Duality and Derivative Pricing with Time-Changed Lévy Processes Jos´ e Fajardo Ernesto Mordecki IBMEC Business School Universidad de La Republica del Uruguay Fourth Bachelier Finance Society Congress. Tokyo, August 20, 2006 – p. 1/3
Outline • Motivation – p. 2/3
Outline • Motivation • Time-Changed Lévy processes – p. 2/3
Outline • Motivation • Time-Changed Lévy processes • Bidimensional Derivative pricing – p. 2/3
Outline • Motivation • Time-Changed Lévy processes • Bidimensional Derivative pricing • Duality and Symmetry – p. 2/3
Outline • Motivation • Time-Changed Lévy processes • Bidimensional Derivative pricing • Duality and Symmetry • Conclusions – p. 2/3
Motivation BiDimensional Derivative Pricing: European type American Perpetual type BM Margrabe (1978) Gerber and Shiu (1996) LP Fajardo and Mordecki (2006) Fajardo and Mordecki (2006) AP Eberlein and Papantaleon (2005b) ? TCLP This Paper ? – p. 3/3
Motivation BiDimensional Derivative Pricing: European type American Perpetual type BM Margrabe (1978) Gerber and Shiu (1996) LP Fajardo and Mordecki (2006) Fajardo and Mordecki (2006) AP Eberlein and Papantaleon (2005b) ? TCLP This Paper ? Duality Put-Call Exotic Derivatives BM Carr and Chesney (1996) Henderson and Wojakowski (2002) LP Fajardo and Mordecki (2006) Eberlein and Papantaleon (2005a) AP Eberlein and Papantaleon (2005b) ? TCLP This Paper ? SM Schroder (1999) ? – p. 3/3
Lévy Processes We say that X = { X t } t ≥ 0 is a Lévy Process, or a process with stationary and independent increments, if: – p. 4/3
Lévy Processes We say that X = { X t } t ≥ 0 is a Lévy Process, or a process with stationary and independent increments, if: • X has paths RCLL – p. 4/3
Lévy Processes We say that X = { X t } t ≥ 0 is a Lévy Process, or a process with stationary and independent increments, if: • X has paths RCLL • X 0 = 0 , and has independent increments, given 0 < t 1 < t 2 < ... < t n , the r.v. X t 1 , X t 2 − X t 1 , · · · , X t n − X t n − 1 are independents. – p. 4/3
Lévy Processes We say that X = { X t } t ≥ 0 is a Lévy Process, or a process with stationary and independent increments, if: • X has paths RCLL • X 0 = 0 , and has independent increments, given 0 < t 1 < t 2 < ... < t n , the r.v. X t 1 , X t 2 − X t 1 , · · · , X t n − X t n − 1 are independents. • The distribution of the increment X t − X s is homogenous in time, that is, depends just on the difference t − s . – p. 4/3
Lévy-Khintchine Formula A key result in the theory of Lévy Processes is the Lévy-Khintchine formula, that computes de characteristic function of X t as: E ( e zX t ) = e tψ ( z ) – p. 5/3
Lévy-Khintchine Formula A key result in the theory of Lévy Processes is the Lévy-Khintchine formula, that computes de characteristic function of X t as: E ( e zX t ) = e tψ ( z ) Where ψ is called characteristic exponent , and is given by: – p. 5/3
Lévy-Khintchine Formula A key result in the theory of Lévy Processes is the Lévy-Khintchine formula, that computes de characteristic function of X t as: E ( e zX t ) = e tψ ( z ) Where ψ is called characteristic exponent , and is given by: � ψ ( z ) = bz + 1 2 σ 2 z 2 + ( e zy − 1 − zy 1 {| y | < 1 } ) Π ( dy ) , I R where b and σ ≥ 0 are real constants, and Π is a positive measure in I R − { 0 } such that � (1 ∧ y 2 )Π( dy ) < ∞ , and is called Lévy measure, – p. 5/3
Remarks • Due to the independence of increments we can not model range dependence. – p. 6/3
Remarks • Due to the independence of increments we can not model range dependence. • Due to the homogeneity in time we can not have flexible representations for the term structure of volatility. – p. 6/3
This Paper • Time-inhomogenous Processes – p. 7/3
This Paper • Time-inhomogenous Processes • Derivative Pricing and Duality Relationship. – p. 7/3
Carr and Wu (2004) • Monroe (1978) – p. 8/3
Carr and Wu (2004) • Monroe (1978) • Stochastic Time change on Lévy Processes → stochastic volatility – p. 8/3
Carr and Wu (2004) • Monroe (1978) • Stochastic Time change on Lévy Processes → stochastic volatility • Correlation between Lévy processes and random clock → leverage effect. – p. 8/3
Carr and Wu (2004) • Monroe (1978) • Stochastic Time change on Lévy Processes → stochastic volatility • Correlation between Lévy processes and random clock → leverage effect. • Original clock as calendar time and new random clock as business time – p. 8/3
Time-Change Now let t �→ T t , , t ≥ 0 , be an increasing RCLL process, such that for each fixed t , T t is a stopping time with respect to F . – p. 9/3
Time-Change Now let t �→ T t , , t ≥ 0 , be an increasing RCLL process, such that for each fixed t , T t is a stopping time with respect to F . Furthermore, suppose T t is finite P − a.s., ∀ t ≥ 0 and T t → ∞ as t → ∞ . – p. 9/3
Time-Change Now let t �→ T t , , t ≥ 0 , be an increasing RCLL process, such that for each fixed t , T t is a stopping time with respect to F . Furthermore, suppose T t is finite P − a.s., ∀ t ≥ 0 and T t → ∞ as t → ∞ . Then {T t } defines a random change on time, we can also impose E T t = t . – p. 9/3
Time-Changed Lévy processes Then, consider the process Y t defined by: Y t ≡ X T t , t ≥ 0 , using different triplets for X and different time changes T t , we can obtain a good candidate for the underlying asset return process. . – p. 10/3
Time-Changed Lévy processes Then, consider the process Y t defined by: Y t ≡ X T t , t ≥ 0 , using different triplets for X and different time changes T t , we can obtain a good candidate for the underlying asset return process. We know that if T t is an independent subordinator, then Y is a Lévy process. – p. 10/3
Time-Changed Lévy processes Then, consider the process Y t defined by: Y t ≡ X T t , t ≥ 0 , using different triplets for X and different time changes T t , we can obtain a good candidate for the underlying asset return process. We know that if T t is an independent subordinator, then Y is a Lévy process. A more general situation is when T t is modeled by a non-decreasing semimartingale. – p. 10/3
Time-Changed Lévy Processes In that case � t � ∞ T t = b t + yµ ( dy, ds ) 0 0 where b is a drift and µ is the counting measure of jumps of the time change and we can take µ = 0 and just take locally deterministic time changes, so we need to specify the local intensity ν : � t T t = ν ( s − ) ds (1) 0 where ν is the instantaneous activity rate, observe that ν must be non-negative. – p. 11/3
Time-Changed Lévy Processes In that case � t � ∞ T t = b t + yµ ( dy, ds ) 0 0 where b is a drift and µ is the counting measure of jumps of the time change and we can take µ = 0 and just take locally deterministic time changes, so we need to specify the local intensity ν : � t T t = ν ( s − ) ds (2) 0 where ν is the instantaneous activity rate, observe that ν must be non-negative. – p. 11/3
Remarks • When X t is the Brownian motion , ν is proportional to the instantaneous variance rate of the Brownian motion – p. 12/3
Remarks • When X t is the Brownian motion , ν is proportional to the instantaneous variance rate of the Brownian motion • When X t is a pure jump Lévy process, ν is proportional to the Lévy intensity of jumps. – p. 12/3
Time-Changed Lévy Processes We can obtain the characteristic function of Y t : � � �� φ Y t ( z ) = E ( e z ′ X T t ) = E e z ′ X u / T t = u . E – p. 13/3
Time-Changed Lévy Processes We can obtain the characteristic function of Y t : � � �� φ Y t ( z ) = E ( e z ′ X T t ) = E e z ′ X u / T t = u . E If T t and X t were independent, then: φ Y t ( z ) = L T t ( ψ ( z )) where L T t is the Laplace transform of T t . – p. 13/3
Time-Changed Lévy Processes So if the Laplace transform of T and the characteristic exponent of X have closed forms, we can obtain a closed form for φ Y t . Using equation (1) we have: R t L T t ( λ ) = E ( e − λ 0 ν ( s − ) ds ) (3) From bond pricing literature Carr and Wu (2004) obtained a closed forms for φ Y t . – p. 14/3
Time-Changed Lévy Processes • When T is a Lévy processes the characteristics of Y are known explicitly. – p. 15/3
Time-Changed Lévy Processes • When T is a Lévy processes the characteristics of Y are known explicitly. • When T has independent increments we can obtain a deterministic characteristics for Y . – p. 15/3
Time-Changed Lévy Processes • When T is a Lévy processes the characteristics of Y are known explicitly. • When T has independent increments we can obtain a deterministic characteristics for Y . • When T is just a processes that turns Y in to a general semimartingale we can obtain a predictable version of the characteristics. – p. 15/3
Model Consider a market model with three assets ( S 1 , S 2 , S 3 ) given by t = e Y 1 0 e Y 2 0 e Y 3 t , t , S 1 S 2 t = S 2 S 3 t = S 3 t (4) where ( Y 1 , Y 2 , Y 3 ) is a three dimensional Time-Changed Lévy process. – p. 16/3
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