• 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 Sk. In this case, the process Sk is speci ed as
8
< uSk
Sk+1
=
Sk
u
: Sk
with probability pu;
with probability pd;
with probability 1 pu pd:
Just as with the binomial tree whose leaves can be combined to form the bino-mial lattice, you can get a trinomial lattice from a trinomial tree. To ensure the martingale property
EfSk+1 j Skg = R Sk;
where R = er Tn , as with the Binomial Tree. We need to assign1 p
• =
pu =
e2(T=n)
p
1
2
pu
!
p1u
R
pu
p
p
!
2
R
u
pd =
p
p1u
u
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 f(k + 1; i + 1); (k + 1; i); (k + 1; i 1)g 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
V (k; i) =
maxf0; S0 ui
Kg
V (k+1;i 1)
if (k; i) is a terminal node
) V (k+1;i)+p
pu V (k+1;i+1)+(1 pu pdR
d
otherwise
The option price is V (0; 0).
You are going to do a little more than this for this programming assignment.
• You can verify this at your leisure.
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• Part 1: Pricing an American-Option using a Trinomial Model by Recursion
Using my C++ program american option pricing by binomial model.cpp on Com-pass 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 gure
??
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.).
• 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 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 gure
??
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Figure 2: Sample output (Note: A large number of Divisions should not be a problem when you use Dynamic Programming.)
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