Affine Process Minicourse CompFin

8/9/2019 Affine Process Minicourse CompFin

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Affine Processes

Martin Keller-Ressel

TU [email protected]

Workshop on Interest Rates and Credit Risk 2011

TU Chemnitz23. November 2011

Martin Keller-Ressel Affine Processes


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Outline

Introduction to Affine Processes

Affi

ne Jump-Diff

usionsThe Moment Formula

Bond & Option Pricing in Affine Models

Extensions & Further Topics



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Part I

Introduction to Affine Processes



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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 price

modeling, interest rates, commodities, credit risk, . . . )efficient methods for pricing bonds, options,. . .

dynamics and (some) distributional properties arewell-understood

They include models withmean-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 . . . )



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The mathematical tools used are

characteristic functions (Fourier transforms)

stochastic calculus (with jumps)ordinary differential equations

Markov processes



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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 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.



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

The simplest (continuous-time) stochastic models for

mean-reverting processesUsed for modeling of interest rates, stochastic volatility,default intensity, commodity (spot) prices, etc.

Also the simplest examples of affine processes!



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Ornstein-Uhlenbeck process and Feller Diffusion

Ornstein-Uhlenbeck (OU)-process

dXt= (Xt ) dt+ dWt, X0 R

Feller Diffusion

dXt= (Xt ) dt+

XtdWt, X0 R0. . . long-term mean>0. . . rate of mean-reversion

0. . . volatility parameter

We define (Xt) :=

for the OU-process

Xt for the Feller diffusion.



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

What can be said about the distribution ofXt?

We will try to understand the distribution ofXt through itscharacteristic function

Xt(y) = E

eiyXt



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Let k N. IfE[|X|k]< , then

E[Xk] =ik k

ykX(y)

y=0

.

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

f(x) =

1

2

e

iyx

X(y) dy .



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

Exponentially-Affine characteristic function

E

eiyXt

= E

euXt

= exp ((t, u) + (t, u)X0) (1)

More precisely, if we can find functions (t, u), (t, u) with(t, u) = 0 and (t, u) =u, such that

Mt=f(t,Xt) = exp((T t, u) + (T t, u)Xt)

is a martingale then we have

E

euXT

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

and (1) indeed gives the characteristic function.



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Assume , are sufficiently differentiable and apply the

Ito-formula to

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

The relevant derivatives are

t

f(t,Xt) = (T t, u) +Xt(T t, u) f(t,Xt)

xf(t,Xt) =(T t, u)f(t,Xt)

2

x2 f(t,Xt) =(T t, u)2

f(t,Xt)



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We get:

df(t,Xt)

f(t,Xt)

=

Tt+XtTt dt+ TtdXt+1

2

2Tt2Xtdt=

=

Tt+XtTt

dt+Tt(Xt ) dt+

+ Tt(Xt) dWt+1

22Tt(Xt)

2 dt

f(t,Xt) is local martingale, if

(Tt+XtTt) = Tt(Xt ) +12

2Tt(Xt)2

for all possible states Xt.

Note that both sides are affinein Xt, since

(Xt)2 =

2 for the OU-process

2Xt for the CIR process



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

(s, u) =(s, u) + 2

2 (s, u)

(s, u) = (s, u)

For the CIR process we get

(s, u) =(s, u)

(s, u) = (s, u) + 2

2 (s, u)

These are ordinary differential equations. We also know theinitial conditions

(0, u) = 0, (0, u) =u.



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The OU process

For the OU-process we solve

(s, u) =(s, u) +2

2 (s, u)2, (0, u) = 0

(s, u) = (s, u), (0, u) =uand get

(t, u) =etu

(t, u) =u(1 et) + 24

u2(1 e2t)



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

E

eiyXt

= exp

iy

etX0+ (1 et) y22

2

2(1 e2t)

and we get the following:

Distributional Properties of OU-processLet Xbe an Ornstein-Uhlenbeck process. Then Xt is normallydistributed, with

EXt=+et(X

0 ), VarX

t=

2

2 1 e2t ,Q: Can you think of a simpler way to obtain the above result?



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The CIR process

For the CIR-process we solve

(s, u) =(s, u), (0, u) = 0

(s, u) = (s, u) + 2

2 (s, u)2, (0, u) =u.

and get

(t, u) = uet

1 22u(1 et)(2)

(t, u) = 22

log1 22

u(1 et) (3)The differential equation for is called a Riccati equation.Q: How was the solution of the Riccati equation determined?



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

E

eiyXt= 1 2

2(1 et)iy 22 exp etiy

1 22(1 et)iy

and we get the following:

Distributional Properties of the Feller Diffusion

Let Xbe an Feller-diffusion, and define b(t) = 2

4(1 et). ThenXtb(t) has non-central

2-distribution, with parameters

k=42

, = et

b(t),

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



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Summary

The key assumption was that the characteristic function ofXtis of exponentially-affine form

E

eiyXt

= exp ((t, iy) +X0(t, iy))

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

(t, u) =F((t, u)), (0, u) = 0

(t, u) =R((t, u)), (0, u) =u

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

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



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Part II

Affine Jump-Diffusions



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Jump Diffusions

We consider a jump-diffusion on D= Rm0Rn

Jump-Diffusion

dXt=(Xt) dt+ (Xt) dWt

diffusion part+ dZt

jump part(4)

where

Wtis a Brownian motion in Rd;

: D Rd, :D Rdd, andZis a right-continuous pure jump process, whose jump

heights have a fixed distribution (dx) and arrive withintensity(Xt), for some :D [0,).The Brownian motion W, the jump heights ofZ, and thejump times ofZare assumed to be independent.



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Jump Diffusions (2)



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Some elementary properties and notation for the jump process Zt:

Ztis RCLL (right continuous with left limits)Zt:= limst,stZs and Zt :=Zt Zt.Zt=Zt if and only Zt= 0 if and only a jump occurs attime t.

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

0st

f(Zs) :=

0(i)t

f(Zs)

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



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Ito formula for jump-diffusions

Ito formula for jump diffusions

Let Xbe a jump-diffusion with diffusion part Dtand jump part Zt.Assume that f : Rd R is a C1,2-function and that Zt is a purejump process of finite variation. Then

f(t, Xt) =f(0, X0) + t0

f

t(s,Xs) ds+ t

0

f

x(s, Xs) dDs+

+1

2

t0

tr

2f

x2(s,Xs)(Xs)(Xs)

ds+

+ 0st f(s,Xs).Here f

x =

fx1

, . . . , fxd

denotes the gradient off, and

2fx2

= 2fxixjis the Hessian matrix of the second derivatives off.Martin Keller-Ressel Affine Processes


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Affine Jump-Diffusion

Affine Jump-DiffusionWe call the jump diffusion X(defined in (4)) affine, if the drift(Xt), the diffusion matrix (Xt)(Xt)

and the jump intensity(Xt) are affine functions ofXt.

More precisely, assume that

(x) =b+ 1x1+ + dxd

(x)(x) =a+ 1x1+ + dxd

(x) =m+1x1+ dxd

where b,i Rd; a,i Rdd and m, i [0,).Note: (d+ 1) 3 parameters for a d-dimensional process.



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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 Xbe an affine jump-diffusion on D= Rm0 Rn. Then

E euXT

Ft= exp ((T t, u) +Xt (T t, u))for all u=iz iRd and 0 t T, where and solve thesystem of differential equations

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

(t, u) =R((t, u)), (0, u) =u (6)

with...



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Proof (sketch:)Show that the generalized Riccati equations have uniqueglobal solutions , (This is the hard part, and here theassumption that D= Rm0 Rn enters!)Fix T

0, define

Mt=f(t,Xt) = exp((T t, u) + (T t, u)Xt)

and show that Mt remains bounded.

Apply Itos formula to Mt:



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The relevant quantities for Itos formula are

t

f(t,Xt) = (T t, u) +Xt (T t, u) f(t,Xt)

xf(t,Xt) =(T t, u)f(t,Xt)

2

x2 f(t,Xt) =(T t, u) (T t, u)

f(t,Xt)

f(t,Xt) =

e(Tt,u)Xt 1

f(t,Xt)

Also define the cumulant generating function of the jump measure:

(u) =Rd

(eux 1)(dx).



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

f(

t,

Xt) = local martingale t

0

(T s, u) +Xs (T s, u)

f(s,Xs) ds+

+ t

0(T s, u)(Xs)f(s,Xs) ds+

+1

2

t0

(T s, u)(Xs)(Xs)(T s, u)f(s, Xs) ds+

+

t

0

(T s, u)

(Xs)f(s, Xs) ds

Inserting the definitions of(Xs), (Xs)(Xs) and (Xs)

and using the generalized Riccati equations we obtain the localmartingale property ofM.



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

E

euXT

Ft= E [ MT| Ft] =

=Mt=exp ((T t, u) + (T t, u)Xt) ,

showing desired form of the conditional characteristic function.


E l Th H d l


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Example: The Heston model

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

Heston model

dSt=

VtStdW1t

dVt= (Vt ) dt+

Vt

dW1t +

1 2 dW2t

where Wt= (W1t,W2t) is two-dimensional Brownian motion.1

We assume here that the interest rate r = 0


Th H d l (2)


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The Heston model (2)

The parameters have the following interpretation:

. . . mean-reversion rate of the variance process. . . long-term average ofVt. . . vol-of-var: the volatility of the variance process. . . leverage: correlation bet. moves in stock price and invariance.


Th H d l (3)


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

dLt= Vt2

dt+

VtdW1t

dVt= (Xt ) dt+

Vt

dW1t +

1 2 dW2t

which is a two dimensional a

ffine di

ffusion!WritingXt= (Lt,Vt) we find

(Xt) =

0

b

+ 0

1Lt+

1/2

2

Vt

(Xt)(Xt) = 0

a

+ 01

Lt+

1 2

2

Vt


Th H t d l (4)


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Thus, the characteristic function of log-price Ltand stochasticvarianceVtof the Heston model can be calculated from

(

t,

u) =

2(

t,

u)

2(t, u) =1

2

u21 u1

2(t, u) + 22

22(t, u) + u12(t, u)

with initial conditions (0, u) = 0, 2(t, u) =u2.

Note that 1(t, u) = 0 and thus 1(t, u) =u1 for all t 0.


D ffi G l d f lt i t it


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Duffie-Garleanu default intensity process

Duffie and Garleanu propose to use the following process (taking

values in D= R0) as a model for default intensities:Duffie-Garleanu model

dXt= (Xt ) dt+

XtdWt+dZt

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

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

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

where

F(u) =u+ cu

u, R(u) = u+ u

2

22


Parameter Restrictions


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Parameter Restrictions

Revisit the Feller Diffusion

Feller Diffusion

dXt= (Xt ) dt+

XtdWt, X0 R0

Can we allow


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(continued)

a,kare positive semi-definite matrices and j= 0 for allj

J.

aek= 0 for all k Iiek= 0 for all k I and i I\ {k}j= 0 for all j J

b Di ek 0 for all k I and i I\ {k}j ek= 0 for all k I and j Jj= 0 for all j J

supp D .Conversely, if the parameters a,k, b,k,m, k, (dx) satisfy theabove conditions, then an affine jump-diffusion X with state spaceD= Rn Rm0 exists.


Illustration of the parameter conditions


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Illustration of the parameter conditions

a=

00 0

j(j J)

= 0 i(i I)

=

...

0...0

0 0 iii 0 00...0

whereiii 0

b=

...

...

j

(j J) =

...

0...0

i

(i I) =

...

...

ii...

wherei

iR

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



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We sketch a proof of the conditions necessity:

(x)(x) =a+ 1x1+ dxdhas to be positivesemidefinite for all x D

= a, aiare positive semidefinite for i I and j= 0 forj

J.

(x) =m+1x1 +dxdhas to be non-negative for allx D

=j= 0 for j J.

The process must not move outside Dby jumping=supp D.



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Part III

The Moment Formula


The Moment formula


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The Moment formula

Let Xbe an affine jump-diffusion on D= Rm0 Rn. We haveshown that

E euXTFt= exp ((T t, u) +Xt (T t, u))for all u iRd where and solve the generalized Riccatiequations.

What can be said about general u

Cd and in particular about

the moment generating function E eXT with Rd?



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In general we should expect that

The exponential moment EeuXT may be finite or infinite

depending on the value ofu 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)



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Moment formula

Let Xbe an affine jump-diffusion on D= Rm0Rn with X0

D

and assume that dom Rd is open. Let

t(t, u) =F((t, u)), (0, u) = 0 (7)

t(t, u) =R((t, u)), (0, u) =u (8)

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

S(dom ) := u Cd : Re u dom .Then the following holds. . . ,



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Moment formula (contd.)

(a) Let u Cd

and suppose that E euXT


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Sketch of the proof of (a) (for real arguments

Rd):

Show by analytic extension that there exist functions (t, )and (t, ) such that

Mt := E

eXT

Ft

= exp ((T t, ) + (T t, )Xt) .

By the assumption of (a) M is a martingale.

Show that and are differentiable in t(This is the hardpart!)

Use the Ito-formula to show that the martingale property ofM

implies that and solve the generalized Riccati equations



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Sketch of the proof of (b):

Let dom . DefineMt= exp ((T t, ) + (T t, )Xt)

Use the Ito-formula and the generalized Riccati equations to

show that M is a localmartingaleSince Mis positive, it is a supermartingale and

E

e,XT

= E [MT] M0


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Some consequences (we still assume that dom is open)

Exponential Martingales: t eXt is a martingale if and only if

dom and F() =R() = 0.Exponential Measure Change: Let Xbe an affine jump diffusion

and dom . Then there exists a measure P Psuch that X is an affine jump-diffusion under P with

F

(u) =F(u+ ) F()R(u) =R(u+ ) R().

Exponential Family: The measures (P)dom form a curvedexponential family with likelihood process

Lt = dP

dP = exp

Xt F()t R()

t0

Xsds

.


Proof: Extension of state-space approach


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Consider the process (Xt, Yt=t

0 Xs). The process (X,Y) isagain an affine jump-diffusion (note: dYt=Xtdt)

DefineLt= exp(Xt F()t R()Yt)

Applying the moment formula to find the exponential momentof order (,R()) of the extendedprocess (X,Y) we get

E LTFt== exp (p(T t) +q(T t)Xt)exp(F()T R()Yt)

where

tp(t) =F(q(t)), p(0) = 0

tq(t) =R(q(t)) R(), q(0) =.



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is a stationary pointof the second Riccati equation. Hence,the (global) solutions are q(t) = and p(t) =tF() for all

t 0Inserting the solution yields

E

LT

Ft

= exp (Xt F()t R()Yt) =Lt,

and hence t Lt is a martingale.Define the measure P by

dP

dP Ft =Lt.A similar calculation yields F(u) and R(u) for the process Xunder P.



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Part IV

Bond and Option Pricing in Affine Models



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To allow for analytical calculations we make the following

assumption: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+ Xt, St=e

Xt

for some fixed parameters r, 0 and dom .This setup includes the combination of many important short rate

and stock price models: Vasicek, Cox-Ingersoll-Ross,Black-Scholes, Heston, Heston with jumps,. . .



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

Discounted moment generating function

Let u S(dom ) and (t, u) =MtEQ

M1T euXT

Ft. Supposethe differential equations

(t, u) =F((t, u)), (0, u) = 0 (10)

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

withF

(u

) =F

(u

)r, and

R

(u

) =R

(u

),

or more precisely . . .


( i d)


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(continued)

F(u) =bu+12

uau+m(u) r

R1 (u) =1 u+

1

2u1u+1(u) 1,

..

.Rd(u) =

du+

1

2udu+d(u) d.

have solutions t (t, u) and t (t, u) up to time T, then

(t, u) = exp ((T t, u) + (T t, u)Xt)

for all t T.


Bond Pricing in Affine Jump Diffusion models


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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 have

solutions up to time T for the initial value u= 0. Then the priceat time tof a (unit-notional) zero-coupon bond Pt(T) maturing attime T is given by

Pt(T) = exp ((T

t, 0) +Xt

(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:


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The martingale assumption

EQ

M1T ST

Ft

= M1t St

leads to the following no-arbitrage constraints on F and R:

No-arbitrage constraints

F() =F() r= 0R() =R() = 0.


Pricing of European Options


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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 Kand denote its value at time t by

Ct(y,T).The goal is to derive a pricing formula based on ourknowledge of the discounted moment generating function

(t, u) =MtEQ

M1T e

uXTFt



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

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

Solution 1: Use the exponentially dampenedcall priceCt(y,T) =eyCt(y,T) where >0.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. . .



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Fourier pricing formula for European put options:

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

Pt(y,T) = ey

2

eiy(T t, (+ 1 +i))

(+i)(+ 1 +i) d (13)

where is chosen such that > 1and the generalized Riccatiequations starting at (+ 1) have solutions up to time T.

(This formula is obtained by exponential dampening)

Note: the required can always be found, since dom is open andcontains 0 and .



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Part V

Extensions and further topics


Extensions


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Allow jumps with infinite activity, superpositions ofd+ 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



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Thank you for your attention!


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