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

Equivalent Measure Changes for Jump- Diffusions D. Filipovi´ c Equivalent Measure Changes for Problem Jump-Diffusions Result Applications CIR Short Rate Model Damir Filipovi´ c Stochastic Volatility Model 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 Outline Changes for Jump- Diffusions D. Filipovi´ c Problem Result 1 Problem Applications CIR Short Rate Model Stochastic Volatility Model 2 Result 3 Applications CIR Short Rate Model Stochastic Volatility Model

Equivalent Measure Outline Changes for Jump- Diffusions D. Filipovi´ c Problem Result 1 Problem Applications CIR Short Rate Model Stochastic Volatility Model 2 Result 3 Applications CIR Short Rate Model Stochastic Volatility Model

Equivalent Measure Ingredients Changes for Jump- Diffusions D. Filipovi´ c Problem • m , d ∈ N Result • State space (open or closed) E ⊆ R m Applications CIR Short Rate • Locally bounded measurable mappings Model Stochastic Volatility Model b : E → R m × 1 , σ : E → R m × d • Transition kernel ν from E to R m such that � R m � ξ � ∧ � ξ � 2 ν ( x , d ξ ) x �→ is locally bounded on E

Equivalent Measure Ingredients Changes for Jump- Diffusions D. Filipovi´ c Problem • m , d ∈ N Result • State space (open or closed) E ⊆ R m Applications CIR Short Rate • Locally bounded measurable mappings Model Stochastic Volatility Model b : E → R m × 1 , σ : E → R m × d • Transition kernel ν from E to R m such that � R m � ξ � ∧ � ξ � 2 ν ( x , d ξ ) x �→ is locally bounded on E

Equivalent Measure Ingredients Changes for Jump- Diffusions D. Filipovi´ c Problem • m , d ∈ N Result • State space (open or closed) E ⊆ R m Applications CIR Short Rate • Locally bounded measurable mappings Model Stochastic Volatility Model b : E → R m × 1 , σ : E → R m × d • Transition kernel ν from E to R m such that � R m � ξ � ∧ � ξ � 2 ν ( x , d ξ ) x �→ is locally bounded on E

Equivalent Measure Ingredients Changes for Jump- Diffusions D. Filipovi´ c Problem • m , d ∈ N Result • State space (open or closed) E ⊆ R m Applications CIR Short Rate • Locally bounded measurable mappings Model Stochastic Volatility Model b : E → R m × 1 , σ : E → R m × d • Transition kernel ν from E to R m such that � R m � ξ � ∧ � ξ � 2 ν ( x , d ξ ) x �→ is locally bounded on E

Equivalent Measure Special Semimartingale Changes for Jump- Diffusions D. Filipovi´ c Problem • Filtered probability space (Ω , F , ( F t ) t ≥ 0 , P ) Result • Carrying d -dimensional Brownian motion W , and Applications CIR Short Rate • Random measure µ ( dt , d ξ ) associated to the jumps of . . . Model Stochastic Volatility Model • . . . the special (for simplicity) semimartingale X with canonical decomposition � t � t X t = X 0 + b ( X s ) ds + σ ( X s ) dW s 0 0 � t � + R m ξ ( µ ( ds , d ξ ) − ν ( X s , d ξ ) ds ) 0

Equivalent Measure Special Semimartingale Changes for Jump- Diffusions D. Filipovi´ c Problem • Filtered probability space (Ω , F , ( F t ) t ≥ 0 , P ) Result • Carrying d -dimensional Brownian motion W , and Applications CIR Short Rate • Random measure µ ( dt , d ξ ) associated to the jumps of . . . Model Stochastic Volatility Model • . . . the special (for simplicity) semimartingale X with canonical decomposition � t � t X t = X 0 + b ( X s ) ds + σ ( X s ) dW s 0 0 � t � + R m ξ ( µ ( ds , d ξ ) − ν ( X s , d ξ ) ds ) 0

Equivalent Measure Special Semimartingale Changes for Jump- Diffusions D. Filipovi´ c Problem • Filtered probability space (Ω , F , ( F t ) t ≥ 0 , P ) Result • Carrying d -dimensional Brownian motion W , and Applications CIR Short Rate • Random measure µ ( dt , d ξ ) associated to the jumps of . . . Model Stochastic Volatility Model • . . . the special (for simplicity) semimartingale X with canonical decomposition � t � t X t = X 0 + b ( X s ) ds + σ ( X s ) dW s 0 0 � t � + R m ξ ( µ ( ds , d ξ ) − ν ( X s , d ξ ) ds ) 0

Equivalent Measure Special Semimartingale Changes for Jump- Diffusions D. Filipovi´ c Problem • Filtered probability space (Ω , F , ( F t ) t ≥ 0 , P ) Result • Carrying d -dimensional Brownian motion W , and Applications CIR Short Rate • Random measure µ ( dt , d ξ ) associated to the jumps of . . . Model Stochastic Volatility Model • . . . the special (for simplicity) semimartingale X with canonical decomposition � t � t X t = X 0 + b ( X s ) ds + σ ( X s ) dW s 0 0 � t � + R m ξ ( µ ( ds , d ξ ) − ν ( X s , d ξ ) ds ) 0

Equivalent Measure Density Process Heuristics I Changes for Jump- Diffusions • Measurable mappings . . . D. Filipovi´ c κ : E × R m → (0 , ∞ ) λ : E → R d × 1 , Problem Result • . . . such that the local martingale L is well defined: Applications � t CIR Short Rate Model λ ( X s ) ⊤ dW s L t = Stochastic Volatility Model 0 � t � + R m ( κ ( X s − , ξ ) − 1) ( µ ( ds , d ξ ) − ν ( X s , d ξ ) ds ) 0 • Assume its stochastic exponential � � t L t − 1 � λ ( X s ) � 2 ds E t ( L ) = exp 2 0 � t � � + R m (log κ ( X s − , ξ ) − κ ( X s − , ξ ) + 1) µ ( ds , d ξ ) 0 is a true martingale

Equivalent Measure Density Process Heuristics I Changes for Jump- Diffusions • Measurable mappings . . . D. Filipovi´ c κ : E × R m → (0 , ∞ ) λ : E → R d × 1 , Problem Result • . . . such that the local martingale L is well defined: Applications � t CIR Short Rate Model λ ( X s ) ⊤ dW s L t = Stochastic Volatility Model 0 � t � + R m ( κ ( X s − , ξ ) − 1) ( µ ( ds , d ξ ) − ν ( X s , d ξ ) ds ) 0 • Assume its stochastic exponential � � t L t − 1 � λ ( X s ) � 2 ds E t ( L ) = exp 2 0 � t � � + R m (log κ ( X s − , ξ ) − κ ( X s − , ξ ) + 1) µ ( ds , d ξ ) 0 is a true martingale

Equivalent Measure Density Process Heuristics I Changes for Jump- Diffusions • Measurable mappings . . . D. Filipovi´ c κ : E × R m → (0 , ∞ ) λ : E → R d × 1 , Problem Result • . . . such that the local martingale L is well defined: Applications � t CIR Short Rate Model λ ( X s ) ⊤ dW s L t = Stochastic Volatility Model 0 � t � + R m ( κ ( X s − , ξ ) − 1) ( µ ( ds , d ξ ) − ν ( X s , d ξ ) ds ) 0 • Assume its stochastic exponential � � t L t − 1 � λ ( X s ) � 2 ds E t ( L ) = exp 2 0 � t � � + R m (log κ ( X s − , ξ ) − κ ( X s − , ξ ) + 1) µ ( ds , d ξ ) 0 is a true martingale

Equivalent Measure Heuristics II Changes for Jump- Diffusions D. Filipovi´ c • Finite time horizon T Problem • Define equivalent probability measure Q ∼ P on F T by Result Applications d Q CIR Short Rate d P = E T ( L ) Model Stochastic Volatility Model • Girsanov’s theorem implies that � t � W t = W t − λ ( X s ) ds , t ∈ [0 , T ] 0 is a Q -Brownian motion, and the compensator of µ ( dt , d ξ ) under Q becomes ν ( X t , d ξ ) dt = κ ( X t , ξ ) ν ( X t , d ξ ) dt , � t ∈ [0 , T ] .

Equivalent Measure Heuristics II Changes for Jump- Diffusions D. Filipovi´ c • Finite time horizon T Problem • Define equivalent probability measure Q ∼ P on F T by Result Applications d Q CIR Short Rate d P = E T ( L ) Model Stochastic Volatility Model • Girsanov’s theorem implies that � t � W t = W t − λ ( X s ) ds , t ∈ [0 , T ] 0 is a Q -Brownian motion, and the compensator of µ ( dt , d ξ ) under Q becomes ν ( X t , d ξ ) dt = κ ( X t , ξ ) ν ( X t , d ξ ) dt , � t ∈ [0 , T ] .

Equivalent Measure Heuristics II Changes for Jump- Diffusions D. Filipovi´ c • Finite time horizon T Problem • Define equivalent probability measure Q ∼ P on F T by Result Applications d Q CIR Short Rate d P = E T ( L ) Model Stochastic Volatility Model • Girsanov’s theorem implies that � t � W t = W t − λ ( X s ) ds , t ∈ [0 , T ] 0 is a Q -Brownian motion, and the compensator of µ ( dt , d ξ ) under Q becomes ν ( X t , d ξ ) dt = κ ( X t , ξ ) ν ( X t , d ξ ) dt , � t ∈ [0 , T ] .

Equivalent Measure Heuristics III Changes for Jump- Diffusions D. Filipovi´ c Problem • Canonical decomposition of X under Q reads Result � t � t Applications � σ ( X s ) d � CIR Short Rate X t = X 0 + b ( X s ) ds + W s Model Stochastic 0 0 � t � Volatility Model + R m ξ ( µ ( ds , d ξ ) − � ν ( X s , d ξ ) ds ) 0 • With modified drift function defined as � � b ( x ) = b ( x ) + σ ( x ) λ ( x ) + R m ξ ( κ ( x , ξ ) − 1) ν ( x , d ξ ) .

Equivalent Measure Heuristics III Changes for Jump- Diffusions D. Filipovi´ c Problem • Canonical decomposition of X under Q reads Result � t � t Applications � σ ( X s ) d � CIR Short Rate X t = X 0 + b ( X s ) ds + W s Model Stochastic 0 0 � t � Volatility Model + R m ξ ( µ ( ds , d ξ ) − � ν ( X s , d ξ ) ds ) 0 • With modified drift function defined as � � b ( x ) = b ( x ) + σ ( x ) λ ( x ) + R m ξ ( κ ( x , ξ ) − 1) ν ( x , d ξ ) .