We will derive a two-state call option value in this problem. Data: S0 = 250; X

Question

We will derive a two-state call option value in this problem. Data: S0 = 250; X = 260; 1 + r = 1.1. The two possibilities for ST are 280 and 180. a. The range of S is 100 while that of C is 20 across the two states. What is the hedge ratio of the call? (Round your answer to 2 decimal places.) Hedge ratio             b. Calculate the value of a call option on the stock with an exercise price of 260. (Do not use continuous compounding to calculate the present value of X in this example, because the interest rate is quoted as an effective per-period rate.) (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Explanation / Answer

a) Hedge ratio

Hedge ratio = (Return to option Buyer if Price goes up - Return to option buyer if Price goes down) / (Upper Price - Lower Price)

Return would be zero in case price goes down as the option buyer would not exercise the option.

Return if price goes up = stock price on maturity - strike price = 280 - 260 = 20

Hedge ratio = (20 - 0) / (280 - 180) = 1 / 5

We will buy the shares equal to the numerator, i.e., 1 and we will write / sell call options in number of shares equal to the denominator, i.e., 5.

b) First Compute Loss in both situations

In case price goes up, the option buyer for our 5 short options will exercise the options. So, we will buy@market price and sell them to the option buyer @strike price. The one share bough will be sold @ market price.

Loss (if price goes up) = loss on options + Profit Sale of shares = (260 - 280) x 5 - (280 - 250) x 1 = (-)70

In case price goes down, the option buyer will not exercise the options. The one share bought will be sold @market price.

Loss (if price goes up) = (180 - 250) x 1 = (-)70

Loss is same in both situations as it was a hedge.

Next step - Option value

we compute the option value using the equation -

(Vs - n x Vo) (1+rt) =Net Investment

Where, Vs = Value of shares bought = 1 x 250 = 250, n = no. of options sold, r = rate of interest, t = time to expiration in years, Net Investment = Initial Investment - Loss = 250 - 70 = 180

We have -

(250 - 5 x Vo) (1.1) = 180

Or, Vo = 17.273

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