I.
Generalities
Market Skews
Dominating fact since 1987 crash: strong negative skew on
Equity Markets
σ impl
K
Not a general phenomenon
Gold: σ impl
FX: σ impl
K
K
We focus on Equity Markets
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Skews
• Volatility Skew: slope of implied volatility as a
function of Strike
• Link with Skewness (asymmetry) of the Risk
Neutral density function ϕ ?
Moments
1
2
3
4
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Statistics
Expectation
Variance
Skewness
Kurtosis
Finance
FWD price
Level of implied vol
Slope of implied vol
Convexity of implied vol
3
Why Volatility Skews?
• Market prices governed by
– a) Anticipated dynamics (future behavior of volatility or jumps)
– b) Supply and Demand
σ impl
Sup
Market Skew
ply
and
Dem
and
Th. Skew
K
• To “ arbitrage” European options, estimate a) to capture
risk premium b)
• To “arbitrage” (or correctly price) exotics, find Risk
Neutral dynamics calibrated to the market
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Modeling Uncertainty
Main ingredients for spot modeling
• Many small shocks: Brownian Motion
(continuous prices) S
t
• A few big shocks: Poisson process (jumps)
S
t
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2 mechanisms to produce Skews (1)
• To obtain downward sloping implied volatilities
σimp
K
– a) Negative link between prices and volatility
• Deterministic dependency (Local Volatility Model)
• Or negative correlation (Stochastic volatility Model)
– b) Downward jumps
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2 mechanisms to produce Skews (2)
– a) Negative link between prices and volatility
S1
S2
– b) Downward jumps
S1
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S2
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Model Requirements
• Has to fit static/current data:
– Spot Price
– Interest Rate Structure
– Implied Volatility Surface
• Should fit dynamics of:
– Spot Price (Realistic Dynamics)
– Volatility surface when prices move
– Interest Rates (possibly)
• Has to be
– Understandable
– In line with the actual hedge
– Easy to implement
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Beyond initial vol surface fitting
• Need to have proper dynamics of implied volatility
– Future skews determine the price of Barriers and
OTM Cliquets
– Moves of the ATM implied vol determine the ∆ of
European options
• Calibrating to the current vol surface do not impose
these dynamics
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Barrier options as Skew trades
• In Black-Scholes, a Call option of strike K extinguished
at L can be statically replicated by a Risk Reversal
L
K
• Value of Risk Reversal at L is 0 for any level of (flat) vol
• Pb: In the real world, value of Risk Reversal at L
depends on the Skew
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II.
A Brief History of Volatility
A Brief History of Volatility (1)
–
dSt = σ dWt Q
: Bachelier 1900
–
dS t
= r dt + σ dWt Q
St
: Black-Scholes 1973
–
dSt
= r (t ) dt + σ (t ) dWt Q
St
–
dSt
= (r − λk ) dt + σ dWt Q + dq : Merton 1976
St
–
 dSt
Q
=
+
r
dt
σ
dW
t
t
S
 t
dσ 2 = a(V − V )dt + ξ σ α dZ
L
t
 t
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: Merton 1973
: Hull&White 1987
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A Brief History of Volatility (2)
dS t
= σ t dWt Q
St
∂ 2 LT (t )
Q
dσ = 2
dt
+
α
dZ
t
∂T 2
2
t
dSt
= r (t ) dt + σ (S , t ) dWt Q
St
∂C
∂C
+ rK
2
σ (K , T ) = 2 ∂T 2 ∂K
∂ CK ,T
K2
∂K 2
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Dupire 1992, arbitrage model
which fits term structure of
volatility given by log contracts.
Dupire 1993, minimal model
to fit current volatility surface
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A Brief History of Volatility (3)
 dSt
 S = r dt + σ t dWt
 t
dσ 2 = b(σ 2 − σ 2 )dt + βσ dZ
∞
t
t
t
 t
dV K ,T = α K ,T dt + bK ,T dZ tQ
VK ,T
: instantane ous forward variance
conditiona l to
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ST = K
Heston 1993,
semi-analytical formulae.
Dupire 1996 (UTV),
Derman 1997,
stochastic volatility model
which fits current volatility
surface HJM treatment.
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A Brief History of Volatility (4)
– Bates 1996, Heston + Jumps:
 dSt
 S = r dt + σ t dZ t + dq
 t
dσ 2 = b(σ 2 − σ 2 )dt + βσ dW
∞
t
t
t
 t
– Local volatility + stochastic volatility:
• Markov specification of UTV
• Reech Capital Model: f is quadratic
• SABR: f is a power function
dS t
= r dt + σ t f (S , t ) dZ tQ
St
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A Brief History of Volatility (5)
– Lévy Processes
– Stochastic clock:
• VG (Variance Gamma) Model:
– BM taken at random time g(t)
• CGMY model:
– same, with integrated square root process
–
–
–
–
–
–
Jumps in volatility (Duffie, Pan & Singleton)
Path dependent volatility
Implied volatility modelling
Incorporate stochastic interest rates
n dimensional dynamics of s
n assets stochastic correlation
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III. Local Volatility Model
From Simple to Complex
• How to extend Black-Scholes to make it
compatible with market option prices?
– Exotics are hedged with Europeans.
– A model for pricing complex options has to
price simple options correctly.
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Black-Scholes assumption
• BS assumes constant volatility
=> same implied vols for all options.
dS
= µ dt + σ dW
S
(instantaneous vol)
σ
CALL PRICES
σ
Strike
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Black-Scholes assumption
• In practice, highly varying.
Implied
Vol
Implied
Vol
Strike
Maturity
Nikkei
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Strike
Maturity
Japanese Government Bonds
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Modeling Problems
• Problem: one model per option.
– for C1 (strike 130)
–for C2 (strike 80)
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σ = 10%
σ = 20%
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One Single Model
• We know that a model with σ(S,t) would
generate smiles.
– Can we find σ(S,t) which fits market smiles?
– Are there several solutions?
ANSWER: One and only one way to do it.
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Interest rate analogy
• From the current Yield Curve, one can
compute an Instantaneous Forward Rate.
13
11
9
7
5
0
1
2
3
4
5
6
7
8
9
10
13
12
11
10
9
8
7
6
5
0
1
2
3
4
5
6
7
8
9
10
– Would be realized in a world of certainty,
– Are not realized in real world,
– Have to be taken into account for pricing.
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31
21
27
L ocal vol
Im plied vol
Volatility
19
17
15
19
15
11
13
Strike
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Maturity
Dream: from Implied Vols
Strike
Maturity
read Local (Instantaneous Forward) Vols
How to make it real?
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Discretization
• Two approaches:
– to build a tree that matches European options,
– to seek the continuous time process that matches
European options and discretize it.
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Tree Geometry
Binomial
Trinomial
To discretize σ(S,t) TRINOMIAL is more adapted
Example:
20%
20%
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5%
5%
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Tango Tree
• Rules to compute connections
– price correctly Arrow-Debreu associated with nodes
– respect local risk-neutral drift
• Example
.1
40
20
.25
25
50
1
.5
40
40
25
30
.2
40
30
20
.5
40
.1
.2
.25
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Continuous Time Approach
Call Prices
Distributions
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Exotics
?
Diffusion
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Distributions - Diffusion
Distributions
Diffusion
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Distributions - Diffusion
• Two different diffusions may generate the same
distributions
x
x
time
dx = −λ x dt + σ dWt
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time
dx = b (t ) dWt
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The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
Risk
Neutral
Processes
Diffusions
Compatible
with Smile
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sought diffusion
(obtained by integrating twice
Fokker-Planck equation)
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Continuous Time Analysis
Implied Volatility
Strike
Strike
Maturity
Call Prices
Maturity
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Local Volatility
Densities
Maturity
Strike
Maturity
Strike
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Implication : risk management
Implied
volatility
Perturbation
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Black box
Price
Sensitivity
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Forward Equations (1)
• BWD Equation:
price of one option C (K 0 ,T0 ) for different (S, t )
• FWD Equation:
price of all options C (K , T ) for current (S 0 ,t0 )
• Advantage of FWD equation:
– If local volatilities known, fast computation of implied
volatility surface,
– If current implied volatility surface known, extraction of
local volatilities,
– Understanding of forward volatilities and how to lock
them.
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Forward Equations (2)
• Several ways to obtain them:
– Fokker-Planck equation:
• Integrate twice Kolmogorov Forward Equation
– Tanaka formula:
• Expectation of local time
– Replication
• Replication portfolio gives a much more financial
insight
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Fokker-Planck
• If dx = b( x, t )dW
∂ϕ 1 ∂ 2 (b 2ϕ )
• Fokker-Planck Equation:
=
∂t 2 ∂x 2
∂ 2C
• Where ϕ is the Risk Neutral density. As ϕ =
∂K 2
2
2



∂
∂
C
C
2
2
2  ∂C 

∂  b
∂ 
 ∂ 2 
2 
 ∂t  =  ∂K  = 1  ∂K 
∂x 2
∂t
∂x 2
2
• Integrating twice w.r.t. x: ∂C b2 ∂2C
=
2
∂t 2 ∂K
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FWD Equation: dS/S = σ(S,t) dW
δT
Define CS K ,T ≡
C K ,T + δ T − C K ,T
δT
CK ,T +δT
CK ,T
ST
K
CS Kδ T,T at T
dT
dT/2
σ 2 (K , T )
δT → 0
2
ST
ST
K
K 2δ K ,T
K
2
2
(
)
C
K
,
T
C
∂
σ
∂
2
Equating prices at t0:
K
=
∂T
2
∂K 2
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FWD Equation: dS/S = r dt + σ(S,t) dW
TV
CS Kδ T,T at T = Time Value + Intrinsic Value
S T − Ke − rδ T
IV
ST − K
(Strike Convexity)
(Interest on Strike)
CK ,T
CK ,T +δT
Ke−rδT
ST
K
σ 2 (K , T )
2
TV
Ke−rδT K
IV
δT → 0
rK
rKDigK ,T
ST
Equating prices at t0:
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K 2δ K ,T
K
ST
∂C σ 2 (K , T ) 2 ∂ 2C
∂C
K
=
− rK
2
∂T
2
∂K
∂K
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FWD Equation: dS/S = (r-d) dt + σ(S,t) dW
ST − Ke−rδT
TVIV
CS KδT,T at T =
ST e − dδT − Ke − rδT
CK,T+δT
(d−r)δT
Ke
CK,T
TV + Interests on K
– Dividends on S
ST
K
σ 2 (K , T )
2
δT → 0
(r − d )K
Ke(d −r )δT K
K
− d ⋅ CK ,T
∂T
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(r − d ) K DigK ,T
ST
Equating prices at t0: ∂C
=
K 2δ K ,T
σ 2 (K , T )
2
ST
− d ⋅ CK ,T
∂ 2C
∂C
− (r − d )K
K
− d ⋅C
2
∂K
∂K
2
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