60
Equivalent Measure Changes for Jump- Diffusions D. Filipovi´ c Problem Result Applications CIR Short Rate Model Stochastic Volatility Model Equivalent Measure Changes for Jump-Diffusions Damir Filipovi´ c Swiss Finance Institute Ecole Polytechnique F´ ed´ erale de Lausanne (joint with Patrick Cheridito and Marc Yor) Analysis, Stochastics, and Applications Vienna, 13 July 2010

Equivalent Measure Changes for Jump-Diffusionsanstap10/slides/Filipovic.pdf · Equivalent Measure Changes for Jump-Di usions D. Filipovi c Problem Result Applications CIR Short Rate

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EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Equivalent Measure Changes forJump-Diffusions

Damir Filipovic

Swiss Finance InstituteEcole Polytechnique Federale de Lausanne

(joint with Patrick Cheridito and Marc Yor)

Analysis, Stochastics, and ApplicationsVienna, 13 July 2010

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Outline

1 Problem

2 Result

3 ApplicationsCIR Short Rate ModelStochastic Volatility Model

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Outline

1 Problem

2 Result

3 ApplicationsCIR Short Rate ModelStochastic Volatility Model

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Ingredients

• m, d ∈ N• State space (open or closed) E ⊆ Rm

• Locally bounded measurable mappings

b : E → Rm×1, σ : E → Rm×d

• Transition kernel ν from E to Rm such that

x 7→∫Rm

‖ξ‖ ∧ ‖ξ‖2 ν(x , dξ)

is locally bounded on E

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Ingredients

• m, d ∈ N• State space (open or closed) E ⊆ Rm

• Locally bounded measurable mappings

b : E → Rm×1, σ : E → Rm×d

• Transition kernel ν from E to Rm such that

x 7→∫Rm

‖ξ‖ ∧ ‖ξ‖2 ν(x , dξ)

is locally bounded on E

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Ingredients

• m, d ∈ N• State space (open or closed) E ⊆ Rm

• Locally bounded measurable mappings

b : E → Rm×1, σ : E → Rm×d

• Transition kernel ν from E to Rm such that

x 7→∫Rm

‖ξ‖ ∧ ‖ξ‖2 ν(x , dξ)

is locally bounded on E

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Ingredients

• m, d ∈ N• State space (open or closed) E ⊆ Rm

• Locally bounded measurable mappings

b : E → Rm×1, σ : E → Rm×d

• Transition kernel ν from E to Rm such that

x 7→∫Rm

‖ξ‖ ∧ ‖ξ‖2 ν(x , dξ)

is locally bounded on E

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Special Semimartingale

• Filtered probability space (Ω,F , (Ft)t≥0,P)

• Carrying d-dimensional Brownian motion W , and

• Random measure µ(dt, dξ) associated to the jumps of . . .

• . . . the special (for simplicity) semimartingale X withcanonical decomposition

Xt = X0 +

∫ t

0b(Xs) ds +

∫ t

0σ(Xs) dWs

+

∫ t

0

∫Rm

ξ (µ(ds, dξ)− ν(Xs , dξ)ds)

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Special Semimartingale

• Filtered probability space (Ω,F , (Ft)t≥0,P)

• Carrying d-dimensional Brownian motion W , and

• Random measure µ(dt, dξ) associated to the jumps of . . .

• . . . the special (for simplicity) semimartingale X withcanonical decomposition

Xt = X0 +

∫ t

0b(Xs) ds +

∫ t

0σ(Xs) dWs

+

∫ t

0

∫Rm

ξ (µ(ds, dξ)− ν(Xs , dξ)ds)

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Special Semimartingale

• Filtered probability space (Ω,F , (Ft)t≥0,P)

• Carrying d-dimensional Brownian motion W , and

• Random measure µ(dt, dξ) associated to the jumps of . . .

• . . . the special (for simplicity) semimartingale X withcanonical decomposition

Xt = X0 +

∫ t

0b(Xs) ds +

∫ t

0σ(Xs) dWs

+

∫ t

0

∫Rm

ξ (µ(ds, dξ)− ν(Xs , dξ)ds)

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Special Semimartingale

• Filtered probability space (Ω,F , (Ft)t≥0,P)

• Carrying d-dimensional Brownian motion W , and

• Random measure µ(dt, dξ) associated to the jumps of . . .

• . . . the special (for simplicity) semimartingale X withcanonical decomposition

Xt = X0 +

∫ t

0b(Xs) ds +

∫ t

0σ(Xs) dWs

+

∫ t

0

∫Rm

ξ (µ(ds, dξ)− ν(Xs , dξ)ds)

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Density Process Heuristics I

• Measurable mappings . . .

λ : E → Rd×1, κ : E × Rm → (0,∞)

• . . . such that the local martingale L is well defined:

Lt =

∫ t

0λ(Xs)>dWs

+

∫ t

0

∫Rm

(κ(Xs−, ξ)− 1) (µ(ds, dξ)− ν(Xs , dξ)ds)

• Assume its stochastic exponential

Et(L) = exp

(Lt −

1

2

∫ t

0‖λ(Xs)‖2 ds

+

∫ t

0

∫Rm

(log κ(Xs−, ξ)− κ(Xs−, ξ) + 1)µ(ds, dξ)

)is a true martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Density Process Heuristics I

• Measurable mappings . . .

λ : E → Rd×1, κ : E × Rm → (0,∞)

• . . . such that the local martingale L is well defined:

Lt =

∫ t

0λ(Xs)>dWs

+

∫ t

0

∫Rm

(κ(Xs−, ξ)− 1) (µ(ds, dξ)− ν(Xs , dξ)ds)

• Assume its stochastic exponential

Et(L) = exp

(Lt −

1

2

∫ t

0‖λ(Xs)‖2 ds

+

∫ t

0

∫Rm

(log κ(Xs−, ξ)− κ(Xs−, ξ) + 1)µ(ds, dξ)

)is a true martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Density Process Heuristics I

• Measurable mappings . . .

λ : E → Rd×1, κ : E × Rm → (0,∞)

• . . . such that the local martingale L is well defined:

Lt =

∫ t

0λ(Xs)>dWs

+

∫ t

0

∫Rm

(κ(Xs−, ξ)− 1) (µ(ds, dξ)− ν(Xs , dξ)ds)

• Assume its stochastic exponential

Et(L) = exp

(Lt −

1

2

∫ t

0‖λ(Xs)‖2 ds

+

∫ t

0

∫Rm

(log κ(Xs−, ξ)− κ(Xs−, ξ) + 1)µ(ds, dξ)

)is a true martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Heuristics II

• Finite time horizon T

• Define equivalent probability measure Q ∼ P on FT by

dQdP

= ET (L)

• Girsanov’s theorem implies that

Wt = Wt −∫ t

0λ(Xs) ds, t ∈ [0,T ]

is a Q-Brownian motion, and the compensator ofµ(dt, dξ) under Q becomes

ν(Xt , dξ)dt = κ(Xt, ξ)ν(Xt , dξ)dt, t ∈ [0,T ].

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Heuristics II

• Finite time horizon T

• Define equivalent probability measure Q ∼ P on FT by

dQdP

= ET (L)

• Girsanov’s theorem implies that

Wt = Wt −∫ t

0λ(Xs) ds, t ∈ [0,T ]

is a Q-Brownian motion, and the compensator ofµ(dt, dξ) under Q becomes

ν(Xt , dξ)dt = κ(Xt, ξ)ν(Xt , dξ)dt, t ∈ [0,T ].

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Heuristics II

• Finite time horizon T

• Define equivalent probability measure Q ∼ P on FT by

dQdP

= ET (L)

• Girsanov’s theorem implies that

Wt = Wt −∫ t

0λ(Xs) ds, t ∈ [0,T ]

is a Q-Brownian motion, and the compensator ofµ(dt, dξ) under Q becomes

ν(Xt , dξ)dt = κ(Xt, ξ)ν(Xt , dξ)dt, t ∈ [0,T ].

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Heuristics III

• Canonical decomposition of X under Q reads

Xt = X0 +

∫ t

0b(Xs) ds +

∫ t

0σ(Xs) dWs

+

∫ t

0

∫Rm

ξ (µ(ds, dξ)− ν(Xs , dξ)ds)

• With modified drift function defined as

b(x) = b(x) + σ(x)λ(x) +

∫Rm

ξ (κ(x, ξ)− 1) ν(x , dξ).

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Heuristics III

• Canonical decomposition of X under Q reads

Xt = X0 +

∫ t

0b(Xs) ds +

∫ t

0σ(Xs) dWs

+

∫ t

0

∫Rm

ξ (µ(ds, dξ)− ν(Xs , dξ)ds)

• With modified drift function defined as

b(x) = b(x) + σ(x)λ(x) +

∫Rm

ξ (κ(x, ξ)− 1) ν(x , dξ).

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Heuristics IV

• In other words: infinitesimal generator of X under Q is

Af (x) =m∑i=1

bi (x)∂f (x)

∂xi+

1

2

m∑i ,j=1

(σ σ>)ij(x)∂2f (x)

∂xi∂xj

+

∫Rm

(f (x + ξ)− f (x)−

m∑i=1

∂f (x)

∂xiξi

)ν(x , dξ)

• Ito’s lemma implies: for any f ∈ C 2c (E ),

f (Xt)− f (X0)−∫ t

0Af (Xs) ds

is a Q-martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Heuristics IV

• In other words: infinitesimal generator of X under Q is

Af (x) =m∑i=1

bi (x)∂f (x)

∂xi+

1

2

m∑i ,j=1

(σ σ>)ij(x)∂2f (x)

∂xi∂xj

+

∫Rm

(f (x + ξ)− f (x)−

m∑i=1

∂f (x)

∂xiξi

)ν(x , dξ)

• Ito’s lemma implies: for any f ∈ C 2c (E ),

f (Xt)− f (X0)−∫ t

0Af (Xs) ds

is a Q-martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Question

• QUESTION: when is E(L) a true martingale??

• EQUIVALENTLY: when is

E[ET (L)] = 1 ?

• Note: this does not depend on the filtration, but only onthe law of X !

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Question

• QUESTION: when is E(L) a true martingale??

• EQUIVALENTLY: when is

E[ET (L)] = 1 ?

• Note: this does not depend on the filtration, but only onthe law of X !

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Question

• QUESTION: when is E(L) a true martingale??

• EQUIVALENTLY: when is

E[ET (L)] = 1 ?

• Note: this does not depend on the filtration, but only onthe law of X !

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Outline

1 Problem

2 Result

3 ApplicationsCIR Short Rate ModelStochastic Volatility Model

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Martingale Problem

• Canonical basis: Ω = space of cadlag paths in E ,

Xt(ω) = ω(t), Ft = FXt

Definition 2.1.A probability measure Q on (Ω,FX ) is a solution of themartingale problem for A if for all f ∈ C 2

c (E ),

f (Xt)− f (X0)−∫ t

0Af (Xs) ds

is a Q-martingale. The martingale problem for A is well-posedif for every probability distribution η on E there exists a uniquesolution Q with Q X−1

0 = η.

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Martingale Problem

• Canonical basis: Ω = space of cadlag paths in E ,

Xt(ω) = ω(t), Ft = FXt

Definition 2.1.A probability measure Q on (Ω,FX ) is a solution of themartingale problem for A if for all f ∈ C 2

c (E ),

f (Xt)− f (X0)−∫ t

0Af (Xs) ds

is a Q-martingale. The martingale problem for A is well-posedif for every probability distribution η on E there exists a uniquesolution Q with Q X−1

0 = η.

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Main Result

Theorem 2.2.Assume that x 7→ λ(x) and

x 7→∫Rm

(κ(x, ξ) log κ(x, ξ)− κ(x, ξ) + 1) ν(x , dξ)

are locally bounded on E, and that the martingale problem forA is well-posed. Then E(L) is a true martingale.

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Proof I: Lepingle and Memin [3]

• Localizing sequence of bounded stopping timesS1 ≤ S2 ≤ · · · ↑ ∞ such that

Λn :=1

2

∫ Sn

0‖λ(Xs)‖2 ds

+

∫ Sn

0

∫Rd

(κ(Xs , ξ) log κ(Xs , ξ)− κ(Xs , ξ) + 1) ν(Xs , dξ) ds

is uniformly bounded

• Lepingle and Memin [3, Theoreme IV.3]:

Et∧Sn(L) is a martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Proof I: Lepingle and Memin [3]

• Localizing sequence of bounded stopping timesS1 ≤ S2 ≤ · · · ↑ ∞ such that

Λn :=1

2

∫ Sn

0‖λ(Xs)‖2 ds

+

∫ Sn

0

∫Rd

(κ(Xs , ξ) log κ(Xs , ξ)− κ(Xs , ξ) + 1) ν(Xs , dξ) ds

is uniformly bounded

• Lepingle and Memin [3, Theoreme IV.3]:

Et∧Sn(L) is a martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Proof II: Stopped MartingaleProblem

• Girsanov’s theorem implies that for any f ∈ C 2c (E ):

f (X Snt )− f (X0)−

∫ t∧Sn

0Af (X Sn

s ) ds

is a ESn(L) · P-martingale

• Uniqueness of the stopped martingale problem (Ethier andKurtz [2, Theorem 4.6.1]) implies that

ESn(L) · P = Q on FXSn

where Q is the solution of the martingale problem for Awith Q = P on FX

0

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Proof II: Stopped MartingaleProblem

• Girsanov’s theorem implies that for any f ∈ C 2c (E ):

f (X Snt )− f (X0)−

∫ t∧Sn

0Af (X Sn

s ) ds

is a ESn(L) · P-martingale

• Uniqueness of the stopped martingale problem (Ethier andKurtz [2, Theorem 4.6.1]) implies that

ESn(L) · P = Q on FXSn

where Q is the solution of the martingale problem for Awith Q = P on FX

0

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Proof III: Limit

• Monotone convergence theorem, and sinceT < Sn ∈ FX

T∧Sn :

1 = limn→∞

Q[T < Sn]

= limn→∞

EP[ET∧Sn(L) 1T<Sn]

= limn→∞

EP[ET (L) 1T<Sn]

= EP[ET (L)]

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Outline

1 Problem

2 Result

3 ApplicationsCIR Short Rate ModelStochastic Volatility Model

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Outline

1 Problem

2 Result

3 ApplicationsCIR Short Rate ModelStochastic Volatility Model

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Cox–Ingersoll–Ross (CIR) Model

• Model for short rate under P: square root (“CIR”) process

dXt = (b + βXt) dt + σ√

Xt dWt

• State space E = (0,∞)

• Feller condition: 0 not attained iff b ≥ σ2/2

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Cox–Ingersoll–Ross (CIR) Model

• Model for short rate under P: square root (“CIR”) process

dXt = (b + βXt) dt + σ√

Xt dWt

• State space E = (0,∞)

• Feller condition: 0 not attained iff b ≥ σ2/2

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Cox–Ingersoll–Ross (CIR) Model

• Model for short rate under P: square root (“CIR”) process

dXt = (b + βXt) dt + σ√

Xt dWt

• State space E = (0,∞)

• Feller condition: 0 not attained iff b ≥ σ2/2

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Market Price of Risk Specification

• Aim: MPR specification that preserves affine structure:

dXt = (bQ + βQXt) dt + σ√

Xt

(dWt +

`+ λXt

σ√Xt

)︸ ︷︷ ︸

=dWQt

• MPR parameters:

` = b − bQ, λ = β − βQ

• Formal density process E(− `+λX

σ√X•W

)• Novikov condition not satisfied, since

E[e

12

∫ T0

1Xt

dt]

=∞, E[e

12

∫ T0 Xt dt

]=∞

for T large enough in general

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Market Price of Risk Specification

• Aim: MPR specification that preserves affine structure:

dXt = (bQ + βQXt) dt + σ√

Xt

(dWt +

`+ λXt

σ√Xt

)︸ ︷︷ ︸

=dWQt

• MPR parameters:

` = b − bQ, λ = β − βQ

• Formal density process E(− `+λX

σ√X•W

)• Novikov condition not satisfied, since

E[e

12

∫ T0

1Xt

dt]

=∞, E[e

12

∫ T0 Xt dt

]=∞

for T large enough in general

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Market Price of Risk Specification

• Aim: MPR specification that preserves affine structure:

dXt = (bQ + βQXt) dt + σ√

Xt

(dWt +

`+ λXt

σ√Xt

)︸ ︷︷ ︸

=dWQt

• MPR parameters:

` = b − bQ, λ = β − βQ

• Formal density process E(− `+λX

σ√X•W

)• Novikov condition not satisfied, since

E[e

12

∫ T0

1Xt

dt]

=∞, E[e

12

∫ T0 Xt dt

]=∞

for T large enough in general

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Market Price of Risk Specification

• Aim: MPR specification that preserves affine structure:

dXt = (bQ + βQXt) dt + σ√

Xt

(dWt +

`+ λXt

σ√Xt

)︸ ︷︷ ︸

=dWQt

• MPR parameters:

` = b − bQ, λ = β − βQ

• Formal density process E(− `+λX

σ√X•W

)• Novikov condition not satisfied, since

E[e

12

∫ T0

1Xt

dt]

=∞, E[e

12

∫ T0 Xt dt

]=∞

for T large enough in general

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

CFY Condition

• Assume that Feller condition is also satisfied for bQ:

bQ ≥ σ2/2

• Then the martingale problem for

Af (x) =(bQ + βQx

)f ′(x) +

1

2σ2xf ′′(x)

is well-posed in E = (0,∞)

• CFY Theorem implies that E(− `+λX

σ√X•W

)is a true

martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

CFY Condition

• Assume that Feller condition is also satisfied for bQ:

bQ ≥ σ2/2

• Then the martingale problem for

Af (x) =(bQ + βQx

)f ′(x) +

1

2σ2xf ′′(x)

is well-posed in E = (0,∞)

• CFY Theorem implies that E(− `+λX

σ√X•W

)is a true

martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

CFY Condition

• Assume that Feller condition is also satisfied for bQ:

bQ ≥ σ2/2

• Then the martingale problem for

Af (x) =(bQ + βQx

)f ′(x) +

1

2σ2xf ′′(x)

is well-posed in E = (0,∞)

• CFY Theorem implies that E(− `+λX

σ√X•W

)is a true

martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Outline

1 Problem

2 Result

3 ApplicationsCIR Short Rate ModelStochastic Volatility Model

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Stochastic Volatility

• Model for volatility: GARCH diffusion

dXt = (b + βXt) dt + Xt dW1t

• State space E = (0,∞); that is, b ≥ 0

• Model for discounted S&P 500 index process:

dStSt

= Xt

(ρ dW 1

t +√

1− ρ2 dW 2t

)• Leverage effect: non-positive correlation ρ ≤ 0 between

d [X , log S ]t = X 2t ρ ≤ 0

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Stochastic Volatility

• Model for volatility: GARCH diffusion

dXt = (b + βXt) dt + Xt dW1t

• State space E = (0,∞); that is, b ≥ 0

• Model for discounted S&P 500 index process:

dStSt

= Xt

(ρ dW 1

t +√

1− ρ2 dW 2t

)• Leverage effect: non-positive correlation ρ ≤ 0 between

d [X , log S ]t = X 2t ρ ≤ 0

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Stochastic Volatility

• Model for volatility: GARCH diffusion

dXt = (b + βXt) dt + Xt dW1t

• State space E = (0,∞); that is, b ≥ 0

• Model for discounted S&P 500 index process:

dStSt

= Xt

(ρ dW 1

t +√

1− ρ2 dW 2t

)• Leverage effect: non-positive correlation ρ ≤ 0 between

d [X , log S ]t = X 2t ρ ≤ 0

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Stochastic Volatility

• Model for volatility: GARCH diffusion

dXt = (b + βXt) dt + Xt dW1t

• State space E = (0,∞); that is, b ≥ 0

• Model for discounted S&P 500 index process:

dStSt

= Xt

(ρ dW 1

t +√

1− ρ2 dW 2t

)• Leverage effect: non-positive correlation ρ ≤ 0 between

d [X , log S ]t = X 2t ρ ≤ 0

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Martingality of S

• Question: is S a true martingale? (vital for pricing!)

• Write S as stochastic exponential

St = S0 Et(λ(X)> •W

)with

λ(x) = x

(ρ√

1− ρ2

)• Novikov condition fails:

E[e

12

∫ T0 X 2

t dt]

=∞

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Martingality of S

• Question: is S a true martingale? (vital for pricing!)

• Write S as stochastic exponential

St = S0 Et(λ(X)> •W

)with

λ(x) = x

(ρ√

1− ρ2

)• Novikov condition fails:

E[e

12

∫ T0 X 2

t dt]

=∞

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

Martingality of S

• Question: is S a true martingale? (vital for pricing!)

• Write S as stochastic exponential

St = S0 Et(λ(X)> •W

)with

λ(x) = x

(ρ√

1− ρ2

)• Novikov condition fails:

E[e

12

∫ T0 X 2

t dt]

=∞

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

CFY Condition

• Apply auxiliary change of measure with density process

StS0

= Et(λ(X)> •W

)• Formally, the generator of X becomes

Af (x) =(b + βx + ρx2

)f ′(x) +

1

2xf ′′(x)

• Inspection shows: the martingale problem for A iswell-posed in E = (0,∞)

• CFY Theorem implies that S is a true martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

CFY Condition

• Apply auxiliary change of measure with density process

StS0

= Et(λ(X)> •W

)• Formally, the generator of X becomes

Af (x) =(b + βx + ρx2

)f ′(x) +

1

2xf ′′(x)

• Inspection shows: the martingale problem for A iswell-posed in E = (0,∞)

• CFY Theorem implies that S is a true martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

CFY Condition

• Apply auxiliary change of measure with density process

StS0

= Et(λ(X)> •W

)• Formally, the generator of X becomes

Af (x) =(b + βx + ρx2

)f ′(x) +

1

2xf ′′(x)

• Inspection shows: the martingale problem for A iswell-posed in E = (0,∞)

• CFY Theorem implies that S is a true martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

CFY Condition

• Apply auxiliary change of measure with density process

StS0

= Et(λ(X)> •W

)• Formally, the generator of X becomes

Af (x) =(b + βx + ρx2

)f ′(x) +

1

2xf ′′(x)

• Inspection shows: the martingale problem for A iswell-posed in E = (0,∞)

• CFY Theorem implies that S is a true martingale

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

References

P. Cheridito, D. Filipovic, and M. Yor.Equivalent and absolutely continuous measure changes forjump-diffusion processes.Ann. Appl. Probab., 15(3):1713–1732, 2005.

S. N. Ethier and T. G. Kurtz.Markov processes.Wiley Series in Probability and Mathematical Statistics:Probability and Mathematical Statistics. John Wiley &Sons Inc., New York, 1986.Characterization and convergence.

D. Lepingle and J. Memin.Sur l’integrabilite uniforme des martingales exponentielles.Z. Wahrsch. Verw. Gebiete, 42(3):175–203, 1978.

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

References

P. Cheridito, D. Filipovic, and M. Yor.Equivalent and absolutely continuous measure changes forjump-diffusion processes.Ann. Appl. Probab., 15(3):1713–1732, 2005.

S. N. Ethier and T. G. Kurtz.Markov processes.Wiley Series in Probability and Mathematical Statistics:Probability and Mathematical Statistics. John Wiley &Sons Inc., New York, 1986.Characterization and convergence.

D. Lepingle and J. Memin.Sur l’integrabilite uniforme des martingales exponentielles.Z. Wahrsch. Verw. Gebiete, 42(3):175–203, 1978.

EquivalentMeasure

Changes forJump-

Diffusions

D. Filipovic

Problem

Result

Applications

CIR Short RateModel

StochasticVolatility Model

References

P. Cheridito, D. Filipovic, and M. Yor.Equivalent and absolutely continuous measure changes forjump-diffusion processes.Ann. Appl. Probab., 15(3):1713–1732, 2005.

S. N. Ethier and T. G. Kurtz.Markov processes.Wiley Series in Probability and Mathematical Statistics:Probability and Mathematical Statistics. John Wiley &Sons Inc., New York, 1986.Characterization and convergence.

D. Lepingle and J. Memin.Sur l’integrabilite uniforme des martingales exponentielles.Z. Wahrsch. Verw. Gebiete, 42(3):175–203, 1978.