9. (The Leisen–Reimer Method)

It is well known that convergence of the solution of the binomial method to the true solution is not monotone in general and it tends to oscillate around the exact value.

Decreasing the step size (or equivalently, increasing the number of time steps NT) does not necessarily give better accuracy when using the CRR method, for example.

We consider a method to price one-factor options (Leisen and Reimer, 1996). The method involves the computation of the up and down parameters u and d in such a way that the tree centres around the strike price. The method is an improvement on other methods such as CRR because of the less jagged convergence properties. The up and down parameters are defined by (Haug, 2007, pp. 290–292):
u = ebΔt h(d1)
h(d2)
, d = ebΔt − pu 1 − p , p = h(d2) (11.6)
where:
b = cost-of-carry Δt = T∕NT d1 = log(S∕K) + (b + ????
2∕2)T ????
√
T d2 = d1 − ????
√
T.
Finally, h = h(x) is a specially chosen function, for example the Preitzer–Pratt inversion formula. Details can be found in Haug (2007).