Fourier Transform Methods for Regime-Switching Jump-Diffusions and the Pricing of Forward Starting Options

In this paper we consider a jump-diffusion dynamic whose parameters are driven by a continuous time and stationary Markov Chain on a finite state space as a model for the underlying of European contingent claims. For this class of processes we firstly outline the Fourier transform method both in log-price and log-strike to efficiently calculate the value of various types of options and as a concrete example of application, we present some numerical results within a two-state regime switching version of the Merton jump-diffusion model. Then we develop a closed-form solution to the problem of pricing a Forward Starting Option and use this result to approximate the value of such a derivative in a general stochastic volatility framework.Comment: 25 pages, 6 figure

Mixture dynamics and regime switching diffusions with application to option pricing

In this paper we present a class of regime switching diffusion models described by a pair (X(t),Y(t)) ∈ Rn × S, S = {1, 2, . . . , N}, Y(t) being a Markov chain, for which the marginal probability of the diffusive component X(t) is a given mixture. Our main motivation is to extend to a multivariate setting the class of mixture models proposed by Brigo and Mercurio in a series of papers. Furthermore, a simple algorithm is available for simulating paths through a thinning mechanism. The application to option pricing is considered by proposing a mixture version for theMargrabe Option formula and the Heston stochastic volatility formula for a plain vanilla

On the regularity of American options with regime-switching uncertainty

We study the regularity of the stochastic representation of the solution of a class of initial-boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transition rates are known to lie within level-dependent compact sets.Comment: 22 pages, to appear in Stochastic Processes and their Application

Interest rate models with Markov chains

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A high order finite element scheme for pricing options under regime switching jump diffusion processes

This paper considers the numerical pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. In an incomplete market structure and using the no-arbitrage pricing principle, the option pricing problem under the jump modulated regime switching process is formulated as a set of coupled partial integro-differential equations describing different states of a Markov chain. We develop efficient numerical algorithms to approximate the spatial terms of the option pricing equations using linear and quadratic basis polynomial approximations and solve the resulting initial value problem using exponential time integration. Various numerical examples are given to demonstrate the superiority of our computational scheme with higher level of accuracy and faster convergence compared to existing methods for pricing options under the regime switching model

Modelling FX smile : from stochastic volatility to skewness

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Multifrequency Jump-Diffusions: An Equilibrium Approach

This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices, which provides a structural alternative to traditional reduced-form specifications with exogenous discontinuities. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate shocks of heterogeneous durations in consumption and dividends while keeping constant the number of parameters. Equilibrium valuation creates an endogenous relation between a shock's persistence and the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.