Pricing a zero-coupon bond when the term structure is non-affine using numerical methods.

A bond is a certificate of debt by which the bond writer promises to pay the bond holder ran agreed sequence of interest payments during a specified length of time and refund the loan at an agreed maturity time. The simplest form of bonds is the zero-coupon bond. The zero-coupon bond is issued at a price lower than its face value. The price of a zero-coupon bond at any time t∈(0,T)is represented by a function P(t,r(t),T),where r(t) is the short-term interest rate, and Tis the maturity time. Two approaches are used in this study to derive the pricing function P(t,r(t),T)namely, the risk-neutral approach, and the partial differential equation approach. An explicit formula forth e pricing function P(t,r(t),T) is easy to derive when the short-term interest rate is affine (linear). However, this is not the case if r(t)is none-affine (non-linear). In this study, the pricing function P(t,r(t),T) of a zero-coupon bond in which the short-term interest rate r(t) is non-linear is approximated using two types of numerical methods, Monte Carlo methods, and finite difference methods.Both classical Monte Carlo and antithetic variate are applied under the martingale approach. Likewise, the backward in time,central in space (BTCS) implicit, and Crack-Nicolson finite difference methods are applied in approximating the solution of the associated parabolic partial differential equation that governs the pricing function P(t,r(t),T). The results obtained using the two sets of numerical methods indicate that both approaches yield reasonably accurate approximations of the pricing function under the assumption that the short-term interest rate is non-affine.