Question

5. A stock sells at $50. The price will be either $57.5 or $47.5 three months from now. Assume the risk-free rate is 12% per annum with continuous compounding. Consider a call option on the stock that has a strike price of $52.5 and a maturity of 3 months.

a) Find a portfolio of the stock and bonds such that buying the call is equivalent to holding the portfolio. What is the cost of the portfolio? And what is the call price?

b) What position in the stock is necessary to hedge a long position in one call option?

c) What is the risk-neutral probability of the stock rising to $57.5?

Do Problem 5 for a put option with the same strike and time to maturity.

Answer 1

A) The portfolio of stocks and bonds for buying call equivalent could be simplified to understand an a risk free return of 12% plus for the bonds and risk involved expected return of over 20% on stocks. The 20% expectation on stocks is worked keeping in view the strike price of the call $52.5 which as an implied premium of $2.5 or 5% for 3 months maturity. Therefore a portfolio that provides for a return of over 20% annually as a combination of bonds and stocks could equate to the same returns as expected from call buying.

The call price could be worked keeping the risk free free rate as base for arriving at option price. The implied upside at the end of maturity period equals to $51.5 i.e. 12% on $50 for three months. However the strike price of the call is $52.5. Therefore potential upside or downside of the stock to reach $57.5 or $47.5 is 9.5%. Applying a discounting of 9.5% annualized for a 3 months maturity on 52.5 stike price could be $0.623 / option.

B) Hedging position is meant to counter the original position in order to protect from the volatile movements. To hedge on a long call position from a stock hedging point of view should be short. That is stocks should be on the sell side / short side to hedge for long call position.

C)Risk neutral probability can be worked based on the Binomial Option Pricing model.