Portal de Periódicos da CAPES

Pricing American options written on two underlying assets

2013; Taylor & Francis; Volume: 14; Issue: 3 Linguagem: Inglês

10.1080/14697688.2013.810811

1469-7696

Carl Chiarella, Jonathan Ziveyi,

Insurance, Mortality, Demography, Risk Management

Abstract This paper extends the integral transform approach of McKean [Ind. Manage. Rev., 1965, 6, 32–39] and Chiarella and Ziogas [J. Econ. Dyn. Control, 2005, 29, 229–263] to the pricing of American options written on more than one underlying asset under the Black and Scholes [J. Polit. Econ., 1973, 81, 637–659] framework. A bivariate transition density function of the two underlying stochastic processes is derived by solving the associated backward Kolmogorov partial differential equation. Fourier transform techniques are used to transform the partial differential equation to a corresponding ordinary differential equation whose solution can be readily found by using the integrating factor method. An integral expression of the American option written on any two assets is then obtained by applying Duhamel’s principle. A numerical algorithm for calculating American spread call option prices is given as an example, with the corresponding early exercise boundaries approximated by linear functions. Numerical results are presented and comparisons made with other alternative approaches. Keywords: American optionsFourier transformMultiple underlying assetsJEL Classification: C63G13 Notes This is an option to exchange one asset for another asset. This is the general pay-off function for any option contract which is not path dependent written on two underlying assets. By path dependency, we mean options with pay-off functions that depend on the entire history of the underlying asset dynamics such as barrier, lookback and Asian options among others. We will provide the spread option example in this paper. This system of equation is obtained after applying a Cholesky decomposition where and are correlated Wiener processes with correlation . The early exercise boundaries are pay-off specific. For example, if we are dealing with the American spread call option case, Broadie and Detemple (Citation1997) show that the early exercise boundary point above which the call option can be exercised is represented as . By making use of the value-matching condition, we end up with the equation . Sometimes called the Green’s function. Here and are the values of and at maturity, respectively. Details about Duhamel’s principle have can found in Appendix A: See Davis and Rabinowitz (Citation2007) on how weights of the Simpson’s Rule are generated. The subscript in the two functions and represents the number of iterations required for convergence of the iterative process at time step . All the source codes have been run on the UTS HPC Linux Cluster with a graphics user interface consisting of eight nodes running Red Hat Enterprise Linux 4.0 (64 bit) with GHz 4MB cache Xeon 5160 dual core Processors, 8GB 667 MHz DDR2-RAM. We make use of the following key result for complex functions (See for example Abramowitz and Stegun (Citation1964)). Let p and q be complex variables independent of the integration variable , with Re. Also let n be a positive integer. Then, the integral of the exponential quadratic function with respect to is given by (54)