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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.