Kelly F526 Key Notes

Put–call parity, Greeks (finance), Black–Scholes model

F526 Mid-term Exam Key
Fall 2010

(7.5 points) 1. Suppose that the current 3 month interest rate is 0.25% and the current dividend yield on the stocks in the S&P 500 Index is 3%. Suppose that the current level of the S&P 500 stock index is 1140. How does the S&P 500 Index future price for delivery in three months compare to the current index level? It is (circle one): less than

Why? Time value of money (the ability to delay the purchase of the asset) tends to push the future price up while the loss of dividends tends to push the future price down. In this case, the short-term interest rate is less than the dividend yield.

Exam B had the opposite answer as the interest rate was given as 3.25%.

(10 points) 2. Consider the following prices. The current stock price is $50. A put and a call option exist. Both options have strike prices of $50 and one year until expiration. The call option’s price is $3 and the put option’s price is $2. The current interest rate is 4% per year. Demonstrate a risk-free arbitrage is available given these prices and show how much money is to be made.

Put call parity: +S+P = +C+PV(E) means 50+2 should equal 3+48.08 but 52 > 51.08. So the put is expensive. Sell the put and buy a synthetic put (-S+C+lend).

At t = 0 –P-S+C+PV(E) = +2+50-3-48.08 = +0.92 profit.
At expiration if the stock price is greater than $50 use your call to cover your short position at $50. If the stock price is less than $50 the put owner will force you to buy the stock at $50. The $50 from the lending ($48.08+interest) will just cover the stock purchase.

(7.5 points) 3. Suppose I am trying to compute the implied volatility of a call option using the Black-Scholes formula. The actual value of the call option in the market place is $5.25. My initial guess at the implied volatility is 30% per year. Using that guess, I compute a theoretical Black-Scholes price of $6.00.

Is the implied volatility greater than, less than, or equal...
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