Binomial Methods
Paul Wilmott, Sam Howison, Jeff Dewynne · Cambridge University Press eBooks · 1995
Introduction Binomial methods for valuing options and other derivative securities arise from discrete random walk models of the underlying security. They rely only indirectly on the Black-Scholes analysis through the assumption of risk neutrality; their relation to the partial differential equation and inequality models described and derived earlier in this book becomes evident only when it is seen that binomial methods are particular cases of the explicit finite-difference method described in Chapter 8 (see Exercise 5). There are two main ideas underlying the binomial methods. The first of these is that the continuous random walk (2.1) may be modelled by a discrete random walk with the following properties: The asset price S changes only at the discrete times δ t , 2δ t , 3δ t , …, up to M δ t = T , the expiry date of the derivative security. We use δ t instead of dt to denote the small but non-infinitesimal time-step between movements in the asset price. If the asset price is S m at time m δ t then at time ( m + 1) δ t it may take one of only two possible values, u S m > S m or ∂ S m 0 and ∂ - 1 < 0, and that these two returns are the same for all time-steps. […]