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Posted: January 31st, 2022
Math 3589 · Homework #2
Due: February 5, 2022
This assignment is to be written out legibly by hand or by using typesetting software (for example,
LaTeX or MS Word). You must leave at least one inch margins on all sides of your paper, and
you may not use a “two-column”-format. If a proof is requested, you must write a coherent and
logically correct mathematical proof.
2.1. Prove directly, using the definition from class, that the function f : R → R defined by
f(x) = x
2
is convex.
Hint: it may be helpful to relate this to the fact that Var(X) ≥ 0 for any random variable X.
This can be done directly, or by repurposing the algebra used in the proof of Var(X) ≥ 0.
2.2. You have a stock in the three-period binomial model such that S0 = 4, S1(H) = 8, S1(T) = 2,
and r = 0.25. Give the full value trees for each of the following derivative securities. You
may use a calculator for this problem to help it go faster, but make sure to indicate what
formulas you are using and to maintain sufficient numerical precision.
1. A 3-period strike-$6 European call. Use Cn instead of Vn for the value of the call.
2. A 3-period strike-$6 European put. Use Pn instead of Vn for the value of the put.
3. A 3-period forward contract* with delivery price $6. Use Fn instead of Vn for the value
of the forward contract.
4. Verify that Fn = Cn − Pn in all states.
Note that in all cases, the value you are calculating is the value to the individual in the long
position (call holder, put holder, counterparty to receive delivery of stock for the forward
contract).
* A forward contract with delivery price K is a contract that requires that the counterparties
to exchange one share of stock for K in currency at expiration, with the party in the long
position receiving the stock and paying K in currency. This is often called the “receive
delivery” position, with the opposite counterparty in the “make delivery” position.
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