Problem 1:

There are two stocks with prices and . The call option on these two stocks with strike prices and respectively are valued at and . A trader constructs a basketof these two stocks with weights and such that . If the strike price on the basket is and the corresponding call option on the basket is valued at then prove that in equilibrium

Problem 2:

There are three strike prices , and such that andis a weighted average of and .

If , and are three call values corresponding to the three strike prices then prove that:

Problem 3

Suppose you want to invest $100 in a risky asset (such as a stock index) for one year so that you can capture the upside. But at the same time you don't want to lose more than 10% should the risky asset produce poor performance or go down in value. If the asset can only go up to $125 or go down to $75 in a year's time (only two outcomes are permitted) and the riskfree rate is 5% then estimate the cost of a protective put option that you should buy (without using any Black-Scholes type solution) to protect your portfolio from going down more than 10%. Will it be a feasible solution? Discuss.

Problem 4

A derivative contract has a payoff of the form

where is the value of the contract at the origin and is the value of the contract now. Plot the payoff of the contract as a function of . Is the function uniformly convex or is it uniformly concave? What is the gamma of the contract? (Hint: the payoff is very similar to the one that was seen first in a LIBOR squared swap sold by Bankers Trust in the early nineties).

Problem 5

An asset price path is governed by a stochastic differential equation of the form

where, is a Weiner process and z is a random normal variable such that it has a mean of zero and a standard deviation of one.

1.   Plot the value of as a function of the random variable on your Excel sheet; what happens when the asset price gets close to zero?
2.   If we make a transformation of the form then what would be the corresponding stochastic differential equation for . Under such a transformation what would be the nature of the 'drift' term.
3.   Discuss the nature of the random walk in detail for the process .
4.   What kind of assets (or asset classes) may follow a random walk such as S and J. Discuss.

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