Math 3589 \u00b7 Homework #2
\nDue: February 5, 2022
\nThis assignment is to be written out legibly by hand or by using typesetting software (for example,
\nLaTeX or MS Word). You must leave at least one inch margins on all sides of your paper, and
\nyou may not use a \u201ctwo-column\u201d-format. If a proof is requested, you must write a coherent and
\nlogically correct mathematical proof.
\n2.1. Prove directly, using the definition from class, that the function f : R \u2192 R defined by
\nf(x) = x
\n2
\nis convex.
\nHint: it may be helpful to relate this to the fact that Var(X) \u2265 0 for any random variable X.
\nThis can be done directly, or by repurposing the algebra used in the proof of Var(X) \u2265 0.
\n2.2. You have a stock in the three-period binomial model such that S0 = 4, S1(H) = 8, S1(T) = 2,
\nand r = 0.25. Give the full value trees for each of the following derivative securities. You
\nmay use a calculator for this problem to help it go faster, but make sure to indicate what
\nformulas you are using and to maintain sufficient numerical precision.
\n1. A 3-period strike-$6 European call. Use Cn instead of Vn for the value of the call.
\n2. A 3-period strike-$6 European put. Use Pn instead of Vn for the value of the put.
\n3. A 3-period forward contract* with delivery price $6. Use Fn instead of Vn for the value
\nof the forward contract.
\n4. Verify that Fn = Cn \u2212 Pn in all states.
\nNote that in all cases, the value you are calculating is the value to the individual in the long
\nposition (call holder, put holder, counterparty to receive delivery of stock for the forward
\ncontract).
\n* A forward contract with delivery price K is a contract that requires that the counterparties
\nto exchange one share of stock for K in currency at expiration, with the party in the long
\nposition receiving the stock and paying K in currency. This is often called the \u201creceive
\ndelivery\u201d position, with the opposite counterparty in the \u201cmake delivery\u201d position.
\n1<\/p>\n","protected":false},"excerpt":{"rendered":"
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