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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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Introduction
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100 200 300 400 500 600 700
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.
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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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1000
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4000
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100 200 300 400 500 600 700
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
4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 60000
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1200
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Implied volatility surface
4320
4340
4360
4380
4400
4420
4440
4460
3*10^5*sigma
4000
4400
4800
5200
5600
6000
K
1000
1200
1400
16001800
2000
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.
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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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