Methods for Pricing American Options under Regime Switching

Abstract

Abstract. We analyze a number of techniques for pricing American options under a regime switching stochastic process. The techniques analyzed include both explicit and implicit discretizations with the focus being on methods which are unconditionally stable. In the case of implicit methods we also compare a number of iterative procedures for solving the associated nonlinear algebraic equations. Numerical tests indicate that a fixed point policy iteration, coupled with a direct control formulation, is a reliable general purpose method. Finally, we remark that we formulate the American problem as an abstract optimal control problem; hence our results are applicable to more general problems as well. Key words. regime switching, American options, iterative methods AMS subject classifications. 65N06, 93C20DOI. 10.1137/1108209201. Introduction. The standard approach to valuation of contingent claims (also known as derivatives) is to specify a stochastic process for the underlying asset and then construct a dynamic, self-financing hedging portfolio to minimize risk. The initial cost of constructing the portfolio is then considered to be the fair value of the contingent claim. This has been used with great success in the case of stochastic processes having constant volatility in the case of both European and (the more difficult) American options.However, it is well known that a financial model which follows a stochastic process having constant volatility is not consistent with market prices. Recent research has shown that models based on stochastic volatility, jump diffusion, and regime switching processes produce better fits to market data. A nonexhaustive list of regime switching applications includes insurance [22] [3], and interest rate dynamics [27]. Regime switching models are intuitively appealing, and computationally inexpensive compared to a stochastic volatility jump diffusion model.In this paper we study numerical techniques for the solution of American option contracts under regime switching. While our examples focus on problems with constant properties in each regime, the numerical methods developed can easily be applied to cases where the properties in each regime are more complex. An example would be the use of price-dependent regime switching (i.e., default hazard rates) in convertible bond pricing [4]. A number of different methods has been proposed for handling American options under regime switching models. Semianalytic approaches have been

A new efficient numerical method for solving American option under regime switching model

“…Finally, in the last case we assign toũ n li,j = 0 due to condition (18). From the properties of the model for any regime i one gets…”

“…Using the absence of arbitrage principle, the initial cost of constructing the portfolio, typically given by a partial differential equation (PDE), is then considered to be the fair value of the derivative, [18].…”

“…Lattice methods [19,26] are popular for practitioners because they are easy to implement, but they have the drawback of the absence of numerical analysis and subsequent unreliability, because the lack of numerical analysis 35 may waste the best model. The penalty method [18,22,23,34] uses a coupling of the penalty term and the regime coupling terms. Both, the lattice and penalty methods do not calculate the optimal stopping boundary that has interest from the practitioners point of view.…”

Finite Difference Methods for nonlinear American Option Pricing models: Numerical Analysis and Computing

“…Using the absence of arbitrage principle, the initial cost of constructing the portfolio, typically given by a partial differential equation (PDE), is then considered to be the fair value of the derivative, [57].…”

“…Lattice methods [87,114] are popular for practitioners because they are easy to implement, but they have the drawback of the absence of numerical analysis and subsequent unreliability, because the lack of numerical analysis may waste the best model. The penalty method [57,70,71,116] uses a coupling of the penalty term and the regime coupling terms. Both, the lattice and penalty methods do not calculate the optimal stopping boundary that has interest from the practitioners point of view.…”