A new finite difference method for pricing and hedging fixed income derivatives: Comparative analysis and the case of an Asian option

Abstract

We propose a second order accurate numerical finite difference method to replace the classical schemes used to solve PDEs in financial engineering. We name it Modified Fully Implicit method. The motivation for doing so stems from the accuracy loss while trying to stabilize the solution via the up-wind scheme in the convective term as well as the fact that spurious oscillations solutions occur when volatilities are low (this is actually the range that is commonly observed in interest rate markets). Unlike the classical schemes, our method covers the whole spectrum of volatilities in the interest rate dynamics.We obtain analytical and numerical results for pricing and hedging a zero-coupon bond and an Asian interest rate option. In the case of the Asian option, we compare the realistic discrete compounding interest rate scheme (associated with the Modified Fully Implicit method) with the continuous compounding scheme (often exploited in the literature), obtaining relative discrepancies between prices exceeding 50%. This indicates that the former scheme is more appropriate then the latter to price more complicate derivatives than straight bonds.