Jump Telegraph Processes and Financial Markets with Memory

Abstract

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.

Kac’s rescaling for jump-telegraph processes

“…A sample path of X = T + J. Ratanov (2007b) and Di Crescenzo and Martinucci (2011) (telegraph processes with jumps), Stadje and Zacks (2004) (the case of random velocities), Zacks (2004) and Bshouty et al (in press) (the processes with general distributions of inter-arrival times τ n+1 − τ n ).…”

Telegraph Processes with Random Jumps and Complete Market Models

“…The explicit formulae for HV i (t) are rather cumbersome, even if the case of constant and deterministic velocities and jumps. Nevertheless, it is easy to compute the limits of HV i (t) as t → 0 and as t → ∞: , i = 0, 1, see (4.5)-(4.6) in [22]. Here the jump-telegraph process X is defined with the constant velocities c 0 , c 1 , c 0 > c 1 and with the constant jumps h 0 , h 1 > −1; λ = (λ 0 + λ 1 )/2, B = (h 0 + h 1 )/2 and c = (c 0 − c 1 )/2; the subscript i = ε(0) indicates the initial market state.…”

“…where b = (h 0 − h 1 )/2, γ i = −2c(c/λ + (−1) i h i ), i = 0, 1, see formula (4.2) in [22]. In particular, if in this symmetric case the jumps are also symmetric, h 0 = −h 1 = h, and X is the martingale, c + λ h = 0, then B = 0, c + λ b = 0 and γ 0 = γ 1 = 0.…”

Generalized Telegraph Process with Random Jumps

“…Moreover, a geometric telegraph process was proposed by Di Crescenzo and Pellerey [9] as a model to describe the dynamics of the price of risky assets. Aiming to refine such processes and looking for a model which is free of arbitrage and is complete, the jump-telegraph process has been recently introduced and studied by Ratanov [20], [21], and [22] and Ratanov and Melnikov [23]. This process includes the occurrence of a deterministic jump along the alternating direction at each velocity reversal.…”