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Affine Processes

Martin Keller-ResselTU Berlin

[email protected]

Workshop on Interest Rates and Credit Risk 2011TU 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


Part I

Introduction to Affine Processes


Affine Processes

Affine Processes are a class of stochastic processes. . .

with good analytic tractability(= explicit calculations and/or efficient numerical methodsoften available)

that can be found in every corner of finance (stock pricemodeling, interest rates, commodities, credit risk, . . . )

efficient methods for pricing bonds, options,. . .

dynamics and (some) distributional properties arewell-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 . . . )


The mathematical tools used are

characteristic functions (Fourier transforms)

stochastic calculus (with jumps)

ordinary differential equations

Markov 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, DamirFilipovic and Walter Schachermayer, The Annals of AppliedProbability, 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, DamirFilipovic and Eberhard Mayerhofer, Radon Series Comp. Appl.Math 8, 1-40, 2009.


We start by looking at the Ornstein-Uhlenbeck process and theFeller Diffusion.

The simplest (continuous-time) stochastic models formean-reverting processes

Used for modeling of interest rates, stochastic volatility,default intensity, commodity (spot) prices, etc.

Also the simplest examples of 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 meanλ > 0. . . rate of mean-reversionσ ≥ 0. . . volatility parameter

We define σ(Xt) :=

�σ for the OU-process

σ√

Xt for the Feller diffusion.


An important difference: The OU-process has support R, whilethe Feller diffusion stays non-negative

What can be said about the distribution of Xt?We will try to understand the distribution of Xt through itscharacteristic function

ΦXt (y) = E�e iyXt

�


Characteristic Function

Characteristic Function

For y ∈ R, the characteristic function ΦX (y) of a random variableX is defined as

ΦX (y) := E�e iyX

�=

� ∞

−∞e iyx dF (x) .

Properties:

ΦX (0) = 1, ΦX (−y) = ΦX (y), and |ΦX (y)| ≤ 1 for all y ∈ R.

ΦX (y) = ΦY (y) for all y ∈ R, if and only if Xd= Y .

Let X and Y be independent random variables. Then

ΦX+Y (y) = ΦX (y) · ΦY (y) .


Let k ∈ N. If E[|X |k ] <∞, then

E[X k ] = i−k ∂k

∂ykΦX (y)

��y=0

.

If the characteristic function ΦX (y) of a random variable Xwith density f (x) is known, then f (x) can be recovered by aninverse Fourier transform:

f (x) =1

2π

� ∞

−∞e−iyxΦX (y) dy .


Back to the OU and CIR processes: We write u = iy and make theansatz that the characteristic function of Xt is ofexponentially-affine form:

Exponentially-Affine characteristic function

E�e iyXt

�= E

�euXt

�= exp (φ(t, u) + ψ(t, u)X0) (1)

More precisely, if we can find functions φ(t, u), ψ(t, u) withφ(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�euXT

�= E [MT ] = M0 = exp (φ(T , u) + ψ(T , u)X0) ,

and (1) indeed gives the characteristic function.


Assume φ, ψ are sufficiently differentiable and apply theIto-formula to

f (t,Xt) = exp (φ(T − t, u) + Xtψ(T − t, u)) .

The relevant derivatives are

∂

∂tf (t,Xt) = −

�φ(T − t, u) + Xtψ(T − t, u)

�f (t,Xt)

∂

∂xf (t,Xt) = ψ(T − t, u)f (t,Xt)

∂2

∂x2f (t,Xt) = ψ(T − t, u)2f (t,Xt)


We get:

df (t,Xt)

f (t,Xt)= −

�φT−t + XtψT−t

�dt + ψT−t dXt +

1

2ψ2

T−tσ2Xt dt =

= −�φT−t + XtψT−t

�dt +−ψT−tλ(Xt − θ) dt+

+ ψT−tσ(Xt) dWt +1

2ψ2

T−tσ(Xt)2 dt

f (t,Xt) is local martingale, if

(φT−t + XtψT−t) = −ψT−tλ(Xt − θ) +1

2ψ2

T−tσ(Xt)2

for all possible states Xt .

Note that both sides are affine in Xt , since

σ(Xt)2 =

�σ2 for the OU-process

σ2Xt for the CIR process


We can ‘collect coefficients’:For the OU-process this yields

φ(s, u) = θλψ(s, u) +σ2

2ψ(s, u)

ψ(s, u) = −λψ(s, u)

For the CIR process we get

φ(s, u) = θλψ(s, u)

ψ(s, u) = −λψ(s, u) +σ2

2ψ(s, u)

These are ordinary differential equations. We also know theinitial conditions

φ(0, u) = 0, ψ(0, u) = u .


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) + Xoψ(t, iy))

is the characteristic function of Xt .


The OU process

For the OU-process we solve

φ(s, u) = θλψ(s, u) +σ2

2ψ(s, u)2, φ(0, u) = 0

ψ(s, u) = −λψ(s, u), ψ(0, u) = u

and get

ψ(t, u) = e−λtu

φ(t, u) = θu(1− e−λt) +σ2

4λu2(1− e−2λt)


Thus the characteristic function of the OU-process is given by

E�e iyXt

�= exp

�iy

�e−λtX0 + θ(1− e−λt)

�− y2

2

σ2

2λ(1− e−2λt)

�

and we get the following:

Distributional Properties of OU-process

Let X be an Ornstein-Uhlenbeck process. Then Xt is normallydistributed, with

EXt = θ + e−λt(X0 − θ), VarXt =σ2

2λ

�1− e−2λt

�,

Q: Can you think of a simpler way to obtain the above result?


The CIR process

For the CIR-process we solve

φ(s, u) = θλψ(s, u), φ(0, u) = 0

ψ(s, u) = −λψ(s, u) +σ2

2ψ(s, u)2, ψ(0, u) = u .

and get

ψ(t, u) =ue−λt

1− σ2

2λu(1− e−λt)(2)

φ(t, u) = −2λθ

σ2log

�1− σ2

2λu(1− e−λt)

�(3)

The differential equation for ψ is called a Riccati equation.Q: How was the solution of the Riccati equation determined?


Thus the characteristic function of the CIR-process is given by

E�e iyXt

�=

�1− σ2

2λ(1− e−λt)iy

�− 2λθσ2

exp

�e−λt iy

1− σ2

2λ(1− e−λt)iy

�

and we get the following:

Distributional Properties of the Feller Diffusion

Let X be an Feller-diffusion, and define b(t) = σ2

4λ(1− e−λt). ThenXt

b(t) has non-central χ2-distribution, with parameters

k =4λθ

σ2, α =

e−λt

b(t),

Q: Does there exist a limiting distribution? What is it?


Summary

The key assumption was that the characteristic function of Xt

is of exponentially-affine form

E�e iyXt

�= exp (φ(t, iy) + X0ψ(t, iy))

We derived that φ(t, u) and ψ(t, u) satisfy ordinarydifferential equations of the form

φ(t, u) = F (ψ(t, u)), φ(0, u) = 0

ψ(t, u) = R(ψ(t, u)), ψ(0, u) = u

Solving the differential equation gave φ(t, u) and ψ(t, u) inexplicit form.

The same approach works if the coefficients of the SDEs aretime-dependent; ODEs become time-dependent too.


Part II

Affine Jump-Diffusions


Jump Diffusions

We consider a jump-diffusion on D = Rm�0 × Rn

Jump-Diffusion

dXt = µ(Xt) dt + σ(Xt) dWt� �� diffusion part

+ dZt��jump part

(4)

where

Wt is a Brownian motion in Rd ;

µ : D → Rd , σ : D → Rd×d , and

Z is a right-continuous pure jump process, whose jumpheights have a fixed distribution ν(dx) and arrive withintensity λ(Xt−), for some λ : D → [0,∞).

The Brownian motion W , the jump heights of Z , and thejump times of Z are assumed to be independent.


Jump Diffusions (2)


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 attime t.

Let τ(i) be the time of the i-th jump of Zt . Let f be afunction such that f (0) = 0. Then

�

0≤s≤t

f (∆Zs) :=�

0≤τ(i)≤t

f (∆Zs)

is a well-defined sum, that runs only over finitely many values(a.s.)


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 purejump process of finite variation. Then

f (t,Xt) = f (0,X0) +

� t

0

∂f

∂t(s,Xs−) ds +

� t

0

∂f

∂x(s,Xs−) dDs+

+1

2

� t

0tr

�∂2f

∂x2(s,Xs−)σ(Xs−)σ(Xs−)�

�ds+

+�

0≤s≤t

∆ f (s,Xs) .

Here ∂f∂x =

�∂f∂x1

, . . . , ∂f∂xd

�denotes the gradient of f , and

∂2f∂x2 =

�∂2f

∂xi∂xj

�is the Hessian matrix of the second derivatives of f .


Affine Jump-Diffusion

Affine Jump-Diffusion

We call the jump diffusion X (defined in (4)) affine, if the driftµ(Xt), the diffusion matrix σ(Xt)σ(Xt)� and the jump intensityλ(Xt−) are affine functions of Xt .

More precisely, assume that

µ(x) = b + β1x1 + · · · + βdxd

σ(x)σ(x)� = a + α1x1 + · · · + αdxd

λ(x) = m + µ1x1 + · · ·µdxd

where b, βi ∈ Rd ; a, αi ∈ Rd×d and m, µi ∈ [0,∞).

Note: (d + 1)× 3 parameters for a d-dimensional process.


We want to show that an affine jump-diffusion has a (conditional)characteristic function of exponentially-affine form:

Characteristic function of Affine Jump Diffusion

Let X be an affine jump-diffusion on D = Rm�0 × Rn. Then

E�eu·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 thesystem of differential equations

φ(t, u) = F (ψ(t, u)), φ(0, u) = 0 (5)

ψ(t, u) = R(ψ(t, u)), ψ(0, u) = u (6)

with. . . �


(continued)

κ(u) =�

Rd (eu·x − 1) ν(dx), and

F (u) = b�u +1

2u�au + mκ(u)

R1(u) = β�1 u +1

2u�α1u + µ1κ(u),

...

Rd(u) = β�d u +1

2u�αdu + µdκ(u).

The differential equations satisfied by φ(t, u) and ψ(t, u) are calledgeneralized Riccati equations.The functions F (u),R1(u), . . . ,Rd(u) are of Levy-Khintchine form.


Proof (sketch:)

Show that the generalized Riccati equations have uniqueglobal solutions φ, ψ (This is the hard part, and here theassumption that D = Rm

�0 × Rn 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 :


The relevant quantities for Ito’s formula are

∂

∂tf (t,Xt−) = −

�φ(T − t, u) + Xt · ψ(T − t, u)

�f (t,Xt−)

∂

∂xf (t,Xt−) = ψ(T − t, u)f (t,Xt−)

∂2

∂x2f (t,Xt−) = ψ(T − t, u) · ψ(T − t, u)�f (t,Xt−)

∆ f (t,Xt) =�eψ(T−t,u)·∆Xt − 1

�f (t,Xt−)

Also define the cumulant generating function of the jump measure:

κ(u) =

�

Rd(eu·x − 1)ν(dx).


We can write f (t,Xt) as...

f (t,Xt) = ‘local martingale’−

−� t

0

�φ(T − s, u) + Xs− · ψ(T − s, u)

�f (s,Xs−) ds+

+

� t

0ψ(T − s, u) · µ(Xs−)f (s,Xs−) ds+

+1

2

� t

0ψ(T − s, u)�σ(Xs−)σ(Xs−)�ψ(T − s, u)f (s,Xs−) ds+

+

� t

0κ�ψ(T − s, u)

�λ(Xs−)f (s,Xs−) ds

Inserting the definitions of µ(Xs−), σ(Xs−)σ(Xs−)� and λ(Xs−)and using the generalized Riccati equations we obtain the localmartingale property of M.


Since M is bounded it is a true martingale and it holds that

E�euXT

��Ft

�= E [MT | Ft ] =

= Mt = exp (φ(T − t, u) + ψ(T − t, u) · Xt) ,

showing desired form of the conditional characteristic function.


Example: The Heston model

Heston proposes the following model for a stock St and its(mean-reverting) stochastic variance Vt (under the risk-neutralmeasure Q)1:

Heston model

dSt =�

VtSt dW 1t

dVt = −λ(Vt − θ) dt + η�

Vt

�ρ dW 1

t +�

1− ρ2 dW 2t

�

where Wt = (W 1t ,W 2

t ) is two-dimensional Brownian motion.

1We assume here that the interest rate r = 0Martin Keller-Ressel Affine Processes

The Heston model (2)

The parameters have the following interpretation:

λ. . . mean-reversion rate of the variance processθ. . . long-term average of Vt

η. . . ‘vol-of-var’: the volatility of the variance processρ. . . ‘leverage’: correlation bet. moves in stock price and invariance.



Transforming to the log-price Lt = log(St) we get

dLt = −Vt

2dt +

�Vt dW 1

t

dVt = −λ(Xt − θ) dt + η�

Vt

�ρ dW 1

t +�

1− ρ2 dW 2t

�

which is a two dimensional affine diffusion!Writing Xt = (Lt ,Vt) we find

µ(Xt) =

�0λθ

�

� �� b

+ 0��β1

Lt +

�−1/2−λ

�

� �� β2

Vt

σ(Xt)σ(Xt)� = 0��

a

+ 0��α1

Lt +

�1 ηρηρ η2

�

� �� α2

Vt



Thus, the characteristic function of log-price Lt and stochasticvariance Vt of the Heston model can be calculated from

φ(t, u) = λθψ2(t, u)

ψ2(t, u) =1

2

�u21 − u1

�− λψ2(t, u) +

η2

2ψ2

2(t, u) + ηρu1ψ2(t, u)

with initial conditions φ(0, u) = 0, ψ2(t, u) = u2.

Note that ψ1(t, u) = 0 and thus ψ1(t, u) = u1 for all t ≥ 0.


Duffie-Garleanu default intensity process

Duffie and Garleanu propose to use the following process (takingvalues in D = R�0) as a model for default intensities:

Duffie-Garleanu model


Xt dWt + dZt

where Zt is a pure jump process with constant intensity c , whosejumps are exponentially distributed with parameter α.

The above process is an affine jump diffusion, whose characteristicfunction can be calculated from the generalized Riccati equations

φ(t, u) = F (ψ(t, u)), ψ(t, u) = R(ψ(t, u))

where

F (u) = λθu +cu

α− u, R(u) = −λu +

u2

2σ2


Parameter Restrictions

Revisit the Feller Diffusion

Feller Diffusion


Xt dWt , X0 ∈ R�0

Can we allow θ < 0?

When Xt = 0, then Xt+∆t ≈ λθ < 0 and�

Xt+∆t is notwell-defined.

=⇒ Parameter restrictions are necessary.

Ideally, we can find necessary & sufficient parameterrestrictions.


Characterization of affine jump-diff. on D = Rn × Rm�0

Duffie, Filipovic & Schachermayer (2003) derive the necessary &sufficient parameter restrictions (‘admissibility conditions’) for allaffine jump-diffusions on the state space D = Rn ×Rm

�0 ⊂ Rd . Wewrite

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 followingconditions: �


(continued)

a, αk are positive semi-definite matrices and αj = 0 for allj ∈ J.

aek = 0 for all k ∈ I

αiek = 0 for all k ∈ I and i ∈ I \ {k}αj = 0 for all j ∈ J

b ∈ D

β�i ek ≥ 0 for all k ∈ I and i ∈ I \ {k}β�j 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 theabove conditions, then an affine jump-diffusion X with state spaceD = Rn × Rm

�0 exists.


Illustration of the parameter conditions

a =

≥ 00 0

αj

(j ∈ J)= 0 αi

(i ∈ I )=

�

≥ ...�0...0

� · · · � 0 · · · 0 αiii 0 · · · 00...0

where αiii ≥ 0

b =

�...�≥...≥

βj

(j ∈ J)=

�...�0...0

βi

(i ∈ I )=

�...�≥...≥βi

i≥...≥

where βii ∈ R

Stars denote arbitrary real numbers; the small ≥-signs denote non-negative real numbers and the big ≥-sign apositive semi-definite matrix.


We sketch a proof of the conditions’ necessity:

σ(x)σ(x)� = a + α1x1 + · · ·αdxd has to be positivesemidefinite for all x ∈ D

=⇒ a, ai are positive semidefinite for i ∈ I and αj = 0 forj ∈ J.

λ(x) = m + µ1x1 · · · + µdxd has to be non-negative for allx ∈ D

=⇒ µj = 0 for j ∈ J.

The process must not move outside D by jumping

=⇒ supp ν ⊂ D.


Assume that Xt has reached the boundary of D, that is Xt = xwith xk = 0 for some k ∈ I . The following conditions have to hold,such that Xt does not cross the boundary:

inward pointing drift: 0 ≤ e�k µ(x) = e�k

�b +

�i �=k βixi

�

=⇒ b ∈ D, β�i ek ≥ 0 for all i ∈ I \ {i}, and β�j ek = 0for all j ∈ J.

diffusion parallel to the boundary:

0 = e�k σ(x) = e�k

�a +

�i �=k αixi

�

=⇒ aek = 0 and αiek = 0 for all i ∈ I \ {k}.

(ek denotes the k-th unit vector.)


Part III

The Moment Formula


The Moment formula

Let X be an affine jump-diffusion on D = Rm�0 × Rn. We have

shown that

E�eu·XT

��Ft

�= exp (φ(T − t, u) + Xt · ψ(T − t, u))

for all u ∈ iRd where φ and ψ solve the generalized Riccatiequations.

What can be said about general u ∈ Cd and in particular aboutthe moment generating function θ �→ E

�eθ·XT

�with θ ∈ Rd?


In general we should expect that

The exponential moment E��eu·XT

�� may be finite or infinitedepending on the value of u ∈ Cd and on the distribution ofXT

The generalized Riccati equations no longer have globalsolutions for arbitrary starting values u ∈ Cd (blow-up ofsolutions may appear)


Moment formula

Let X be an affine jump-diffusion on D = Rm�0 × Rn with X0 ∈ D◦

and assume that dom κ ⊆ Rd is open. Let

∂

∂tφ(t, u) = F (ψ(t, u)), φ(0, u) = 0 (7)

∂

∂tψ(t, u) = R(ψ(t, u)), ψ(0, u) = u (8)

be the associated generalized Riccati equations, with F and Ranalytically extended to

S(dom κ) :=�

u ∈ Cd : Re u ∈ dom κ�

.

Then the following holds. . . , �


Moment formula (contd.)

(a) Let u ∈ Cd and suppose that E��eu·XT

�� <∞. Thenu ∈ S(dom κ) and there exists unique solutions φ, ψ of thegen. Riccati equations such that

E�eu·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 equationshave solutions φ, ψ that start at u and exist up to T . ThenE

��eu·XT�� <∞ and (9) holds for all t ∈ [0,T ].

Essentially: Solution to gen. Riccati equation exists ⇐⇒Exponential Moment exists.


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�eθ·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 hardpart!)

Use the Ito-formula to show that the martingale property of Mimplies that φ and ψ solve the generalized Riccati equations


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 toshow 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 ofthe gen. Riccati equations are unique.


Some consequences (we still assume that dom κ is open)

Exponential Martingales: t �→ eθ·Xt is a martingale if and only ifθ ∈ dom κ and F (θ) = R(θ) = 0.

Exponential Measure Change: Let X be an affine jump diffusionand θ ∈ dom κ. Then there exists a measure Pθ ∼ Psuch that X is an affine jump-diffusion under Pθ with

F θ(u) = F (u + θ)− F (θ)

Rθ(u) = R(u + θ)− R(θ).

Exponential Family: The measures (Pθ)θ∈dom κ form a curvedexponential family with likelihood process

Lθt =

dPθ

dP = exp

�θ · Xt − F (θ)t − R(θ) ·

� t

0Xsds

�.


Proof: Extension of state-space approach

Consider the process (Xt ,Yt =� t0 Xs). The process (X ,Y ) is

again an affine jump-diffusion (note: dYt = Xtdt)

DefineLθ

t = exp (θ · Xt − F (θ)t − R(θ) · Yt)

Applying the moment formula to find the exponential momentof order (θ,−R(θ)) of the extended process (X ,Y ) we get

E�Lθ

T

��Ft

�=

= exp (p(T − t) + q(T − t) · Xt)·exp (−F (θ)T − R(θ) · Yt)

where

∂

∂tp(t) = F (q(t)), p(0) = 0

∂

∂tq(t) = R(q(t))− R(θ), q(0) = θ.


θ is a stationary point of the second Riccati equation. Hence,the (global) solutions are q(t) = θ and p(t) = tF (θ) for allt ≥ 0

Inserting the solution yields

E�Lθ

T

��Ft

�= exp (θ · Xt − F (θ)t − R(θ) · Yt) = Lθ

t ,

and hence t �→ Lθt is a martingale.

Define the measure Pθ by

dPθ

dP

��Ft

= Lt .

A similar calculation yields F θ(u) and Rθ(u) for the process Xunder Pθ.


Part IV

Bond and Option Pricing in Affine Models


Pricing of Derivatives

We consider the following setup:

The goal is to price a European claim on some underlyingasset St , which has payoff f (ST ) at time T . We denote thevalue of the claim at time t by Vt .

As numeraire asset, we use the money market account

Mt = exp�� t

0 R(Xs) ds�

determined by the short rate process

R(Xs).

Under the assumption of no-arbitrage, there exists amartingale measure Q for the discounted asset price processM−1

t St , such that

Vt = MtEQ �M−1

T f (ST )��Ft

�.


To allow for analytical calculations we make the followingassumption:

Both the short rate process R(Xt) and the asset St are modelledunder the risk-neutral measure Q through an affine jump-diffusionprocess Xt in the following way:

R(Xt) = r + ρ�Xt , St = eϑ�Xt

for some fixed parameters r , ρ ≥ 0 and ϑ ∈ dom κ.

This setup includes the combination of many important short rateand stock price models: Vasicek, Cox-Ingersoll-Ross,Black-Scholes, Heston, Heston with jumps,. . .


Extension-of-state-space-approach and moment formula yield thefollowing:

Discounted moment generating function

Let u ∈ S(dom κ) and Φ(t, u) = MtEQ �M−1

T eu·XT��Ft

�. Suppose

the differential equations

φ∗(t, u) = F ∗(ψ∗(t, u)), φ∗(0, u) = 0 (10)

ψ∗(t, u) = R∗(ψ∗(t, u)), ψ∗(0, u) = u (11)

withF ∗(u) = F (u)− r , and R∗(u) = R(u)− ρ,

or more precisely . . . �


(continued)

F ∗(u) = b�u +1

2u�au + mκ(u)− r

R∗1 (u) = β�1 u +1

2u�α1u + µ1κ(u)− ρ1,

...

R∗d(u) = β�d u +1

2u�αdu + µdκ(u)− ρd .

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 .


Bond Pricing in Affine Jump Diffusion models

As an immediate application we derive the following formula forpricing of zero-coupon bonds:

Bond Pricing

Suppose the gen. Riccati equations for the discounted mgf havesolutions up to time T for the initial value u = 0. Then the priceat time t of a (unit-notional) zero-coupon bond Pt(T ) maturing attime 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 theCIR-Model as special cases.


No-arbitrage constraints on F ∗ and R∗:

The martingale assumption

EQ �M−1

T ST

��Ft�

= M−1t 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 .


Pricing of European Options

A European call option with strike K and time-to-maturity Tpays (ST − K )+ at time T . We will parameterize the optionby the log-strike y = log K and denote its value at time t byCt(y ,T ).

The goal is to derive a pricing formula based on ourknowledge of the discounted moment generating function

Φ(t, u) = MtEQ�M−1

T eu·XT

��Ft

�


Idea: Calculate the Fourier transform of Ct(y ,T ) (regardedas a function of y), and hope that it is a niceexpression involving Φ(T − t, u).

Problem: Ct(y ,T ) may not be integrable, and thus may haveno Fourier transform.

Solution 1: Use the exponentially dampened call price�Ct(y ,T ) = eyζCt(y ,T ) where ζ > 0.

Solution 2: Replace the call option by a ‘covered call’ with payoffST − (ST − K )+ = min(ST ,K ).

Several other (related) solutions can be found in the literature. . .


Fourier pricing formula for European call options:

Let Ct(y ,T ) be the price of a European call option with log-strikey and maturity T . Then Ct(y ,T ) is given by the inverse Fouriertransform

Ct(y ,T ) =e−ζy

2π

� ∞

−∞e−iωy Φ(T − t, (ζ + 1 + iω)ϑ)

(ζ + iω)(ζ + 1 + iω)dω (12)

where ζ is chosen such that ζ > 0 and the generalized Riccatiequations 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 andcontains 0 and θ.


Fourier pricing formula for European put options:

Let Pt(y ,T ) be the price of a European put option with log-strikey and maturity T . Then Pt(y ,T ) is given by the inverse Fouriertransform

Pt(y ,T ) =e−ζy

2π

� ∞

−∞e−iωy Φ(T − t, (ζ + 1 + iω)ϑ)

(ζ + iω)(ζ + 1 + iω)dω (13)

where ζ is chosen such that ζ > −1 and the generalized Riccatiequations 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 andcontains 0 and θ.


Part V

Extensions and further topics


Extensions

Allow jumps with infinite activity, superpositions of d + 1different jump measures and killing.

These are the ‘affine processes’ in the sense of Duffie, Filipovicand Schachermayer (2003))This definition includes all Levy process and all so-calledcontinuous-state branching processes with immigration.

Consider other state spaces:Positive semidefinite matrices: Wishart process, etc.Polyhedral and symmetric conesQuadratic state spaces (level sets of quadratic polynomials)

Time-inhomogeneous processes


Further Topics/Current Research

Utility maximization and variance-optimal hedging in affinemodels (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 characteristicfunction (Christa Cuchiero, Josef Teichmann et al.)

Relation to branching processes and superprocesses,infinite-dimensional generalizations

Large-deviations and stationary distributions of affineprocesses

Interaction between state-space geometry and distributional-or path-properties


Further Topics/Current Research

Statistical estimation, density approximations, spectralapproximations

State-space-independent classification and/or characterizationresults

Affine processes as finite-dimensional realizations of HJM-typemodels

Applications, applications, applications:Affine term structure models (ATSMs)Affine stochastic volatility models (ASVMs)Credit risk models, . . .


Thank you for your attention!


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