Module Code: MATH532001
Module Title: Discrete Time Finance
c UNIVERSITY OF LEEDS
School of Mathematics Semester One 2019/20
Assessed Assignment (30% of the total mark)
Hand in date: Tuesday 3/12/2019, 12 noon
Consider a binomial market model with one stock and one bank account. Trading takes place
in discrete time and one period’s length is 1 month. The stock has an initial price of S0.
Each month, the stock price can go up by a factor u with probability pu or down by a factor
d = 1/u with probability pd = 1 − pu. The monthly risk-free interest rate is r (per month).
In the above market, consider the down and out barrier Put option defined by the following
payoff:
X = 1{St≥K1, t=1,2,…,T}[(K2 − ST )
+], K1 < K2.
1. Estimate numerically, using Monte Carlo simulations of 1000 price paths, the expected
discounted payoff of the above option. For your simulations use the following parameter
values:
r p S0 u K1 K2
0.01 0.6 100 1.2 50 70
and take three different values for the maturity time T = 3 and T = 10. Do the
simulation of 1000 price paths in each case 10 times and determine the mean and
standard deviation of the obtained expected discounted payoffs. Interpret your findings.
2. Repeat the procedure above using the risk neutral probability measure instead of the
physical measure. Estimate the expected discounted payoff for the two different maturity
times given in the previous subquestion.
3. Calculate the risk neutral expected discounted payoff for T = 3 manually.
4. How many different values can the payoff take for T = 40? Present your calculations
and arguments.
5. Modify your program in such a way that you can change the value of u. Simulate the
expected discounted payoff under the risk-neutral measure for values of 1.2 ≥ u ≥ 1.1
such that you cover the given interval with small steps. Plot the graph of the expected
discounted payoff as a function of u. Discuss your result.
6. Consider now the American version of the above option, that is when the owner of the
option is allowed to exercise it at time 0 < t ≤ T. The corresponding payoff will be
Xt = 1{Ss≥K1, 0<s≤t}[(K2 − St)
+], K1 < K2.
Compute manually the price of the option with T = 3. Explain the procedure you used.
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Module Code: MATH532001
There is no word count for this assignment. The report should typed as one whole document
(no hand-written parts will be accepted). Code should be included in the appendix and not
in the main report. One hard copy of the assignment must be submitted to G. Aivaliotis’s
letterbox on level 8 School of Mathematics building and an electronic copy must be uploaded
on Minerva by Tuesday 3/12/2019, 12 noon. Failure to meet this initial deadline will
result in a reduction of marks. Assignments submitted later than Friday 6/12/19, 12 noon,
will receive a mark of 0. Faxed or emailed copies of the report will not be accepted. A signed
statement on plagiarism has to be enclosed with your hard copy.
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