Mixture dynamics and option pricing: a regime switching model

In this work I extend the mixture model proposed by Brigo and Mercurio (2000, 2001) as an alternative of the well-known Black-Scholes asset price model, by using a regime-switching diffusion framework

First passage of a Markov additive process and generalized Jordan chains

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique, which can be used to derive various further identities.Lévy processes, Fluctuation theory, Markov Additive Processes

Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations

Positive results are derived concerning the long time dynamics of numerical simulations of stochastic differential equation systems with Markovian switching. Euler-Maruyama discretizations are shown to capture almost sure and momente xponential stability for all sufficiently small timesteps under appropriate conditions

First passage process of a Markov additive process, with applications to reflection problems

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. Importantly, our result also provides us with a technique, which can be used to derive various further identities. We then proceed to show how to compute the stationary distribution associated with a one-sided reflected (at zero) MAP for both the spectrally positive and spectrally negative cases as well as for the two sided reflected Markov-modulated Brownian motion; these results can be interpreted in terms of queues with MAP input.Comment: 16 page

The optimal hedging in a semi-Markov modulated market

This paper includes an original self contained proof of well-posedness of an initial-boundary value problem involving a non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. We call this market model a semi-Markov modulated market. Although a wellposedness result of that problem is available in the literature, but this recent paper has a different proof. Here the existence of solution is established without invoking mild solution technique. We study the well-posedness of the initial-boundary value problem via a Volterra integral equation of second kind. The method of conditioning on stopping times was used only for showing uniqueness. Furthermore, in the present study we find an integral representation of the PDE problem which enables us to find a robust numerical scheme to compute derivative of the solution. This study paves for addressing many other interesting problems involving this new set of PDEs. Some derivations of external cash flow corresponding to an optimal strategy are presented. These quantities are extremely important when dealing with an incomplete market. Apart from these, the risk measures for discrete trading are formulated which may be of interest to the practitioners.Comment: 23 pages, 4 figure

Monotonicity of the value function for a two-dimensional optimal stopping problem

We consider a pair $(X,Y)$ of stochastic processes satisfying the equation $dX=a(X)Y\,dB$ driven by a Brownian motion and study the monotonicity and continuity in $y$ of the value function $v(x,y)=\sup_{\tau}E_{x,y}[e^{-q\tau}g(X_{\tau})]$, where the supremum is taken over stopping times with respect to the filtration generated by $(X,Y)$. Our results can successfully be applied to pricing American options where $X$ is the discounted price of an asset while $Y$ is given by a stochastic volatility model such as those proposed by Heston or Hull and White. The main method of proof is based on time-change and coupling.Comment: Published in at http://dx.doi.org/10.1214/13-AAP956 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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. Such a model captures well the stock price dynamics under periodic financial cycles. The distribution of this process is described in detail.We also provide a closed form of the structure of risk-neutral measures. This incomplete model can be completed by adding another asset based on the same sources of randomness. For completed market model we obtain explicit formulae for call prices. © 2010, Brazilian Statistical Association. All rights reserved