The Change of Measure Problem for Tempered Stable Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
The Change of Measure Problem for Tempered Stable Processes Michele Leonardo Bianchi1,2 (joint work with Y.S.Kim, S.T.Rachev and F.J.Fabozzi) 1 Ph.D. student, Department of Mathematics, Statistics, Computer Science and Applications University of Bergamo 2 Visiting Ph.D. student, Institute of Statistic and Economics, University of Karlsruhe
Radon Workshop on Financial and Actuarial Mathematics for Young Researchers, Linz, May 30 - 31, 2007 Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
History In 1994, the truncated Le´vy flight model is introduced in statistical physics (Mantegna and Stanley, Novikov, Koponen) In 2000, truncated Le´vy processes are considered for pricing options (Boyarchenko et al.) In 2002, CGMY model appears in mathematical finance literature (Carr et al.) In 2006, GARCH model with modified tempered stable (MTS) innovation. See (Kim et al.). In 2007, Kim-Rachev (KR) tempered stable distribution is introduced (Kim, Rachev, Bianchi, Fabozzi)
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Motivations Heavy tails Gain/loss asymmetry Aggregational normality Equivalent change of measure based on historical estimates
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
The importance of tempered stable processes comes from the fact that they combine both α-stable and Gaussian trends. The Lévy measure M0 of a α-stable distribution on Rd in polar coordinates is of the form M0 (dv , du) = v −α−1 dv σ(du) where σ is a finite measure on S d−1 .
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
´ Definition (Rosinski) Let α ∈ (0, 2), σ is a finite measure on S d−1 . A probability measure on Rd is called tempered α-stable (denoted as TαS) if is infinitely divisible without Gaussian part and whose Lévy measure M can be written in polar coordinates as M(dv , du) = v −α−1 q(v,u)dv σ(du). where q : (0, ∞) × S d−1 7→ (0, ∞) is a Borel function such that q(·, u) is completely monotone with q(∞, u) = 0 for each u ∈ S d−1 . A TαS distribution is called a proper TαS distribution if limv →0+ q(v , u) = 1 for each u ∈ S d−1 .
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
A function q(·, u) is completely monotone if (−1)n
d q(v , u) > 0 dv
for all v > 0, u ∈ S d−1 , and n = 0, 1, 2, · · · . In particular q(·, u) is stictly decreasing and convex. The tempering function q can be represented as the Laplace transform Z ∞ q(v , u) = e−vs Q(ds|u) 0
where {Q(·|u)}u∈S d−1 is a measurable family of Borel measures on (0, ∞).
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
´ Theorem (Rosinski) Lévy measure M of TαS distribution can be written in the form Z Z ∞ M(A) = IA (tx)αt −α−1 e−t dtR(dx), A ∈ B(Rd ). Rd0
(1)
0
where R is a unique measure on Rd such that Z R({0}) = 0 and (||x||2 ∧ ||x||α )R(dx) < ∞.
(2)
Rd
Conversely, if R is a measure satisfying (2) then (1) defines Lévy measure of a TαS distribution. M corresponds to a proper TαS distribution if and only if Z ||x||α R(dx) < ∞. Rd
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
´ R is call the spectral measure (or Rosinski measure) of the corresponding TαS distribution Using the measure R, Characteristic function Change of measure can be found explicitly.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
KR tempered stable distribution
In the following we will define a r.v. X with KR Tempered Stable distribution (shortly KR distribution) with parameters α, k+ , k− , r+ , r− , p+ , p− , m with α ∈ (0, 2), k+ , k− , r+ , r− > 0, p+ , p− ∈ (−α, ∞) \ {−1, 0}
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
Let σ be a finite measure on S 0 σ(A) =
α k− r− k+ r+α IA (1) + IA (−1), α + p+ α + p−
A ⊂ S0,
and q(·, u) a completely monotone function Z r+ q(v , 1) = (α + p+ )r+−α−p+ e−v /s sα+p+ −1 ds 0 Z r− −α−p− e−v /s sα+p− −1 ds, q(v , −1) = (α + p− )r− 0
with α ∈ (0, 2), k+ , k− , r+ , r− > 0 and p+ , p− > −α.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
Then the spectral measure R corresponding to the Lévy measure M can be deduced as −p−
R(dx) = (k+ r+−p+ I(0,r+ ) (x)|x|p+ −1 + k− r−
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
I(−r− ,0) (x)|x|p− −1 ) dx.
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
If α 6= 1 , E[e
iuX
] = exp Hα (u; k+ , r+ , p+ ) + Hα (−u; k− , r− , p− ) k+ r+ k− r− + iu m + αΓ(−α) − , p+ + 1 p− + 1
where Hα (u; a, h, p) =
aΓ(−α) (2 F1 (p, −α; 1 + p; ihu) − 1) . p
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
−1 E[eθX ] < ∞ iff −r− ≤ θ ≤ r+−1 Infinitely divisible, therefore a Lévy process can be considered.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
Proposition The KR distribution with parameters (α, k+ , k− , r+ , r− , p+ , p− , m) converges weakly to the CGMY distribution as p± → ∞ provided that α 6= 1 and −α k± = c(α + p± )h±
for c > 0.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
8 CGMY KR p+=p−=−0.2
7
KR p+=p−=1 KR p+=p−=10
6 5 4 3 2 1 0
C = 0.01 −Y k+ = C(Y + p)r+
−0.2
G = 2 k− = C(Y + p)r
−Y −
−0.1
0
0.1
M = 10
Y = 1.25
r+ = 1/M
r− = 1/G
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
0.2
0.3
p+ = p− = { −0.25, 1, 10,}
The Change of Measure Problem for TS Processes
α = Y
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
E[X ] = c1 (X ) = m ! 2 k− r− k+ r+2 Var(X ) = c2 (X ) = Γ(2 − α) + p+ + 2 p− + 2 3 k r− k r+3 Γ(3 − α) p+++3 − p−−+3 c3 (X ) s(X ) = = 2 2 3/2 c2 (X )3/2 k− r− k r+ Γ(2 − α)3/2 p+++2 + p− +2 4 k r− k r+4 Γ(4 − α) p+++4 + p−−+4 c4 (X ) k (X ) = = 2 2 2 c2 (X )2 k− r− k r+ Γ(2 − α)2 p+++2 + p− +2
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
Tail Behavior
Upper and lower bounds 1 P(|X − m| ≥ λ) ≤ 2 Γ(2 − α) λ
2 k− r− k+ r+2 + p+ + 2 p− + 2
!
2λ
P(|X − m| ≥ λ) ≥ C
e− ¯r λα+2
where ¯r = max(r+ , r− ).
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
The market stock price process St = S0 eXt ,
t ∈ [0, T ]
where Xt is a KR Lévy process Under the market measure P, Xt is a KR process with parameters (α, k+ , k− , r+ , r− , p+ , p− , µ − ωP ) Under the risk neutral measure Q, Xt is a KR process with ˜+ , p ˜ − , r − ωQ ) parameters (α, k˜+ , k˜− , ˜r+ , ˜r− , p the convexity corrections ωP and ωQ are added, in order to obtain EP [St ] = S0 EP [eXt ] = S0 eµt and EQ [St ] = S0 EQ [eXt ] = S0 ert
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Let us consider a canonical process (Xt )t≥0 on (Ω, F, (Ft )t≥0 ), where Ω is the set of cadlag functions on [0, ∞) into R and F = σ{Xs ; s ≥ 0} Ft = ∩s>t σ{Xu : u ≤ s},
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
t ≥ 0.
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Theorem Consider two probability measure P1 , P2 and the canonical process (Xt )t≥0 on (Ω, F, (Ft )t≥0 ), given above. For each j = 1, 2, suppose (Xt )t≥0 is the KR tempered stable process under Pj with parameters (αj , kj,+ , kj,− , rj,+ , rj,− , pj,+ , pj,− , mj ) and pj,± > 12 − αj , αj ∈ (0, 1) . Then P1 |Ft and P2 |Ft are equivalent pj,± > 1 − αj , αj ∈ [1, 2) for every t > 0 if and only if (3)
α := α1 = α2 , α k1,+ r1,+
α + p1,+
α k2,+ r2,+
=
α + p2,+
,
α k1,− r1,−
α + p1,+
=
α k2,− r2,−
(4)
α + p2,+
and m2 −m1 = Γ(1−α)
X
(−1)j
j=1,2
kj,+ rj,+ kj,− rj,− − pj,+ + 1 pj,− + 1
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
if α 6= 1 (5)
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Proof Without consider all details, we focus our attention on the following condition Z Z 1 (1 − qj (v , u))2 v −αj −1 dv σ(du) < ∞. S0
0
The complete proof can be found in Kim et al.[1, Theorem 4.3]. ´ It comes from results of Sato [3, Theorem 33.1] and Rosinski [5, Theorem 4.1].
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Change of measure - KR case Theorem Assume that (St )t∈[0,T ] is the the KR price process with parameters (α, k+ , k− , r+ , r− , p+ , p− , µ) under the market measure P, and with ˜+ , p ˜− , r ) under a measure Q. Then Q parameters (˜ α, k˜+ , k˜− , ˜r+ , ˜r− , p is an EMM of P if and only if α=α ˜, k˜+˜r+α k+ r+α = , ˜+ α + p+ α+p
α α k− r− k˜−˜r− = ˜− α + p− α+p
and µ − r = Hα (−i; k+ , r+ , p+ ) + Hα (i; k− , r− , p− ) ˜+ ) − Hα (i; k˜− , ˜r− , p ˜− ). − Hα (−i; k˜+ , ˜r+ , p Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Change of measure - CGMY case
Theorem (Kim and Lee, 2006) Assume that (St )t∈[0,T ] is the CGMY stock price process with parameters (C, G, M, Y , µ) under the market measure P, and with ˜ G, ˜ M, ˜ Y ˜ , r ) under a measure Q. Then Q is an EMM of parameters (C, P if and only if ˜ = C, C ˜ = Y, Y and ˜ M, ˜ Y ) = µ − Ψ0 (−i; C, G, M, Y ). r − Ψ0 (−i; C, G,
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Why do we consider this so complicated KR distribution? CGMY model 4 parameters (+ r ) - 3 equality constrains = 1 free parameter (in the estimation of risk neutral parameters) KR model 7 parameters (+ r ) - 4 equality constrains = 3 free parameters (in the estimation of risk neutral parameters) With KR model a better fit in the tails can be obtained (in the market parameters estimation)
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Why do we consider this so complicated KR distribution? CGMY model 4 parameters (+ r ) - 3 equality constrains = 1 free parameter (in the estimation of risk neutral parameters) KR model 7 parameters (+ r ) - 4 equality constrains = 3 free parameters (in the estimation of risk neutral parameters) With KR model a better fit in the tails can be obtained (in the market parameters estimation)
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Why do we consider this so complicated KR distribution? CGMY model 4 parameters (+ r ) - 3 equality constrains = 1 free parameter (in the estimation of risk neutral parameters) KR model 7 parameters (+ r ) - 4 equality constrains = 3 free parameters (in the estimation of risk neutral parameters) With KR model a better fit in the tails can be obtained (in the market parameters estimation)
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
60 Market data Normal CGMY KR
50
40
30
20
10
0 −0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
S&P 500 Index (from January 1st, 1992 to April 18th, 2002) MLE density fit.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
χ2 , KS, AD and AD2 statistics (degrees of freedom in round brackets). Monte Carlo P-values were obtained via 1000 simulations.
Model Normal CGMY KR
Model Normal CGMY KR
S&P 500 Index from January 1st, 1992 to April 18th, 2002 χ2 KS AD 546.49(288) 0.0663 2180.7 273.4(255) 0.0103 0.2945 268.91(252) 0.0109 0.2315
P-value Theoretical χ2 KS AD2 0 0 0 0.2045 0.9450 0.6356 0.2216 0.9165 0.9082
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
χ2 0 0.43 0.53
AD2 23.762 0.6130 0.3367
Monte Carlo KS AD 0 0 0.908 0.098 0.875 0.242
AD2 0 0.656 0.916
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
70 Market data B−S CGMY KR
60
50
40
30
20
10
0 −0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
S&P 500 Index (from January 1st, 1984 to January 1st, 1994) MLE density fit.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
χ2 , KS, AD and AD2 statistics (degrees of freedom in round brackets). Monte Carlo P-values were obtained via 1000 simulations. S&P 500 Index from January 1st, 1984 to January 1th, 1994 Model χ2 KS AD Normal 482.39(202) 0.0699 3.9e+6 CGMY 191.68(179) 0.0191 0.1527 KR 180.07(181) 0.0107 0.1302
Model Normal CGMY KR
P-value Theoretical χ2 KS AD2 0 0 0 0.2451 0.3180 0.0865 0.5055 0.9343 0.3723
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
χ2 0 0.893 0.974
AD2 33.654 2.0475 0.9719
Monte Carlo KS AD 0 0 0.305 0.696 0.875 0.872
AD2 0 0.086 0.361
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
min θ˜
N X
˜
ˆ i − C θ (Ti , Ki ))2 (C
i=1
ˆ i is the price of an option as observed in the market, C ˜
Ciθ the price computed according to a pricing formula in a chosen ˜ model with a parameter set θ.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
S&P 500 Index MLE density fit and EMM estimations
Normal CGMY KR
µ 0.096364 C 10.161 k+ 3286.1
S&P 500 Index from January 1st, 1992 to April 18th, 2002 Parameters σ 0.15756 G M Y m 97.455 98.891 0.5634 0.1135 k− r+ r− p+ p− 2124.8 0.0090 0.0113 17.736 17.736
CGMY T 0.0880 0.1840 0.4360 0.6920 0.9360 1.1920 1.7080
˜ M 106.5827 103.4463 92.4701 89.4576 90.0040 82.6216 77.3594
˜ G 96.1341 93.3887 83.7430 81.0851 81.5675 75.0354 70.3609
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
α 0.5103
µ 0.1252
KR k˜+ 5325.8 9126.3 4757.3 3866.4 6655.4 9896.7 10000
k˜− 33.727 33.024 31.327 30.776 30.78 29.483 28.468
˜r+ 0.0065 0.0066 0.0074 0.0076 0.0075 0.0079 0.0084
˜r− 0.0330 0.034 0.0381 0.0395 0.03953 0.0430 0.046
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Error Estimators of Option Prices fit. Option data April 18, 2002 T 0.0880
Model
APE
AEE
RMSE
ARPE
CGMY KR
0.0149 0.0030
0.4019 0.0826
0.4613 0.1023
0.0175 0.0035
CGMY KR
0.0341 0.0234
1.0998 0.7541
1.4270 0.9937
0.0442 0.0295
CGMY KR
0.0437 0.0361
3.1727 2.6249
3.5159 2.8972
0.0788 0.0651
CGMY KR
0.0577 0.0503
4.4063 3.8468
5.0448 4.4086
0.1093 0.0953
CGMY KR
0.0802 0.0717
4.4772 4.0071
5.2826 4.7401
0.1378 0.1233
CGMY KR
0.0898 0.0820
6.7185 6.1366
7.5797 6.9289
0.2003 0.1825
CGMY KR
0.1238 0.1156
9.0494 8.4512
9.8394 9.1809
0.2588 0.2409
0.1840
0.4360
0.6920
0.9360
1.1920
1.7080
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
References
K IM , Y.S., R ACHEV, S.T., B IANCHI , M.L. AND FABOZZI , F.J. A New Tempered Stable Distribution and Its Application to Finance forthcoming in G. Bol, et al, (eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007. K IM , Y. S. AND J. H. L EE (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, Mathematical Methods of Operations Research. S ATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press. S CHOUTENS , W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons. ´ R OSI NSKI , J. (2006).
Tempering stable processes, Working Paper, http://www.math.utk.edu/˜rosinski/Manuscripts/tstableF.pdf.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
References
K IM , Y.S., R ACHEV, S.T., B IANCHI , M.L. AND FABOZZI , F.J. A New Tempered Stable Distribution and Its Application to Finance forthcoming in G. Bol, et al, (eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007. K IM , Y. S. AND J. H. L EE (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, Mathematical Methods of Operations Research. S ATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press. S CHOUTENS , W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons. ´ R OSI NSKI , J. (2006).
Tempering stable processes, Working Paper, http://www.math.utk.edu/˜rosinski/Manuscripts/tstableF.pdf.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
References
K IM , Y.S., R ACHEV, S.T., B IANCHI , M.L. AND FABOZZI , F.J. A New Tempered Stable Distribution and Its Application to Finance forthcoming in G. Bol, et al, (eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007. K IM , Y. S. AND J. H. L EE (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, Mathematical Methods of Operations Research. S ATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press. S CHOUTENS , W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons. ´ R OSI NSKI , J. (2006).
Tempering stable processes, Working Paper, http://www.math.utk.edu/˜rosinski/Manuscripts/tstableF.pdf.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
References
K IM , Y.S., R ACHEV, S.T., B IANCHI , M.L. AND FABOZZI , F.J. A New Tempered Stable Distribution and Its Application to Finance forthcoming in G. Bol, et al, (eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007. K IM , Y. S. AND J. H. L EE (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, Mathematical Methods of Operations Research. S ATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press. S CHOUTENS , W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons. ´ R OSI NSKI , J. (2006).
Tempering stable processes, Working Paper, http://www.math.utk.edu/˜rosinski/Manuscripts/tstableF.pdf.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
The Change of Measure Problem for Tempered Stable Processes Michele Leonardo Bianchi1,2 (joint work with Y.S.Kim, S.T.Rachev and F.J.Fabozzi) 1 Ph.D. student, Department of Mathematics, Statistics, Computer Science and Applications University of Bergamo 2 Visiting Ph.D. student, Institute of Statistic and Economics, University of Karlsruhe
Radon Workshop on Financial and Actuarial Mathematics for Young Researchers, Linz, May 30 - 31, 2007 Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
History In 1994, the truncated Le´vy flight model is introduced in statistical physics (Mantegna and Stanley, Novikov, Koponen) In 2000, truncated Le´vy processes are considered for pricing options (Boyarchenko et al.) In 2002, CGMY model appears in mathematical finance literature (Carr et al.) In 2006, GARCH model with modified tempered stable (MTS) innovation. See (Kim et al.). In 2007, Kim-Rachev (KR) tempered stable distribution is introduced (Kim, Rachev, Bianchi, Fabozzi)
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Motivations Heavy tails Gain/loss asymmetry Aggregational normality Equivalent change of measure based on historical estimates
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Outline Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
The importance of tempered stable processes comes from the fact that they combine both α-stable and Gaussian trends. The Lévy measure M0 of a α-stable distribution on Rd in polar coordinates is of the form M0 (dv , du) = v −α−1 dv σ(du) where σ is a finite measure on S d−1 .
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
´ Definition (Rosinski) Let α ∈ (0, 2), σ is a finite measure on S d−1 . A probability measure on Rd is called tempered α-stable (denoted as TαS) if is infinitely divisible without Gaussian part and whose Lévy measure M can be written in polar coordinates as M(dv , du) = v −α−1 q(v,u)dv σ(du). where q : (0, ∞) × S d−1 7→ (0, ∞) is a Borel function such that q(·, u) is completely monotone with q(∞, u) = 0 for each u ∈ S d−1 . A TαS distribution is called a proper TαS distribution if limv →0+ q(v , u) = 1 for each u ∈ S d−1 .
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
A function q(·, u) is completely monotone if (−1)n
d q(v , u) > 0 dv
for all v > 0, u ∈ S d−1 , and n = 0, 1, 2, · · · . In particular q(·, u) is stictly decreasing and convex. The tempering function q can be represented as the Laplace transform Z ∞ q(v , u) = e−vs Q(ds|u) 0
where {Q(·|u)}u∈S d−1 is a measurable family of Borel measures on (0, ∞).
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
´ Theorem (Rosinski) Lévy measure M of TαS distribution can be written in the form Z Z ∞ M(A) = IA (tx)αt −α−1 e−t dtR(dx), A ∈ B(Rd ). Rd0
(1)
0
where R is a unique measure on Rd such that Z R({0}) = 0 and (||x||2 ∧ ||x||α )R(dx) < ∞.
(2)
Rd
Conversely, if R is a measure satisfying (2) then (1) defines Lévy measure of a TαS distribution. M corresponds to a proper TαS distribution if and only if Z ||x||α R(dx) < ∞. Rd
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
´ R is call the spectral measure (or Rosinski measure) of the corresponding TαS distribution Using the measure R, Characteristic function Change of measure can be found explicitly.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
KR tempered stable distribution
In the following we will define a r.v. X with KR Tempered Stable distribution (shortly KR distribution) with parameters α, k+ , k− , r+ , r− , p+ , p− , m with α ∈ (0, 2), k+ , k− , r+ , r− > 0, p+ , p− ∈ (−α, ∞) \ {−1, 0}
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
Let σ be a finite measure on S 0 σ(A) =
α k− r− k+ r+α IA (1) + IA (−1), α + p+ α + p−
A ⊂ S0,
and q(·, u) a completely monotone function Z r+ q(v , 1) = (α + p+ )r+−α−p+ e−v /s sα+p+ −1 ds 0 Z r− −α−p− e−v /s sα+p− −1 ds, q(v , −1) = (α + p− )r− 0
with α ∈ (0, 2), k+ , k− , r+ , r− > 0 and p+ , p− > −α.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
Then the spectral measure R corresponding to the Lévy measure M can be deduced as −p−
R(dx) = (k+ r+−p+ I(0,r+ ) (x)|x|p+ −1 + k− r−
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
I(−r− ,0) (x)|x|p− −1 ) dx.
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
If α 6= 1 , E[e
iuX
] = exp Hα (u; k+ , r+ , p+ ) + Hα (−u; k− , r− , p− ) k+ r+ k− r− + iu m + αΓ(−α) − , p+ + 1 p− + 1
where Hα (u; a, h, p) =
aΓ(−α) (2 F1 (p, −α; 1 + p; ihu) − 1) . p
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
−1 E[eθX ] < ∞ iff −r− ≤ θ ≤ r+−1 Infinitely divisible, therefore a Lévy process can be considered.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
Proposition The KR distribution with parameters (α, k+ , k− , r+ , r− , p+ , p− , m) converges weakly to the CGMY distribution as p± → ∞ provided that α 6= 1 and −α k± = c(α + p± )h±
for c > 0.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
8 CGMY KR p+=p−=−0.2
7
KR p+=p−=1 KR p+=p−=10
6 5 4 3 2 1 0
C = 0.01 −Y k+ = C(Y + p)r+
−0.2
G = 2 k− = C(Y + p)r
−Y −
−0.1
0
0.1
M = 10
Y = 1.25
r+ = 1/M
r− = 1/G
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
0.2
0.3
p+ = p− = { −0.25, 1, 10,}
The Change of Measure Problem for TS Processes
α = Y
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
E[X ] = c1 (X ) = m ! 2 k− r− k+ r+2 Var(X ) = c2 (X ) = Γ(2 − α) + p+ + 2 p− + 2 3 k r− k r+3 Γ(3 − α) p+++3 − p−−+3 c3 (X ) s(X ) = = 2 2 3/2 c2 (X )3/2 k− r− k r+ Γ(2 − α)3/2 p+++2 + p− +2 4 k r− k r+4 Γ(4 − α) p+++4 + p−−+4 c4 (X ) k (X ) = = 2 2 2 c2 (X )2 k− r− k r+ Γ(2 − α)2 p+++2 + p− +2
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Definition Properties
Tail Behavior
Upper and lower bounds 1 P(|X − m| ≥ λ) ≤ 2 Γ(2 − α) λ
2 k− r− k+ r+2 + p+ + 2 p− + 2
!
2λ
P(|X − m| ≥ λ) ≥ C
e− ¯r λα+2
where ¯r = max(r+ , r− ).
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
The market stock price process St = S0 eXt ,
t ∈ [0, T ]
where Xt is a KR Lévy process Under the market measure P, Xt is a KR process with parameters (α, k+ , k− , r+ , r− , p+ , p− , µ − ωP ) Under the risk neutral measure Q, Xt is a KR process with ˜+ , p ˜ − , r − ωQ ) parameters (α, k˜+ , k˜− , ˜r+ , ˜r− , p the convexity corrections ωP and ωQ are added, in order to obtain EP [St ] = S0 EP [eXt ] = S0 eµt and EQ [St ] = S0 EQ [eXt ] = S0 ert
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Let us consider a canonical process (Xt )t≥0 on (Ω, F, (Ft )t≥0 ), where Ω is the set of cadlag functions on [0, ∞) into R and F = σ{Xs ; s ≥ 0} Ft = ∩s>t σ{Xu : u ≤ s},
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
t ≥ 0.
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Theorem Consider two probability measure P1 , P2 and the canonical process (Xt )t≥0 on (Ω, F, (Ft )t≥0 ), given above. For each j = 1, 2, suppose (Xt )t≥0 is the KR tempered stable process under Pj with parameters (αj , kj,+ , kj,− , rj,+ , rj,− , pj,+ , pj,− , mj ) and pj,± > 12 − αj , αj ∈ (0, 1) . Then P1 |Ft and P2 |Ft are equivalent pj,± > 1 − αj , αj ∈ [1, 2) for every t > 0 if and only if (3)
α := α1 = α2 , α k1,+ r1,+
α + p1,+
α k2,+ r2,+
=
α + p2,+
,
α k1,− r1,−
α + p1,+
=
α k2,− r2,−
(4)
α + p2,+
and m2 −m1 = Γ(1−α)
X
(−1)j
j=1,2
kj,+ rj,+ kj,− rj,− − pj,+ + 1 pj,− + 1
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
if α 6= 1 (5)
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Proof Without consider all details, we focus our attention on the following condition Z Z 1 (1 − qj (v , u))2 v −αj −1 dv σ(du) < ∞. S0
0
The complete proof can be found in Kim et al.[1, Theorem 4.3]. ´ It comes from results of Sato [3, Theorem 33.1] and Rosinski [5, Theorem 4.1].
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Change of measure - KR case Theorem Assume that (St )t∈[0,T ] is the the KR price process with parameters (α, k+ , k− , r+ , r− , p+ , p− , µ) under the market measure P, and with ˜+ , p ˜− , r ) under a measure Q. Then Q parameters (˜ α, k˜+ , k˜− , ˜r+ , ˜r− , p is an EMM of P if and only if α=α ˜, k˜+˜r+α k+ r+α = , ˜+ α + p+ α+p
α α k− r− k˜−˜r− = ˜− α + p− α+p
and µ − r = Hα (−i; k+ , r+ , p+ ) + Hα (i; k− , r− , p− ) ˜+ ) − Hα (i; k˜− , ˜r− , p ˜− ). − Hα (−i; k˜+ , ˜r+ , p Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Change of measure - CGMY case
Theorem (Kim and Lee, 2006) Assume that (St )t∈[0,T ] is the CGMY stock price process with parameters (C, G, M, Y , µ) under the market measure P, and with ˜ G, ˜ M, ˜ Y ˜ , r ) under a measure Q. Then Q is an EMM of parameters (C, P if and only if ˜ = C, C ˜ = Y, Y and ˜ M, ˜ Y ) = µ − Ψ0 (−i; C, G, M, Y ). r − Ψ0 (−i; C, G,
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Why do we consider this so complicated KR distribution? CGMY model 4 parameters (+ r ) - 3 equality constrains = 1 free parameter (in the estimation of risk neutral parameters) KR model 7 parameters (+ r ) - 4 equality constrains = 3 free parameters (in the estimation of risk neutral parameters) With KR model a better fit in the tails can be obtained (in the market parameters estimation)
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Why do we consider this so complicated KR distribution? CGMY model 4 parameters (+ r ) - 3 equality constrains = 1 free parameter (in the estimation of risk neutral parameters) KR model 7 parameters (+ r ) - 4 equality constrains = 3 free parameters (in the estimation of risk neutral parameters) With KR model a better fit in the tails can be obtained (in the market parameters estimation)
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Why do we consider this so complicated KR distribution? CGMY model 4 parameters (+ r ) - 3 equality constrains = 1 free parameter (in the estimation of risk neutral parameters) KR model 7 parameters (+ r ) - 4 equality constrains = 3 free parameters (in the estimation of risk neutral parameters) With KR model a better fit in the tails can be obtained (in the market parameters estimation)
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
60 Market data Normal CGMY KR
50
40
30
20
10
0 −0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
S&P 500 Index (from January 1st, 1992 to April 18th, 2002) MLE density fit.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
χ2 , KS, AD and AD2 statistics (degrees of freedom in round brackets). Monte Carlo P-values were obtained via 1000 simulations.
Model Normal CGMY KR
Model Normal CGMY KR
S&P 500 Index from January 1st, 1992 to April 18th, 2002 χ2 KS AD 546.49(288) 0.0663 2180.7 273.4(255) 0.0103 0.2945 268.91(252) 0.0109 0.2315
P-value Theoretical χ2 KS AD2 0 0 0 0.2045 0.9450 0.6356 0.2216 0.9165 0.9082
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
χ2 0 0.43 0.53
AD2 23.762 0.6130 0.3367
Monte Carlo KS AD 0 0 0.908 0.098 0.875 0.242
AD2 0 0.656 0.916
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
70 Market data B−S CGMY KR
60
50
40
30
20
10
0 −0.04
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
0.04
S&P 500 Index (from January 1st, 1984 to January 1st, 1994) MLE density fit.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
χ2 , KS, AD and AD2 statistics (degrees of freedom in round brackets). Monte Carlo P-values were obtained via 1000 simulations. S&P 500 Index from January 1st, 1984 to January 1th, 1994 Model χ2 KS AD Normal 482.39(202) 0.0699 3.9e+6 CGMY 191.68(179) 0.0191 0.1527 KR 180.07(181) 0.0107 0.1302
Model Normal CGMY KR
P-value Theoretical χ2 KS AD2 0 0 0 0.2451 0.3180 0.0865 0.5055 0.9343 0.3723
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
χ2 0 0.893 0.974
AD2 33.654 2.0475 0.9719
Monte Carlo KS AD 0 0 0.305 0.696 0.875 0.872
AD2 0 0.086 0.361
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
min θ˜
N X
˜
ˆ i − C θ (Ti , Ki ))2 (C
i=1
ˆ i is the price of an option as observed in the market, C ˜
Ciθ the price computed according to a pricing formula in a chosen ˜ model with a parameter set θ.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
S&P 500 Index MLE density fit and EMM estimations
Normal CGMY KR
µ 0.096364 C 10.161 k+ 3286.1
S&P 500 Index from January 1st, 1992 to April 18th, 2002 Parameters σ 0.15756 G M Y m 97.455 98.891 0.5634 0.1135 k− r+ r− p+ p− 2124.8 0.0090 0.0113 17.736 17.736
CGMY T 0.0880 0.1840 0.4360 0.6920 0.9360 1.1920 1.7080
˜ M 106.5827 103.4463 92.4701 89.4576 90.0040 82.6216 77.3594
˜ G 96.1341 93.3887 83.7430 81.0851 81.5675 75.0354 70.3609
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
α 0.5103
µ 0.1252
KR k˜+ 5325.8 9126.3 4757.3 3866.4 6655.4 9896.7 10000
k˜− 33.727 33.024 31.327 30.776 30.78 29.483 28.468
˜r+ 0.0065 0.0066 0.0074 0.0076 0.0075 0.0079 0.0084
˜r− 0.0330 0.034 0.0381 0.0395 0.03953 0.0430 0.046
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
Error Estimators of Option Prices fit. Option data April 18, 2002 T 0.0880
Model
APE
AEE
RMSE
ARPE
CGMY KR
0.0149 0.0030
0.4019 0.0826
0.4613 0.1023
0.0175 0.0035
CGMY KR
0.0341 0.0234
1.0998 0.7541
1.4270 0.9937
0.0442 0.0295
CGMY KR
0.0437 0.0361
3.1727 2.6249
3.5159 2.8972
0.0788 0.0651
CGMY KR
0.0577 0.0503
4.4063 3.8468
5.0448 4.4086
0.1093 0.0953
CGMY KR
0.0802 0.0717
4.4772 4.0071
5.2826 4.7401
0.1378 0.1233
CGMY KR
0.0898 0.0820
6.7185 6.1366
7.5797 6.9289
0.2003 0.1825
CGMY KR
0.1238 0.1156
9.0494 8.4512
9.8394 9.1809
0.2588 0.2409
0.1840
0.4360
0.6920
0.9360
1.1920
1.7080
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
References
K IM , Y.S., R ACHEV, S.T., B IANCHI , M.L. AND FABOZZI , F.J. A New Tempered Stable Distribution and Its Application to Finance forthcoming in G. Bol, et al, (eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007. K IM , Y. S. AND J. H. L EE (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, Mathematical Methods of Operations Research. S ATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press. S CHOUTENS , W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons. ´ R OSI NSKI , J. (2006).
Tempering stable processes, Working Paper, http://www.math.utk.edu/˜rosinski/Manuscripts/tstableF.pdf.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
References
K IM , Y.S., R ACHEV, S.T., B IANCHI , M.L. AND FABOZZI , F.J. A New Tempered Stable Distribution and Its Application to Finance forthcoming in G. Bol, et al, (eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007. K IM , Y. S. AND J. H. L EE (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, Mathematical Methods of Operations Research. S ATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press. S CHOUTENS , W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons. ´ R OSI NSKI , J. (2006).
Tempering stable processes, Working Paper, http://www.math.utk.edu/˜rosinski/Manuscripts/tstableF.pdf.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
References
K IM , Y.S., R ACHEV, S.T., B IANCHI , M.L. AND FABOZZI , F.J. A New Tempered Stable Distribution and Its Application to Finance forthcoming in G. Bol, et al, (eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007. K IM , Y. S. AND J. H. L EE (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, Mathematical Methods of Operations Research. S ATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press. S CHOUTENS , W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons. ´ R OSI NSKI , J. (2006).
Tempering stable processes, Working Paper, http://www.math.utk.edu/˜rosinski/Manuscripts/tstableF.pdf.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
Introduction Tempered stable distributions KR tempered stable distribution KR tempered stable market model Change of measure Estimation results
References
K IM , Y.S., R ACHEV, S.T., B IANCHI , M.L. AND FABOZZI , F.J. A New Tempered Stable Distribution and Its Application to Finance forthcoming in G. Bol, et al, (eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007. K IM , Y. S. AND J. H. L EE (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, Mathematical Methods of Operations Research. S ATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press. S CHOUTENS , W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons. ´ R OSI NSKI , J. (2006).
Tempering stable processes, Working Paper, http://www.math.utk.edu/˜rosinski/Manuscripts/tstableF.pdf.
Y.S.Kim, S.T.Rachev, M.L.Bianchi, F.J.Fabozzi
The Change of Measure Problem for TS Processes
