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 all work and also how you would hedge it

Answer 1

Payoff table of Put option with strike price of $49:

		Price after 6-months	Value of put option [Strike price ($49) – price]
	Move up (u=$56/$50 =1.12)	$56	$0(price is above the strike price so option will not exercise)
Current Price of stock($50)
	Move down (d=$45/$50=0.9)	$45	$4

Probability of moving up, P = e ^r*t – d / (u-d)

Where

Risk free rate, r = 3% per year or 0.03

Time period, t = 6-months or ½ years

Factor of moving up, u = 1.12

Factor of moving down, d = 0.9

Therefore,

P = [e^ (0.03*1/2) – 0.9] / (1.12 -0.9)

P = 0.52324

And (1- P) = 1- 52324 = 0.47676 (probability of moving down)

Now the expected value of put option

= e^ - (0.03*1/2) * (1-P) * $ 4

= 0.9851 *0.47676 *$4

= $ 1.8786

Therefore the value of the put option is $1.88

The hedge ratio of the put can be calculated in following manner -

When uS0 = $56 then Pu = $ 0

When d S0 = $45 then Pd = $4 (as the strike price is X = $49)

Now Hedge ratio H = (Pu – Pd) / (uS0 – dS0)

= (0 - $4) / ($56 - $45) = - 4 / 11

Form a portfolio of four shares of stock and 11 puts.

Portfolio (riskless)

	ST = $45	ST = $56
Value of 4 shares	$180	$224
Value of 11 puts	$44	$ 0
Total	$224	$224

Therefore payoff to the portfolio is $224 in both situations