Università degli Studi di Milano. Jump Telegraph-Diffusion Option Pricing

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1 Università degli Studi di Milano Statistics and Mathematics Year 28 Paper 33 Jump Telegraph-Diffusion Option Pricing Niita Ratanov Universidad del Rosario This woring paper site is hosted by The Bereley Electronic Press (bepress) and may not be commercially reproduced without the publisher s permission. Copyright c 28 by the author.

2 Jump Telegraph-Diffusion Option Pricing Abstract The paper develops a class of Financial maret models with jumps based on a Brownian motion, and inhomogeneous telegraph processes: random motions with alternating velocities. We assume that jumps occur when the velocities are switching. The distribution of such a process is described in detail. For this model we obtain the structure of the set of martingale measures. The model can be completed adding another asset based on the same sources of randomness. Explicit formulae for prices of standard European options in completed maret are obtained.

3 Jump Telegraph-Diffusion Option Pricing Niita Ratanov University of Rosario, Bogotá, Colombia Abstract The paper develops a class of financial maret models with jumps based on a Brownian motion, and inhomogeneous telegraph processes: random motions with alternating velocities. We assume that jumps occur when the velocities are switching. The distribution of such a process is described in detail. For this model we obtain the structure of the set of martingale measures. The model can be completed adding another asset based on the same sources of randomness. Explicit formulae for prices of standard European options in completed maret are obtained. 1. Introduction Option pricing models based on the geometric Brownian motion, e.g. Blac-Scholes model, S(t) = S e µt+vw(t), t T, have well nown generic limitations and shortages. These models (Blac-Scholes and its derivatives) have infinite propagation velocities, independent log-returns increments on separated time intervals among others. To overcome these limitations various approaches are exploited. Among them one could mention a model based on so called jump telegraph processes [1]-[13]. This model presumes that the log-prices of risy asset moves with pair of constant velocities alternating one to another at Poisson times. To mae the model more adequate and to avoid arbitrage opportunities the log-return movement should be supplied with jumps occurring at times of the tendency switchings. The jump telegraph model is free of arbitrage opportunities and it is complete. Moreover it permits exact standard option pricing formulae similar to the classic Blac-Scholes model. Telegraph processes have been studied before in different probabilistic aspects (see, for instance, Goldstein [4], Kac [7] and Zacs [14]). These processes have been exploited for stochastic volatility modelling (Di Masi et al [3]), as well as for obtaining a telegraph analog of the Blac-Scholes model (Di Crescenzo and Pellerey [2]). Recently the telegraph processes was applied to actuarial problems [9]. Another simple approach of similar features is based on Marov modulated diffusion processes ([5], [6]). This approach supplies alternating tendencies of the telegraph process with a diffusion process of alternating diffusion coefficients. This paper combines both approaches. We consider the asset price which moves according with Marov modulated diffusion process supplied with alternating jumps occur- 1

4 ring at times of the state switchings. We assume the bond price to be random and it moves with geometric telegraph process. The model is incomplete. Section 2 is devoted to detailed definition and description of the underlying processes and their distributions. In this section we obtain the Girsanov theorem for the jump telegraph-diffusion processes. In Section 3 we describe the set of ris-neutral measures as well as the distribution of underlying processes. Also we consider a completion of the model by adding another asset driven by the same sources of randomness. In the completed maret we obtain exact option pricing formulae for the standard call option. This formulae is a mix of Blac-Scholes function and densities of spending times of the driving Marov flow. 2. Jump telegraph processes and jump diffusions with Marov switching Let (Ω, F, P) be a complete probability space. Denote ε i (t), t, i =, 1 Marov processes with two states {, 1}, subscript i indicates the initial state: ε i () = i. Assume that T j the time to leave state j =, 1, is exponentially distributed, P{T j > t} = e λjt, i =, 1. Equivalently, P{ε i (t + t) = j ε i (t) = j} = 1 λ j t + o( t), t, j {, 1}. Let τ 1, τ 2,... are switching times, τ =. The time intervals τ j τ j 1, j = 1, 2,... (τ = ), separated by instants of value changes τ j = τj, i j = 1, 2,... are independent. Also, we denote P i the conditional probability with respect to the initial state i =, 1, and E i the expectation with respect to P i. Denote by N i (t) = max{j : τ j < t}, t a number of switchings of ε i till time t, t. It is clear that N i, i =, 1 are the Poisson processes with alternating intensities λ, λ 1 >. The distribution πn(t) i = P{N i (t) = n}, n =, 1, 2,..., i =, 1, t of the counting process N i = N i (t) can be calculated as follows. Proposition 2.1. Functions π i n(t) follow the equations and π i (t) = e λ it. dπ i n dt = λ i π i n(t) + λ i π 1 i n 1(t), i =, 1, n 1, (2.1) Proof. It is sufficient to notice, that conditioning on the Poisson event on time interval (, t) we have π i n(t + t) = (1 λ i t)π n (t) + λ i tπ 1 i n 1(t) + o( t), t, which leads to (2.1). Let c, c 1, c > c 1 ; h, h 1 ; σ, σ 1 be real numbers. Let w = w(t), t be a standard Brownian motion independent of ε i. We consider X i (t) = X i (t; c, c 1 ) = c εi (τ)dτ, J i (t) = J i (t; h, h 1 ) = 2 h εi (τ)dn i (τ) = N i (t) j=1 h εi (τ j ),

5 D i (t) = D i (t; σ, σ 1 ) = σ εi (τ)dw(τ). Here X, X 1 are telegraph processes with the states < c, λ > and < c 1, λ 1 >, J, J 1 are pure jump processes, and D, D 1 have a sense of diffusion process with Marov switching. The sum X i (t) + J i (t) + D i (t), t, i =, 1 is called jump telegraph-diffusion (JTD) process with the states < c, h, σ, λ > and < c 1, h 1, σ 1, λ 1 >. Further, we will assume all processes to be adapted to the filtration F i = (F i t) t (F i = {, Ω}), generated by ε i (t), t, and w(t), t. We suppose that the filtration satisfies the usual conditions (see e. g. [8]). Let us notice that the stochastic exponential of JTD-process has the form where E t (X i + J i + D i ) = exp κ i (t) = { X i (t) + D i (t) 1 2 N i (t) j=1 σε 2 i (τ)dτ } κ i (t), (2.2) ( ) 1 + hεi (τ j ). (2.3) Denote by p i (x, t, n) (generalized) probability densities with respect to the measure P i of the telegraph-diffusion variable X i (t) + D i (t) (without jump component), which has n turns up to time t: P i {X i (t) + D i (t), N i (t) = n} = p i (x, t, n)dx, i =, 1, t, n =, 1, 2,... (2.4) Remar 2.1. The densities p i (x, t, n) of JTD-process can be expressed as follows: p i (x, t, n) = p i (x j i n, t, n), where j i n = [(n + 1/2]h i + [n/2]h 1 i, n =, 1,... The distribution of X i (t) + D i (t) can be found directly. We derive first the PDEs which describe densities p i (x, t, n). Theorem 2.1. Densities p i, i =, 1 satisfy the following PDE-system p i t (x, t, n) + c p i i x (x, t, n) σ2 i 2 Moreover where 2 p i x 2 (x, t, n) = λ ip i (x, t, n) + λ i p 1 i (x, t, n 1), (2.5) i =, 1, n 1. p i (x, t, ) = e λ it ψ i (x, t), ψ i (x, t) = 1 e (x c i t) 2 2σ i 2t. σ i 2πt 3

6 Proof. Let t >. Let τ is the r. v. uniformly distributed on [, t] and independent of X i + D i. Denote Z i = (c i t + σ i w( t)) 1 {Ni ( t)=}+(c i τ + c 1 i ( t τ) + σ i w(τ) + σ 1 i w( t τ)) 1 {Ni ( t)=1}. Notice that P{N i ( t) > 1} = o( t), t. Hence X i ( t) + D i ( t) d = Z i + ξ i, where ξ i = o( t), t, i. e. ξ i is the r.v. which satisfies P i {ξ i } = o( t) t. Thus X i (t + t) + D i (t + t) d = Z i + X i (t) + D i (t) + o( t). Here X i + D i is the telegraph-diffusion process independent of X i + D i. Conditioning on a jump in (, t) we have p i (x, t+ t, n) = (1 λ i t)p i (, t, n) ψ i (, t)(x)+λ i tp 1 i (, t, n 1) ψ i (, t)(x)+o( t), i =, 1, t. Here ψ i is the distribution density of c i t + σ i w( t), ψi is the distribution density of c i τ + c 1 i ( t τ) + σ i w(τ) + σ 1 i w( t τ); the notation is used for the convolution in spacial variables. It is easy to see, that ψ i (x, t), ψi (x, t) δ(x) as t. Hence as t. Then, p i (, t, n) ψ i (, t)(x), p i (, t, n) ψ i (, t)(x) p i (x, t, n) 1 t [p i(, t, n) ψ i (, t)(x) p i (x, t, n)] = 1 t p i (x y, t, n)ψ i (y, t)dy p i (x, t, n) = 1 t [ ] p i (x c i t yσ i t, t, n) pi (x, t, n) ψ(y)dy, where ψ = ψ( ) is N (, 1)-density. The latter value equals to 1 t [ pi ψ(y) = 1 t x (x, t, n)( c i t yσ i t) ] 2 p i x (x, t, n)( c i t yσ 2 i t) 2 + o( t) dy [ pi ψ(y) x (x, t, n)( c i t) + 1 ] 2 p i 2 x (x, t, 2 n)y2 σi 2 t + o( t) dy System (2.5) is obtained. p i c i x (x, t, n) + σ2 i 2 2 p i (x, t, n). x2 4

7 It is easy to solve system (2.5). First consider (2.5) without diffusion part, i. e. for σ = σ 1 =, and of velocities c = 1, c 1 = 1. In this case p () (x, t) = e λt δ(x c t), p () 1 (x, t) = e λ1t δ(x c 1 t). { } Setting θ(x, t) = exp λ 1 c c 1 (c t x) λ c c 1 (x c 1 t) 1 {c1 t<x<c t}, for n 1 we find and for n p (x, t, 2n) = p 1 (x, t, 2n) = λ n λ n 1 (c c 1 ) (c t x) n 1 (x c 1 t) n θ(x, t), 2n (n 1)!n! λ n λ n 1 (c c 1 ) (c t x) n (x c 1 t) n 1 θ(x, t), (2.6) 2n n!(n 1)! λ n+1 λ n 1 p (x, t, 2n + 1) = (c c 1 ) (c t x) n (x c 1 t) n θ(x, t), 2n+1 (n!) 2 p 1 (x, t, 2n + 1) = λ n λ n+1 1 (c c 1 ) 2n+1 (c t x) n (x c 1 t) n (n!) 2 θ(x, t). (2.7) Conditioning on the number of switches we get the probability density of the telegraph process which is described by parameters < c, λ > and < c 1, λ 1 >: p i (x, t) = p i (x, t, n), (2.8) n= For the general case of JTD-process the respective densities has the same form (2.8), but with the convolution p i (, t, n) ψ (n) i (, t) instead of p i (x, t, n). Here p i (, t, n) is the densities respected to jump telegraph process X i (t) + J i (t) (see Remar 2.1) and ψ (n) i (, t) is the density of N (, σn) i where σn i = [(n + 1)/2]σ i + [n/2]σ 1 i. Formulae (2.8)-(2.7) give the following rules of changes in the intensities λ i : if λ is changed to λ and λ 1 is changed to λ 1, the probability densities p i will be changed to: p i (x, t) = p i(x, t, n) (2.9) { where p i(x, t, n) = p i (x, t, n) exp λ 1 λ 1 c c 1 (c t x) λ λ c c 1 (x c 1 t) n= κ (2n) λ /λ,i = (λ /λ ) n (λ 1/λ 1 ) n } κ (n) λ /λ,i with κ (2n+1) λ /λ, = (λ /λ ) n+1 (λ 1/λ 1 ) n, κ (2n+1) λ /λ,1 = (λ /λ ) n (λ 1/λ 1 ) n+1, n =, 1,... (2.1) Remar 2.2. Formulae (2.8)-(2.7) in particular case B = h + h 1 = becomes [ ( ) p i (x, t) = e λ it δ(x c i t)+ e Λt λx λ i I λ λ 1 (c t x + h i )(x h i c 1 t)/(c c 1 ) θ(x h i, t) c c 1 + ( ) ( 1) 1 i c t x 2 ( ) λ λ 1 I 1 λ λ 1 (c t x)(x c 1 t)/(c c 1 ) θ(x, t), x c 1 t and I 1 (z) = I (z) are usual modified Bessel functions. Com- where I (z) = (z/2) 2n n= (n!) 2 pare with [1]. 5

8 We apply previous results to obtain the distributions of times which the process ε i spends in the certain state. Let T i = T 1 {ε i (t)=}dt, i =, 1 be the total time between and T spending by the process ε i in the state starting form the state i. If we consider a standard telegraph processes with velocities c = 1, c 1 = 1, X (t) = ( 1)N (τ) dτ and X 1 (t) = ( 1)N 1(τ) dτ, then X (T ) = T (T T ) = 2T T and X 1 (T ) = 2T 1 T. (2.11) Let f i (t, T, n), t T denote the density of T i : for all Υ [ T, T ] f i (t, T, n)dt = P i {T i Υ, N i (T ) = n} (2.12) Applying (2.11) we can notice that Υ f (t, T, n) = 2 p (2t T, T, n), f 1 (t, T, n) = 2 p 1 (2t T, T, n), (2.13) where p and p 1 are the densities of the standard telegraph process (with c = 1 and c 1 = 1) defined in (2.8)-(2.7). Using formulae for densities p i, which are obtained in (2.8)-(2.7), from (2.13) we have f (t, T, ) = e λ T δ(t T ), f 1 (t, T, ) = e λ 1T δ(t). For n 1 and for n f (t, T, 2n) = λ n λ n 1 f 1 (t, T, 2n) = λ n λ n 1 (T t) n 1 t n e λ t λ 1 (T t) 1 { t T }, (2.14) (n 1)!n! (T t) n t n 1 e λ t λ 1 (T t) 1 { t T }, (2.15) (n 1)!n! f (t, T, 2n + 1) = λ n+1 λ n (T t) n t n 1 e λ t λ 1 (T t) 1 (n!) 2 { t T }, (2.16) f 1 (t, T, 2n + 1) = λ n λ n+1 (T t) n t n 1 e λ t λ 1 (T t) 1 (n!) 2 { t T }. (2.17) Summarizing we have the following expressions for the densities f i (t, T ) of the spending time of the the process X i (t), t T in state : [ f (t, T ) = e λt δ(t T ) + e λ t λ 1 (T t) λ I (2 λ λ 1 t(t t)) + t λ λ 1 T t I 1(2 ] λ λ 1 t(t t)), (2.18) [ f 1 (t, T ) = e λ1t δ(t) + e λ t λ 1 (T t) λ 1 I (2 λ λ 1 t(t t)) + T t λ λ 1 I 1 (2 ] λ λ 1 t(t t)). (2.19) t Next we describe in this framewor martingales and martingale measures. The next theorem could be considered as a version of the Doob-Meyer decomposition for telegraphdiffusion processes with alternating intensities. 6

9 Theorem 2.2. JTD-process X i +J i +D i, i =, 1 is a martingale if and only if c = λ h and c 1 = λ 1 h 1. Proof. The processes σ εi = σ εi (s), s t are F t -measurable. Hence the processes D i = D i (t) = σ ε i (τ)dw(τ), t, i =, 1 are F t -martingales. Now, the result follows from Theorem 2.1 [11]. Corollary 2.1. The process exp{x i (t) + D i (t)}κ i (t) is a martingale if and only if c i + σ 2 i /2 = λ i h i, i =, 1. Proof. It is sufficient to notice that exp{x i (t) + D i (t)}κ i (t) = E t ( X i + J i + D i ), where X i (t) = X i (t) σ 2 ε i (τ) dτ. Now we study the properties of JTD-processes under a change of measure. Let Xi, i =, 1 be the telegraph processes with states < c, λ > and < c 1, λ 1 >, and Ji = N i (t) c ε i (τ j ) /λ ε i (τ j ), i =, 1 be the jump processes with jump values h i = c ε i /λ εi > j=1 1, which let the sum X i +J i to be a martingale. Let D i = σε i (τ) dw(τ) be the diffusion with alternating diffusion coefficients σ i, i =, 1. Consider a probability measure P i with a local density with respect to P i : ( Z i (t) = P i t = E t (Xi + Ji + Di ) = exp Xi (t) + Di (t) 1 P i 2 ) (σε i (s)) 2 ds κ i (t), (2.2) where κ i (t) is defined in (2.3) with h i instead of h i. Notice that Z i (t) is stochastic exponential of JTD-process with the states < c i, h i, σ i, λ i >, i =, 1. Theorem 2.3 (Girsanov theorem). Under the probability measure P i 1) process w(t) := w(t) σε i (τ) dτ is a standard Brownian motion; 2) counting Poisson process N i (t) has intensities λ i := λ i (1 + h i ) = λ i c i. Proof. Let U i (t) := exp{z w(t)} = exp{z(w(t) show that for any t 1 < t Seeing for simplicity we prove it for t 1 =. Notice that σ ε i (τ) E i {Z i (t)u i (t) F t1 } = e z2 (t t 1 )/2 Z i (t 1 )U i (t 1 ). Z i (t)u i (t) = exp Xi (t) + Di (t) 1 2 = E t X i 1 2 (σ ε i (τ)) 2 dτ z (σ ε i (τ)) 2 dτ + zw(t) z σ ε i (τ)dτ + D i + zw dτ)}. For 1) it is sufficient to σ ε i (τ)dτ κ i (t) (σε i (τ) + z) 2 dτ + Ji 7

10 = E t (X i + D i + J i + zw) exp(z 2 t/2). Thus E i (Z i (t)u i (t)) = exp(z 2 t/2). To prove the second part of the theorem we denote π,n i = P i {N i (t) = n} = E i (Z i (t)1 {Ni (t)=n}) = κ i,n ex p i (x, t, n)dx, where p i = p i (x, t, n) are (generalized) probability densities of telegraph-diffusion process Xi (t)+di (t) (σ ε i (τ) )2 dτ/2. Notice that functions p i (x, t, n) satisfy the system (2.5) with c i (σi ) 2 /2 and σi instead of c i and σ i respectively. Therefore dπ i,n (t) = (c i λ i )π i dt,n(t) + λ i (1 + h i )π,n 1(t). 1 i Next notice that λ i c i = λ i + λ i h i := λ i and, thus dπ i,n (t) = λ i π i dt,n(t) + λ i π,n 1(t). 1 i The theorem follows from Proposition Jump telegraph-diffusion model Let ε i = ε i (t) =, 1, t be a Marov switching process defined in Section 2 which indicates two of possible maret states. First we consider the maret with one risy asset. Assume the price of the risy asset which moves initially at the state i, follows the equation ds(t) = S(t )d(x i (t) + J i (t) + D i (t)), i =, 1, (3.1) where (X i, J i, D i ) is the JDT-process based on ε i. As observed in Section 2, ( S(t) = S E t (X i + J i + D i ) = S exp X i (t) + D i (t) 1 2 σε 2 i (τ)dτ ) κ i (t). (3.2) Let r i, r i is the interest rate of the maret which is in the state i, i =, 1. Let us consider the geometric telegraph process of the form B(t) = exp r εi (τ)dτ (3.3) as a numeraire. This model is incomplete: there are many equivalent ris-neutral measures. Due to simplicity of proposed model (3.2)-(3.3) one can describe the set M of such measures. These measures depend on two positive numbers: θ, θ 1 >. Consider c = λ θ, c 1 = λ 1 θ 1, h = 1 + θ /λ, h 1 = 1 + θ 1 /λ 1 and arbitrary σ, σ1. Consider the process Z i (t) = E t (Xi + Ji + Di ), t (as in (2.2)) using this set of parameters. We define measure P i by means of the density Z i (t), t. Notice that under this measure the driving process has intensities λ i = θ i, i =, 1 (see Theorem 2.3). 8

11 Proposition 3.1. Let probability measure P i be defined by means of the density Z i (t), t. The process B(t) 1 S(t) is a P i -martingale if and only if σ and σ 1 are as follows, σ = (r c h θ )/σ and σ 1 = (r 1 c 1 h 1 θ 1 )/σ 1, θ, θ 1 >. Proof. Indeed, where Z i (t)b(t) 1 S(t) = S exp{y i (t)} κ i (t), Y i (t) = X i (t) + Xi (t) + D i (t) + Di (t) 1 (σε 2 2 i (τ) + σε (τ)2 i )dτ r εi (τ)dτ and κ i (t) is defined as in (2.3) with h i = θ i (1 + h i )/λ i 1 instead of h i. Using Corollary 2.1 we see that Z i (t)b(t) 1 S(t) is the P i -martingale, if { c r + σ σ = θ h c 1 r 1 + σ 1 σ 1 = θ 1 h 1. To complete the model we can add new assets. Consider the maret of two assets which are driven by common Brownian motion w and counting Poisson processes N i : ds (m) (t) = S (m) (t )d(x (m) i As usual i =, 1 denotes the initial maret state. Denote (t) + J (m) i (t) + D (m) i (t)), m = 1, 2. (3.4) = σ (1) h (2) σ (2) h (1), 1 = σ (1) 1 h (2) 1 σ (2) 1 h (1) 1. Theorem 3.1. Both processes B(t) 1 S (m) (t), t, m = 1, 2 are P i -martingales if and only if the measure P i is defined by (2.2) with the following parameters: for =, 1 and σ = (r c (1) )h(2) (r c (2) )h(1) c = λ + (r c (1) i )σ (2) (r c (2) )σ(1), h = c /λ. If the prices of both risy assets are supplied with nonzero jumps, h, h 1, then and σ = α(1) α (2) β (1) β (2) c = λ β(1) α(2) β (2) β (1) β (2) 9 α(1),

12 where Proof. First notice α (m) = r c (m), β (m) h (m) = σ(m) h (m), m = 1, 2, =, 1. Z i (t)b(t) 1 S (m) (t) = S (i) E t exp(xi + Ji + Di ) exp( Y (t))e t (X i + J i + D i ) = exp X (t) + D (t) 1 2 σε (τ)2 i dτ κ (t) exp X(t) + D(t) Y (t) 1 2 σ 2 ε i (τ)dτ κ(t) = E t X + X + D + D Y + σ εi (τ)σ ε i (τ)dτ. It is a martingale if and only if (Theorem 2.2) { c (1) i + c i r i + σ (1) i σi = λ i (h (1) i + h i + h (1) i h i ) c (2) i + c i r i + σ (2) i σi = λ i (h (2) i + h i + h (2) i h i ) Now using the identities c i = λ i h i, i =, 1 it is easy to finish the proof. Now we are ready to obtain price of standard call option for completed maret. Let Z be a r.v. with normal distribution N (, σ 2 ). We denote ϕ(x, K, σ) = E[xe Z σ2 /2 K] + = xf ( ln(x/k) + σ2 /2 σ where F (x) is the distribution function of standard normal law:. ) KF ( ln(x/k) σ2 /2 ), (3.5) σ F (x) = 1 2π x e y2 /2 dy. Let the maret contains two risy assets (3.4). Consider the standard call option on the first asset with the claim ( S (1) (T ) K ) +. Therefore the call-price is c i = E i B(T ) 1 (S (1) i (T ) K) +, (3.6) where E i is the expectation with respect to the martingale measure P i which is constructed in Theorem 3.1. Under measure P the process w (t) = w(t) σ ε i (τ) dτ is the Brownian motion. Hence { T B(T ) 1 S (1) i (T ) = S (1) i () exp X (1) i (T ) + σ εi (τ)dw(τ) 1 T } σε 2 2 i (τ)dτ Y (T ) κ (1) i (T ) { T = S (1) i () exp X (1) i (T ) + σ εi (τ)dw (τ) + T σ εi (τ)σ ε i (τ)dτ 1 2 T } σε 2 i (τ)dτ Y (T ) κ (1) i (T ). 1

13 Then notice that c i r i + σ i σi = λ i h (1) i. Thus B(T ) 1 S (1) i (T ) = S (1) i X (1) i { () exp where λ i and it has the velocities c i = λ i h (1) i c i = n= T X (1) i (T ) + σ εi (τ)dw (τ) 1 2 T σε 2 i (τ)dτ } κ (1) i (T ), is the telegraph process which is driven by Poisson process with parameters. Therefore T f i (t, T, n)ϕ(x i (t, T, n), Ke r t r 1 (T t), σ 2 t + σ 2 1(T t))dt, i =, 1, (3.7) where x i (t, T, n) = S i ()κ i (n)e c t+ c 1 (T t) and κ i (2n) = (1 + h ) n (1 + h 1 ) n, i =, 1, κ (2n + 1) = (1 + h ) n+1 (1 + h 1 ) n, κ 1 (2n + 1) = (1 + h 1 ) n+1 (1 + h ) n, n =, 1, 2,... In particular, if h = h 1 = we can summarize in (3.7) applying (2.14)-(2.17): c i = T f i (t, T )ϕ(s e c t+ c 1 (T t), Ke r t r 1 (T t), σ 2 t + σ 2 1(T t))dt, i =, 1, (3.8) where f i (t, T ) are defined in (2.18) and (2.19) (cf. [5]). References [1] Beghin, L., Nieddu, L., Orsingher, E. (21). Probabilistic analysis of the telegrapher s process with drift by mean of relativistic transformations. J. Appl. Math. Stoch. Anal. 14, [2] Di Crescenzo, A. and Pellerey, F. (22). On prices evolutions based on geometric telegrapher s process. Appl. Stoch. Models Bus. Ind. 18, [3] Di Masi, G., Kabanov, Y. and Runggaldier, W. (1994). Mean-variance hedging of options on stocs with Marov volatilities. Theor. Prob. Appl. 39, [4] Goldstein, S. (1951). On diffusion by discontinuous movements and on telegraph equation. Quart. J. Mech. Appl. Math. 4, [5] Guo, X. (21). Information and option pricings. Quant. Finance [6] Jobert, A. and Rogers, L.C.G. (26). Option pricing with Marov-modulated dynamics. SIAM J. Control Optim., [7] Kac, M. (1974). A stochastic model related to the telegraph equation. Rocy Mountain J. Math. 4, [8] Karatzas, I. and Shreve, S. E. (1998). Methods of mathematical finance, vol. 39 of Applications of Mathematics. Springer-Verlag, New Yor. 11

14 [9] Mazza, C. and Rullière, D. (24). A lin between wave governed random motions and ruin processes. Insurance: Mathematics and Economics 35, [1] Ratanov, N. (25). Pricing options under telegraph processes, Rev. Econ. Ros [11] Ratanov, N. (27). A jump telegraph model for option pricing. Quant. Finance [12] Ratanov, N. (27). Jump telegraph models and financial marets with memory. J. Appl. Math. Stoch. Anal., vol. 27, Article ID 72326, 19 pages, (27) doi:1.1155/27/72326 [13] Ratanov, N. and Melniov, A. (28) On financial marets based on telegraph processes. Stochastics: An International Journal of Probability and Stochastic Processes 8, No. 2-3, [14] Zacs, S. (24). Generalized integrated telegraph processes and the distribution of related stopping times, J. Appl. Prob. 41,

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