Jump Telegraph Processes and Financial Markets with Memory

The paper develops a new class of financial market models. These models are based on generalized telegraph processes with alternating velocities and jumps occurring at switching velocities. The model under consideration is arbitrage-free and complete if the directions of jumps in stock prices are in a certain correspondence with their velocity and with the behaviour of the interest rate. A risk-neutral measure and arbitrage-free formulae for a standard call option are constructed. This model has some features of models with memory, but it is more simple.


Introduction
It is widely recognized that the dynamics of asset returns cannot be adequately described by geometric Brownian motion with constant volatility.Due to the market efficiency, alternative models are based on random processes with independent increments (Brownian motion, jump diffusions, and the variance gamma process).The development of non-semimartingale models is focused mainly on accounting for the dependence of asset prices on the past (long-term memory processes, fractional Brownian motion, etc.).However, till now there is still no commonly accepted theory on this topic, nor adequate uses of existing theoretical results in practice (see, e.g., [1]).
On the other hand, models which are based on pure jump processes with independent increments recently were widely proposed (see, e.g., [2][3][4]).Carr et al. [2] empirically show that the diffusion component could be ignored, if the pure jump process allows infinite activity.This means that there are infinitely many small jumps which asymptotically model a diffusion component.Usually these models are incomplete.
Let h ± ∈ (−1,∞) and c ± be real numbers.Consider the (right continuous) processes (2. 2) The subscripts ± indicate the initial state of the processes.
Introducing the jumping times τ 1 ,τ 2 ,... of the processes N ± and setting τ 0 = 0, we have the following representation (e.g., for the subscript +): h σ+(τj −) . (2. 3) The processes X ± = (X ± (t), t ≥ 0) are usually referred to as (integrated) telegraph process (see Goldstein [5] and Kac [12,6]).The processes J ± = (J ± (t), t ≥ 0) are pure jump processes with alternating jump sizes h ± .Let us introduce the standard telegraph and jump processes associated with c ± = ±1 and h ± = ±1: (2.4) Proposition 2.1.The processes X ± and J ± are linearly connected with X 0 ± and J 0 ± : where Proof.We only consider the case related to the subscript +.The other case is quite similar.We have For c ± = ±1, we find that On the other hand, ( The next theorem could be considered as a version of the Doob-Meyer decomposition for telegraph processes.
Theorem 2.2.The jump telegraph processes Z + := X + + J + and Z − := X − + J − are martingales if and only if c The proof is based on direct calculations of the conditional expectations E(X ± (t) + J ± (t) | F ± s ) (see Remark 2.10 below).We can obtain the exact distribution of jump telegraph processes Z ± = X ± + J ± in terms of generalized probability densities p ± (x,t), which are defined by for any Borelian set Δ.By generalized densities we mean that the distributions of X ± + J ± are made up of an absolutely continuous part (i.e., a genuine density) and a discrete part.
Proof.First notice that from the properties of counting Poisson process (see, e.g., [13]) for t 2 > t 1 ≥ 0 it follows that where Z ± are copies of the processes Z ± which are independent of Applying (2.10) with the choice t 1 = Δt, t 2 = t + Δt we have which immediately implies the first equation of (2.9).The second equation can be obtained similarly.
Conditioning on the number of switches one can show that system (2.9) has the following solution: where Here q (0) ± (x,t) = δ(x − c ± t) and, setting θ(x,t) = 1 {c−t<x<c+t} , for n ≥ 1 and for n ≥ 0 (2.17) Alternatively one can obtain formulae (2.14)-(2.17)by applying the results of Zacks [8].There, the probability densities p ± (x,t) are expressed in terms of Poisson and Erlang densities.
Formulae (2.14)-(2.17)give the following rules of changes in the intensities λ ± : if λ + is changed to λ + and λ − is changed to λ − , the probability densities p ± will be changed to where Remark 2.5.In particular case, if where 2 and I 1 (z) = I 0 (z) are usual modified Bessel functions.Compare with [14].
form a solution of the system ) Applying (2.9) immediately yields (2.22).
We apply (2.22) to deduce formulae for the mean value and the variance of the jump telegraph process:

.24)
Seeking simplicity it will be done only in the symmetric case.
Remark 2.9.The function m + can be directly evaluated.Indeed, since we have On the other hand, similarly, E(h σ+(τ) ) = B + b e −2λτ and since the increments of N + are independent, we have (2.44)This confirms (2.25).
Remark 2.10.For the process X + + J + being a martingale, it is necessary (but not sufficient) for its expectation to be constant in the course of time.This condition implies, in view of (2.25), that A + λB = a + λb = 0 or, equivalently, that c + + λh + = c − + λh − = 0. Hence, we retrieve the conditions stated in Theorem 2.2 in the symmetric case.
It can be checked that these conditions are sufficient by computing the conditional expectation . Indeed, we have (2.45) Using the Markov property of the processes σ + and N + which can be stated, for τ ≥ 0, as where σ ± and N ± are copies of the processes σ ± and N ± which are independent of F + t1 , we obtain (2.47) The prime superscript in E above means that the expectation applies to the functionals of the process N .So, we have E(X + (t 2 ) + J + (t 2 ) | F t1 ) = X + (t 1 ) + J + (t 1 ) if and only if A + λB = a + λb = 0.

Market model based on jump telegraph processes
We introduce the following market model.The price of a risky asset S(t) follows the equation and the process (S(t), t ≥ 0) is right-continuous.Here X s = (X s (t), t ≥ 0) is a telegraph process with velocity values c ± , and J s = (J s (t), t ≥ 0) is a pure jump process with jump values h ± > −1, which are defined in (2.2), in a complete probability space (Ω,Ᏺ s ,P s ).
As usual (see, e.g., [15]), we define a new probability measure P * s , which is equivalent to the original measure P s .Let be the density of the new measure with respect to measure P s .Here the jump values h * ± are defined as follows: According to Theorem 2.2, for arbitrary c * ± ,c * ± < λ ± , the processes X * s + J * s and (Ᏹ t (X * s + J * s ), t ≥ 0) are martingales.The following theorem describes changes in the distributions with respect to the new measure (see the proof in [16]).
Theorem 3.1 (Girsanov's theorem).Under the probability measure ) is a telegraph process with states (c + ,λ * + ) and (c − ,λ * − ).We now assume that the parameters of the model (3.1)-(3.6)satisfy the conditions Using Theorem 2.2, under such conditions we can find a unique martingale measure in the framework of the market (3.1)- (3.6).Recall that the measure P * s , which is equivalent to P s , is a martingale measure if the process (B(t) −1 S(t)) t≥0 is a P * s -martingale (see, e.g., [15]).As before, s = ± indicates the initial market state.
Theorem 3.2.The measure P * s , defined by the density Z s (t), t ≥ 0, is the martingale measure if and only if the velocity values c * ± satisfy where Moreover, under the probability measure P * s , the process N s is a Poisson process with alternating intensities λ * ± .Proof.According to Theorem 3.1, the process X s − Y s is a telegraph process (with respect to P * s ) with the velocity values c ± − r ± and the alternating intensities , and the theorem is proved.
It is interesting to analyze the problem of convergence to the famous Black-Scholes model.First of all, note that the symmetric telegraph process associated with the parameters λ + = λ − := λ, c + = a, c − = −a converges to the standard Brownian motion (w(t), t ≥ 0) if a,λ → ∞, such that a 2 /λ → 1 (see [12] or [19]).
The following theorem provides a similar connection under respective scaling between market model driven by geometric telegraph processes (with jumps) and geometric Brownian motion.
Suppose that a,λ − ,λ + → ∞, and h − ,h + → 0 such that Although the jump telegraph process is of finite variation and of finite activity around the origin, this scaling agrees with empirical observations of Carr et al. [2].
Let p 0 ± be the (generalized) probability densities of the standard telegraph processes X 0 ± , which are controlled by the Poisson process with alternating intensities λ ± = λ ± (1 + h ± ) z , and set (3.25) Note that under scaling a 2 /λ ± → σ 2 we have the convergence to normal distribution ᏺ(0,tσ 2 ): and one can derive from (3.20)In this case, the volatility of the limit is v 2 = lima 2 /λ ± .Hence in model (3.1)-(3.6)values a/ λ ± can be interpreted as "telegraph" components of the volatility, and λ ± ln(1 + h ± ) are the volatilities engendered by jumps.

Memory effects and historical volatility in the framework of jump telegraph market model
In this section, we explain how the simple model ( For models without memory the historical volatility coincides with the implied volatility.For example, the Black-Scholes model has HV(t) ≡ σ.It is interesting to describe the behavior of HV(t) in our model (3.1).