Computational Methods in Finance, Fall 1998

Assignment 1.

Given September 8, due September 22.

**Objective:** To explore dynamic programming.

This is a qualitative model. It is unrealistic in several ways in order
to be as simple as possible. We suppose that a floating interest
rate fluctuates
between and (annualized). Now it is . Each week
it may move up or
down by , except that it is blocked from moving outside its
range. It moves in either direction with probability 1/3 and
remains unchanged with this probability. All movements are independent.
This interest rate is *R*(*t*).
A loan of is to be repaid in 250 weeks. Each week, we add some
interest and make a payment. If our interest rate at week *t* is
*r*(*t*), and the outstanding balance is *B*(*t*), then, at week *t*
we first add interest, replacing *B*(*t*) by
. Next we make a prorated payment,
. This leaves
. Our interest rate is determined as
follows. At week 0, . However, *r*(*t*) remains
fixed until we ``refinance''. Whenever we refinence, *r*(*t*) is replaced by
*R*(*t*). We are allowed to refinance up to four times during this 250
week period. We want to do so so to minimize our total expected payments,
. Set this up as a dynamic
programming problem. Identify the state space. Write a computer program
to solve it by dynamic programming.

**Programming tip of the day:** Don't hard code numbers such as 52, 4, and
250. It is not more efficient amd makes the code hard to test.

**Guidelines for good homeworks:**

- Explain what steps you took to insure that the answers produced by the program are correct. Did you run the program in a case for which you know the answer? Did you check afterwards that the answer is correct? Are there consistency checks (sums being one or zero, ...).`?
- Hand in a good reader friendly program. It should be modular, clearly domumented, and have logic that is easy to follow.
- Choose output wisely. Don't hand in a tree worth of paper, or just one number. Think of informative graphics.
- Make a brief writeup of results. Hand in more than just the program and the results.

Tue Sep 8 15:54:29 EDT 1998