1 Trinomial Trees and Lattices

Pretty much everything parallels the Binomial Tree from a conceptual sense. If the stock price at discrete time step k is denoted by S_k. In this case, the process S_k is specified as

with probability p_u, with probability p_d, with probability 1 p_u p_d.

Just as with the binomial tree whose leaves can be combined to form the binomial lattice, you can get a trinomial lattice from a trinomial tree. To ensure the martingale property

E{S_k+1 | S_k} = R S_k,

where, as with the Binomial Tree. We need to assign^[1]

Just as with the binomial lattice. you can use (k,i) to denote an arbitrary vertex on the trinomial lattice. The vertex (k,i) is connected to {(k + 1,i + 1),(k + 1,i),(k+1,i1)} with the appropriate probabilities along the appropriate arcs. If V (k,i) is value of the option at (k,i), for the european call option we have the recursion

if (k,i) is a terminal node otherwise

The option price is V (0,0).

You are going to do a little more than this for this programming assignment.

2 Part 1: Pricing an American-Option using a Trinomial Model by Recursion

Using my C++ program american option pricing by binomial model.cpp on Compass as reference, write a C++ program that takes as command-line input the values of T,N,r,,S₀ and K, and presents the value of an American-Option using a trinomial model.

Input: The command-line input is the same as what I have for the C++ code that priced the European option in lesson 6 of my notes.

Output: A sample output that I expect from your code is shown in figure ??

Figure 1: Sample output (Note: 20 divisions will take some time to run on your computer. Be patient. The next part of this assignment will be able to handle a large number of divisions, as you will see.).

3 Part 2: Pricing an American-Option using a Trinomial Model by Dynamic Programming

Using my C++ program american option pricing by dynamic programming.cpp on Compass as reference, write a C++ program that takes as command-line input the values of T,N,r,,S₀ and K, and presents the value of an AmericanOption using a trinomial model.

Input: The command-line input is the same as what I have for the C++ code that priced the European option in lesson 6 of my notes.

Output: A sample output that I expect from your code is shown in figure ??

Figure 2: Sample output (Note: A large number of Divisions should not be a problem when you use Dynamic Programming.)

[1] You can verify this at your leisure.