Option Pricing with Semi-Markov Switching Lévy Process Financial Mathematics Lunch Talk Yi (Ivy) Zhang Department of Mathematics and Statistics University of Calgary February 26, 2019 Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 1 / 35
Markov Chains Markov Chain: a stochastic process that satisfies the "Markov Property" Markov Property: the "memoryless" property - the conditional probability distribution of future states of the process (conditional on both past and present states) depends only upon the present state, not on the sequence of events that preceded it Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 2 / 35
Semi-Markov Chains Consider a state space S = { 1 , 2 , ..., N } and a set of random variables ( X n , τ n ), where τ n are the jump times and X n are the associated states in the Markov Chain. The inter-arrival times are defined by θ n = τ n − τ n − 1 . Then the sequence ( X n , τ n ) is called a Markov Renewal Process if Pr ( θ n +1 ≤ t , X n +1 = j | ( X 0 , τ 0 ) , ( X 1 , τ 1 ) , ..., ( X n = i , τ n )) = Pr ( θ n +1 ≤ t , X n +1 = j | X n = i ) ∀ n ≥ 1 , t ≥ 0 , i , j ∈ S Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 3 / 35
Markov and semi-Markov Processes Semi-Markov Process: Define a new stochastic process Y t := X n for t ∈ [ τ n , τ n +1 ], then Y t is called a semi-Markov process. If the inter-arrival times in a semi-Markov process are exponentially distributed, then we have a continuous-time Markov chain, i.e. Pr ( θ n +1 ≤ t , X n +1 = j | X n = i ) = Pr ( X n +1 = j | X n = i )(1 − e λ i t ) For semi-Markov processes, the inter-arrival times can be from any arbitrary distribution. Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 4 / 35
Why Semi-Markov Switching Process? To price financial derivatives, we often use S t = S 0 e L t where L t is the semi-Markov switching process in our case. Many previous literatures are mainly using Markov switching models, ex. Buffington and Elliott. Because of the memoryless property of a Markov process, it is argued that a semi-Markov model is more adequate to model pricing on a financial market. We will focus on semi-Markov switching models where the distribution of holding times is Gamma distributed. Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 5 / 35
Gamma Distribution k : shape parameter; θ : scale parameter pdf : θ k Γ( k ) x k − 1 e − θ x f G ( x ) = x > 0 characteristic function : 1 − iu � − k � φ Gamma ( u ; k , θ ) = θ Mean and Variance: E [ X ] = k /θ Var ( X ) = k /θ 2 Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 6 / 35
Gamma vs Exponential Distributions Why Gamma? - To model the holding times of semi-Markov regime switching process X ; - Easy to compare with the Markov case where the exponential distribution is just a special case of gamma distribution with k = 1. Figure: PDFs of Gamma vs Exponential Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 7 / 35
Lévy Process A cádlág, adapted, real valued stochastic process L = ( L t ) t ≥ 0 with L 0 = 0 is a Lévy process if the following conditions are satisfied: (i) L has independent increments, i.e. L t − L s is independent of F s for any 0 ≤ s ≤ t ≤ T . (ii) L has stationary increments, i.e. the distribution of L t + s − L t does not depend on t for any s , t ≥ 0. (iii) L is stochastically continuous, i.e. for every t ≥ 0 and ǫ > 0: lim s → t P ( | L t − L s | > ǫ ) = 0 Examples of Lévy processes: Brownian motion, Poisson process, Variance Gamma process Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 8 / 35
Variance Gamma as the Lévy Process We will then introduce the Variance Gamma process as the Lévy process that is dominating the pricing process within each regime. Characteristic function of the VG( σ, ν, θ ) law: 1 − iu θν + 1 2 σ 2 ν u 2 � − 1 /ν � φ ( u ; σ, ν, θ ) = Characteristic function of the VG process: √ E [ e iuL VG t ] = φ VG ( u ; σ t , ν/ t , t θ ) = ( φ VG ( u ; σ, ν, θ )) t 1 − iu θν + 1 2 σ 2 ν u 2 � − t /ν � = E [ L t ] = θ t , Var ( L t ) = σ 2 t + θ 2 ν t . Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy Process February 26, 2019 9 / 35
Characteristic Function under Risk-Neutral measure Introduction Under the risk-neutral measure P : i ∆ ut − t � 1 − iu θν + 1 � �� 2 σ 2 u 2 ν φ RN ( u ) = exp ν ln = e i ∆ ut � 1 − iu θν + 1 � − t /ν 2 σ 2 u 2 ν where ∆ is the risk-neutral drift and equal to ∆ = r + ln(1 − θν − 1 2 σ 2 ν ) ν r: risk-neutral rate Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy ProcessFebruary 26, 2019 10 / 35
Semi-Markov Switching Lévy Processes Introduction The semi-Markov switching Lévy process is defined as L t = L j t , X t = e j , L j t : Lévy process X t : finite state semi-Markov switching process E = { e 1 , e 2 , ..., e m } : state space Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy ProcessFebruary 26, 2019 11 / 35
Semi-Markov Switching Lévy Processes Now, we will introduce some definitions and notations: 0 < t 1 < t 2 < t 3 ... : jump times for X Q i , j ( t ) = P ( X n = j , t n − t n − 1 ≤ t | X n − 1 = i ) p ij = lim t →∞ Q i , j ( t ) F ( t | i , j ) = P ( τ n ≤ t | X n = j , X n − 1 = i ) S ( t | i , j ) = 1 − F ( t | i , j ) − dS dt ( t | i , j ) λ i , j ( t ) = p i , j S ( t − | i ) Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy ProcessFebruary 26, 2019 12 / 35
Semi-Markov Switching Lévy Processes The Lévy-Itô decomposition for L j t has the following form: � � L j t = a j t + σ j W t + zN j ( t , dz ) + z ( N j ( t , dz ) − m j ( dz ) t ) . | z | > 1 | z |≤ 1 W : Brownian motion; N : random measure counting the jumps; m ( dz ): compensated measure/Lévy measure: expected number of jumps. The triplet ( a j , σ j , m j ( dz )) is completely determined by the characteristic function of L j t : t ( u ) = E [ e iuL j φ j t ] = exp { t ( ia j u − 1 � j u 2 + ( e iuz − 1 − iz 1 ( | z | ≤ 1)) m j ( dz )) } 2 σ 2 R Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy ProcessFebruary 26, 2019 13 / 35
Itô’s Formula for Semi-Markov Switching Lévy Processes Considering the following SDE: dx t = x t dL j t , x ( t n ) = x n , t ∈ [ t n , t n +1 ) Also define: � γ t = ( t − t n ) 1 t n ≤ t < t n +1 n ≥ 0 α n : state jump process caused by the semi-Markov process ln( α n ): jump in log price at regime change g : density function of α n g ( z | X n − 1 , X n ) = g ( e z | X n − 1 , X n ) e z ˜ � µ ( t , A , B ) = 1 ( t ≥ t n , ln( α n ) ∈ A , ( X n − 1 , X n ) ∈ B ) n ≥ 1 � t � ν ( t , A , { i , j } ) = ˜ g ( z | i , j ) λ i , j ( γ s ) dzds 0 z ∈ A Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy ProcessFebruary 26, 2019 14 / 35
Itô’s Formula for Semi-Markov Switching Lévy Processes dV ( s , γ s , X s , x s ) ∂ V =( L V )( s , γ s , X s , x s ) ds + σ j x s ∂ x dW s � � � ˜ + V ( s , γ s , X s , x s + x s G j ( z )) − V ( s , γ s , X s , x s ) N j ( ds , dz ) | z |≤ 1 � � � ˜ + V ( s , γ s , X s , x s + x s H j ( z )) − V ( s , γ s , X s , x s ) N j ( ds , dz ) | z | > 1 � � � � V ( s , γ s , j , x s e z ) − V ( s , γ s , X s , x s ) + µ ( ds , dz , { ( X s , j ) } ) ˜ z ∈ R j ∈ E \{ X s } Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy ProcessFebruary 26, 2019 15 / 35
Itô’s Formula for Semi-Markov Switching Lévy Processes where ( L V )( s , γ s , X s , x s ) ∂ 2 V = ∂ V ∂ s + ∂ V ∂γ + a j x s − ∂ V ∂ x + 1 2 σ 2 j x 2 s ∂ x 2 ∂ V � � � + V ( s , γ s , X s , x s + x s G j ( z ) − V ( s , γ s , X s , x s ) − G j ( z ) x s m j ( dz ) ∂ x | z |≤ 1 � � � + V ( s , γ s , X s , x s + x s H j ( z )) − V ( s , γ s , X s , x s ) m j ( dz ) | z | > 1 � � � � V ( s , γ s , j , x s e z ) − V ( s , γ s , X s , x s ) + λ X s , j ( γ s ) ˜ g ( z | X s , j ) dz z ∈ R j � = X s G(z), H(z): smooth functions defining the jump size z . Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy ProcessFebruary 26, 2019 16 / 35
Characteristic Function for Semi-Markov Switching Lévy Processes Our goal here is to derive a closed form expression for the conditional characteristic function � e iu ln( x t ) � � Φ( u , t , γ, j , x ) = E � γ 0 = γ, X 0 = j , x 0 = x � for the log price process below n ( t ) ln( α a ) + L j � ln( x t ) = t . a =1 Yi (Ivy) Zhang (Universities of Calgary) Option Pricing with Semi-Markov Switching Lévy ProcessFebruary 26, 2019 17 / 35
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