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.
Deep Dive into Option Pricing Model Based on a Markov-modulated Diffusion with Jumps.
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.
Beginning with the works of Mandelbrot (1963), Mandelbrot and Taylor (1967), and Clark (1973), it is commonly accepted that the dynamics of asset returns cannot be described by geometric Brownian motion with constant parameters of drift and volatility. A lot of sophisticated constructions have been exploited to capture the features that help to express the reality better than Black-Scholes-Merton model. Merton (1976) which have incorporated jump diffusion model for the asset price was the first. Later on the constructions with random drift and random volatility parameters appeared. A popular approach is to use Lévy processes with stationary independent increments. However, this theoretical behavior does not match empirical observations.
Another approach utilizes markovian dependence on the past and the technique of Markov random processes (see Elliott and van der Hoek (1997)). We deal mainly with this direction. More precisely, the model is based on a standard Brownian motion w = w(t), t ≥ 0 and on a Markov process ε(t), t ≥ 0 with two states 0, 1 and transition probability intensities λ 0 and λ 1 .
Let us define processes c ε(t) , σ ε(t) and r ε(t) , t ≥ 0, where c 0 ≥ c 1 , r 0 , r 1 > 0. Then, we introduce T (t) = t 0 c ε(τ ) dτ , D(t) = t 0 σ ε(τ ) dw(τ ) and a pure jump process J = J (t) with alternating jumps of sizes h 0 and h 1 , h 0 , h 1 > -1.
The continuous time random motion T (t) = t 0 c ε(τ ) dτ, t ≥ 0 with alternating velocities is known as telegraph process. This type of processes have been used before in various probabilistic aspects (see, for instance, Goldstein (1951), Kac (1974) and Zacks (2004)). These processes have been exploited for stochastic volatility modeling (Di Masi et al (1994)), as well as for obtaining a “telegraph analog” of the Black-Scholes model (Di Crescenzo and Pellerey (2002)). The option pricing models based on continuous-time random walks are widely presented in the physics literature (see Masoliver et al (2006) or Montero (2008)). Recently the telegraph processes was applied to actuarial problems, Mazza and Rullière (2004). Markov-modulated diffusion process D(t) = t 0 σ ε(τ ) dw(τ ) was exploited for financial market modeling (see Guo (2001), Jobert and Rogers (2006)), as well as in insurance (see Bäuerle and Kötter (2007)) or in theory of queueing networks (see Ren and Kobayashi (1998)).
This paper deals with the market model which presumes the evolution of risky asset S(t) is given by the stochastic exponential of the sum X = T (t) + D(t) + J (t). The bond price is the usual exponential of the process Y = Y(t) = t 0 r ε(τ ) dτ, t ≥ 0 with alternating interest rates r 0 and r 1 .
This model generalizes classic Black-Scholes-Merton model based on geometric Brownian motion (c Black and Scholes (1973), Merton (1973). Other particular versions of this model was also discussed before: Merton (1976), Cox and Ross (1976); Jobert and Rogers (2006).
The jump-telegraph model, as well as Black-Scholes and Merton model, is free of arbitrage opportunities, and it is complete. Moreover it permits explicit standard option pricing formulae similar to the classic Black-Scholes formula. Under suitable rescaling this model converges to the Black-Scholes (see Ratanov (2007)). First calibration results of the parameters of the telegraph model have been presented in De Gregorio and Iacus (2007). These estimations have been based on the data of Dow-Jones industrial average (July 1971 -Aug 1974). However, a presence of jumps and/or diffusion components has not been estimated. Nevertheless, an implied volatility with respect to a moneyness variable in stochastic volatility models of the Ornstein-Uhlenbeck type (see Nicolato and Venardos (2003)) looks very similar to the volatility smile in jump telegraph model (see Ratanov (2007b)).
In this paper we extend the jump-telegraph market model, presented in Ratanov (2007), by adding the diffusion component with alternating volatility coefficient.
The jump-telegraph model equipped with the diffusion term becomes more realistic. Indeed, the alternating velocities of the telegraph process describe long-term financial trends, and the diffusion summand introduces an uncertainty of current prices. This uncertainty may has different volatilities in the bearish and in the bullish trends (σ 0 = σ 1 ).
The paper is organized as follows: in Section 2 we present the detailed definitions and the description of underlying processes and their distributions. The explicit construction of a measure change is given by the Girsanov theorem for jump telegraph-diffusion processes.
In Section 3 we describe the set of risk-neutral measures for the incomplete jump telegraph-diffusion model. Also we consider a completion of the model by adding another asset driven by the same sources of randomness. For the completed market model we obtain explicit option pricing formulae of the standard call option. These formulae are based on a mix of Black-Scholes function
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