Question

A currency futures price is currently $1.90 and has a volatility of 20%. The domestic and foreign risk-free interest rates are 6% and 3%, respectively. Use a two-step binomial tree to derive a) the value of a three-month European call option on the currency futures with a strike price of $1.91, and b) the currency futures position which will hedge a short position in the European call option today.

Answer 1

So = $1.90

X = $1.91

Rfd = 6%

Rff = 3%

σ = 20%

T = 3/12 = 0.25 years

u = eσ*sqrt(T) = exp(20%*sqrt(0.25)) = 1.105

d = 1/u = 0.905

p = [e(Rfd-Rff)*T - d] / [u - d] = 0.513

1-p = 0.487

t =	0	1	2
			2.32
		2.10
	2		1.9
		1.72
			1.56

(Payoff)t = MAX(Currency price-Strike price, 0)

Mean payoff = p*Up value + (1-p)*Down value

(Payoff)t-1 = (Mean Payoff)t / (1+r)

where r = (Rfd-Rff)/8 = 0.375%

t =	0	1		2
	Put premium	Mean payoffs	Payoffs	Mean payoffs	Payoffs
					0.41
			0.21	0.21
	0.11	0.11			0.00
			0.00	0.00
					0.00

Value of European Call = $0.11

(b) To hedge a short position in the European call, one should long currency futures with the same time to expiration as the option.

A long position in a futures contract plus a short position in a call option (covered call). The long position “covers” the investor from the payoff on writing the short call that becomes necessary if prices increase. Downside risk remains if prices drop.

The other party will exercise the call only if the currency futures price goes above $1.91. In this case, a futures contract will allow the investor to buy currency futures at a price of $1.90. Downside risk remains if prices drop.