13. (Richardson Extrapolation)

The binomial method is first-order accurate and it is possible to expand the binomial price as an asymptotic expansion depending on the number of time steps n as follows:

Xn = BS + A n + 0(n−2)

X2n+1 = BS + A 2n+1 + 0(n−2)

where:

Xn = binomial price with n steps A = constant independent of n BS = exact Black–Scholes price (independent of n)

O(n−2) = big oh notation as discussed in Chapter 17.

We wish to remove the first-order terms in the above asymptotic expansions to get a new second-order approximation of the following form after doing some simple algebra:

(

1 − n 2n + 1

)−1 (

X2n+1 − n 2n + 1 Xn

)

BS + 0(n−2). (11.7)

Answer the following questions:

a) Design and implement the code to realise Richardson extrapolation.

b) Test the code on plain options with different values of the number of time steps. Do you notice erratic behaviour in convergence?

c) Is convergence monotone or do you notice sawtooth behaviour as when using the CRR method? A discussion of this topic is given in M¨uller (2009).

(Caveat: the author is not convinced that using Richardson extrapolation is a good approach for this class of problems.)