Question

A futures price is currently 120. It is known that at the end of three months the price will be either 100 or 140. What is the value of a three-month European call option on the futures with a strike price of 122 if the risk-free interest rate is 5% per annum (continuously compounded)? How would you hedge this option if you sold it?

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Answer 1

Current Futures Price = $ 120 , Probable Prices = $ 100 or $ 140, Option Tenure = 3 months and Contnuously Compounded Risk Free Rate = 5 % per annum, Strike Price = $ 122

Upward Movement % = u = [(140 - 120) / 120] x 100 = 16.667% and Downward Movement = [(120-100) / 120] x 100 = d = 16.667 %

Let the risk-neutral probability of an upward movement be denoted by P

Therefore, P = [ e^(0.05 x 0.25) - 0.83333] / [1.1667 - 0.8333] = 0.5377 approximately

Payoff if price rises = P(u) = 140 -122 = $ 18 and Payoff if price falls = P(d) = $ 0

Therefore, Expected Payoff = 0.5377 x 18 + (1-0.5377) x 0 = $ 9.6786

Option Price = PV of Expected Payoff at 5% per annum C.C = 9.6786 / e^(0.05 x 0.25) = $ 9.55837 or $9.558 approximately.

If the investor sells the call option he/she is obliged to sell the futures at $ 122 if the option is exercised. Such a scenario is loss-making when the futures price rises and hence to guard against the same, the investor can acquire the future (to be sold under the initial call option) under an arrangement involving a sold put option and a bought call option. If the futures price rises, the call option will be exercised by the investor to purchase the futures at $122 and if the futures price falls, the put option will be exercised by the person who had bought the put option, thereby obliging the investor to purchase the futures again at $ 122. This implies that the futures purchase price is locked at $122 and when the initially sold call option is exercised the investor does not faces a loss as he/she acquires the futures under this arrangement of a bought call and a sold put.