Financial derivative pricing under stochastic market environments

Abstract

In this paper, we discuss the valuation of options and other derivatives under stochastic market environments. Empirical evidence on underlying asset prices strongly suggests that risk-free interest rates and asset price volatility are stochastically changed, depending on the macroeconomic market circumstances. The pricing of options under this condition is an important problem. In this paper, we explore especially two parts of this area; valuation of interest rate derivatives in the market with stochastic interest rate and option pricing in the market having two-state regimes. First, we provide a analytic valuation method for interest rate derivatives under the affine term structure model. As an example, we give the pricing formulas for a range accrual note and a spread range accrual note. Using these formulas, we provide several numerical implications. We confirm that the choice of model significantly affects the price and hedging ratio of range accrual notes, and that the price of range accrual notes is sensitive on the intensity of market jumps. Second, we develop a lattice method for exotic options under the regime-switching volatility model and apply it to a lookback option. We exhibit the value of lookback put calculated by constructing a pentanomial tree with reflecting barrier. In order to show the efficiency of our method, we compare the results with those from the partial differential equation method and from the Monte-Carlo simulation.