Question

A futures price is currently $3000 and its volatility is 25%. The risk-free interest rate is 5% per

annum.

a) Use a two-step binomial tree to derive the value today of a one-year European put option

with a strike price of $2900 written on the futures contract.

b) Use put-call parity to value the one-year European call option with a strike price of $2900

written on the futures contract.

c) How would you hedge today a short position in the put option? Derive the futures

position you would take.

Answer 1

Spot Price (So)	3000
Strike Price (K)	3250
Interest Rate ( r )	5.00%
Time to maturity (T)	1.00
Annualized Implied Volatility (σ)	25.00%

U = size of the up move factor=eσ√t, and

D = size of the down move factor=e−σ√t=1/eσ√t=1/U; where t = 6 months = 0.5 year

Assuming the continuously compounded risk-free rate is 5% per annum.

πu = probability of an up move=[ert − D]/[U−D] , where r = 5%, t = 6 months = 0.5 year

πd = probability of a down move=1−πu

u	1.193
d	0.838
πu	0.527
πd	0.473

Possible stock prices:

t =	0	1	2
			4272.36
		3580.09
	3000.00		3000
		2513.90
			2106.57

Value of Put Option:

t =	0	1		2
	Put premium	Mean payoffs	Payoffs	Mean payoffs	Payoffs
					0.00
			0.00	0.00
	168.75	173.02			0.00
			365.91	375.17
					793.43

(b) Using Put-Call parity

Put=Call−stock+Xe^−rT

Put = 168.75

Stock = 3000

X = 2900

e^−rT = e^−5%*1 = 0.9512

Call = Put + Stock - Xe^−rT

Call = 168.75 + 3000 - 2900*0.9512 = $410.18

(c) There are many hedging strategies that can be adopted to hedge a short position in a Put option:

Covered Put - Short the underlying asset, along with selling a put option on the same number of shares. By doing this, the trader is able to generate income in the form of premium for writing the put option.

The delta, Δ, of a stock option is the ratio of the change in the price of the stock option to the change in the price of the underlying stock. It is the number of units of the stock an investor/trader should hold for each option shorted in order to create a riskless portfolio. This process is called delta-hedging.

	Stock price = 2900	Stock price = 3000
1 Put option	168.75	190.02

Option delta = (168.75 - 190.02)/(3000-2900) = -0.22

If the investor is short one put on the stock (with a delta of -0.22, or -22 since options have a multiplier of 100), they would maintain a delta neutral position by selling 22 shares of the underlying stock.