Portfolio Choice and Option Pricing in Financial Markets Assignment

Assignment Task

Question 1

Consider the following portfolio choice problem. The investor has initial wealth w and utility u ( x ) = ^x _n ⁿ . There is a safe asset (such as a US government bond) that has net real return of zero. There is also a risky asset with a random net return that has only two possible returns, R ₁ with probability 1 −qand R ₀ with probabilityq . We assume R ₁ < 0, R ₀ > 0. Let A be the amount invested in the risky asset, so that w-Ais invested in the safe asset.

1. What are risk preferences of this investor, are they risk-averse, riskneutral or risk-loving?

2. Find A as a function of w .

3. Does the investor put more or less of his portfolio into the risky asset as his wealth increases?

4. Now find the share of wealth, a , invested in the risky asset. How does α change with wealth?

5. Calculate relative risk aversion for this investor. How does relative risk aversion depend on wealth?

Question 2

In a financial market a stock is traded with a current price of 50. Next period the price of the stock can either go up with 30 per cent or go down with 25 per cent. Risk-free debt is available with an interest rate of 8 per cent. Also traded are European options on the stock with an exercise price of 45 and a time to maturity of 1, i.e. they mature next period.

1. Find prices of Arrow-Debreu securities.

2. Calculate the price of a call option by constructing and pricing a replicating portfolio.

3. Calculate the price of a put option by RNVR.

4. Does put-call parity hold? Explain.

5. Construct one long strap and one long straddle using any options from previous part. Sketch the profit and loss graph for each of your portfolios separately and explain the similarity and difference between two positions.

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