Lecture 3: Meromorphic L evy Processes and Wiener-Hopf Monte-Carlo - PowerPoint PPT Presentation

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods A. E. Kyprianou Department of Mathematical Sciences, University of Bath 1/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Lecture 1 & 2 recap: 2/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Lecture 1 & 2 recap: L´ evy process. A (one dimensional) process with stationary and independent increments which has paths which are right continuous with left limits. 2/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Lecture 1 & 2 recap: L´ evy process. A (one dimensional) process with stationary and independent increments which has paths which are right continuous with left limits. A popular model in mathematical finance for the evolution of a risky asset is S t := e X t , t ≥ 0 where { X t : t ≥ 0 } is a L´ evy process. 2/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Lecture 1 & 2 recap: L´ evy process. A (one dimensional) process with stationary and independent increments which has paths which are right continuous with left limits. A popular model in mathematical finance for the evolution of a risky asset is S t := e X t , t ≥ 0 where { X t : t ≥ 0 } is a L´ evy process. Barrier options: The value of up-and-out barrier option with expiry date T and barrier b is typically priced as E x ( f ( S T ) 1 { S T ≤ b } ) where S T = sup u ≤ T S u = exp { sup u ≤ T X u } , f is some nice function. 2/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Lecture 1 & 2 recap: L´ evy process. A (one dimensional) process with stationary and independent increments which has paths which are right continuous with left limits. A popular model in mathematical finance for the evolution of a risky asset is S t := e X t , t ≥ 0 where { X t : t ≥ 0 } is a L´ evy process. Barrier options: The value of up-and-out barrier option with expiry date T and barrier b is typically priced as E x ( f ( S T ) 1 { S T ≤ b } ) where S T = sup u ≤ T S u = exp { sup u ≤ T X u } , f is some nice function. One is fundamentally interested in the joint distribution P ( X t ∈ d x , X t ∈ d y ) for any L´ evy process ( X , P ) , where X t = sup s ≤ t X s . 2/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Fourier methods 3/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Fourier methods Theoretically, things are already difficult enough if considering P ( X t ∈ d y ) or P ( X t ∈ d x ) , especially in the former case. 3/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Fourier methods Theoretically, things are already difficult enough if considering P ( X t ∈ d y ) or P ( X t ∈ d x ) , especially in the former case. For the case of P ( X t ∈ d x ) one is sometimes lucky and knows this in explicit form. But usually one only knows something about − 1 t log E (e i θ X t ) Ψ( θ ) := a i θ + 1 � 2 σ 2 θ 2 + (1 − e i θ x + i θ x 1 {| x |≤ 1 } )Π(d x ) = R where a ∈ R , σ ∈ R and Π is a measure concentrated on R \{ 0 } satisfying R (1 ∧ x 2 )Π(d x ) < ∞ . � 3/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Fourier methods Theoretically, things are already difficult enough if considering P ( X t ∈ d y ) or P ( X t ∈ d x ) , especially in the former case. For the case of P ( X t ∈ d x ) one is sometimes lucky and knows this in explicit form. But usually one only knows something about − 1 t log E (e i θ X t ) Ψ( θ ) := a i θ + 1 � 2 σ 2 θ 2 + (1 − e i θ x + i θ x 1 {| x |≤ 1 } )Π(d x ) = R where a ∈ R , σ ∈ R and Π is a measure concentrated on R \{ 0 } satisfying R (1 ∧ x 2 )Π(d x ) < ∞ . � ...in which case there are fast-Fourier methods for inverting exp {− Ψ( θ ) t } to give P ( X t ∈ d x ) . 3/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Fourier methods Theoretically, things are already difficult enough if considering P ( X t ∈ d y ) or P ( X t ∈ d x ) , especially in the former case. For the case of P ( X t ∈ d x ) one is sometimes lucky and knows this in explicit form. But usually one only knows something about − 1 t log E (e i θ X t ) Ψ( θ ) := a i θ + 1 � 2 σ 2 θ 2 + (1 − e i θ x + i θ x 1 {| x |≤ 1 } )Π(d x ) = R where a ∈ R , σ ∈ R and Π is a measure concentrated on R \{ 0 } satisfying R (1 ∧ x 2 )Π(d x ) < ∞ . � ...in which case there are fast-Fourier methods for inverting exp {− Ψ( θ ) t } to give P ( X t ∈ d x ) . For the case of P ( X t ∈ d x ) , recent methods have concentrated on Fourier inversion of the, so called, Wiener-Hopf factors. 3/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Wiener-Hopf-Fourier methods 4/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Wiener-Hopf-Fourier methods Recall that it turns out that one may always uniquely decompose q q + Ψ( θ ) = E (e i θ X e q ) × E (e i θ X e q ) where e q is an independent and exponentially distributed random variable with rate q > 0 and X t = inf s ≤ t X s . 4/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Wiener-Hopf-Fourier methods Recall that it turns out that one may always uniquely decompose q q + Ψ( θ ) = E (e i θ X e q ) × E (e i θ X e q ) where e q is an independent and exponentially distributed random variable with rate q > 0 and X t = inf s ≤ t X s . If one is in possession of close analytical expressions for these factors , Fourier inversion, first in θ and then in q would be an option for accessing the law of X t and X t . However one is rarely in possession of the factors (even after 60 years of research into this topic), and even then there is the issue of the double Fourier inversion. 4/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Wiener-Hopf-Fourier methods Recall that it turns out that one may always uniquely decompose q q + Ψ( θ ) = E (e i θ X e q ) × E (e i θ X e q ) where e q is an independent and exponentially distributed random variable with rate q > 0 and X t = inf s ≤ t X s . If one is in possession of close analytical expressions for these factors , Fourier inversion, first in θ and then in q would be an option for accessing the law of X t and X t . However one is rarely in possession of the factors (even after 60 years of research into this topic), and even then there is the issue of the double Fourier inversion. There are no convenient formulae which contain both X t and X t which could be Fourier inverted. 4/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Some new ideas 5/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Some new ideas Suppose that e (1) , e (2) , · · · are a sequence of i.i.d unit mean exponentially distributed random variables. 5/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Some new ideas Suppose that e (1) , e (2) , · · · are a sequence of i.i.d unit mean exponentially distributed random variables. Note that n 1 � q e ( i ) g ( n , q ) := i =1 is a Gamma (Erlang) distribution with parameters n and q and by the strong law of Large numbers, for t > 0 , n t n e ( i ) → t � g ( n , n / t ) = i =1 almost surely. 5/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Some new ideas Suppose that e (1) , e (2) , · · · are a sequence of i.i.d unit mean exponentially distributed random variables. Note that n 1 � q e ( i ) g ( n , q ) := i =1 is a Gamma (Erlang) distribution with parameters n and q and by the strong law of Large numbers, for t > 0 , n t n e ( i ) → t � g ( n , n / t ) = i =1 almost surely. Hence for a suitably large n , we have in distribution ( X g ( n , n / t ) , X g ( n , n / t ) ) ≃ ( X t , X t ) . Indeed since t is not a jump time with probability 1, we have that ( X g ( n , n / t ) , X g ( n , n / t ) ) → ( X t , X t ) almost surely. 5/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Some new ideas 6/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Some new ideas A reformulation of the Wiener-Hopf factorization states that d X e q = S q + I q where S q is independent of I q and they are respectively equal in distribution to X e q and X e q . 6/ 31

Lecture 3: Meromorphic L´ evy Processes and Wiener-Hopf Monte-Carlo simulation methods Some new ideas A reformulation of the Wiener-Hopf factorization states that d X e q = S q + I q where S q is independent of I q and they are respectively equal in distribution to X e q and X e q . d = X Xe - X e q e q q X e q e q 6/ 31