Equivalent Measure Changes for Problem Jump-Diffusions Result - - PowerPoint PPT Presentation

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- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Outline

1 Problem 2 Result 3 Applications

CIR Short Rate Model Stochastic Volatility Model

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Outline

1 Problem 2 Result 3 Applications

CIR Short Rate Model Stochastic Volatility Model

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 →

• Rm ξ ∧ ξ2 ν(x, dξ)

is locally bounded on E

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 →

• Rm ξ ∧ ξ2 ν(x, dξ)

is locally bounded on E

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 →

• Rm ξ ∧ ξ2 ν(x, dξ)

is locally bounded on E

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 →

• Rm ξ ∧ ξ2 ν(x, dξ)

is locally bounded on E

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 with

canonical decomposition Xt = X0 + t b(Xs) ds + t σ(Xs) dWs + t

• Rm ξ (µ(ds, dξ) − ν(Xs, dξ)ds)

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 with

canonical decomposition Xt = X0 + t b(Xs) ds + t σ(Xs) dWs + t

• Rm ξ (µ(ds, dξ) − ν(Xs, dξ)ds)

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 with

canonical decomposition Xt = X0 + t b(Xs) ds + t σ(Xs) dWs + t

• Rm ξ (µ(ds, dξ) − ν(Xs, dξ)ds)

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 with

canonical decomposition Xt = X0 + t b(Xs) ds + t σ(Xs) dWs + t

• Rm ξ (µ(ds, dξ) − ν(Xs, dξ)ds)

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 λ(Xs)⊤dWs + t

• Rm (κ(Xs−, ξ) − 1) (µ(ds, dξ) − ν(Xs, dξ)ds)
• Assume its stochastic exponential

Et(L) = exp

• Lt − 1

2 t λ(Xs)2 ds + t

• Rm (log κ(Xs−, ξ) − κ(Xs−, ξ) + 1) µ(ds, dξ)
• is a true martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 λ(Xs)⊤dWs + t

• Rm (κ(Xs−, ξ) − 1) (µ(ds, dξ) − ν(Xs, dξ)ds)
• Assume its stochastic exponential

Et(L) = exp

• Lt − 1

2 t λ(Xs)2 ds + t

• Rm (log κ(Xs−, ξ) − κ(Xs−, ξ) + 1) µ(ds, dξ)
• is a true martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 λ(Xs)⊤dWs + t

• Rm (κ(Xs−, ξ) − 1) (µ(ds, dξ) − ν(Xs, dξ)ds)
• Assume its stochastic exponential

Et(L) = exp

• Lt − 1

2 t λ(Xs)2 ds + t

• Rm (log κ(Xs−, ξ) − κ(Xs−, ξ) + 1) µ(ds, dξ)
• is a true martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Heuristics II

• Finite time horizon T
• Define equivalent probability measure Q ∼ P on FT by

dQ dP = ET(L)

• Girsanov’s theorem implies that
• Wt = Wt −

t λ(Xs) ds, t ∈ [0, T] is a Q-Brownian motion, and the compensator of µ(dt, dξ) under Q becomes

• ν(Xt, dξ)dt = κ(Xt, ξ)ν(Xt, dξ)dt,

t ∈ [0, T].

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Heuristics II

• Finite time horizon T
• Define equivalent probability measure Q ∼ P on FT by

dQ dP = ET(L)

• Girsanov’s theorem implies that
• Wt = Wt −

t λ(Xs) ds, t ∈ [0, T] is a Q-Brownian motion, and the compensator of µ(dt, dξ) under Q becomes

• ν(Xt, dξ)dt = κ(Xt, ξ)ν(Xt, dξ)dt,

t ∈ [0, T].

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Heuristics II

• Finite time horizon T
• Define equivalent probability measure Q ∼ P on FT by

dQ dP = ET(L)

• Girsanov’s theorem implies that
• Wt = Wt −

t λ(Xs) ds, t ∈ [0, T] is a Q-Brownian motion, and the compensator of µ(dt, dξ) under Q becomes

• ν(Xt, dξ)dt = κ(Xt, ξ)ν(Xt, dξ)dt,

t ∈ [0, T].

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Heuristics III

• Canonical decomposition of X under Q reads

Xt = X0 + t

• b(Xs) ds +

t σ(Xs) d Ws + t

• 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).

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Heuristics III

• Canonical decomposition of X under Q reads

Xt = X0 + t

• b(Xs) ds +

t σ(Xs) d Ws + t

• 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).

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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) ∂xi ξi

• ν(x, dξ)
• Itˆ
• ’s lemma implies: for any f ∈ C 2

c (E),

f (Xt) − f (X0) − t

• Af (Xs) ds

is a Q-martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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) ∂xi ξi

• ν(x, dξ)
• Itˆ
• ’s lemma implies: for any f ∈ C 2

c (E),

f (Xt) − f (X0) − t

• Af (Xs) ds

is a Q-martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 on

the law of X !

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 on

the law of X !

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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 on

the law of X !

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Outline

1 Problem 2 Result 3 Applications

CIR Short Rate Model Stochastic Volatility Model

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Martingale Problem

• Canonical basis: Ω = space of c`

adl` ag paths in E, Xt(ω) = ω(t), Ft = FX

t

Definition 2.1.

A probability measure Q on (Ω, FX) is a solution of the martingale problem for A if for all f ∈ C 2

c (E),

f (Xt) − f (X0) − t

• Af (Xs) ds

is a Q-martingale. The martingale problem for A is well-posed if for every probability distribution η on E there exists a unique solution Q with Q ◦ X −1 = η.

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Martingale Problem

• Canonical basis: Ω = space of c`

adl` ag paths in E, Xt(ω) = ω(t), Ft = FX

t

Definition 2.1.

A probability measure Q on (Ω, FX) is a solution of the martingale problem for A if for all f ∈ C 2

c (E),

f (Xt) − f (X0) − t

• Af (Xs) ds

is a Q-martingale. The martingale problem for A is well-posed if for every probability distribution η on E there exists a unique solution Q with Q ◦ X −1 = η.

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Main Result

Theorem 2.2.

Assume that x → λ(x) and x →

• 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 for

• A is well-posed. Then E(L) is a true martingale.

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Proof I: L´ epingle and M´ emin [3]

• Localizing sequence of bounded stopping times

S1 ≤ S2 ≤ · · · ↑ ∞ such that Λn := 1 2 Sn λ(Xs)2 ds + Sn

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

is uniformly bounded

• L´

epingle and M´ emin [3, Th´ eor` eme IV.3]: Et∧Sn(L) is a martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Proof I: L´ epingle and M´ emin [3]

• Localizing sequence of bounded stopping times

S1 ≤ S2 ≤ · · · ↑ ∞ such that Λn := 1 2 Sn λ(Xs)2 ds + Sn

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

is uniformly bounded

• L´

epingle and M´ emin [3, Th´ eor` eme IV.3]: Et∧Sn(L) is a martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Proof II: Stopped Martingale Problem

• Girsanov’s theorem implies that for any f ∈ C 2

c (E):

f (X Sn

t ) − f (X0) −

t∧Sn

• Af (X Sn

s ) ds

is a ESn(L) · P-martingale

• Uniqueness of the stopped martingale problem (Ethier and

Kurtz [2, Theorem 4.6.1]) implies that ESn(L) · P = Q

• n FX

Sn

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

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Proof II: Stopped Martingale Problem

• Girsanov’s theorem implies that for any f ∈ C 2

c (E):

f (X Sn

t ) − f (X0) −

t∧Sn

• Af (X Sn

s ) ds

is a ESn(L) · P-martingale

• Uniqueness of the stopped martingale problem (Ethier and

Kurtz [2, Theorem 4.6.1]) implies that ESn(L) · P = Q

• n FX

Sn

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

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Proof III: Limit

• Monotone convergence theorem, and since

{T < Sn} ∈ FX

T∧Sn:

1 = lim

n→∞ Q[T < Sn]

= lim

n→∞ EP[ET∧Sn(L) 1{T<Sn}]

= lim

n→∞ EP[ET(L) 1{T<Sn}]

= EP[ET(L)]

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Outline

1 Problem 2 Result 3 Applications

CIR Short Rate Model Stochastic Volatility Model

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Outline

1 Problem 2 Result 3 Applications

CIR Short Rate Model Stochastic Volatility Model

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Market Price of Risk Specification

• Aim: MPR specification that preserves affine structure:

dXt = (bQ + βQXt) dt + σ

• Xt
• dWt + ℓ + λXt

σ√Xt

• =dW Q

t

• MPR parameters:

ℓ = b − bQ, λ = β − βQ

• Formal density process E
• − ℓ+λX

σ √ X • W

• Novikov condition not satisfied, since

E

• e

1 2

T

1 Xt dt

• = ∞,

E

• e

1 2

T

0 Xt dt

= ∞ for T large enough in general

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Market Price of Risk Specification

• Aim: MPR specification that preserves affine structure:

dXt = (bQ + βQXt) dt + σ

• Xt
• dWt + ℓ + λXt

σ√Xt

• =dW Q

t

• MPR parameters:

ℓ = b − bQ, λ = β − βQ

• Formal density process E
• − ℓ+λX

σ √ X • W

• Novikov condition not satisfied, since

E

• e

1 2

T

1 Xt dt

• = ∞,

E

• e

1 2

T

0 Xt dt

= ∞ for T large enough in general

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Market Price of Risk Specification

• Aim: MPR specification that preserves affine structure:

dXt = (bQ + βQXt) dt + σ

• Xt
• dWt + ℓ + λXt

σ√Xt

• =dW Q

t

• MPR parameters:

ℓ = b − bQ, λ = β − βQ

• Formal density process E
• − ℓ+λX

σ √ X • W

• Novikov condition not satisfied, since

E

• e

1 2

T

1 Xt dt

• = ∞,

E

• e

1 2

T

0 Xt dt

= ∞ for T large enough in general

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Market Price of Risk Specification

• Aim: MPR specification that preserves affine structure:

dXt = (bQ + βQXt) dt + σ

• Xt
• dWt + ℓ + λXt

σ√Xt

• =dW Q

t

• MPR parameters:

ℓ = b − bQ, λ = β − βQ

• Formal density process E
• − ℓ+λX

σ √ X • W

• Novikov condition not satisfied, since

E

• e

1 2

T

1 Xt dt

• = ∞,

E

• e

1 2

T

0 Xt dt

= ∞ for T large enough in general

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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
• − ℓ+λX

σ √ X • W

• is a true

martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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
• − ℓ+λX

σ √ X • W

• is a true

martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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
• − ℓ+λX

σ √ X • W

• is a true

martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Outline

1 Problem 2 Result 3 Applications

CIR Short Rate Model Stochastic Volatility Model

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Stochastic Volatility

• Model for volatility: GARCH diffusion

dXt = (b + βXt) dt + Xt dW 1

t

• State space E = (0, ∞); that is, b ≥ 0
• Model for discounted S&P 500 index process:

dSt St = Xt

• ρ dW 1

t +

• 1 − ρ2 dW 2

t

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

d[X, log S]t = X 2

t ρ ≤ 0

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Stochastic Volatility

• Model for volatility: GARCH diffusion

dXt = (b + βXt) dt + Xt dW 1

t

• State space E = (0, ∞); that is, b ≥ 0
• Model for discounted S&P 500 index process:

dSt St = Xt

• ρ dW 1

t +

• 1 − ρ2 dW 2

t

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

d[X, log S]t = X 2

t ρ ≤ 0

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Stochastic Volatility

• Model for volatility: GARCH diffusion

dXt = (b + βXt) dt + Xt dW 1

t

• State space E = (0, ∞); that is, b ≥ 0
• Model for discounted S&P 500 index process:

dSt St = Xt

• ρ dW 1

t +

• 1 − ρ2 dW 2

t

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

d[X, log S]t = X 2

t ρ ≤ 0

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

Stochastic Volatility

• Model for volatility: GARCH diffusion

dXt = (b + βXt) dt + Xt dW 1

t

• State space E = (0, ∞); that is, b ≥ 0
• Model for discounted S&P 500 index process:

dSt St = Xt

• ρ dW 1

t +

• 1 − ρ2 dW 2

t

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

d[X, log S]t = X 2

t ρ ≤ 0

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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

1 2

T

0 X 2 t dt

= ∞

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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

1 2

T

0 X 2 t dt

= ∞

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility 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

1 2

T

0 X 2 t dt

= ∞

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

CFY Condition

• Apply auxiliary change of measure with density process

St S0 = 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 is well-posed in E = (0, ∞)

• CFY Theorem implies that S is a true martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

CFY Condition

• Apply auxiliary change of measure with density process

St S0 = 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 is well-posed in E = (0, ∞)

• CFY Theorem implies that S is a true martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

CFY Condition

• Apply auxiliary change of measure with density process

St S0 = 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 is well-posed in E = (0, ∞)

• CFY Theorem implies that S is a true martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

CFY Condition

• Apply auxiliary change of measure with density process

St S0 = 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 is well-posed in E = (0, ∞)

• CFY Theorem implies that S is a true martingale

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

References

• P. Cheridito, D. Filipovi´

c, and M. Yor. Equivalent and absolutely continuous measure changes for jump-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. L´

epingle and J. M´ emin. Sur l’int´ egrabilit´ e uniforme des martingales exponentielles.

• Z. Wahrsch. Verw. Gebiete, 42(3):175–203, 1978.

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

References

• P. Cheridito, D. Filipovi´

c, and M. Yor. Equivalent and absolutely continuous measure changes for jump-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. L´

epingle and J. M´ emin. Sur l’int´ egrabilit´ e uniforme des martingales exponentielles.

• Z. Wahrsch. Verw. Gebiete, 42(3):175–203, 1978.

Equivalent Measure Changes for Jump- Diffusions

• D. Filipovi´

c Problem Result Applications

CIR Short Rate Model Stochastic Volatility Model

References

• P. Cheridito, D. Filipovi´

c, and M. Yor. Equivalent and absolutely continuous measure changes for jump-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. L´

epingle and J. M´ emin. Sur l’int´ egrabilit´ e uniforme des martingales exponentielles.

• Z. Wahrsch. Verw. Gebiete, 42(3):175–203, 1978.