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

Pricing Options under Telegraph Processes

In this paper we introduce a financial market model based on continuous time random motions with alternating constant velocities and jumps, which occur with velocity switches. Given that jump directions match velocity directions of the underlying random motion properly in relation to interest rates, in this setting will be free of arbitrage. Additionally, we suppose also the interest rate depending on the market state. The replicating strategies for options are constructed in detail, and closed form formulas for option prices are obtained.jump telegraph process, European option pricing, perfect hedging, selffinancing strategy, fundamental equation

On jump-diffusion processes with regime switching: martingale approach

We study jump-diffusion processes with parameters switching at random times. Being motivated by possible applications, we characterise equivalent martingale measures for these processes by means of the relative entropy. The minimal entropy approach is also developed. It is shown that in contrast to the case of L\'evy processes, for this model an Esscher transformation does not produce the minimal relative entropy.Comment: 23 pages, 2 figure

Quantil Hedging for telegraph markets and its applications to a pricing of equity-linked life insurance contracts

In this paper we develop a financial market model based on continuous time random motions with alternating constant velocities and with jumps occurrng when the velocity switches. If jump directions are in the certain correspondence with the velocity directions of the underlyig random motion with respect to the interest rate, the model is free of arbitrage and complete. Closed form formulas for the option prices and perfect hedging strategies are obtained.The quantile hedging strategies for options are constructed. This methodology is applied to the pricing and risk control of insurance instruments.************************************************************************************************************En este documento está desarrollado un modelo de mercado financiero basado en movimientos aleatorios con tiempo continuo, con velocidades constantes alternates y saltos cuando hay cambios en la velocidad. Si los saltos en la dirección tienen correspondencia con la dirección de la velocidad del comportamiento aleatorio subyacente, con respecto a la tasa interés, el modelo no presenta arbitraje y es completo. Se contruye en detalle las estrategias replicables para opciones y se obtiene una representación cerrada para el precio de las opciones.Las estrategias de cubrimiento quantile para opciones son construidas. Esta metodología es aplicada al control de riesgo y fijación de precios de instrumentos de seguros.jump telegraph model, perfect hedging, quantile hedging, pure endowment, equity-linked life insurance

Damped jump-telegraph processes

We study a one-dimensional Markov modulated random walk with jumps. It is assumed that amplitudes of jumps as well as a chosen velocity regime are random and depend on a time spent by the process at a previous state of the underlying Markov process. Equations for the distribution and equations for its moments are derived. We characterise the martingale distributions in terms of observable proportions between jump and velocity regimes

Martingale approach to optimal portfolio-consumption problems in Markov-modulated pure-jump models

We study optimal investment strategies that maximize expected utility from consumption and terminal wealth in a pure-jump asset price model with Markov-modulated (regime switching) jump-size distributions. We give sufficient conditions for existence of optimal policies and find closed-form expressions for the optimal value function for agents with logarithmic and fractional power (CRRA) utility in the case of two-state Markov chains. The main tools are convex duality techniques, stochastic calculus for pure-jump processes and explicit formulae for the moments of telegraph processes with Markov-modulated random jumps

Option Pricing Model Based on Telegraph Processes with Jumps

In this paper we overcome a lacks of Black-Scholes model, i.e. the infinite propagation velocity, the infinitely large asset prices etc. The proposed model is based on the telegraph process with jumps. The option price formula is derived.Telegraph Processes, option pricing

Jump-telegraph models for the short rate: pricing and convexity adjustments of zero coupon bonds

In this article, we consider a Markov-modulated model with jumps for short rate dynamics. We obtain closed formulas for the term structure and forward rates using the properties of the jump-telegraph process and the expectation hypothesis. The results are compared with the numerical solution of the corresponding partial differential equation

On Financial Markets Based on Telegraph Processes

The paper develops a new class of financial market models. These models are based on generalized telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black-Scholes fundamental differential equation is derived, but, in contrast with the Black-Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging.Comment: To appear in a Special Volume of Stochastics: An International Journal of Probability and Stochastic Processes (http://www.informaworld.com/openurl?genre=journal%26issn=1744-2508) edited by N.H. Bingham and I.V. Evstigneev which will be reprinted as Volume 57 of the IMS Lecture Notes Monograph Series (http://imstat.org/publications/lecnotes.htm

Ornstein-Uhlenbeck processes of bounded variation

Ornstein-Uhlenbeck process of bounded variation is introduced as a solution of an analogue of the Langevin equation with an integrated telegraph process replacing a Brownian motion. There is an interval $I$ such that the process starting from the internal point of $I$ always remains within $I$. Starting outside, this process a. s. reaches this interval in a finite time. The distribution of the time for which the process falls into this interval is obtained explicitly. The certain formulae for the mean and the variance of this process are obtained on the basis of the joint distribution of the telegraph process and its integrated copy. Under Kac's rescaling, the limit process is identified as the classical Ornstein-Uhlenbeck process.Comment: 23 page