, Tj , Tj+1 )
P
0,Tj
?
-forward measure
?
P0,T1 -forward measure
Cross-currency
-forward measure
F i (·, Tj )
X
-
P
i,Tj
?
-forward measure
Credit risk
Duality theory
Capital
Requirements
?
Pi,T1 -forward measure
Relationship between domestic and foreign fixed income markets in a discrete-tenor framework.
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 116
Libor rates in a cross currency
setting
Introduction
Lévy processes
Financial
modeling
∗
Discrete tenor structure T0 < T1 < · · · < Tn < Tn+1 = T
Accrual periods
δ = Tj+1 − Tj
Valuation
Risk Managem.
Interest rate
Calibration
T0
T1
T2
Tj−1
TN
Tj
T*
Lévy LIBOR
Cross-currency
(m + 1) markets i = 0, . . . , m
0 = domestic market
Range options
Credit risk
i
Want to model the dynamics of the Libor rate L (t,Tj−1 ) which applies to
time period [Tj−1 ,Tj ] in market i (i = 0, . . . , m)
We target at the form
Li (t,Tj−1 ) = Li (0,Tj−1 ) exp
Duality theory
Capital
Requirements
Spread Option
t
Z
i,Tj
λi (s,Tj−1 ) dLs
Net Asset Value
Reserve Capital
0
References
c Eberlein, Uni Freiburg, 117
The driving process
Introduction
Lévy processes
T∗
T∗
) is a d-dimensional
L
, . . . , L0,
= (L0,
1
d
T∗
is given by
process. The law of L0,
t
0,T ∗
Financial
modeling
time-inhomogeneous Lévy
Valuation
Risk Managem.
Interest rate
T∗
IE[exp(iu > L0,
)]
t
t
Z
= exp
∗
θs0,T (iu) ds
Calibration
with
Lévy LIBOR
0
∗
θs0,T (z) = z > bs0,T
∗
1
+ z > Cs z +
2
Z
ez
>
x
∗
− 1 − z > x λ0,T
(dx),
s
Rd
∗
where bt0,T ∈ Rd , Cs is a symmetric nonnegative-definite
∗
is a Lévy measure.
d × d-matrix and λ0,T
s
Integrability assumptions
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 118
Description in terms of modern
stochastic analysis
Introduction
Lévy processes
Financial
modeling
∗
∗
T
L0,T = (L0,
) is a special semimartingale with canonical representation
t
Valuation
Risk Managem.
∗
T
L0,
=
t
Z
t
∗
bs0,T ds +
0
∗
t
Z
∗
cs dWs0,T +
0
Interest rate
Z tZ
x(µ − ν0,T ∗ )(ds, dx)
0
Rd
Lévy LIBOR
∗
(Wt0,T ) is a P0,T -standard Brownian motion with values in Rd
ct is a measurable version of the square∗ root of Ct
T
µ the random measure of jumps of (L0,
)
t
∗
Calibration
Cross-currency
Range options
Credit risk
Duality theory
∗
T
ν0,T ∗ (ds, dx) = λ0,
(dx) ds is the P0,T -compensator of µ
s
∗
0,T
(Lt ) is also called a process with independent increments and
absolutely continuous characteristics (PIIAC).
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 119
The foreign forward exchange
rate for date T ∗ (1)
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Assumptions
Interest rate
(FXR.1): For every market i ∈ {0, . . . , m} there are strictly decreasing
and strictly positive zero-coupon bond prices B i (0,Tj )(j = 0, . . . , N + 1)
and for every foreign economy i ∈ {1, . . . , m} there are spot exchange
rates X i (0) given.
Consequently the initial foreign forward exchange rate corresponding to
T ∗ is
B i (0,T ∗ )X i (0)
FX i (0,T ∗ ) =
B 0 (0,T ∗ )
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 120
The foreign forward exchange
rate for date T ∗ (2)
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Assumptions
Interest rate
(FXR.2): For every foreign market i ∈ {1, . . . , m} there is a continuous
deterministic function ξ i (·,T ∗ ) : [0,T ∗ ] → Rd+ .
Calibration
For every coordinate 1 ≤ k ≤ d we assume
Cross-currency
Lévy LIBOR
Range options
(ξ i (s,T ∗ ))k ≤ M
where M <
(s ∈ [0,T ∗ ], 1 ≤ i ≤ m)
Credit risk
Duality theory
M
.
N +2
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 121
The foreign forward exchange
rate for date T ∗ (3)
Introduction
Lévy processes
Assumptions
Financial
modeling
(FXR.3): For every i ∈ {1, . . . , m} the foreign forward exchange rate for
date T ∗ is given by
Z t
Z t
∗
∗
i
∗
i
∗ >
0,T ∗
FX i (t,T ) = FX i (0,T ) exp
γ (s,T ) ds +
ξ (s,T ) dLs
Valuation
0
where
0
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
i
∗
γ (s,T ) = −ξ
i
2
1
− ξ i (s,T ∗ )> cs 2
i
∗ >
T∗
eξ (s,T ) x − 1 − ξ i (s,T ∗ )> x λ0,
(dx)
s
∗
(s,T ∗ )> bs0,T
Z
−
Rd
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Equivalently
Z ·
∗
FX i (t,T ∗ ) = FX i (0,T ∗ )Et
ξ i (s,T ∗ )> cs dWs0,T
0
Z ·Z +
exp ξ i (s,T ∗ )> x − 1 (µ − ν0,T ∗ )(ds, dx)
0
Rd
c Eberlein, Uni Freiburg, 122
Spread Option
Net Asset Value
Reserve Capital
References
The foreign forward exchange
rate for date T ∗ (4)
Consequences:
0,T ∗
∗
FX i (·, T ) is a P
-martingale
FX i (t,T ∗ )
=1
EP0,T ∗
FX i (0,T ∗ )
Define
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
∗ F i (t,T ∗ )
dPi,T = X
∗ 0,
T
FX i (0,T ∗ )
dP
Ft
By Girsanov’s theorem we get a P
Introduction
Lévy LIBOR
Cross-currency
i,T ∗
-standard Brownian motion
Range options
Credit risk
∗
∗
Wti,T = Wt0,T −
Z
t
cs ξ i (s,T ∗ ) ds
Duality theory
Capital
Requirements
0
∗
and a Pi,T -compensator
Spread Option
νi,T ∗ (dt, dx) = exp(ξ i (t,T ∗ )> x)ν0,T ∗ (dt, dx)
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 123
The Lévy Libor model
Introduction
as in Eberlein–Özkan (2005)
Lévy processes
Tenor structure T0 < T1 < · · · < TN < TN+1 = T ∗
Financial
modeling
with Tj+1 − Tj = δ, set Tj∗ = T ∗−jδ for j = 1, . . . , N
TN*
*
TN−1
T1
T2
T2 *
Valuation
T1 *
Risk Managem.
Interest rate
T0
T3
TN−1
TN
T*
Calibration
Lévy LIBOR
Assumptions
Cross-currency
Range options
(CLM.1): For every market i and every maturity Tj there is a bounded
deterministic function λi (·,Tj ), which represents the volatility
of the forward Libor rate process Li (·,Tj ) in market i.
(CLM.2): The initial term structure of forward Libor rates in market i
is given by
B i (0,Tj )
1
Li (0,Tj ) =
−
1
δ B i (0,Tj + δ)
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 124
Backward Induction
Introduction
Lévy processes
∗
Given a stochastic basis (Ω, FT ∗ , P0,T , (Ft )0≤t≤T ∗ )
TN*
*
TN−1
T1
T2
T0
T3
We postulate that under Pi,T
Financial
modeling
T2 *
T1 *
TN−1
TN
Valuation
Risk Managem.
T*
Interest rate
Calibration
∗
Lévy LIBOR
Li (t, T1∗ ) = Li (0, T1∗ ) exp
t
Z
λi (s, T1∗ ) dLi,sT
∗
0
Cross-currency
Range options
Credit risk
where
Duality theory
∗
Li,t T =
t
Z
0
∗
bsi,T ds +
t
Z
∗
cs dWsi,T +
0
Z tZ
x(µ − νi,T ∗ )(ds, dx)
0
Rd
Capital
Requirements
Spread Option
∗
Net Asset Value
with W i,T and νi,T ∗ as given before.
Reserve Capital
References
c Eberlein, Uni Freiburg, 125
Cross-currency Lévy market model
Introduction
Foreign Market
Domestic Market
∗
P0,T -forward measure
F i (·, T ∗ )
X
∗
Pi,T -forward measure
-
?
?
P0,TN -forward measure
Pi,TN -forward measure
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
?
P
0,TN−1
?
-forward measure
P
i,TN−1
Calibration
-forward measure
Lévy LIBOR
?
P
0,Tj+1
?
-forward measure
P
i,Tj+1
Range options
F i (·, Tj , Tj+1 )
B
FB (·, Tj , Tj+1 )
P
0,Tj
?
-forward measure
?
P0,T1 -forward measure
Cross-currency
-forward measure
F i (·, Tj )
X
-
P
i,Tj
?
-forward measure
Credit risk
Duality theory
Capital
Requirements
?
Pi,T1 -forward measure
Relationship between domestic and foreign fixed income markets in a discrete-tenor framework.
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 126
Relationship between the
domestic and the foreign
market
The forward exchange rates in the i-th foreign market are related by
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
F i (t,Tj ,Tj+1 )
FX i (t,Tj ) = FX i (t,Tj+1 ) B
FB0 (t,Tj ,Tj+1 )
Interest rate
Calibration
Lévy LIBOR
From this one gets the dynamics of FX i (t,Tj )
Z
dFX i (t,Tj )
0,T
= ζ i (t,Tj ,Tj+1 ) dWt j +
(ζ i (t, x,Tj ,Tj+1 )−1)(µ−ν0,Tj )(dt, dx)
dFX i (t−,Tj )
Rd
Cross-currency
Range options
Credit risk
Duality theory
where the coefficients are given recursively
ζ i (t,Tj ,Tj+1 ) = αi (t,Tj ,Tj+1 ) − α0 (t,Tj ,Tj+1 ) + ζ i (t,Tj+1 ,Tj+2 )
β i (t, x,Tj ,Tj+1 ) i
ζ (t, x,Tj ,Tj+1 ) = 0
ζ (t, x,Tj+1 ,Tj+2 )
β (t, x,Tj ,Tj+1 )
i
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 127
Pricing cross-currency
derivatives (1)
Introduction
Lévy processes
Financial
modeling
Foreign forward caps and floors
Valuation
δX [Li (Tj−1 ,Tj−1 ) − K i ]+
Risk Managem.
Time-0-value of a foreign TN -maturity cap
i
FC (0, TN ) = δ
N+1
X
Interest rate
Calibration
i
B (0,Tj )EPi,Tj
+ Li (Tj−1 ,Tj−1 ) − K i
Lévy LIBOR
Cross-currency
j=1
e i = 1 + δK i (forward process approach)
Alternatively if we define K
FCi (0, TN ) =
N+1
X
B i (0,Tj )EPi,Tj
ei
1 + δLi (Tj−1 ,Tj−1 ) − K
+ =
Credit risk
Duality theory
,
j=1
N+1
X
Range options
Capital
Requirements
Spread Option
i
ei)
C (0,Tj , K
Net Asset Value
j=1
Reserve Capital
References
c Eberlein, Uni Freiburg, 128
Pricing cross-currency
derivatives (2)
Introduction
Lévy processes
Financial
modeling
Numerical evaluation of the cap price
Z t
1 + δLi (t,Tj−1 )
i,T
Define XTi j−1 (t) =
λi (s,Tj−1 ) dLs j = ln
1 + δLi (0,Tj−1 )
0
Valuation
Risk Managem.
Interest rate
Calibration
and let χi,Tj−1 (z) be its characteristic function, then
ei
C i (0,Tj , K̃ i ) = B i (0,Tj )K
exp(ξ˜ji R)
2π
Z
∞
exp(iu ξ˜ji )
−∞
Lévy LIBOR
χi,Tj−1 (iR − u)
du
(R + iu)(1 + R + iu)
Cross-currency
Range options
Credit risk
e i ) − ln(1 + δLi (0,Tj−1 )) and R is s.t. χi,Tj−1 (iR) < ∞.
where ξ˜ji = ln(K
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 129
Pricing cross-currency
derivatives (3)
Introduction
Lévy processes
Financial
modeling
Cross-currency swaps
Valuation
Floating-for-floating cross-currency (i; `; 0) swap
Risk Managem.
Libor rate Li (Tj−1 ,Tj−1 ) of currency i is received at each date Tj
Interest rate
Calibration
Lévy LIBOR
`
Libor rate L (Tj−1 ,Tj−1 ) of currency ` is paid
Cross-currency
Payments are made in units of the domestic currency
Range options
Credit risk
Thus the cashflow at time point Tj is (notional = 1)
Duality theory
δ(Li (Tj−1 ,Tj−1 ) − L` (Tj−1 ,Tj−1 ))
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 130
Pricing cross-currency
derivatives (4)
The time-0-value of a floating-for-floating (i; `; 0) cross-currency forward
swap in units of the domestic currency is

N+1
X
B i (0,Tj−1 )
0
CCFS[i;`;0] (0) = B (0,Tj ) 
exp (Di (0,Tj−1 ,Tj ))
B i (0,Tj )
j=1

N+1
X
B ` (0,Tj−1 )
`
exp (D (0,Tj−1 ,Tj ))
−
B ` (0,Tj )
j=1
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
where
Credit risk
Duality theory
Di (0,Tj−1 ,Tj )
Z Tj−1
= −
λi (s, Tj−1 )> cs ζ i (s,Tj ,Tj+1 ) ds
Capital
Requirements
Spread Option
0
Tj−1Z
Z
−
0
Rd
Net Asset Value
exp λi (s,Tj−1 )> x − 1 ζ i (s, x,Tj ,Tj+1 ) − 1 ν0,Tj (ds, dx) Reserve Capital
References
c Eberlein, Uni Freiburg, 131
Pricing cross-currency
derivatives (5)
Introduction
Lévy processes
Financial
modeling
Valuation
A quanto caplet with strike K i , which expires at time Tj−1 , pays at time Tj
QCpl i (Tj ,Tj , K i ) = δX i (Li (Tj−1 ,Tj−1 ) − K i )+
i
where X is the preassigned foreign exchange rate
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Time-0-value
i
Range options
i
0
i
i
i +
QCpl (0,Tj , K ) = B (0,Tj )IEP0,Tj [δX (L (Tj−1 ,Tj−1 ) − K ) ]
= B 0 (0,Tj )X i IEP0,Tj [(1 + δLi (Tj−1 ,Tj−1 ) − (1 + δK i ))+ ]
(forward process approach)
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 132
Pricing cross-currency
derivatives (6)
Numerical evaluation of quanto caplets. Write
Z
1 + δLi (Tj−1 ,Tj−1 ) = (1 + δLi (0,Tj−1 )) exp
Introduction
Lévy processes
Financial
modeling
Tj−1
i,Tj
λi (s,Tj−1 ) dLs
Valuation
Risk Managem.
0
= (1 + δLi (0,Tj−1 )) exp Mi (0,Tj−1 ,Tj ) + Di (0,Tj−1 ,Tj )
{z
} |
{z
}
|
random
non-random
Interest rate
Calibration
Lévy LIBOR
Cross-currency
then for v (x) = (e−x − 1)+
Range options
i
i
0
i
i
QCpl (0,Tj , K ) = B (0,Tj )X (1 + δK )(v ∗ %)(ξj )
Credit risk
Duality theory
Finally we get
QCpl i (0,Tj , K i ) = B 0 (0,Tj )X i (1 + δK i )
Z
i
exp(ξj R) ∞
χM ,Tj−1 (iR − u)
·
exp(iuξj )
du
2π
(R + iu)(R + 1 + iu)
−∞
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 133
Absolute errors of EUR caplet
calibration
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
3.0%
Range options
1
2.0%
2
3
1.0%
4
5
6
0.0%
1.50%
7
1.75%
2.00%
Strike Rate
Maturity (years)
9
2.50%
Duality theory
Capital
Requirements
Spread Option
8
2.25%
Credit risk
Net Asset Value
10
Reserve Capital
References
c Eberlein, Uni Freiburg, 134
Absolute errors of USD caplet
calibration
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
3.0%
1
2.0%
3
4
1.0%
5
6
0.0%
1.00%
1.25%
1.50%
2.00%
3.00%
Strike Rate
Credit risk
2
7
Maturity (years)
Duality theory
Capital
Requirements
Spread Option
8
9
Net Asset Value
10
Reserve Capital
References
c Eberlein, Uni Freiburg, 135
Basic interest rates
Introduction
Lévy processes
P(t,T ): price at time t ∈ [0, T ] of a default-free zero coupon bond
with maturity T ∈ [0,T ∗ ]
(P(T,T ) = 1)
f (t,T ): instantaneous forward rate
Financial
modeling
Valuation
Risk Managem.
R
T
P(t,T ) = exp − t f (t,u) du
L(t,T ): default-free forward Libor rate for the interval T to T + δ as
of time t ≤ T (δ-forward Libor rate)
P(t,T )
L(t,T ) := δ1 P(t,T
−1
+δ)
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Digital options
Range notes
FP (t,T ,U): forward price process for the two maturities T and U
FP (t,T ,U) :=
=⇒
P(t,T )
P(t,U)
P(t,T )
1 + δL(t,T ) =
= FP (t,T ,T + δ)
P(t,T + δ)
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Pricing of options on bonds
Introduction
Lévy processes
P(t,T ): underlying
Financial
modeling
w(P(t,T ), K ): payoff of a European option with maturity t and strike K
V (0, t,T , w): time-0-price of the option
V (0, t,T , w) =
Valuation
Risk Managem.
IEP [Bt−1 w(P(t,T ), K )]
Interest rate
Calibration
Lévy LIBOR
Caps, Floors, Swaptions, Digitals, Range options
Cross-currency
Range options
Turnbull (1995): floating range notes in 1-factor Gaussian HJM
Digital options
Range notes
Credit risk
Navatte and Quittard-Pinon (1999): delayed digital options
Nunes (2004): multifactor Gaussian HJM
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Forward measure and adjusted
forward measure
Introduction
Lévy processes
Financial
modeling
Valuation
Forward martingale measure for settlement day T
dPT
1
:=
= exp
dP
BT P(0,T )
Z
T
−
Risk Managem.
T
Z
A(s,T ) ds +
0
Σ(s,T ) dLs
0
Interest rate
Calibration
Lévy LIBOR
0
Adjusted forward measure PT 0,T for T < T
dPT 0,T
F (T 0,T 0,T )
P(0,T )
:=
=
dPT
F (0,T 0,T )
P(0,T 0 )P(T 0,T )
For 0 ≤ t ≤ T 0 :
dPT 0,T dPT 0 =
dP Ft
dP Ft
Cross-currency
Range options
Digital options
Range notes
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Digital options (1)
Introduction
Lévy processes
Standard European interest rate digital call (put) with strike rK
Financial
modeling
Valuation
SD(Θ)T [rn (T, T + δ); rk ; T ] := 1{Θrn (T,T +δ)>Θrk } ,
Risk Managem.
Interest rate
where rn (T, T + δ) is the reference rate (Libor)
1
1
rn (T, T + δ) =
−1
δ P(T, T + δ)
Calibration
Lévy LIBOR
Cross-currency
Range options
and Θ = 1 for a digital call, Θ = −1 for a digital put
Digital options
Range notes
Credit risk
Delayed digital option for maturity T and payment date T1
DD(Θ)T1 [rn (T, T + δ); rk ; T1 ] := 1{Θrn (T,T +δ)>Θrk }
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Digital options (2)
Introduction
Lévy processes
Delayed range digital options (T ≤ T1 )
DRDT1 [rn (T, T + δ); r` ; ru ; T1 ] := 1{rn (T,T +δ)∈[r` ,ru ]}
Obvious relationship for time-t prices
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
DRDt [rn (T, T + δ); r` ; ru ; T1 ] = P(t, T1 )
Lévy LIBOR
− DD(1)t [rn (T, T + δ); ru ; T1 ]
Cross-currency
− DD(−1)t [rn (T, T + δ); r` ; T1 ].
Range options
Digital options
Call-put parity only when L(P(T, T + δ)) without point masses
Range notes
Credit risk
DD(1)t [rn (T, T + δ); rk ; T1 ] = P(t, T1 )
− DD(−1)t [rn (T, T + δ); rk ; T1 ]
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Pricing formulae for delayed
digital options (1)
Introduction
Lévy processes
Financial
modeling
Dt := DD(1)t [rn (T, T + δ); rk ; T1 ]
1
= Bt IEP
1{rn (T,T +δ)>rk } Ft
BT1
h
i
= P(t,T1 )IET1 1{rn (T,T +δ)>rk } Ft
h
i
= P(t,T1 )IET1 1nP(T,T +δ)< 1 o Ft
δr +1
k
P(t,T + δ)
= P(t,T1 )h
P(t,T )
h
i
where h(y ) = IET1 1ny exp − R T A(s,T,T +δ) ds+R T Σ(s,T,T +δ) dL < 1 o
[ t
s ] δr +1
t
k
and
A(s, T, T + δ) = A(s,T + δ) − A(s,T ),
Σ(s, T ,T + δ) = Σ(s,T + δ) − Σ(s,T )
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Digital options
Range notes
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Pricing formulae for delayed
digital options (2)
Z
Denote X
Σ(s,T,T + δ) dLs
Z T
1
:=
exp
A(s,T,T + δ) ds
δrK + 1
t
t
PXT1 = distribution of X under PT1
Z
then h(y ) =
1
ex < Ky
dPXT (x)
1
Z
=
Lévy processes
Financial
modeling
T
:=
K
Introduction
fy (−x)ϕ(x) dx = (fy ∗ ϕ)(0) = V (0)
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Digital options
Range notes
Credit risk
for fy (x) = 1
e−x < Ky
and V (ζ) = (fy ∗ ϕ)(ζ)
Denote by MTX1 the moment generating function of X w.r.t. PT1
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Pricing formulae for delayed
digital options (3)
Introduction
Lévy processes
Financial
modeling
Theorem
Valuation
Suppose the distribution of X possesses a Lebesgue density. Choose
an R > 0 such that MTX1 (−R) < ∞. Then
Dt =
1
P(t,T1 )
π
∞
Z
<
0
P(t,T )
K
P(t,T + δ)
R+iu
1
MTX (−R − iu) du
R + iu 1
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Digital options
Proof: L[V ](z) = L[fy ](z)L[ϕ](z)
Range notes
Credit risk
V (0) =
1
lim
2πi Y →∞
L[fy ](R + iu) =
R+iY
Z
L[V ](z) dz
R−iY
1
R+iu
R+iu
K
y
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Pricing range notes (1)
Introduction
Lévy processes
n 0−
T0
t
Financial
modeling
n 0+
n1
T1
Valuation
Risk Managem.
T2
TN
Interest rate
Calibration
Lévy LIBOR
Tj
nj
=
=
δj
Tj,i
δj,i
=
=
=
coupon payment dates
number of days between Tj and Tj+1 based on some day
count convention
number of years between Tj and Tj+1
Tj + i
length (in years) of the compounding period starting at Tj,i
Cross-currency
Range options
Digital options
Range notes
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Pricing range notes (2)
Introduction
Lévy processes
For a floating range note, the coupon at time Tj+1 is
νj+1 (Tj+1 ) :=
rn (Tj ,Tj + δj ) + sj
H(Tj ,Tj+1 )
Dj
Financial
modeling
Valuation
Risk Managem.
Interest rate
where sj is the spread over the reference rate
Calibration
Dj number of days for the (j +1)-th compounding period
nj
X
H(Tj ,Tj+1 ) =
1{r` (Tj,i )≤rn (Tj,i ,Tj,i +δj,i )≤ru (Tj,i )}
i=1
Lévy LIBOR
Cross-currency
Range options
Digital options
Range notes
Time-t value of a flaoting range note
F`RN(t) := P(t,TN ) +
Credit risk
N−1
X
j=0
Duality theory
νj+1 (t)
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Valuation of F`RN coupons (1)
Introduction
Lévy processes
Financial
modeling
Valuation
1 rn (T0 ,T0 + δ0 ) + s0
ν1 (t) = Bt IE
H(T0 ,T1 )Ft
BT1
D0
h
i
rn (T0 ,T0 + δ0 ) + s0
=
P(t,T1 )IET1 H(T0 ,T1 )Ft
D0
rn (T0 ,T0 + δ0 ) + s0
=
P(t,T1 )H(T0 , t)
D0
n0
i
h
X
+
P(t, T1 ) IET1 1{r` (T0,i )≤rn (T0,i ,T0,i +δ0,i )≤ru (T0,i )} Ft
{z
}
|
i=n0− +1
=DRDt [rn (T0,i ,T0,i +δ0,i );r` (T0,i );ru (T0,i );T1 ]
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Digital options
Range notes
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Valuation of F`RN coupons (2)
Introduction
νj+1 (t) = P(t,Tj+1 )IETj+1
=
sj
1
−
Dj
δj Dj
rn (Tj ,Tj+1 ) + sj
H(Tj ,Tj+1 )Ft
Dj
nj
P(t,Tj+1 )
Risk Managem.
P(t,Tj+1 ) X
1
IETj+1
1{r` (Tj,i )≤rn (Tj,i ,Tj,i +δj,i )≤ru (Tj,i )} Ft
δj Dj
P(Tj ,Tj+1 )
i=1
=:
1
νj+1
(t)
+
2
νj+1
(t).
Calibration
Lévy LIBOR
Range options
Digital options
=
Interest rate
Cross-currency
Note that
1
νj+1
(t)
Financial
modeling
i
h
Valuation
IETj+1 1{r` (Tj,i )≤rn (Tj,i ,Tj,i +δj,i )≤ru (Tj,i )} Ft
i=1
nj
+
X
Lévy processes
sj
1
−
Dj
δj Dj
nj
X
Range notes
DRDt [rn (Tj,i ,Tj,i + δj,i ); r` (Tj,i ); ru (Tj,i );Tj+1 ]
Credit risk
Duality theory
i=1
Capital
Requirements
and
2
νj+1
(t) =
nj
X
P(t,Tj )
IETj ,Tj+1 1{r` (Tj,i )≤rn (Tj,i ,Tj,i +δj,i )≤ru (Tj,i )} Ft
δj Dj |
{z
}
i=1
j,i
=:Dt
Spread Option
Net Asset Value
Reserve Capital
References
Valuation of F`RN coupons (3)
Introduction
Neglect now the indices i, j
Lévy processes
Theorem
Financial
modeling
Suppose the distribution of X possesses a Lebesgue density. Choose
an R > 0 such that M X (−R) < ∞. Then
!
R+iu
Z
P(t,T )
1 ∞
1
X
Dt =
<
K
M (−R − iu) du
π 0
P(t,T + δ)
R + iu
!
R+iu
Z
P(t,T )
1 ∞
1
−
<
K
M X (−R − iu) du
π 0
P(t,T + δ)
R + iu
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Digital options
Range notes
with
Credit risk
K :=
K :=
T
1
exp
δr` (T ) + 1
Z
1
exp
δru (T ) + 1
Z
A(s,T, T + δ) ds
Capital
Requirements
t
T
A(s,T, T + δ) ds
t
Duality theory
Spread Option
Net Asset Value
Reserve Capital
References
Lévy credit risk model
Introduction
0.20
Lévy processes
Caa
Financial
modeling
Valuation
0.15
B3
Risk Managem.
Interest rate
0.10
B1
Calibration
Lévy LIBOR
Cross-currency
Ba3
Ba2
Ba1
0.05
Baa3
Baa1
A1
Aaa
Range options
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Government−Bond
Spread Option
0.0
Interest rate
B2
Net Asset Value
0
2
4
6
8
10
Reserve Capital
Maturities in years
References
c Eberlein, Uni Freiburg, 149
Basic interest rates
Introduction
Lévy processes
B(t,T ): price at time t ∈ [0, T ] of a default-free zero coupon bond
with maturity T ∈ [0, T ∗ ]
(B(T ,T ) = 1)
f (t,T ): instantaneous forward rate
Financial
modeling
Valuation
Risk Managem.
R
T
B(t,T ) = exp − t f (t,u) du
L(t,T ): default-free forward Libor rate for the interval T to T + δ as
of time t ≤ T (δ-forward Libor rate)
B(t,T )
L(t,T ) := δ1 B(t,T
−1
+δ)
FB (t,T ,U): forward price process for the two maturities T < U
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Credit derivatives
Duality theory
FB (t,T ,U) :=
=⇒
1 + δL(t,T ) =
B(t,T )
B(t,U)
Capital
Requirements
B(t,T )
= FB (t,T ,T + δ)
B(t,T + δ)
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 150
The Lévy Libor model with
default risk
(Eberlein, Kluge, Schönbucher 2006)
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
B 0 (t,Tk ): time-t price of a defaultable zero coupon bond with zero
recovery and maturity Tk
τ : time of default
Interest rate
Calibration
Lévy LIBOR
Cross-currency
B(t,Tk ): pre-default value of the defaultable bond
Range options
=⇒
B 0 (t,Tk ) = 1{τ >t} B(t,Tk ), B(Tk ,Tk ) = 1 (k = 1, . . . , n)
Credit risk
Credit derivatives
Terminal value of the defaultable bond
Duality theory
B 0 (Tk ,Tk ) = 1{τ >Tk } B(Tk ,Tk ) = 1{τ >Tk }
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 151
The Lévy Libor model with
default risk (2)
• The defaultable forward Libor rates for the interval [Tk , Tk +1 ] are
given by
Introduction
Lévy processes
Financial
modeling
Valuation
1
L(t,Tk ) :=
δk
B(t,Tk )
−1 .
B(t,Tk +1 )
• The forward Libor spreads are given by
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
S(t,Tk ) := L(t,Tk ) − L(t,Tk ).
• The default risk factors or forward survival processes are given by
Cross-currency
Range options
Credit risk
Credit derivatives
B(t,Tk )
.
D(t,Tk ) :=
B(t,Tk )
Duality theory
• The discrete-tenor forward default intensities are given by
Capital
Requirements
Spread Option
1
H(t,Tk ) :=
δk
D(t,Tk )
S(t,Tk )
−1 =
.
D(t,Tk +1 )
1 + δL(t,Tk )
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 152
Canonical construction of the
time of default
Introduction
Lévy processes
Let Γ = (Γt )t≥0 be an (Fet )-adapted, right-continuous, increasing
e F,
e PT ∗ ), Γ0 = 0, limt→∞ Γt = ∞.
process on (Ω,
b F,
b uniformly distributed on [0, 1].
b P)
Let η be a random variable on (Ω,
Financial
modeling
b
b
G := Fe ⊗ F,
QT ∗ := PT ∗ ⊗ P
e
(Ft ) trivial extension of (Ft ) to (Ω, G, QT ∗ )
Interest rate
e × Ω,
b
Define Ω := Ω
−Γt
τ := inf{t ∈ R+ : e
≤ η}
Denote Ht := σ 1{τ ≤u} | 0 ≤ u ≤ t , Gt := Ft ∨ Ht
Valuation
Risk Managem.
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Credit derivatives
=⇒
τ is a (Gt )-stopping time
QT ∗ {τ > s|FT ∗ } = QT ∗ {τ > s|Fs } = e−Γs
Duality theory
(0 ≤ s ≤ T ∗ )
Capital
Requirements
Spread Option
=⇒ (Γt ) is the (Ft )-hazard process of τ under QT ∗
(and also under all QTk )
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 153
Consequences for the price of
a defaultable bond
Introduction
Lévy processes
Payoff at maturity: B 0 (Tk ,Tk ) = 1{τ >Tk }
=⇒
Financial
modeling
0
B (t,Tk ) = B(t,Tk )IEQTk [1{τ >Tk } |Gt ]
= B(t,Tk )1{τ >t}
Risk Managem.
e−Γt
Interest rate
Calibration
Therefore, define
B(t,Tk ) := B(t,Tk )
=⇒
Valuation
IEQTk [1{τ >Tk } |Ft ]
H(t,Tk ) =
1
δk
IEQTk [1{τ >Tk } |Ft ]
Lévy LIBOR
e−Γt
Cross-currency
−ΓT
Range options
IEQTk [e k Ft ]
−
1
−Γ
IEQTk +1 [e Tk +1 Ft ]
Credit risk
Credit derivatives
(ΓTk )k=1,...,n can be chosen such that H(t,Tk ) has the form
Z t
Z t
T
1/2
H(t,Tk ) = H(0, Tk ) exp
bH (s,Tk ,Tk +1 ) ds +
cs γ(s,Tk ) dWs k+1
0
0
Z tZ
hγ(s, Tk ), xi(µ − ν
+
0
Tk +1
)(ds, dx) .
Rd
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 154
Defaultable forward measures
Introduction
Lévy processes
The defaultable forward measure (or survival measure) QTi for the
settlement day Ti is defined on (Ω, GTi ) by
dQTi
B(0,Ti ) 0
B(0,Ti )
:= 0
B (Ti ,Ti ) =
1{τ >Ti } .
dQTi
B (0,Ti )
B(0,Ti )
=⇒
QTi (A) = QTi (A|{τ > Ti })
(A ∈ GTi ),
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
forward measure conditioned on survival until Ti −→ survival measure
Cross-currency
Range options
Denote PTi := QTi |FTi
The restricted defaultable forward measure PTi for the settlement day Ti
is defined on (Ω, FTi ) by
i−1
dPTi
B(0,Ti )
B(0,Ti ) Y
1
=
QTi ({τ > Ti }|FTi ) =
.
dPTi
B(0,Ti )
B(0,Ti ) k =0 1 + δk H(Tk ,Tk )
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 155
Successive restricted
defaultable forward measures
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
The defaultable Libor rate (L(t, Ti ))0≤t≤Ti turns out to be a
PTi+1 -martingale and
dPTi
dPTi+1
=
Ft
B(0, Ti+1 )
1 + δi L(t, Ti )
(1 + δi L(t, Ti )) =
B(0, Ti )
1 + δi L(0, Ti )
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 156
Pricing contingent claims with
defaultable forward measures
X promised payoff at day Ti with zero recovery upon default
Lévy processes
Financial
modeling
Valuation
πtX its price at time t ∈ [0, Ti ]
πtX
Introduction
Risk Managem.
= 1{τ >t} B(t,Ti )IEQTi [X 1{τ >Ti } |Gt ]
(t ∈ [0, Ti ])
Interest rate
Calibration
The defaultable forward measures QTi and PTi are the appropriate tools.
Lévy LIBOR
If X is GTi -measurable
Range options
πtX = 1{τ >t} B(t,Ti )IEQT [X |Gt ] = B 0 (t,Ti )IEQT [X |Gt ].
i
Cross-currency
Credit risk
Credit derivatives
i
Duality theory
If X is FTi -measurable
Capital
Requirements
πtX = 1{τ >t} B(t,Ti )IEPT [X |Ft ] = B 0 (t,Ti )IEPT [X |Ft ].
i
i
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 157
Recovery rules and bond
prices
Defaultable zero coupon bonds
−→ fractional recovery of treasury value scheme
At maturity of the bond
B π (T , T ) = 1{τ >T } + π1{τ ≤T } = π + (1 − π)1{τ >T }
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Time-t value (t ∈ [0, T ])
B π (t,T ) = πB(t,T ) + (1 − π)1{τ >t} B(t,T )
Defaultable coupon bonds −→ recovery of par scheme
Recovery of par: If default occurs in the time interval (Tk , Tk+1 ],
recovery is given by the recovery rate π times the sum of the notional
and the accrued interest over (Tk , Tk +1 ]. It is paid at Tk +1 .
Corresponding cashflow pattern
• at Tk +1 (k = 0, . . . , m − 1):
• at Tm :
Introduction
Lévy LIBOR
Cross-currency
Range options
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
c1{τ >Tk +1 } + π(1 + c)1{Tk <τ ≤Tk +1 }
Net Asset Value
Reserve Capital
1{τ >Tm }
References
c Eberlein, Uni Freiburg, 158
Default payments
Introduction
Lévy processes
Financial
modeling
Denote by ekX (t) the time-t value of receiving an amount of X at Tk +1
⇔ default occured in period (Tk , Tk+1 ]
Valuation
Risk Managem.
Interest rate
Calibration
Lemma
Lévy LIBOR
If X is FTk -measurable, then for t ≤ Tk
Cross-currency
Range options
ekX (t) = 1{τ >t} B(t, Tk +1 )δk IEPT
k+1
[XH(Tk , Tk )|Ft ]
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 159
Pricing of defaultable coupon
bonds
Introduction
Lévy processes
Financial
modeling
Fixed coupon of c to be paid at dates T1 , . . . , Tm
π
Bfixed
(0) = B(0,Tm ) +
m−1
X
Valuation
B(0,Tk+1 ) c + π(1 + c)δk IEPT
k +1
k =0
[H(Tk ,Tk )] .
Interest rate
Calibration
Floating coupon bond that pays Libor plus a constant spread x
Promised payoff at the date Tk +1 : δk (L(Tk ,Tk ) + x)
π
Bfloating
(0)
Risk Managem.
= B(0,Tm ) +
m−1
X
Lévy LIBOR
Cross-currency
Range options
δk B(0,Tk +1 ) x + IEPT
k +1
k =0
+ π(1 + δk x)IEPT
+ πδk IEPT
k +1
k +1
[L(Tk ,Tk )]
[H(Tk ,Tk )]
[H(Tk ,Tk )L(Tk ,Tk )] .
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 160
Numerical aspects
Introduction
Lévy processes
Financial
modeling
Z
t
t
Z
H
Valuation
T
1/2
cs γ(s, Tk ) dW s k +1
b (s, Tk , Tk +1 ) ds +
H(t, Tk ) = H(0, Tk ) exp
0
0
Z tZ
hγ(s, Tk ), xi(µ − ν
+
0
Tk +1
(ds, dx)
Rd
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Drift coefficient bH (s, Tk , Tk +1 ) to be approximated
Range options
Credit risk
IEPT
[H(Tk , Tk )L(Tk , Tk )]
k +1
1
=
L(0, Tk ) − IEPT [L(Tk , Tk )] − IEPT [H(Tk , Tk )]
k +1
k +1
δk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 161
Credit default swaps (CDS)
Introduction
Lévy processes
Financial
modeling
Valuation
Standard default swap: Default of a coupon bond
A receives:
1 − π(1 + c)
Risk Managem.
(fixed coupon)
Interest rate
1 − π(1 + δk (L(Tk ,Tk ) + x)) (floating coupon)
P
Time-0 value of the fee payments: s m
k=1 B(0,Tk−1 )
s default swap rate
Calibration
Lévy LIBOR
Cross-currency
Range options
m X
B(0,Tk )δk−1 IEPT [H(Tk −1 ,Tk−1 )]
1 − π(1 + c)
sfixed = Pm
k
k =1 B(0,Tk −1 ) k =1
m
X
1
B(0,Tk )δk −1 (1 − π(1 + δk−1 x))
sfloating = Pm
k=1 B(0,Tk −1 ) k =1
× IEPT [H(Tk −1 ,Tk −1 )]−πδk −1 IEPT
k
k
[H(Tk−1 ,Tk −1 )L(Tk −1 ,Tk−1 )]
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 162
Credit default swaptions (1)
Introduction
Assumption: The volatility structures factorize in the following way:
λ(s, Ti ) = λi σ(s)
and
Lévy processes
(0 ≤ s ≤ Ti ).
γ(s, Ti ) = γi σ(s)
Financial
modeling
Valuation
Payoff of a credit default swaption that is knocked out at default with
strike S and maturity Ti on a CDS that terminates at Tm :
!
m−1
X
+
B(Ti , Tk )
1{τ >Ti } (s(Ti ; Ti , Tm ) − S)
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
k =i
Cross-currency
where s(Ti ; Ti , Tm ) denotes the default swap rate at Ti .
Range options
Credit risk
Price at time 0:
Credit derivatives
"
π0CDS = B(0, Ti )IEPT
i
Duality theory
(1 − π(1 + c))δm−1 C i,m−1 H(Ti , Tm−1 )
Qm−1
l=i (1 + δl L(Ti , Tl ))(1 + δl H(Ti , Tl ))
+
m−2
X
k =i
(1 − π(1 + c))δk C i,k H(Ti , Tk ) − S
−S
Qk
l=i (1 + δl L(Ti , Tl ))(1 + δl H(Ti , Tl ))
Capital
Requirements
!+ #
.
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 163
Credit default swaptions (2)
Forward Libor rates and default intensities can be written as
λ
l
XTi + BlL ,
L(Ti , Tl ) = L(0, Tl ) exp
σsum
γl
H(Ti , Tl ) = H(0, Tl ) exp
XTi + BlH
σsum
R Ti
Pm−1
∗
with σsum := l=i (λl +γl ), XTi := σsum 0 σ(s) dLTs and constants BlL ,
BlH .
Assume the distribution of XTi w.r.t. PTi has a Lebesgue-density ϕ, then
Z
π0CDS = B(0, Ti ) g(−x)ϕ(x) dx = B(0, Ti )(g ∗ ϕ)(0)
R
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Credit derivatives
for some (explicitly given) function g.
Duality theory
Performing Laplace and inverse Laplace transformations and denoting
Capital
Requirements
XT
by M Ti i the PTi -moment generating function of XTi yields
π0CDS
1
= B(0, Ti )
π
∞
Z
0
XT
< L[g](R + iu)M Ti i (−R − iu) du.
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 164
Options on defaultable bonds
(1)
Introduction
Lévy processes
Financial
modeling
Payoff of a call with maturity Ti and strike K ∈ (0, 1) on a defaultable
zero coupon bond with maturity Tm (i < m) which is knocked out at
default
πTCO
(K , Ti , Tm ) = 1{τ >Ti } (B π (Ti , Tm ) − K )+
i
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Price at time 0:
π0CO
Cross-currency
= B(0, Tm )IEPT
m
m−1
Y
π
1 + δ` H(Ti , T` ) + (1 − π)
Credit risk
Credit derivatives
`=i
−K
Range options
m−1
Y
+
1 + δ` L(Ti , T` ) 1 + δ` H(Ti , T` )
Duality theory
Capital
Requirements
`=i
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 165
Options on defaultable bonds
(2)
Introduction
Lévy processes
Financial
modeling
Valuation
Using again a convolution representation
Z
π0CO = B(0, Tm ) g(−x)ϕ(x) dx = B(0, Tm )(g ∗ ϕ)(0)
R
XT
one gets for an R > 0 such that M Tmi (−R) < ∞ the following
(approximate) formula
Z
XT
1 ∞
< L[g](R + iu) · M Tmi (−R − iu) du
π0CO (K , Ti , Tm ) = B(0, Tm )
π 0
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 166
Further credit derivatives
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
• Total rate of return swaps
Calibration
• Asset package swaps
Lévy LIBOR
Cross-currency
• Credit spread options
Range options
Credit risk
Credit derivatives
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 167
The Theme
Introduction
Call option in FX market:
Lévy processes
Euro/Dollar
Gives you the right to buy Euros paying in Dollars.
At the same time a right to sell Dollars getting Euros.
Financial
modeling
Valuation
Risk Managem.
Payoff
(ST − K )+
(St ) exchange rate, K strike
Interest rate
Calibration
Lévy LIBOR
(ST − K )
+
= KST
= KST
1
1 +
−
K
ST
+
K 0 − ST0
↑
Dollar/Euro rate
Cross-currency
Range options
Credit risk
Duality theory
Exponential
semimartingale models
Call-Put Duality
Call price = K · put price (in the dual market)
Multiasset setting
Capital
Requirements
−→ duality principle
Spread Option
Net Asset Value
Reserve Capital
Brief literature survey
•
•
•
•
•
•
•
•
•
Carr (1994) put-call duality in BS-setting and for diffusions
Chesney and Gibson (1995) two-factor diffusion model
Bates (1997) diffusions and jump-diffusions
Introduction
Lévy processes
Financial
modeling
Schroder (1999) various payoffs in diffusions and jump-diffusions
Valuation
Carr, Ellis, and Gupta (1998) static hedging strategies for exotic
derivatives
Risk Managem.
Carr and Chesney (1996) put-call for American options
Detemple (2001) American options with general payoffs
Henderson and Wojakowski (2002) Asian options
Eberlein and Papapantoleon (2005a,b) Exotic options for Lévy and
time-inhomogeneous Lévy models
• Vanmaele, Deelstra, Liinev, Dhaene, Goovaerts (2006) Forward
start Asian options
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
•
•
•
•
Fajardo and Mordecki (2006a,b) Lévy models
Vecer (2002), Vecer and Xu (2004) Asian options (PIDE)
Eberlein, Kluge, and Papapantoleon (2006) Interest rate options
Eberlein, Papapantoleon, Shiryaev (2006) Semimartingales
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Exponential semimartingale models
Introduction
Lévy processes
Let B = (Ω, F, F, P) be a stochastic basis, where F = FT and
F = (Ft )0≤t≤T . We model the price process of a financial asset as an
exponential semimartingale
Ht
St = e ,
Financial
modeling
Valuation
Risk Managem.
Interest rate
0 ≤ t ≤ T.
Calibration
H = (Ht )0≤t≤T is a semimartingale with canonical representation
H = H0 + B + H c + h(x) ∗ (µH − ν) + (x − h(x)) ∗ µH
or, in detail
Ht = H0 + Bt + Htc +
Lévy LIBOR
Cross-currency
Range options
Credit risk
Z tZ
0
h(x)d(µH − ν) +
R
Duality theory
Z tZ
0
(x − h(x))dµH ,
R
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
where
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
• h = h(x) is a truncation function; canonical choice
h(x) = x1{|x|≤1} ;
Introduction
Lévy processes
• B = (Bt )0≤t≤T is a predictable process of bounded variation;
• H c = (Htc )0≤t≤T is the continuous martingale part of H;
• ν = ν(ω; dt, dx) is the predictable compensator of the random
measure of jumps µH = µH (ω; dt, dx) of H.
c
For the processes B, C = hH i, and the measure ν we use the notation
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
T(H|P) = (B, C, ν)
which will be called the triplet of predictable characteristics of the
semimartingale H with respect to the measure P.
Cross-currency
Range options
Credit risk
Duality theory
Assumption: The truncation function satisfies the antisymmetry
property
h(−x) = −h(x).
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Alternative model description
Introduction
E(X ) = (E(X )t )0≤t≤T
stochastic exponential
Lévy processes
Financial
modeling
Valuation
e t, 0 ≤ t ≤ T
St = E(H)
et
dSt = St− d H
Risk Managem.
Interest rate
Calibration
where
e t = Ht + 1 hH c it +
H
2
Z tZ
0
(ex − 1 − x)µH (ds, dx)
Lévy LIBOR
Cross-currency
R
Range options
Note
Credit risk
e t = exp(H
e t − 1 hH
e c it )
E(H)
2
Y
0<s≤t
e s ) exp(−∆H
es )
(1 + ∆H
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
e > −1.
Asset price positive only if ∆H
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Martingale and dual martingale
measures
Introduction
Lévy processes
Assumption (ES)
Financial
modeling
The process 1{x>1} ex ∗ ν has bounded variation.
Valuation
Risk Managem.
Interest rate
Then, H is exponentially special and
S = eH ∈ Mloc (P) ⇔ B +
C
+ (ex − 1 − h(x)) ∗ ν H = 0.
2
Moreover, we assume that S ∈ M(P), therefore EST = 1. Define on
(Ω, F, (Ft )0≤t≤T ) a new probability measure P 0 with
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
dP 0
= ST .
dP
Since S > 0 (P-a.s.), we have P P 0 and
1
dP
=
.
dP 0
ST
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Introduce the process
1
.
S
Then, denoting by H 0 the dual of the semimartingale H, i.e. H 0 = −H,
we have
0
S 0 = eH .
S0 =
Introduction
Lévy processes
Financial
modeling
Valuation
Proposition
Risk Managem.
H
Suppose S = e ∈ M(P) i.e. S is a P-martingale. Then the process
S 0 ∈ M(P 0 ) i.e. S 0 is a P 0 -martingale.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Lemma
Range options
Let f be a predictable, bounded process. The triplet ofR predictable
·
characteristics of the stochastic integral process J = 0 f dH, denoted by
T(J|P) = (BJ , CJ , νJ ), is
BJ = f · B + [h(fx) − fh(x)] ∗ ν
CJ = f 2 · C
1A (x) ∗ νJ = 1A (fx) ∗ ν,
Credit risk
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
A ∈ B(R).
Spread Option
Net Asset Value
Reserve Capital
Theorem
The triplet T(H 0 |P 0 ) = (B 0 , C 0 , ν 0 ) can be expressed via the triplet
T(H|P) = (B, C, ν) by the following formulae:
B 0 = −B − C − h(x)(ex − 1) ∗ ν
Lévy processes
Financial
modeling
0
C = C
1A (x) ∗ ν 0 = 1A (−x)ex ∗ ν,
Introduction
Valuation
A ∈ B(R).
Risk Managem.
Interest rate
Calibration
Structure of the proof:
Lévy LIBOR
(G)
5
0
Cross-currency
T(H|P )
(a)
(c)
T(H|P)
(b)
(−)
(d)
)
Range options
(−)
Credit risk
)
5
T(H 0 |P 0 )
(G)
T(H 0 |P)
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
(G)
/ : Girsanov’s theorem,
(−)
/ : dual of a semimartingale.
Net Asset Value
Reserve Capital
Symmetry of markets
Introduction
Lévy processes
If the original market (S, P) and the dual market (S 0 , P 0 ) satisfy
Law(S|P) = Law(S 0 |P 0 )
Financial
modeling
Valuation
Risk Managem.
Interest rate
then we say these markets are symmetric.
Calibration
In cases where the triplets T(H|P) and T(H 0 |P 0 ) determine these laws
completely (e.g. for Lévy processes H and H 0 )
0
symmetry holds iff ν = ν
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
The equation in the Theorem is then
Exponential
semimartingale models
x
1A (x) ∗ ν = 1A (−x)e ∗ ν,
A ∈ B(R)
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Introduction
Example 1: Diffusion models
Lévy processes
Financial
modeling
dSt = St σ(t, St ) dWt ,
S0 = 1
Valuation
Risk Managem.
local volatility models (Dupire (1994), Skiadopoulos (2001))
Interest rate
Calibration
t
Z
1 t 2
σ(u, e ) dWu −
Ht =
σ (u, eHu ) du
2 0
0
Z
Z ·
1 · 2
σ (u, eHu ) du, C =
σ 2 (u, eHu ) du,
⇒
B=−
2 0
0
Z
Lévy LIBOR
Hu
Cross-currency
ν≡0
Range options
Credit risk
Duality theory
Exponential
semimartingale models
0
Theorem ⇒ B = −B − C =
− 21
R·
0
2
Hu
0
0
σ (u, e ) du, C = C, ν ≡ 0
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Example 2: Purely discontinuous Lévy models
Introduction
Lévy processes
St = eHt ,
T(H, P) = (B, 0, ν)
local characteristics: Bt (ω) = bt, ν(ω; dt, dx) = dtF (dx),
F Lévy measure
Z
S ∈ Mloc (P) ⇔ b = − (eX − 1 − h(x))F (dx)
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
R
Lévy LIBOR
Actually: S ∈ M(P)
Cross-currency
ϑx
Parametric models: F (dx) = e f (x) dx
f even
Generalized hyperbolic (includes hyperbolic, NIG, VG, . . . )
CGMY, Meixner
Z
Z
Dual process H 0 : 1A (x)F 0 (dx) = 1A (−x)e(1+ϑ)x f (x) dx
Z
b0 = − (ex − 1 − h(x))F 0 (dx)
R
Range options
Credit risk
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
European options (1)
Introduction
Lévy processes
Theorem
Financial
modeling
The prices of standard call and put options satisfy the following duality
relations:
CT (S; K ) = K P0T (K 0 ; S 0 )
Valuation
and
Calibration
Risk Managem.
Interest rate
PT (K ; S) = K C0T (S 0 ; K 0 ).
Lévy LIBOR
Cross-currency
Proof: Using the dual measure
Range options
Credit risk
h (S − K )+ i
h (S − K )+ i
T
T
CT (S; K ) = E ST
= E0
= E 0 [(1 − KST0 )+ ]
ST
ST
h 1
+ i
= KE 0
− ST0
= KE 0 [(K 0 − ST0 )+ ],
K
where K 0 =
1
K
.
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
European options (2)
Introduction
Lévy processes
Financial
modeling
Corollary
Valuation
Call and put prices in a dual pair of markets (S, P) and (S 0 , P 0 ) satisfy a
call-call parity
CT (S; K ) = K C0T (S 0 ; K 0 ) + 1 − K
and a put-put parity
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
PT (K ; S) = K
P0T (K 0 ; S 0 )
Range options
+K −1
Credit risk
Duality theory
Proof: Combine with classical call-put parity
CT (S; K ) = PT (K ; S) + 1 − K
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Floating strike lookback options (1)
Introduction
Lévy processes
Payoff of a call:
0
Assume H =
ST − α inf St
0≤t≤T
+
for an α ≥ 1
Valuation
(Ht0 )0≤t≤T
satisfies the reflection principle
Law sup Ht0 − HT0 |P 0 = Law(− inf Ht0 |P 0 )
t≤T
t≤T
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
(holds for Lévy processes), then
CT (S; α inf S) = αP0T
Financial
modeling
Cross-currency
1
α
; inf S 0
Range options
Credit risk
Duality theory
Exponential
semimartingale models
Value of a floating strike lookback call
→ value of a fixed strike lookback put
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Floating strike lookback options (2)
Introduction
Lévy processes
Payoff of a put:
+
β sup St − ST
for a 0 < β ≤ 1
Valuation
0≤t≤T
0
Assume H =
(Ht0 )0≤t≤T
Financial
modeling
Risk Managem.
satisfies
Interest rate
Law HT0 − inf Ht0 |P 0 = Law sup Ht0 |P 0
t≤T
t≤T
Calibration
Lévy LIBOR
(holds for Lévy processes), then
Cross-currency
PT (β sup S; S) = βC0T sup S 0 ;
1
Range options
β
Credit risk
Duality theory
Exponential
semimartingale models
Value of a floating strike lookback put
→ value of a fixed strike lookback call
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Floating strike Asian options
Introduction
Lévy processes
Payoff of a call:
1
ST −
T
T
Z
St dt
+
0
Assume H 0 = (Ht0 )0≤t≤T satisfies
Financial
modeling
Valuation
Risk Managem.
0
0
0
0
Law(HT0 − H(T
−t)− ; 0 ≤ t < T |P ) = Law(Ht ; 0 ≤ t < T |P )
(holds for Lévy processes), then
Z
1 Z
1
S) = P0T 1;
S0
CT (S;
T
T
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Value of a floating strike Asian call
→ value of a fixed strike Asian put
1 Z
1 Z
Similarly PT
S; S = C0T
S0, 1
T
T
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Forward-start options
Introduction
Lévy processes
Payoff of a call:
(ST − St )+
Financial
modeling
Payoff of a put:
(St − ST )+
Valuation
Risk Managem.
0
Assume H =
(Ht0 )0≤t≤T
Interest rate
satisfies
0
0
0
0
Law(HT0 − H(T
−t)− ; 0 ≤ t < T |P ) = Law(Ht ; 0 ≤ t < T |P )
Calibration
Lévy LIBOR
Cross-currency
then
Ct,T (S; S) =
P0T −t (1; S 0 )
Pt,T (S; S) =
C0T −t (S 0 ; 1)
Range options
Credit risk
and
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Value of a forward-start call
→ value of a plain vanilla put
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Multiasset price model
Introduction
Lévy processes
Financial
modeling
Each component S i of the vector of asset price processes
S = (S 1 , . . . , S d )> is an exponential time-inhomogeneous Lévy process
Sti = S0i exp Lit ,
0 ≤ t ≤ T.
d
The driving process L = (Lt )0≤t≤T is an R -valued time-inhomogeneous
Lévy process that satisfies Assumption (EM), with canonical
decomposition
Z t
Z t
Z tZ
1/2
Lt =
bs ds +
cs dWs +
x(µL − ν)(ds, dx).
0
0
0
Rd
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Multiasset price model (2)
Introduction
Theorem
Lévy processes
d
Let L = (Lt )0≤t≤T be an R -valued PIIAC that satisfies Assumption
(EM), with characteristics T(L|P) = (B, C, ν). Let u, v be vectors in Rd
such that v ∈ (−M, M)d and u + v ∈ [−M, M]d . Define the measure P 0
dP 0
ehv ,LT i
.
=
dP
E[ehv ,LT i ]
Valuation
Risk Managem.
Interest rate
Calibration
Then, the process Lu = (Lut )0≤t≤T , where Lut := hu, Lt i, is a
1-dimensional PIIAC with characteristics T(Lu |P 0 ) = (B u , C u , ν u ) with
Z
bsu = hu, bs i + hu, cs v i +
hu, xi ehv ,xi − 1 λs (dx)
Rd
csu = hu, cs ui
Financial
modeling
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
λus (E) = λ0s ({x ∈ Rd : hu, xi ∈ E}),
E ∈ B(R),
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
where λ0s is a measure defined
by
Z
λ0s (A) =
ehv ,xi λs (dx),
A
Application: Multiasset options
A ∈ B(Rd ).
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Example: Swap option
(Margrabe)
Introduction
Lévy processes
Financial
modeling
Theorem
Valuation
We can relate the value of a swap, with payoff (ST1 − ST2 )+ , and a plain
vanilla option via the following duality:
M S01 , S02 ; C, ν = S01 P 1, S02 /S01 ; C 0 , ν 0
where the characteristics (C 0 , ν 0 ) are given in the previous Theorem for
v = (1, 0)> and u = (−1, 1)> .
1
Proof: Using asset S to form the Radon–Nikodym derivative
"
+ #
+ ST1
ST2
1
2
1
M = E ST − ST
= S0 E
1− 1
S01
ST
#
"
"
+ #
+
1
S2
S2
1 − T1
= S01 E eLT 1 − T1
= S01 E 0
,
ST
ST
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Introduction
Lévy processes
>
where v = (1, 0) . Now, note that
Financial
modeling
2
S 2 eLt
S2
St2
= 01 1 = 01 ehu,Lt i ,
1
L
St
S0 e t
S0
0≤t ≤T
Valuation
Risk Managem.
where u = (−1, 1)> and
ehu,Li ∈ M(P 0 )
Interest rate
since
Calibration
2
ehu,Li ehv ,Li = eL ∈ M(P).
Lévy LIBOR
Cross-currency
Then, we have that
Range options
M=
S01 E 0
h
1−
+
ST0
i
Credit risk
Duality theory
where S 0 is an exponential PIIAC with characteristics C 0 and ν 0 .
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Model with interest rates
Introduction
Lévy processes
Asset price processes
Sti = S0i exp[(r − δ i )t + Lit ]
Financial
modeling
Valuation
1
d
where L = (L , . . . , L ) is a PIIAC with triplet (B, C, ν)
Risk Managem.
Interest rate
Calibration
payoff of a Margrabe option: (ST1 − ST2 )+
Lévy LIBOR
Cross-currency
value
M(S01 , S02 ; r , δ, C, ν)
−rT
=e
E[(ST1
−
ST2 )+ ]
Range options
Credit risk
then
Duality theory
M(S01 , S02 ; r , δ, C, ν) = E[ST1 ]eCT P(K , S02 /S01 , δ 1 , r , C 0 , ν 0 )
Exponential
semimartingale models
Call-Put Duality
where K = e
−CT
Multiasset setting
and CT is a constant.
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Duality in the Lévy forward rate model
Denote the value of a call option on a zero coupon bond by
B(0, U)
1
+
(B(T , U) − K ) ,
, K ; C, ν = IE
Vc B(0, T );
B(0, T )
BT
and similarly for a put option
B(0, U)
1
+
(K − B(T , U)) .
Vp B(0, T );
, K ; C, ν = IE
B(0, T )
BT
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Theorem
Cross-currency
Range options
Assume that bond prices are modeled according to the Lévy forward
rate model. Then, the value of a call and a put option on a bond are
related via:
B(0, U)
B(0, U)
, K ; C, ν = Vp B(0, T ); K ,
; C, −f ν
Vc B(0, T );
B(0, T )
B(0, T )
where f (s, x) = exp (Σ(s, U) + Σ(s, T ))x .
Credit risk
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Idea of proof: Define the constant D := IE
via
h
B(T ,U)
(BT )2
i
e
and the measure P
Valuation
Risk Managem.
B(T ,U) F
D(BT )2 t
Girsanov’s theorem for semimartingales we deduce the
e
P-characteristics
of the driving process L. Now,
Vc = IE
"
= IE
1
(B(T , U) − K )+
BT
Lévy processes
Financial
modeling
e
B(T , U)
dP
:=
= ηT .
dP
D (BT )2
e and the density process (ηt )t∈[0,T ] is ηt = IE
P∼P
Introduction
. Using
Interest rate
Calibration
Lévy LIBOR
Cross-currency
B(T , U)
KDBT (K −1 − B(T , U)−1 )+
D (BT )2
Range options
Credit risk
#
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Introduction
Lévy processes
Financial
modeling
e we get that
and changing measure from P to P,
Valuation
h
i
e KDBT (K −1 − B(T , U)−1 )+ .
Vc = E
Risk Managem.
Interest rate
This can be re-written as
Calibration
Lévy LIBOR
+ b − B(T
b , U)
e 1 K
Vc = E
,
bT
B
bT )−1 :=
for (B
B(0,T )
DBT ,
B(0,U)
b :=
K
B(0,U)
B(0,T )
Cross-currency
Range options
b , U) := K B(0,U) B(T , U)−1 .
and B(T
B(0,T )
Duality theory
bT and B(T
b , U) have dynamics analogous to that of BT
Showing that B
and B(T , U) concludes the proof.
Credit risk
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Equivalent formulation (1)
Introduction
time-Ti+1 payoff of a caplet:
Recall
1 + δL(Ti , Ti ) =
Nδ(L(Ti , Ti ) − K )
+
B(Ti ,Ti )
B(Ti ,Ti+1 )
Lévy processes
Financial
modeling
Valuation
Risk Managem.
δ(L(Ti , Ti ) − K )+ = (1 + δL(Ti , Ti ) − (1 + δK ))
1
=
−K
B(Ti , Ti+1 )
time-Ti value of this payoff
+
1
+
1
−K
=K
− B(Ti , Ti+1 )
B(Ti , Ti+1 )
B(Ti , Ti+1 )
K
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Exponential
semimartingale models
Call-Put Duality
1
→ payoff of a put option on a bond with strike
1 + δK
Analogously for a floorlet.
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Equivalent formulation (2)
Introduction
Lévy processes
Value of a floorlet with strike K maturing at time Ti that settles in arrears
at time Ti+1
i
h 1
δ(K − L(Ti , Ti ))+
FL(L(0, Ti ), K ; C, ν) = E
BTi+1
i
h 1
(B(Ti , Ti+1 ) − K)+
= (1 + δK )IE
BTi
B(0,Ti )
where L(0, Ti ) = δ1 B(0,T
−1
initial forward Libor rate
)
i+1
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Therefore
Duality theory
FL(L(0, Ti ), K ; C, ν) = C CL(K , L(0, Ti ); C, −f ν)
Exponential
semimartingale models
Call-Put Duality
1 + δK
where C =
1 + δL(0, Ti )
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Duality in the Lévy Libor model
Introduction
Lévy processes
Value of a caplet with strike K maturing at time Ti that settles in arrears
at time Ti+1
CL(L(0, Ti ), K ; C, ν
Ti+1
+
) = B(0, Ti+1 )EPTi+1 [δ(L(Ti , Ti ) − K ) ]
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Duality result
CL(L(0, Ti ), K ; C, ν
Cross-currency
Ti+1
) = FL(K , L(0, Ti ); C, −f ν
Ti+1
)
Range options
Credit risk
where f (s, x) = exp(λ(s, Ti )x)
Duality theory
Exponential
semimartingale models
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Duality in the Lévy forward
process model
Introduction
Lévy processes
Financial
modeling
Valuation
Value of a call option on the forward process with strike K which is
settled in arrears at time Ti+1
Risk Managem.
Interest rate
Calibration
C(F (0, Ti , Ti+1 ), K ; C, ν
Ti+1
+
) = B(0, Ti+1 )EPTi+1 [(F (Ti , Ti , Ti+1 ) − K ) ]
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality for call and put options on the forward process
Duality theory
Exponential
semimartingale models
C(F (0, Ti , Ti+1 ), K ; C, ν
Ti+1
) = P(K , F (0, Ti , Ti+1 ); C, −f ν
Ti+1
)
Call-Put Duality
Multiasset setting
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
Financial statement Deutsche Bank
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
1,623,811
1,378,011
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
2,202,423
1,925,003
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 197
Financial statement Deutsche Bank
Introduction
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 198
Introduction
Lévy processes
1,333,765
870,085
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
2,202,423
1,925,003
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 199
Asset prices
Introduction
Lévy processes
150
Financial
modeling
145
Valuation
140
Risk Managem.
Interest rate
135
Calibration
130
Lévy LIBOR
Cross-currency
125
Range options
120
Credit risk
Duality theory
115
Capital
Requirements
110
Spread Option
105
100
Oct 1997
Net Asset Value
Reserve Capital
Oct 1998
Oct 1999
Oct 2000
Oct 2001
Oct 2002
Oct 2003
Oct 2004
c Eberlein, Uni Freiburg, 200
References
Decomposition of the balance sheet
Introduction
Lévy processes
Financial
modeling
Cash + Risky Assets = Equity + Risky Debt + Risky Liabilities
M(t) + A(t) = J(t) + D(t) + L(t)
Valuation
Risk Managem.
Interest rate
M(t): Cash + short term investments (cash equivalent reserve)
relatively nonrandom: M(t) = Mert
L(t):
Calibration
Lévy LIBOR
Cross-currency
Short positions in stocks
Negative side of a swap contract
Payouts on writing credit protections
Payouts on selling options
Short positions in variance swaps
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
−→ potentially unbounded
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 201
Equity as a Spread Option
Introduction
Lévy processes
Company set up with limited liability
Financial
modeling
Valuation
At debt maturity (face value F )
J(T ) = (MerT + A(T ) − L(T ) − F )+
Risk Managem.
Interest rate
Calibration
Debt holders receive
Lévy LIBOR
D(T ) = (MerT + A(T ) − L(T ))+ ∧ F
Cross-currency
Range options
Consequently: Initial equity and debt value
h
+ i
J = E0Q e−rT A(T ) − L(T ) − (F − MerT )
h
i
D = E0Q e−rT (MerT + A(T ) − L(T ))+ ∧ F
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 202
Equity as a Spread Option (2)
Introduction
Value of the limited liability firm at debt maturity
(Me
−→ call option struck at
rT
Lévy processes
+
Financial
modeling
+ A(T ) − L(T ))
−MerT
Valuation
Risk Managem.
Value of the firm at time 0
Interest rate
h
V = J + D = E0Q e−rT MerT + A(T ) − L(T )
+ i
Calibration
Lévy LIBOR
Cross-currency
Range options
Negative part of this variable:
Credit risk
put option on A(T ) − L(T ) struck at − Me
rT
Duality theory
Capital
Requirements
Value
h
+ i
P = E0Q e−rT − MerT − (A(T ) − L(T ))
Spread Option
Net Asset Value
Reserve Capital
Capital requirements set by external regulators: M =?
c Eberlein, Uni Freiburg, 203
References
Equity as a Spread Option (3)
Introduction
Lévy processes
Architecture of this approach:
Financial
modeling
Model A(T ) − L(T ) as the difference of two exponential Lévy processes
Valuation
Risk Managem.
Compute equity prices
h
+ i
J(t) = EtQ e−r (T −t) A(T ) − L(T ) − (F − MerT )
Derive prices of equity options for strike K and maturity t
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
W (K , t) = e
−rt
E0Q (J(t)
− K)
+
(compound option)
Credit risk
Duality theory
Capital
Requirements
Calibration to the observed option price surface
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 204
Volatility smile and surface
Introduction
Lévy processes
Financial
modeling
Valuation
14
30.0
13.5
Risk Managem.
28.0
26.0
13
implied vol (%)
Interest rate
24.0
12.5
22.0
12
Calibration
20.0
18.0
11.5
Lévy LIBOR
16.0
11
14.0
10.5
10
10
Cross-currency
12.0
10.0
20
30
40
delta (%) or strike
50
60
70
80
90
1
2
3
4
5
6
7
maturity
8
9
10
2.5
4.0
6.0
8.0
Strike rate (in %)
10.0
10
8
6
4
2
0
Maturity (in years)
Range options
Credit risk
Duality theory
Volatility surfaces
Capital
Requirements
Spread Option
• Volatilities vary in strike (smile)
• Volatilities vary in time to maturity (term structure)
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 205
Net Asset Value Process
Introduction
Model for the risky asset
A(t) = A(0) exp(X (t) + (r + ω X )t)
Model for the risky liability
L(t) = L(0) exp(Y (t) + (r + ω Y )t)
Lévy processes
Financial
modeling
Valuation
Risk Managem.
In order to create the right level of dependence between X (t) and Y (t)
−→ linear mixture of 4 independent VG Lévy processes


U1 (t)

X (t)
cos(η1 ) cos(η2 ) cos(η3 ) cos(η4 ) 
U2 (t)
=
Y (t)
sin(η1 ) sin(η2 ) sin(η3 ) sin(η4 ) U3 (t)
U4 (t)
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Characteristic function of VG(σ, θ, ν)
χVG(σ,θ,ν) (u) =
1
1 − iθνu + (σ 2 ν/2)u 2
1/ν
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 206
Net Asset Value Process (2)
Introduction
Joint characteristic function
=
4
Y


1

j=1
Lévy processes
E [exp (iuX (t) + ivY (t))] = φ(u, v )
1 − i(u cos(ηj ) + v sin(ηj ))θj νj +
t
νj
Financial
modeling
Valuation
σj2 νj
2

(u cos(ηj ) + v sin(ηj
))2
Risk Managem.
Interest rate
The values for the exponential compensators are
σj2 νj cos2 (ηj )
1
ln 1 − cos(ηj )θj νj −
νj
2
j=1
!
4
X
σj2 νj sin2 (ηj )
1
ln 1 − sin(ηj )θj νj −
=
νj
2
ωX =
ωY
Calibration
!
4
X
j=1
Consequently
h
E e
iu ln(A(t))+iv ln(L(t))
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
i
Spread Option
= φ(u, v ) exp(iu ln(A(0)) + iv ln(L(0)) + iu(r + ωX )t + iv (r + ωY )t)
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 207
Balance Sheet data
Introduction
Lévy processes
Balance sheet data for six major US banks from Wharton Research
Data Service (end of 2008 in millions of dollars)
Financial
modeling
Valuation
M
A
L
in millions of dollars
D
N
millions
S
dollars
JPM
368149
1806903
1009277
633474
3732
31.59
MS
210519
448293
181159
392266
1047
15.16
GS
244425
640122
298546
498416
443
82.24
BAC
124905
1693038
882997
632946
5017
13.93
WFC
72092
1237547
781402
375232
4228
29.86
325681
1612789
769572
720317
5450
6.88
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
C
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 208
6
3
Risky Assets
x 10
Introduction
DB
Lévy processes
Financial
modeling
2.5
BAC WFC
Valuation
Risk Managem.
2
JPM
AT less CHE
Interest rate
MS
1.5
WB
Calibration
Lévy LIBOR
C
Cross-currency
GS
Range options
1
Credit risk
Duality theory
Capital
Requirements
0.5
Spread Option
LEH
0
1
2
3
4
5
6
7
Years since 1997
8
9
10
Net Asset Value
MER
11
Reserve Capital
References
c Eberlein, Uni Freiburg, 209
5
12
Risky Liabilities
x 10
Introduction
Lévy processes
WFC
BAC
10
Financial
modeling
C
MS
Valuation
8
Risk Managem.
JPM
DB
WB
Interest rate
AP
GS
Calibration
6
Lévy LIBOR
Cross-currency
Range options
4
Credit risk
Duality theory
2
Capital
Requirements
LEH
Spread Option
MER
Net Asset Value
0
1
2
3
4
5
6
7
Years since 1997
8
9
10
11
Reserve Capital
References
c Eberlein, Uni Freiburg, 210
Required Reserve Capital
Introduction
X random variable: outcome (cashflow) of a risky position
For setting capital requirements: non-dynamic
Lévy processes
Financial
modeling
In complete markets: unique pricing kernel given by a probability
measure Q
Valuation
Risk Managem.
Interest rate
Q
value of the position:
E [X ]
Calibration
position is acceptable if:
E Q [X ] ≥ 0
Lévy LIBOR
company’s objective is:
maximizing E Q [X ]
Cross-currency
Range options
Credit risk
Real markets:
Duality theory
incomplete
Instead of a unique probability measure Q we have to consider a set of
probability measures Q ∈ M
Capital
Requirements
Spread Option
Net Asset Value
E Q [X ] ≥ 0
for all Q ∈ M
or
inf E Q [X ] ≥ 0
Q∈M
Reserve Capital
References
c Eberlein, Uni Freiburg, 211
Required Reserve Capital (2)
Specification of M (test measures, generalized scenarios)
Axiomatic theory of risk measures:
Monotonicity:
desirable properties
X ≥ Y =⇒ %(X ) ≤ %(Y )
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Cash invariance: %(X + c) = %(X ) − c
Interest rate
Scale invariance: %(λX ) = λ%(X ), λ ≥ 0
Subadditivity:
Introduction
Calibration
%(X + Y ) ≤ %(X ) + %(Y )
Lévy LIBOR
Cross-currency
Examples:
Value at Risk (VaR)
Tail-VaR (expected shortfall)
Z 1
General risk measure: %m (X ) = −
qu (X )m(du)
0
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Any coherent risk measure has a representation
Net Asset Value
%(X ) = − inf E Q [X ]
Reserve Capital
Q∈M
References
c Eberlein, Uni Freiburg, 212
Required Reserve Capital (3)
Introduction
Lévy processes
Acceptability of a cash flow?
Financial
modeling
Maybe it exposes the general economy to too much risk of loss
Valuation
Risk Managem.
Business set up with limited liability and insufficient capital
−→ Add capital C such that cash flow C + X is acceptable
Interest rate
Calibration
Lévy LIBOR
Cross-currency
inf E Q [C + X ] ≥ 0
Range options
Q∈M
Credit risk
Smallest such capital
Duality theory
C = − inf E Q [X ]
Capital
Requirements
Q∈M
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 213
Required Reserve Capital (4)
Introduction
Computation of this required reserve capital
Lévy processes
Financial
modeling
Link between acceptability and concave distortions
(Cherny and Madan (2009))
Valuation
Risk Managem.
→ Concave distortions
Interest rate
Assume acceptability is completely defined by the distribution function
of the risk
Calibration
Lévy LIBOR
Cross-currency
Ψ(u): concave distribution function on [0, 1]
Range options
⇒ M the set of supporting measures is given by all measures Q
with density Z = dQ
s.t.
dP
Credit risk
E P [(Z − a)+ ] ≤ sup (Ψ(u) − ua)
for all a ≥ 0
u∈[0,1]
Duality theory
Capital
Requirements
Spread Option
Acceptability of X with distribution function F (x)
Z +∞
xdΨ(F (x)) ≥ 0
Net Asset Value
Reserve Capital
References
−∞
c Eberlein, Uni Freiburg, 214
Distortion
1.0
Introduction
Lévy processes
0.8
Financial
modeling
Valuation
Risk Managem.
0.6
Interest rate
Ψ (x)
Calibration
Lévy LIBOR
0.4
Cross-currency
Range options
0.2
Credit risk
Duality theory
0.0
γ= 2
γ = 10
γ = 20
γ =100
0.0
0.2
0.4
0.6
0.8
Capital
Requirements
Spread Option
1.0
Net Asset Value
Reserve Capital
x
References
c Eberlein, Uni Freiburg, 215
Required Reserve Capital (5)
Introduction
Lévy processes
Consider families of distortions (Ψγ )γ≥0
Financial
modeling
γ stress level
Valuation
Risk Managem.
Example: MIN VaR
Interest rate
Ψγ (x) = 1 − (1 − x)1+γ
(0 ≤ x ≤ 1, γ ≥ 0)
Calibration
Lévy LIBOR
Cross-currency
Statistical interpretation:
Range options
Credit risk
Let γ be an integer, then %γ (X ) = −E(Y ) where
Duality theory
law
Y = min{X1 , . . . , Xγ+1 }
Capital
Requirements
and X1 , . . . , Xγ+1 are independent draws of X
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 216
Required Reserve Capital (6)
Introduction
Further examples: MAX VaR
Ψγ (x) = x
Lévy processes
1
1+γ
(0 ≤ x ≤ 1, γ ≥ 0)
Financial
modeling
Valuation
Statistical interpretation:
%γ (X ) = −E[Y ]
Risk Managem.
where Y is a random variable s.t.
Interest rate
Calibration
law
max{Y1 , . . . , Yγ+1 } = X
Lévy LIBOR
and Y1 , . . . , Yγ+1 are independent draws of Y .
Cross-currency
Range options
Combining MIN VaR and MAX VaR:
Ψγ (x) = (1 − (1 − x)1+γ )
Interpretation:
%γ (X ) = −E[Y ]
MAX MIN VaR
1
1+γ
(0 ≤ x ≤ 1, γ ≥ 0)
Credit risk
Duality theory
Capital
Requirements
Spread Option
with Y s.t.
Net Asset Value
law
max{Y1 , . . . , Yγ+1 } = min{X1 , . . . , Xγ+1 }
Reserve Capital
References
c Eberlein, Uni Freiburg, 217
Required Reserve Capital (7)
Introduction
Distortion used:
MIN MAX VaR
1
Ψγ (x) = 1 − 1 − x 1+γ
Lévy processes
1+γ
(0 ≤ x ≤ 1, γ ≥ 0)
Financial
modeling
Valuation
%γ (X ) = −E[Y ] with Y s.t.
law
Y = min{Z1 , . . . , Zγ+1 },
law
max{Z1 , . . . , Zγ+1 } = X
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Capital required at (stress) level γ
Z ∞
C=−
xdΨγ (FX (x))
Cross-currency
Range options
Credit risk
−∞
Duality theory
Computationally: Let x1 ≤ x2 ≤ · · · ≤ xN be historic or Monte Carlo
realizations of the cashflow X
C≈
N
X
j=1
xj
j
j −1
Ψγ
− Ψγ
N
N
c Eberlein, Uni Freiburg, 218
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
Distortion
1.0
Introduction
Lévy processes
0.8
Financial
modeling
Valuation
Risk Managem.
0.6
Interest rate
Calibration
Ψγ(x)
Lévy LIBOR
0.4
Cross-currency
Range options
Credit risk
0.2
Duality theory
0.0
γ = 0.50
γ = 0.75
γ = 1.0
γ = 5.0
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
0.0
0.2
0.4
0.6
x
0.8
1.0
c Eberlein, Uni Freiburg, 219
References
Computation of
Required Reserve Capital and
the value of the taxpayer put
Introduction
Lévy processes
Financial
modeling
In Billions of US Dollars
Valuation
Risk Managem.
Reserve
Capital
Required
Reserve
Capital
Held
Limited
Liability
Put Value
Required
to Actual
Ratio
Adjustment
Factor
JPM
698.039
368.149
293.96
1.8961
0.3154
MS
116.273
210.519
29.75
0.5523
0.4113
GS
−83.840
244.425
3.37
−0.3430
0.1796
Duality theory
BAC
246.065
124.905
158.17
1.9700
0.2840
Capital
Requirements
WFC
366.832
72.092
220.14
5.0884
0.2107
Spread Option
C
434.596
325.681
156.21
1.3344
0.3984
Net Asset Value
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Reserve Capital
References
c Eberlein, Uni Freiburg, 220
References
Introduction
• E. Eberlein: Fourier based valuation methods in mathematical
finance. to appear in Wolfgang Pauli Proceedings. Springer (2013)
• E. Eberlein, T. Gehrig, D. B. Madan: Pricing to acceptability: With
applications to valuing one’s own credit risk. The Journal of Risk
15(1) (2012) 91–120
• M. Beinhofer, E. Eberlein, A. Janssen, M. Polley: Correlations in
Lévy interest rate models. Quantitative Finance 11 (2011)
1315–1327
• E. Eberlein, K. Glau, A. Papapantoleon: Analyticity of the
Wiener–Hopf factors and valuation of exotic options in Lévy
models. In Advanced Mathematical Methods for Finance, G. Di
Nunno, B. Øksendal (eds.), Springer (2011) pp. 223–245
• E. Eberlein, Z. Grbac: Rating based Lévy LIBOR model. to
appear: Mathematical Finance (2013)
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
• E. Eberlein, D. Madan: Unbounded liabilites, capital reserve
requirements and the taxpayer put option. Quantitative Finance
12(5) (2012) 709–724
c Eberlein, Uni Freiburg, 221
Net Asset Value
Reserve Capital
References
References (cont.)
Introduction
Lévy processes
• E. Eberlein: Jump Processes. In Encylopedia of Quantitative
Finance, R. Cont (ed.), John Wiley & Sons (2010) pp. 990–994
• E. Eberlein, K. Glau, A. Papapantoleon: Analysis of Fourier
transform valuation formulas and applications. Applied
Mathematical Finance 17 (2010) 211–240
• E. Eberlein: Mathematik und die Finanzkrise. Spektrum der
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Wissenschaft 12/09 (2009) 92–100
• E. Eberlein, A. Papapantoleon, A. N. Shiryaev: Esscher transform
Cross-currency
and the duality principle for multidimensional semimartingales.
Annals of Applied Probability 19 (2009) 1944–1971
Range options
• E. Eberlein, D. B. Madan: Sato processes and the valuation of
structured products. Quantitative Finance 9 (1) (2009) 27–42
• E. Eberlein: Jump-type Lévy processes. In Handbook of Financial
Time Series, T. G. Andersen, R. A. Davis, J.-P. Kreiß, T. Mikosch
(eds.), Springer (2009), pp. 439–455
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 222
References (cont.)
Introduction
• E. Eberlein, R. Frey, E. A. von Hammerstein: Advanced credit
portfolio modeling and CDO pricing. In Mathematics – Key
Technology for the Future, W. Jäger and H.-J. Krebs (eds.),
Springer (2008), pp. 253–280
• E. Eberlein, A. Papapantoleon, A. N. Shiryaev: On the duality
principle in option pricing: semimartingale setting. Finance and
Stochastics 12 (2) (2008) 265–292
• E. Eberlein, R. Frey, M. Kalkbrener, L. Overbeck: Mathematics in
Financial Risk Management. Jahresbericht der Deutschen
Mathematiker Vereinigung 109 (2007) 165–193
• E. Eberlein, W. Kluge: Calibration of Lévy term structure models.
In Advances in Mathematical Finance: In Honor of Dilip Madan,
M. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Elliott (eds.), Birkhäuser
(2007), pp. 147–172
• E. Eberlein, J. Liinev: The Lévy swap market model. Applied
Mathematical Finance 14 (2) (2007) 171–196
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 223
References (cont.)
Introduction
• E. Eberlein, W. Kluge, P. J. Schönbucher: The Lévy Libor model
with default risk. Journal of Credit Risk 2 (2) (2006) 3–42
• E. Eberlein, N. Koval: A Cross-Currency Lévy Market Model.
Quantitative Finance 6 (2006) 465–480
Lévy processes
Financial
modeling
Valuation
Risk Managem.
• E. Eberlein, W. Kluge, A. Papapantoleon: Symmetries in Lévy
term structure models. International Journal of Theoretical and
Applied Finance 9 (6) (2006) 967–986
• E. Eberlein, W. Kluge: Exact pricing formulae for caps and
swaptions in a Lévy term structure model. Journal of
Computational Finance 9 (2) (2006) 99–125
• E. Eberlein, A. Papapantoleon: Symmetries and pricing of exotic
options in Lévy models. In Exotic Option Pricing and Advanced
Lévy Models, A. Kyprianou, W. Schoutens, P. Wilmott (eds.), Wiley
(2005), pp. 99–128
• E. Eberlein, W. Kluge: Valuation of floating range notes in Lévy
term structure models. Mathematical Finance 16 (2005) 237–254
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 224
References (cont.)
Introduction
• E. Eberlein, A. Papapantoleon: Equivalence of floating and fixed
strike Asian and lookback options. Stochastic Processes and
Their Applications 115 (2005) 31–40
• E. Eberlein, F. Özkan: The Lévy Libor Model. Finance and
Stochastics 9 (2005) 327–348
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
• E. Eberlein, J. Jacod, S. Raible: Lévy term structure models:
no-arbitrage and completeness. Finance and Stochastics 9 (2005)
67–88
• E. Eberlein, E. A. von Hammerstein: Generalized hyperbolic and
inverse Gaussian distributions: limiting cases and approximation
of processes. In Seminar on Stochastic Analysis, Random Fields
and Applications IV, Progress in Probability 58, R. C. Dalang, M.
Dozzi, F. Russo (eds.), Birkhäuser Verlag (2004) 221–264
• E. Eberlein, G. Stahl: Both sides of the fence: a statistical and
regulatory view of electricity risk. Energy & Power Risk
Management 8 (6) (2003), 34–38
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 225
References (cont.)
Introduction
• E. Eberlein, F. Özkan: Time consistency of Lévy models.
Quantitative Finance 3 (2003) 40–50
• E. Eberlein, J. Kallsen, J. Kristen: Risk management based on
stochastic volatility. Journal of Risk 5 (2) (2003) 19–44
• E. Eberlein, F. Özkan: The defaultable Lévy term structure: ratings
and restructuring. Mathematical Finance 13 (2003) 277–300
• E. Eberlein, K. Prause: The generalized hyperbolic model:
financial derivatives and risk measures. In Mathematical
Finance—Bachelier Congress 2000, H. Geman, D. Madan, S.
Pliska, T. Vorst (eds.), Springer (2002), 245–267
• E. Eberlein, S. Raible: Some analytic facts on the generalized
hyperbolic model. In Proceedings of the 3rd European Meeting of
Mathematics, Progress in Mathematics 202, C. Casacuberta, et al.
(eds.), Birkhäuser Verlag (2001) 367–378
Lévy processes
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
• E. Eberlein: Recent advances in more realistic risk management:
Net Asset Value
the hyperbolic model. In Mastering Risk 2, C. Alexander (Ed.),
Prentice Hall-Financial Times (2001), 56–72
Reserve Capital
c Eberlein, Uni Freiburg, 226
References
References (cont.)
Introduction
Lévy processes
• E. Eberlein: Application of generalized hyperbolic Lévy motions to
finance. In Lévy Processes: Theory and Applications, O.E.
Barndorff-Nielsen, T. Mikosch, and S. Resnick (eds.), Birkhäuser
Verlag (2001) 319–337
• E. Eberlein, S. Raible: Term structure models driven by general
Lévy processes. Mathematical Finance 9 (1999) 31–53
• E. Eberlein, U. Keller, K. Prause: New insights into smile,
mispricing and value at risk: the hyperbolic model. Journal of
Business 71 (1998) 371–405
• E. Eberlein, J. Jacod: On the range of options prices. Finance and
Stochastics 1 (1997) 131–140
• E. Eberlein, U. Keller: Hyperbolic distributions in finance. Bernoulli
1 (1995) 281–299
Financial
modeling
Valuation
Risk Managem.
Interest rate
Calibration
Lévy LIBOR
Cross-currency
Range options
Credit risk
Duality theory
Capital
Requirements
Spread Option
Net Asset Value
Reserve Capital
References
c Eberlein, Uni Freiburg, 227

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