12. (Sawtooth Phenomenon and Smoothing)

One of the issues with some implementations of the binomial method is that the approximate solution oscillates around the true option price as the number of time steps increases. This nonmonotone behaviour is caused by the position of the nodes in relation to the strike price. This is the so-called sawtooth pattern. The solution is to peg the strike price to be at the centre of the final nodes for even values of the number of mesh points. Improved monotonic convergence can be achieved by using the Leisen–Reimer and modified CRR methods.

Answer the following questions:

a) Create a C++ algorithm to show the sawtooth phenomenon: the output is an array of option prices and the input is a sequence of time steps N1 < N2 < ⋯ < Np wherep ≥ 2.

b) Determine how to code the algorithm in step

a) using the Level 1 data structures and Level 2 mechanisms that we introduced in this chapter. All p option prices use the same underlying, thus in the interest of efficiency you might like to consider how to optimise this procedure.

c) Do you see possibilities to improve speedup by using multithreading and parallel programming patterns? Hint: Separate data structures for assets and options as discussed in Section 11.7 to make parallelisation easier. We discuss these issues in more detail in Chapters 28, 29 and 30.

d) Test the algorithm using CRR, modified CRR and Leisen–Reimer methods. What conclusions can you draw?