Affine Processes
Martin Keller-Ressel
TU Berlin
[email protected]
Workshop on Interest Rates and Credit Risk 2011
TU Chemnitz
23. November 2011
Martin Keller-Ressel
Affine Processes
Outline
Introduction to Affine Processes
Affine Jump-Diffusions
The Moment Formula
Bond & Option Pricing in Affine Models
Extensions & Further Topics
Martin Keller-Ressel
Affine Processes
Part I
Introduction to Affine Processes
Martin Keller-Ressel
Affine Processes
Affine Processes
Affine Processes are a class of stochastic processes. . .
with good analytic tractability
(= explicit calculations and/or efficient numerical methods
often available)
that can be found in every corner of finance (stock price
modeling, interest rates, commodities, credit risk, . . . )
efficient methods for pricing bonds, options,. . .
dynamics and (some) distributional properties are
well-understood
They include models with
mean-reversion (important e.g. for interest rates)
jumps in asset prices (may represent shocks, crashes)
correlation and more sophisticated dependency effects
(stochastic volatility, simultaneous jumps, self-excitement . . . )
Martin Keller-Ressel
Affine Processes
The mathematical tools used are
characteristic functions (Fourier transforms)
stochastic calculus (with jumps)
ordinary differential equations
Markov processes
Martin Keller-Ressel
Affine Processes
Recommended Literature
Transform Analysis and Asset Pricing for Affine Jump-Diffusions,
Darrell Duffie, Jun Pan, and Kenneth Singleton, Econometrica,
Vol. 68, No. 6, 2000
Affine Processes and Applications in Finance, Darrell Duffie, Damir
Filipovic and Walter Schachermayer, The Annals of Applied
Probability, Vol. 13, No. 3, 2003
A didactic note on affine stochastic volatility models, Jan Kallsen,
In: From Stochastic Calculus to Mathematical Finance,
pages 343-368. Springer, Berlin, 2006.
Affine Diffusion Processes: Theory and Applications, Damir
Filipovic and Eberhard Mayerhofer, Radon Series Comp. Appl.
Math 8, 1-40, 2009.
Martin Keller-Ressel
Affine Processes
We start by looking at the Ornstein-Uhlenbeck process and the
Feller Diffusion.
The simplest (continuous-time) stochastic models for
mean-reverting processes
Used for modeling of interest rates, stochastic volatility,
default intensity, commodity (spot) prices, etc.
Also the simplest examples of affine processes!
Martin Keller-Ressel
Affine Processes
Ornstein-Uhlenbeck process and Feller Diffusion
Ornstein-Uhlenbeck (OU)-process
dXt = −λ(Xt − θ) dt + σdWt ,
X0 ∈ R
Feller Diffusion
dXt = −λ(Xt − θ) dt + σ
�
Xt dWt ,
X0 ∈ R�0
θ. . . long-term mean
λ > 0. . . rate of mean-reversion
σ ≥ 0. . . volatility parameter
�
σ for the OU-process
We define σ(Xt ) :=
√
σ Xt for the Feller diffusion
Martin Keller-Ressel
Affine Processes
.
An important difference: The OU-process has support R, while
the Feller diffusion stays non-negative
What can be said about the distribution of Xt ?
We will try to understand the distribution of Xt through its
characteristic function
�
�
ΦXt (y ) = E e iyXt
Martin Keller-Ressel
Affine Processes
Characteristic Function
Characteristic Function
For y ∈ R, the characteristic function ΦX (y ) of a random variable
X is defined as
�
� � ∞
iyX
ΦX (y ) := E e
=
e iyx dF (x) .
−∞
Properties:
ΦX (0) = 1, ΦX (−y ) = ΦX (y ), and |ΦX (y )| ≤ 1 for all y ∈ R.
d
ΦX (y ) = ΦY (y ) for all y ∈ R, if and only if X = Y .
Let X and Y be independent random variables. Then
ΦX +Y (y ) = ΦX (y ) · ΦY (y ) .
Martin Keller-Ressel
Affine Processes
Let k ∈ N. If E[|X |k ] < ∞, then
k
E[X ] = i
−k
�
�
∂k
�
Φ
(y
)
.
X
�
∂y k
y =0
If the characteristic function ΦX (y ) of a random variable X
with density f (x) is known, then f (x) can be recovered by an
inverse Fourier transform:
� ∞
1
f (x) =
e −iyx ΦX (y ) dy .
2π −∞
Martin Keller-Ressel
Affine Processes
Back to the OU and CIR processes: We write u = iy and make the
ansatz that the characteristic function of Xt is of
exponentially-affine form:
Exponentially-Affine characteristic function
�
�
�
�
E e iyXt = E e uXt = exp (φ(t, u) + ψ(t, u)X0 )
More precisely, if we can find functions φ(t, u), ψ(t, u) with
φ(t, u) = 0 and ψ(t, u) = u, such that
Mt = f (t, Xt ) = exp(φ(T − t, u) + ψ(T − t, u)Xt )
is a martingale then we have
�
�
E e uXT = E [MT ] = M0 = exp (φ(T , u) + ψ(T , u)X0 ) ,
and (1) indeed gives the characteristic function.
Martin Keller-Ressel
Affine Processes
(1)
Assume φ, ψ are sufficiently differentiable and apply the
Ito-formula to
f (t, Xt ) = exp (φ(T − t, u) + Xt ψ(T − t, u)) .
The relevant derivatives are
�
�
∂
f (t, Xt ) = − φ̇(T − t, u) + Xt ψ̇(T − t, u) f (t, Xt )
∂t
∂
f (t, Xt ) = ψ(T − t, u)f (t, Xt )
∂x
∂2
f (t, Xt ) = ψ(T − t, u)2 f (t, Xt )
∂x 2
Martin Keller-Ressel
Affine Processes
We get:
�
�
df (t, Xt )
1
= − φ̇T −t + Xt ψ̇T −t dt + ψT −t dXt + ψT2 −t σ 2 Xt dt =
f (t, Xt )
2
�
�
= − φ̇T −t + Xt ψ̇T −t dt + −ψT −t λ(Xt − θ) dt+
1
+ ψT −t σ(Xt ) dWt + ψT2 −t σ(Xt )2 dt
2
f (t, Xt ) is local martingale, if
1
(φ̇T −t + Xt ψ̇T −t ) = −ψT −t λ(Xt − θ) + ψT2 −t σ(Xt )2
2
for all possible states Xt .
Note that both sides are affine in Xt , since
�
σ 2 for the OU-process
2
σ(Xt ) =
σ 2 Xt for the CIR process
Martin Keller-Ressel
Affine Processes
We can ‘collect coefficients’:
For the OU-process this yields
φ̇(s, u) = θλψ(s, u) +
σ2
ψ(s, u)
2
ψ̇(s, u) = −λψ(s, u)
For the CIR process we get
φ̇(s, u) = θλψ(s, u)
ψ̇(s, u) = −λψ(s, u) +
σ2
ψ(s, u)
2
These are ordinary differential equations. We also know the
initial conditions
φ(0, u) = 0,
Martin Keller-Ressel
ψ(0, u) = u .
Affine Processes
If φ(t, u) and ψ(t, u) solve the ODEs on the preceding slide,
then Mt is a local martingale.
It is easy to check that in both cases M is also bounded,
hence a true martingale.
If Mt is a martingale, then
�
�
E e iyXt = exp (φ(t, iy ) + Xo ψ(t, iy ))
is the characteristic function of Xt .
Martin Keller-Ressel
Affine Processes
The OU process
For the OU-process we solve
σ2
ψ(s, u)2 , φ(0, u) = 0
2
ψ̇(s, u) = −λψ(s, u), ψ(0, u) = u
φ̇(s, u) = θλψ(s, u) +
and get
ψ(t, u) = e −λt u
φ(t, u) = θu(1 − e −λt ) +
Martin Keller-Ressel
σ2 2
u (1 − e −2λt )
4λ
Affine Processes
Thus the characteristic function of the OU-process is given by
� �
�
�
�
� y 2 σ2
iyXt
−λt
−λt
−2λt
E e
= exp iy e
X0 + θ(1 − e
) −
(1 − e
)
2 2λ
and we get the following:
Distributional Properties of OU-process
Let X be an Ornstein-Uhlenbeck process. Then Xt is normally
distributed, with
EXt = θ + e −λt (X0 − θ),
Var Xt =
�
σ2 �
1 − e −2λt ,
2λ
Q: Can you think of a simpler way to obtain the above result?
Martin Keller-Ressel
Affine Processes
The CIR process
For the CIR-process we solve
φ̇(s, u) = θλψ(s, u),
φ(0, u) = 0
ψ̇(s, u) = −λψ(s, u) +
σ2
ψ(s, u)2 ,
2
ψ(0, u) = u .
and get
ψ(t, u) =
ue −λt
σ2
2λ u(1
− e −λt )
�
�
2λθ
σ2
−λt
φ(t, u) = − 2 log 1 −
u(1 − e
)
σ
2λ
1−
The differential equation for ψ is called a Riccati equation.
Q: How was the solution of the Riccati equation determined?
Martin Keller-Ressel
Affine Processes
(2)
(3)
Thus the characteristic function of the CIR-process is given by
�
�
�− 2λθ2
�
� �
σ
σ2
e −λt iy
iyXt
−λt
E e
= 1−
(1 − e
)iy
exp
2
2λ
1 − σ (1 − e −λt )iy
2λ
and we get the following:
Distributional Properties of the Feller Diffusion
2
σ
Let X be an Feller-diffusion, and define b(t) = 4λ
(1 − e −λt ). Then
Xt
2
b(t) has non-central χ -distribution, with parameters
k=
4λθ
,
σ2
α=
e −λt
,
b(t)
Q: Does there exist a limiting distribution? What is it?
Martin Keller-Ressel
Affine Processes
Summary
The key assumption was that the characteristic function of Xt
is of exponentially-affine form
�
�
E e iyXt = exp (φ(t, iy ) + X0 ψ(t, iy ))
We derived that φ(t, u) and ψ(t, u) satisfy ordinary
differential equations of the form
φ̇(t, u) = F (ψ(t, u)),
φ(0, u) = 0
ψ̇(t, u) = R(ψ(t, u)),
ψ(0, u) = u
Solving the differential equation gave φ(t, u) and ψ(t, u) in
explicit form.
The same approach works if the coefficients of the SDEs are
time-dependent; ODEs become time-dependent too.
Martin Keller-Ressel
Affine Processes
Part II
Affine Jump-Diffusions
Martin Keller-Ressel
Affine Processes
Jump Diffusions
n
We consider a jump-diffusion on D = Rm
�0 × R
Jump-Diffusion
dXt = µ(Xt ) dt + σ(Xt ) dWt + dZt
����
�
��
�
diffusion part
where
jump part
Wt is a Brownian motion in Rd ;
µ : D → Rd , σ : D → Rd×d , and
Z is a right-continuous pure jump process, whose jump
heights have a fixed distribution ν(dx) and arrive with
intensity λ(Xt− ), for some λ : D → [0, ∞).
The Brownian motion W , the jump heights of Z , and the
jump times of Z are assumed to be independent.
Martin Keller-Ressel
Affine Processes
(4)
Jump Diffusions (2)
Martin Keller-Ressel
Affine Processes
Some elementary properties and notation for the jump process Zt :
Zt is RCLL (right continuous with left limits)
Zt− := lims≤t,s→t Zs and ∆Zt := Zt − Zt− .
Zt �= Zt− if and only ∆Zt �= 0 if and only a jump occurs at
time t.
Let τ (i) be the time of the i-th jump of Zt . Let f be a
function such that f (0) = 0. Then
�
�
f (∆Zs ) :=
f (∆Zs )
0≤s≤t
0≤τ (i)≤t
is a well-defined sum, that runs only over finitely many values
(a.s.)
Martin Keller-Ressel
Affine Processes
Ito formula for jump-diffusions
Ito formula for jump diffusions
Let X be a jump-diffusion with diffusion part Dt and jump part Zt .
Assume that f : Rd → R is a C 1,2 -function and that Zt is a pure
jump process of finite variation. Then
� t
� t
∂f
∂f
f (t, Xt ) = f (0, X0 ) +
(s, Xs− ) ds +
(s, Xs− ) dDs +
∂t
∂x
0
0
� 2
�
�
1 t
∂ f
�
+
tr
(s, Xs− )σ(Xs− )σ(Xs− )
ds+
2 0
∂x 2
�
+
∆ f (s, Xs ) .
0≤s≤t
Here
∂2f
∂x 2
∂f
∂x
=
�
=
�
∂f
∂f
∂x1 , . . . , ∂xd
∂2f
∂xi ∂xj
�
�
denotes the gradient of f , and
is the Hessian matrix of the second derivatives of f .
Martin Keller-Ressel
Affine Processes
Affine Jump-Diffusion
Affine Jump-Diffusion
We call the jump diffusion X (defined in (4)) affine, if the drift
µ(Xt ), the diffusion matrix σ(Xt )σ(Xt )� and the jump intensity
λ(Xt− ) are affine functions of Xt .
More precisely, assume that
µ(x) = b + β1 x1 + · · · + βd xd
σ(x)σ(x)� = a + α1 x1 + · · · + αd xd
λ(x) = m + µ1 x1 + · · · µd xd
where b, βi ∈ Rd ; a, αi ∈ Rd×d and m, µi ∈ [0, ∞).
Note: (d + 1) × 3 parameters for a d-dimensional process.
Martin Keller-Ressel
Affine Processes
We want to show that an affine jump-diffusion has a (conditional)
characteristic function of exponentially-affine form:
Characteristic function of Affine Jump Diffusion
n
Let X be an affine jump-diffusion on D = Rm
�0 × R . Then
� �
�
�
E e u·XT � Ft = exp (φ(T − t, u) + Xt · ψ(T − t, u))
for all u = iz ∈ iRd and 0 ≤ t ≤ T , where φ and ψ solve the
system of differential equations
φ̇(t, u) = F (ψ(t, u)),
φ(0, u) = 0
(5)
ψ̇(t, u) = R(ψ(t, u)),
ψ(0, u) = u
(6)
with. . . �
Martin Keller-Ressel
Affine Processes
(continued)
�
κ(u) = Rd (e u·x − 1) ν(dx), and
1
F (u) = b � u + u � au + mκ(u)
2
1
R1 (u) = β1� u + u � α1 u + µ1 κ(u),
2
..
.
1
Rd (u) = βd� u + u � αd u + µd κ(u).
2
The differential equations satisfied by φ(t, u) and ψ(t, u) are called
generalized Riccati equations.
The functions F (u), R1 (u), . . . , Rd (u) are of Lévy-Khintchine form.
Martin Keller-Ressel
Affine Processes
Proof (sketch:)
Show that the generalized Riccati equations have unique
global solutions φ, ψ (This is the hard part, and here the
n
assumption that D = Rm
�0 × R enters!)
Fix T ≥ 0, define
Mt = f (t, Xt ) = exp(φ(T − t, u) + ψ(T − t, u) · Xt )
and show that Mt remains bounded.
Apply Ito’s formula to Mt :
Martin Keller-Ressel
Affine Processes
The relevant quantities for Ito’s formula are
�
�
∂
f (t, Xt− ) = − φ̇(T − t, u) + Xt · ψ̇(T − t, u) f (t, Xt− )
∂t
∂
f (t, Xt− ) = ψ(T − t, u)f (t, Xt− )
∂x
∂2
f (t, Xt− ) = ψ(T − t, u) · ψ(T − t, u)� f (t, Xt− )
∂x 2
�
�
∆ f (t, Xt ) = e ψ(T −t,u)·∆Xt − 1 f (t, Xt− )
Also define the cumulant generating function of the jump measure:
�
κ(u) =
(e u·x − 1)ν(dx).
Rd
Martin Keller-Ressel
Affine Processes
We can write f (t, Xt ) as...
f (t, Xt ) = ‘local martingale’−
� t�
�
−
φ̇(T − s, u) + Xs− · ψ̇(T − s, u) f (s, Xs− ) ds+
�0 t
+
ψ(T − s, u) · µ(Xs− )f (s, Xs− ) ds+
0
� t
1
+
ψ(T − s, u)� σ(Xs− )σ(Xs− )� ψ(T − s, u)f (s, Xs− ) ds+
2 0
� t �
�
+
κ ψ(T − s, u) λ(Xs− )f (s, Xs− ) ds
0
Inserting the definitions of µ(Xs− ), σ(Xs− )σ(Xs− )� and λ(Xs− )
and using the generalized Riccati equations we obtain the local
martingale property of M.
Martin Keller-Ressel
Affine Processes
Since M is bounded it is a true martingale and it holds that
�
E e
� �
� Ft = E [ MT | Ft ] =
uXT �
= Mt = exp (φ(T − t, u) + ψ(T − t, u) · Xt ) ,
showing desired form of the conditional characteristic function.
Martin Keller-Ressel
Affine Processes
Example: The Heston model
Heston proposes the following model for a stock St and its
(mean-reverting) stochastic variance Vt (under the risk-neutral
measure Q)1 :
Heston model
dSt =
�
Vt St dWt1
dVt = −λ(Vt − θ) dt + η
�
�
� �
Vt ρ dWt1 + 1 − ρ2 dWt2
where Wt = (Wt1 , Wt2 ) is two-dimensional Brownian motion.
1
We assume here that the interest rate r = 0
Martin Keller-Ressel
Affine Processes
The Heston model (2)
The parameters have the following interpretation:
λ. . . mean-reversion rate of the variance process
θ. . . long-term average of Vt
η. . . ‘vol-of-var’: the volatility of the variance process
ρ. . . ‘leverage’: correlation bet. moves in stock price and in
variance.
Martin Keller-Ressel
Affine Processes
The Heston model (3)
Transforming to the log-price Lt = log(St ) we get
�
Vt
dt + Vt dWt1
2
�
�
� �
dVt = −λ(Xt − θ) dt + η Vt ρ dWt1 + 1 − ρ2 dWt2
dLt = −
which is a two dimensional affine diffusion!
Writing Xt = (Lt , Vt ) we find
� �
�
�
0
−1/2
µ(Xt ) =
+ ����
0 Lt +
Vt
λθ
−λ
β1
� �� �
� �� �
b
β2
�
1 ηρ
σ(Xt )σ(Xt ) = ����
0 + ����
0 Lt +
Vt
ηρ η 2
a
α1
� �� �
�
�
α2
Martin Keller-Ressel
Affine Processes
The Heston model (4)
Thus, the characteristic function of log-price Lt and stochastic
variance Vt of the Heston model can be calculated from
φ̇(t, u) = λθψ2 (t, u)
�
1� 2
η2
ψ̇2 (t, u) =
u1 − u1 − λψ2 (t, u) + ψ22 (t, u) + ηρu1 ψ2 (t, u)
2
2
with initial conditions φ(0, u) = 0, ψ2 (t, u) = u2 .
Note that ψ̇1 (t, u) = 0 and thus ψ1 (t, u) = u1 for all t ≥ 0.
Martin Keller-Ressel
Affine Processes
Duffie-Garleanu default intensity process
Duffie and Garleanu propose to use the following process (taking
values in D = R�0 ) as a model for default intensities:
Duffie-Garleanu model
dXt = −λ(Xt − θ) dt + σ
�
Xt dWt + dZt
where Zt is a pure jump process with constant intensity c, whose
jumps are exponentially distributed with parameter α.
The above process is an affine jump diffusion, whose characteristic
function can be calculated from the generalized Riccati equations
φ̇(t, u) = F (ψ(t, u)),
where
F (u) = λθu +
cu
,
α−u
Martin Keller-Ressel
ψ̇(t, u) = R(ψ(t, u))
R(u) = −λu +
Affine Processes
u2 2
σ
2
Parameter Restrictions
Revisit the Feller Diffusion
Feller Diffusion
dXt = −λ(Xt − θ) dt + σ
Can we allow θ < 0?
�
Xt dWt ,
When Xt = 0, then Xt+∆t ≈ λθ < 0 and
well-defined.
=⇒ Parameter restrictions are necessary.
X0 ∈ R�0
�
Xt+∆t is not
Ideally, we can find necessary & sufficient parameter
restrictions.
Martin Keller-Ressel
Affine Processes
Characterization of affine jump-diff. on D = Rn × Rm
�0
Duffie, Filipovic & Schachermayer (2003) derive the necessary &
sufficient parameter restrictions (‘admissibility conditions’) for all
d
affine jump-diffusions on the state space D = Rn × Rm
�0 ⊂ R . We
write
J := {1, . . . , n} , I := {n + 1, . . . , n + m}
for indices of the real-valued and the non-negative components.
The following holds:
Characterization of an affine jump-diffusion on Rn × Rm
�0
Let X be an affine jump-diffusion with state space D = Rn × Rm
�0 .
Then the parameters a, αk , b, βk , m, µk , ν(dx) satisfy the following
conditions: �
Martin Keller-Ressel
Affine Processes
(continued)
a, αk are positive semi-definite matrices and αj = 0 for all
j ∈ J.
aek = 0 for all k ∈ I
αi ek = 0 for all k ∈ I and i ∈ I \ {k}
αj = 0 for all j ∈ J
b∈D
βi� ek ≥ 0 for all k ∈ I and i ∈ I \ {k}
βj� ek = 0 for all k ∈ I and j ∈ J
µj = 0 for all j ∈ J
supp ν ⊆ D .
Conversely, if the parameters a, αk , b, βk , m, µk , ν(dx) satisfy the
above conditions, then an affine jump-diffusion X with state space
D = Rn × Rm
�0 exists.
Martin Keller-Ressel
Affine Processes
Illustration of the parameter conditions


a=

≥ 0
0 0





b=





�
.. 
. 

� 

≥ 

... 

≥
αj
=
(j ∈ J)





j
β
=


(j ∈ J)











αi
=

(i ∈ I )

 �





0

�
.. 
. 

� 

0 

... 

0
≥
···

�
..
.
�
0
..
.
� 0 ···
0
0 αiii
0
..
.
0 ···
0










i
β
=

(i ∈ I )








�
.. 
. 


� 
≥ 

.. 
. 
 where βii ∈ R
≥ 

i
βi 

≥ 


... 
≥









 where αiii ≥ 0


0 





Stars denote arbitrary real numbers; the small ≥-signs denote non-negative real numbers and the big ≥-sign a
positive semi-definite matrix.
Martin Keller-Ressel
Affine Processes
We sketch a proof of the conditions’ necessity:
σ(x)σ(x)� = a + α1 x1 + · · · αd xd has to be positive
semidefinite for all x ∈ D
=⇒ a, ai are positive semidefinite for i ∈ I and αj = 0 for
j ∈ J.
λ(x) = m + µ1 x1 · · · + µd xd has to be non-negative for all
x ∈D
=⇒ µj = 0 for j ∈ J.
The process must not move outside D by jumping
=⇒ supp ν ⊂ D.
Martin Keller-Ressel
Affine Processes
Assume that Xt has reached the boundary of D, that is Xt = x
with xk = 0 for some k ∈ I . The following conditions have to hold,
such that Xt does not cross the boundary:
�
�
�
inward pointing drift: 0 ≤ ek� µ(x) = ek� b + i�=k βi xi
=⇒ b ∈ D, βi� ek ≥ 0 for all i ∈ I \ {i}, and βj� ek = 0
for all j ∈ J.
diffusion parallel �to the boundary:
�
�
0 = ek� σ(x) = ek� a + i�=k αi xi
=⇒ aek = 0 and αi ek = 0 for all i ∈ I \ {k}.
(ek denotes the k-th unit vector.)
Martin Keller-Ressel
Affine Processes
Part III
The Moment Formula
Martin Keller-Ressel
Affine Processes
The Moment formula
n
Let X be an affine jump-diffusion on D = Rm
�0 × R . We have
shown that
� �
�
�
E e u·XT � Ft = exp (φ(T − t, u) + Xt · ψ(T − t, u))
for all u ∈ iRd where φ and ψ solve the generalized Riccati
equations.
What can be said about general u ∈ Cd �and in� particular about
the moment generating function θ �→ E e θ·XT with θ ∈ Rd ?
Martin Keller-Ressel
Affine Processes
In general we should expect that
��
��
The exponential moment E �e u·XT � may be finite or infinite
depending on the value of u ∈ Cd and on the distribution of
XT
The generalized Riccati equations no longer have global
solutions for arbitrary starting values u ∈ Cd (blow-up of
solutions may appear)
Martin Keller-Ressel
Affine Processes
Moment formula
n
◦
Let X be an affine jump-diffusion on D = Rm
�0 × R with X0 ∈ D
d
and assume that dom κ ⊆ R is open. Let
∂
φ(t, u) = F (ψ(t, u)),
∂t
∂
ψ(t, u) = R(ψ(t, u)),
∂t
φ(0, u) = 0
(7)
ψ(0, u) = u
(8)
be the associated generalized Riccati equations, with F and R
analytically extended to
�
�
S(dom κ) := u ∈ Cd : Re u ∈ dom κ .
Then the following holds. . . , �
Martin Keller-Ressel
Affine Processes
Moment formula (contd.)
��
��
(a) Let u ∈ Cd and suppose that E �e u·XT � < ∞. Then
u ∈ S(dom κ) and there exists unique solutions φ, ψ of the
gen. Riccati equations such that
� �
�
�
E e u·XT � Ft = exp (φ(T − t, u) + ψ(T − t, u) · Xt ) (9)
for all t ∈ [0, T ].
(b) Let u ∈ S(dom κ) and suppose that the gen. Riccati equations
have
φ, ψ that start at u and exist up to T . Then
��
�� solutions
E �e u·XT � < ∞ and (9) holds for all t ∈ [0, T ].
Essentially: Solution to gen. Riccati equation exists ⇐⇒
Exponential Moment exists.
Martin Keller-Ressel
Affine Processes
Sketch of the proof of (a) (for real arguments θ ∈ Rd ):
Show by analytic extension that there exist functions φ(t, θ)
and ψ(t, θ) such that
� �
�
�
Mt := E e θ·XT � Ft = exp (φ(T − t, θ) + ψ(T − t, θ) · Xt ) .
By the assumption of (a) M is a martingale.
Show that φ and ψ are differentiable in t (This is the hard
part!)
Use the Ito-formula to show that the martingale property of M
implies that φ and ψ solve the generalized Riccati equations
Martin Keller-Ressel
Affine Processes
Sketch of the proof of (b):
Let θ ∈ dom κ. Define
Mt = exp (φ(T − t, θ) + ψ(T − t, θ) · Xt )
Use the Ito-formula and the generalized Riccati equations to
show that M is a local martingale
Since M is positive, it is a supermartingale and
�
�
E e �θ,XT � = E [MT ] ≤ M0 < ∞.
Apply part (a) of the theorem and use that the solutions of
the gen. Riccati equations are unique.
Martin Keller-Ressel
Affine Processes
Some consequences (we still assume that dom κ is open)
Exponential Martingales: t �→ e θ·Xt is a martingale if and only if
θ ∈ dom κ and F (θ) = R(θ) = 0.
Exponential Measure Change: Let X be an affine jump diffusion
and θ ∈ dom κ. Then there exists a measure Pθ ∼ P
such that X is an affine jump-diffusion under Pθ with
F θ (u) = F (u + θ) − F (θ)
R θ (u) = R(u + θ) − R(θ).
Exponential Family: The measures (Pθ )θ∈dom κ form a curved
exponential family with likelihood process
�
�
� t
dPθ
θ
Lt =
= exp θ · Xt − F (θ)t − R(θ) ·
Xs ds .
dP
0
Martin Keller-Ressel
Affine Processes
Proof: Extension of state-space approach
�t
Consider the process (Xt , Yt = 0 Xs ). The process (X , Y ) is
again an affine jump-diffusion (note: dYt = Xt dt)
Define
Lθt = exp (θ · Xt − F (θ)t − R(θ) · Yt )
Applying the moment formula to find the exponential moment
of order (θ, −R(θ)) of the extended process (X , Y ) we get
� � �
�
E LθT � Ft =
= exp (p(T − t) + q(T − t) · Xt )·exp (−F (θ)T − R(θ) · Yt )
where
∂
p(t) = F (q(t)),
p(0) = 0
∂t
∂
q(t) = R(q(t)) − R(θ),
q(0) = θ.
∂t
Martin Keller-Ressel
Affine Processes
θ is a stationary point of the second Riccati equation. Hence,
the (global) solutions are q(t) = θ and p(t) = tF (θ) for all
t≥0
Inserting the solution yields
� � �
�
E LθT � Ft = exp (θ · Xt − F (θ)t − R(θ) · Yt ) = Lθt ,
and hence t �→ Lθt is a martingale.
Define the measure Pθ by
�
dPθ ��
= Lt .
dP �Ft
A similar calculation yields F θ (u) and R θ (u) for the process X
under Pθ .
Martin Keller-Ressel
Affine Processes
Part IV
Bond and Option Pricing in Affine Models
Martin Keller-Ressel
Affine Processes
Pricing of Derivatives
We consider the following setup:
The goal is to price a European claim on some underlying
asset St , which has payoff f (ST ) at time T . We denote the
value of the claim at time t by Vt .
As numeraire
�� asset, we �use the money market account
t
Mt = exp 0 R(Xs ) ds determined by the short rate process
R(Xs ).
Under the assumption of no-arbitrage, there exists a
martingale measure Q for the discounted asset price process
Mt−1 St , such that
� �
�
Vt = Mt EQ MT−1 f (ST )� Ft .
Martin Keller-Ressel
Affine Processes
To allow for analytical calculations we make the following
assumption:
Both the short rate process R(Xt ) and the asset St are modelled
under the risk-neutral measure Q through an affine jump-diffusion
process Xt in the following way:
R(Xt ) = r + ρ� Xt ,
St = e ϑ
�X
t
for some fixed parameters r , ρ ≥ 0 and ϑ ∈ dom κ.
This setup includes the combination of many important short rate
and stock price models: Vasicek, Cox-Ingersoll-Ross,
Black-Scholes, Heston, Heston with jumps,. . .
Martin Keller-Ressel
Affine Processes
Extension-of-state-space-approach and moment formula yield the
following:
Discounted moment generating function
� �
�
Let u ∈ S(dom κ) and Φ(t, u) = Mt EQ MT−1 e u·XT � Ft . Suppose
the differential equations
φ̇∗ (t, u) = F ∗ (ψ ∗ (t, u)),
∗
∗
∗
ψ̇ (t, u) = R (ψ (t, u)),
φ∗ (0, u) = 0
∗
ψ (0, u) = u
with
F ∗ (u) = F (u) − r ,
or more precisely . . . �
Martin Keller-Ressel
and R ∗ (u) = R(u) − ρ,
Affine Processes
(10)
(11)
(continued)
1
F ∗ (u) = b � u + u � au + mκ(u) − r
2
1
∗
�
R1 (u) = β1 u + u � α1 u + µ1 κ(u) − ρ1 ,
2
..
.
1
Rd∗ (u) = βd� u + u � αd u + µd κ(u) − ρd .
2
have solutions t �→ φ∗ (t, u) and t �→ ψ ∗ (t, u) up to time T , then
Φ(t, u) = exp (φ∗ (T − t, u) + ψ ∗ (T − t, u) · Xt )
for all t ≤ T .
Martin Keller-Ressel
Affine Processes
Bond Pricing in Affine Jump Diffusion models
As an immediate application we derive the following formula for
pricing of zero-coupon bonds:
Bond Pricing
Suppose the gen. Riccati equations for the discounted mgf have
solutions up to time T for the initial value u = 0. Then the price
at time t of a (unit-notional) zero-coupon bond Pt (T ) maturing at
time T is given by
Pt (T ) = exp (φ∗ (T − t, 0) + Xt · ψ ∗ (T − t, 0)) .
Yields the well-known pricing formulas for the Vasicek and the
CIR-Model as special cases.
Martin Keller-Ressel
Affine Processes
No-arbitrage constraints on F ∗ and R ∗ :
The martingale assumption
� �
�
EQ MT−1 ST � Ft = Mt−1 St
leads to the following no-arbitrage constraints on F ∗ and R ∗ :
No-arbitrage constraints
F ∗ (ϑ) = F (ϑ) − r = 0
R ∗ (ϑ) = R(ϑ) − ρ = 0 .
Martin Keller-Ressel
Affine Processes
Pricing of European Options
A European call option with strike K and time-to-maturity T
pays (ST − K )+ at time T . We will parameterize the option
by the log-strike y = log K and denote its value at time t by
Ct (y , T ).
The goal is to derive a pricing formula based on our
knowledge of the discounted moment generating function
� �
�
�
Φ(t, u) = Mt EQ MT−1 e u·XT � Ft
Martin Keller-Ressel
Affine Processes
Idea: Calculate the Fourier transform of Ct (y , T ) (regarded
as a function of y ), and hope that it is a nice
expression involving Φ(T − t, u).
Problem: Ct (y , T ) may not be integrable, and thus may have
no Fourier transform.
Solution 1: Use the exponentially dampened call price
�t (y , T ) = e y ζ Ct (y , T ) where ζ > 0.
C
Solution 2: Replace the call option by a ‘covered call’ with payoff
ST − (ST − K )+ = min(ST , K ).
Several other (related) solutions can be found in the literature. . .
Martin Keller-Ressel
Affine Processes
Fourier pricing formula for European call options:
Let Ct (y , T ) be the price of a European call option with log-strike
y and maturity T . Then Ct (y , T ) is given by the inverse Fourier
transform
�
e −ζy ∞ −iωy Φ(T − t, (ζ + 1 + iω)ϑ)
Ct (y , T ) =
e
dω (12)
2π −∞
(ζ + iω)(ζ + 1 + iω)
where ζ is chosen such that ζ > 0 and the generalized Riccati
equations starting at (ζ + 1)θ have solutions up to time T .
(This formula is obtained by exponential dampening)
Note: the required ζ can always be found, since dom κ is open and
contains 0 and θ.
Martin Keller-Ressel
Affine Processes
Fourier pricing formula for European put options:
Let Pt (y , T ) be the price of a European put option with log-strike
y and maturity T . Then Pt (y , T ) is given by the inverse Fourier
transform
�
e −ζy ∞ −iωy Φ(T − t, (ζ + 1 + iω)ϑ)
Pt (y , T ) =
e
dω (13)
2π −∞
(ζ + iω)(ζ + 1 + iω)
where ζ is chosen such that ζ > −1 and the generalized Riccati
equations starting at (ζ + 1)θ have solutions up to time T .
(This formula is obtained by exponential dampening)
Note: the required ζ can always be found, since dom κ is open and
contains 0 and θ.
Martin Keller-Ressel
Affine Processes
Part V
Extensions and further topics
Martin Keller-Ressel
Affine Processes
Extensions
Allow jumps with infinite activity, superpositions of d + 1
different jump measures and killing.
These are the ‘affine processes’ in the sense of Duffie, Filipovic
and Schachermayer (2003))
This definition includes all Lévy process and all so-called
continuous-state branching processes with immigration.
Consider other state spaces:
Positive semidefinite matrices: Wishart process, etc.
Polyhedral and symmetric cones
Quadratic state spaces (level sets of quadratic polynomials)
Time-inhomogeneous processes
Martin Keller-Ressel
Affine Processes
Further Topics/Current Research
Utility maximization and variance-optimal hedging in affine
models (Jan Kallsen, Johannes Muhle-Karbe et al.)
Distributional Properties of affine processes: (non-central)
Wishart distributions, infinite divisibility of marginal laws
(Eberhard Mayerhofer et al.)
Feller property, path regularity, ‘regularity’ of the characteristic
function (Christa Cuchiero, Josef Teichmann et al.)
Relation to branching processes and superprocesses,
infinite-dimensional generalizations
Large-deviations and stationary distributions of affine
processes
Interaction between state-space geometry and distributionalor path-properties
Martin Keller-Ressel
Affine Processes
Further Topics/Current Research
Statistical estimation, density approximations, spectral
approximations
State-space-independent classification and/or characterization
results
Affine processes as finite-dimensional realizations of HJM-type
models
Applications, applications, applications:
Affine term structure models (ATSMs)
Affine stochastic volatility models (ASVMs)
Credit risk models, . . .
Martin Keller-Ressel
Affine Processes
Thank you for your attention!
Martin Keller-Ressel
Affine Processes