Option pricing model based on a Markov-modulated diffusion with jumps

Abstract

The paper proposes a class of financial market models which are based on inhomogeneous telegraph processes and jump diffusions with alternating volatilities. It is assumed that the jumps occur when the tendencies and volatilities are switching. We argue that such a model captures well the stock price dynamics under periodic financial cycles. The distribution of this process is described in detail. For this model we obtain the structure of the set of martingale measures. This incomplete model can be completed by adding another asset based on the same sources of randomness. Explicit closed-form formulae for prices of the standard European options are obtained for the completed market model. 1. c 0 = c 1 , σ 0 = σ 1 = 0, h 0 = h 1 = 0: Merton model, Merton (1976), Cox and Ross (1976); 2. c 0 = c 1 , σ 0 = σ 1 = 0, h 0 = h 1 : jump-telegraph model, Ratanov (2007); 3. c 0 = c 1 , σ 0 = σ 1 , h 0 = h 1 = 0: Markov-modulated dynamics, Guo (2001), Jobert and Rogers (2006).T i (t) = T i (t; c 0 , c 1 ) = t 0 c ε i (τ ) dτ, J i (t) = J i (t; h 0 , h 1 ) = t 0

Double Telegraph Processes and Complete Market Models

“…By Proposition 7.1, this model of the one risky asset is not complete. Nevertheless, the model can be completed, similarly as for the case of the telegraph-diffusion model, which has been studied before, see [16].…”

Self-exciting piecewise linear processes

“…Telegraph-like processes have multiple applications including the applications to financial market modelling, see Di Mazi et al (1994), and then, Ratanov (1999); Di Crescenzo and Pellerey (2002). Nowadays, these applications became the theory of Markov-modulated market models based on telegraph processes with alternating velocities, see e. g. Ratanov (2007Ratanov ( , 2010; López and Ratanov (2014) (see also the survey in Kolesnik and Ratanov, 2013).…”

Option Pricing Under Jump-Diffusion Processes with Regime Switching

“…In general, this model has infinitely many equivalent martingale measures, which makes the market incomplete. The model can be completed by adding other assets; for the jump-diffusion model see Runggaldier (2003), and for the telegraph-jump-diffusion model (hidden Markov model) with constant parameters see Ratanov (2010).…”

“…If d = 2, the market model of asset pricing (with additive jumps superimposed on the diffusion) has been studied before in Ratanov (2010).…”