Question

You are observing the following prices. A put option that expires in six months has an exercise price of $45 and it sells for $5.80. The stock is currently priced at $40, and the risk-free rate is 3.6% per year, compounded continuously.

   1.What is the price of a call option with the same exercise prices and maturity? USE CONTINOUS COMPOUNDING

   2.Suppose you form a portfolio consisting of buying the call and the put options above (Note, they are written on the same stock having the same maturity). The resulting portfolio is known in the market as a “long straddle”. Is this portfolio risk free (perfectly hedged)? NEED DETAILED EXPLANATION AND CALCULATIONS

Answer 1

1. To calculate the price of the call option we would use the formula of put call parity. The formula for put call parity is

Stock price(S₀₎ + cost of put option(P) = Present value of strike price(Xe^rt) + Cost of call option(C).

40 + 5.8 = 45 * e^{-3.6% * 0.5} + C

C = 40 + 5.8 - 44.197246

C = 1.602754

The price of the call option is 1.602754.

2. A 'long straddle' is created by purchasing a call and a put with the same strike price and expiration. This strategy is profitable when the stock price is volatile and could move in either ways. The risk of the portfolio is that the price of the stock would remain constant and the premium on both the options will decrease creating a loss. The stock may not cross the breakeven price, leading to a loss.