Bilateral Gamma Processes in Finance - Slides

8/9/2019 Bilateral Gamma Processes in Finance - Slides

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Bilateral Gamma Processes

in Finance

Stefan TappeVienna Institute of [email protected]

based on joint work with:Uwe Kuchler (Humboldt University Berlin)

PRisMa 2008

One-Day Workshop on Portfolio Risk Management

September 29th, 2008, Vienna


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Stefan Tappe 2008/09/29

Introduction

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Stochastic models for risky assets in financial markets.Bilateral Gamma Processes in Finance 1


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Exponential Levy models

Two financial assets:

St = S0eXt,

Bt = ert.

X is a Levy process.

Example: Black-Scholes model with Xt = Wt + ( 22 )t.

Idea: Take X = X+X, where X+, X are independent subordinators.

Bilateral Gamma Processes in Finance 2


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Bilateral Gamma distributions

Four parameters: Shape parameters: +, > 0,

Scale parameters: +, > 0.

(+, +; , ) is the distribution of X+ X, where X+ and X are independent,

X+ (+, +), X (, ).

Family of bilateral Gamma distributions.



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Bilateral Gamma processes

Characteristic function:

(z) =

+

+ iz+

+ iz

, z R.

Infinitely divisible with Levy measure

F(dx) =

+

xe

+x1 (0,)(x) +

|x|e|x|

1 (,0)(x)

dx.

We call the associated Levy process a bilateral Gamma process.



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Outline of the talk

Bilateral Gamma:1. Bilateral Gamma distributions.

2. Bilateral Gamma processes.

Applications to Finance:1. Option pricing in bilateral Gamma stock models.

2. Illustration of the theory: DAX 1996-1998.



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Related distributions

Generalized tempered stable distributions (Cont and Tankov 2004):

F(dx) =

+

x1++e

+x1 (0,)(x) +

|x|1+e|x|

1 (,0)(x)

dx.

CGMY distributions (Carr, Geman, Madan and Yor 1999):

F(dx) =

C

x1+YeMx 1 (0,)(x) +

C

|x|1+YeG|x|

1 (,0)(x)

dx.

Variance Gamma distributions (Madan, Carr and Chang 1998).



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Characterization of Variance Gamma distributions

Variance Gamma distribution V G(, 2, ):

(z) =

1 iz+

2

2z21

, z R.

Let := (+, +; , ). There is equivalence between:1. is Variance Gamma;

2. + = ;

3. X is a time-changed Brownian motion Xt = WYt;

4. is a limit of GH-distributions.



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Representation of the density

Density function for x > 0:

f(x) =(+)

+()

(+ + )12(

++)(+) x12(++)1 ex2(+)

W1

2

(+),1

2

(++1)(x(+ + )).

W, denotes the Whittaker function

W,(z) =ze

z2

( + 12

)

0

t12et

1 +

t

z

+12dt, z > 0

for > 12

.



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Properties of the density

Unimodality: f is strictly increasing/decreasing on (, x0)/(x0,).

Smoothness: Let N N be such that N < + + N + 1. Then

f CN1

(R) \ CN

(R).

Semi-heavy tails: We have the asymptotic behaviour

f(x)

C1x+1e

+x for x C2|x|

1e

|x| for x .



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Density shapes

alpha-

alpha+

0

1

2

1 2



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Bilateral Gamma processes

X = X+ X, in particular FV process and no Gaussian part.

Infinitely many jumps in each compact interval.

For 0 s < t we have:Xt Xs (+(t s), +; (t s), ).

Efficient methods for simulating bilateral Gamma processes.



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Quick Review

Bilateral Gamma distributions: Simple characteristic function.

Densities: unimodal, semi-heavy tailed.

Bilateral Gamma processes: FV sample paths with infinitely many jumps on every interval.

Easy to simulate.



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Exponential bilateral Gamma models

Two financial assets: St = S0e

Xt,

Bt = ert.

X (+, +; , ) is a bilateral Gamma process.

The market is: free of arbitrage,

but not complete.



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Option Pricing

Price of a European Call Option:

= EQ[(ST K)+],

where QP is a martingale measure.

Fourier transformation: Under appropriate conditions

= erTK

2

i+i

K

S0

izT(z)z(z i)dz,

where T is the characteristic function of XT under Q.



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Requirements on the martingale measure

There are several martingale measures Q P.

Under Q, the characteristic function T should be simple.

Recall that for a bilateral Gamma process:

(z) =

+

+ iz+

+ iz

, z R.

Q should have an economic interpretation.



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Esscher transforms

For (, +) we define P P as

dP

dP Ft:= eXt()t, t 0.

The cumulant generating function of X is given by

(z) = + ln

+

+ z

+ ln

+ z

, z (, +).



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Esscher martingale measure

There exists (, +) such that P is a martingale measure iff

+ + > 1.

In this case, it is the unique solution of the equation

+ + 1

+ +

+ + 1

= erq, (, + 1).

We have X (+, + ; , + ).



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Relative entropy

For each Q P define the relative entropy

HFt(Q |P) = EQ

lndQ

dP

Ft

, t 0.

Find a martingale measure Q P such that

HFt(Q |P) = min

QEMMHFt(Q |P), t 0.

Minimal entropy martingale measure (MEMM).



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Exponential transform

Let Xt := L(eXt(rq)t) be the exponential transform.

X is again a Levy process.

For 0 we define P P asdP

dP

Ft

:= eXt()t, t 0.

denotes the cumulant generating function of

X.



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Minimal entropy martingale measure

If + > 1, there exists 0 such that P is a MEMM iff

+ ln

+

+ 1

+ ln

+ 1

r q.

In this case, it is the unique solution of the equation

+0

e+x

x(ex 1)e(ex1)dx + +

0

ex

x(ex 1)e(ex1)dx

= r q, 0.



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Characteristic function

X is a Levy process under P with

(z) = exp

R

(eizx 1)e(ex1)F(dx)

, z R.

The value of the minimal relative entropy is given by

HF1(P |P) = r q

+

0

e+x

x (e

(ex1)

1)dx

0

e+x

x (e

(ex1)

1)dx.



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Martingale measures considered so far

Esscher martingale measure: Pro: Easy to obtain, X remains bilateral Gamma under P.

Contra: No economic interpretation.

Minimal entropy martingale measure: Pro: Easy to obtain, economic interpretation.

Contra: Characteristic function of X under P not in closed form.



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Bilateral Esscher transforms

Recall that X = X+ X.

For + (, +) and (, ) we define P(+,) P as

dP(+

,

)

dP

Ft

:= e+X+t

+(+)t eXt ()t, t 0.

+, denote the cumulant generating functions of X+, X.



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Bilateral Esscher martingale measures

Define : (, + 1) (, ) as

() :=

+ + 1

+

e(rq) 11

.

Then P(,()) is a martingale measure for each (, + 1).

We have X (+, + ; , ()) under P(,()). Thus:

= erTK

2i+i

KS0iz + + + iz

+T () () izT

dz

z(z i).



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Minimizing the relative entropy

Find (, + 1) such that

HF1(P(,()) |P) = min

(,+1)HF1(P

(,()) |P) minQEMM

HF1(Q |P).

The relative entropy is given by:

HF1(P(,()) |P) = +

+

+ 1 ln

+

+

+

() 1 ln

(), (, + 1).Bilateral Gamma Processes in Finance 25


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Example: DAX 1996-1998

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We assume St = S0eXt, where X (+, +; , ).



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Estimates

Maximum Likelihood Estimate:

(+, +; , ) = (1.55, 133.96;0.94, 88.92).

With = 5.30 we haveHF1(P

(,()) |P) = min(,+1)

HF1(P(,()) |P) = 0.0029411

and X (1.55, 139.26;0.94, 83.65) under P(,()).

Note that minQEMMHF1(Q |P) = 0.0029409.



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Black Scholes and Bilateral Gamma Prices

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Implied volatility surface

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3*10^5*sigma

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10*T



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Relation to the Normal distribution

The Central Limit Theorem yields:

(+, +; , ) N

+

+

,

+

(+)2+

()2

.

Berry-Esseen gives us:

supxR

|Fn(x) n(x)| cn

,

where + n+ and n for some n N.


S f / /


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Conclusion

Stock models based on bilateral Gamma processes.

Option pricing using historical data.

Current Research: Model calibration to option price data.


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Bilateral Gamma Processes in Finance - Slides