Construction of consistent discrete and continuous stochastic models for multiple assets with application to option valuation

Monte Carlo, discrete stochastic, and stochastic differential equation models are constructed from first principles for multiple assets. The different stochastic models are shown to be consistent in the estimation of mutual fund values. The models are applied to the calculation of European call option prices. It is shown that option prices are insensitive to the form of the stochastic model’s diffusion term. In addition, a general nn-dimensional Black–Scholes partial differential equation is derived for option prices. Computational examples illustrate that the Black–Scholes partial differential equation and the stochastic differential equation models are consistent in estimating option prices.