Question

A futures price is currently 50. At the end of six months it will be either 56 or 45. The risk-free interest rate is 3% per annum (continuously compounded). What is the value of a six-month European put option with a strike price of 49? How would you hedge this option if you bought it?

Please show the steps

Answer 1

The put option, written at time t = 0, obliges the writer to buy a unit of stock at time t = 1, which is in six months time, for a price of K 1|0 = 49 if the holder so wishes. This will happen if the actual price is S ^d₁ = 45, in which case p^d_1|0 = 4. If the price is S u 1 = 56, then the put option will not be exercised as it will have no value ( p ^u_1|0 = 0). A portfolio consisting of N units of stock as the assets and one put option as the liability will have the following values at time t = 1:

V1 =	Su1 N ? pu1|0 = 56N, If S1 = Su1 = 56
	Sd1 N ? pd1|0 = 45N ? 4, if S1 = Sd1 = 45

For the risk-free portfolio that is used to value the stock option, these two values must be equal:

V1 = 56 N = 45 N ? 4

N = -0.364

Having a negative quantity of stock might mean that you have borrowed stock and short sold it, and that you must return it to the owner by buying it back on the market.

The solution for the portfolio value is

V1 =	56 * -0.364 = -20.38
	(45 * -0.364) -4 = -20.38

Which is regardless of which is the actual value of the stock at time t = 1.

When this is discounted to its present value, it must be equal to the value of the portfolio at time t = 0:

V₀ = S₀N ? p_1|0 = V_1e^?(r×0.5)

Therefore,

p_1|0 = S₀N ? V₁e^?r

= (49 * - 0.364) ? ( 20.384 * e^?0.03×0.5)

= 2.24