Value a 1-year call option written on £10,000 with an exercise price of $2.00 = £1.00.The spot exchange rate is $2.00 = £1.00; The U.S.risk-free rate is 5% and the U.K.risk-free rate is also 5%.In the next year,the pound will either double in dollar terms or fall by half (i.e.u = 2 and d = ½).Hint: H = ?.
A)$6,349.21
B) $\$ 6,666.67 = \$ 20,000 - ( £ 10,000 - £ 6,666.67 ) \times \frac { \$ 4.00 } { £ 1.00 }$
C) $\$ 6,666.67 = £ 6,666.67 \times \frac { \$ 1 } { £ 1 }$
D)None of the above

Question

Quizplus · Accepted Answer

The formula to value a European call option using the binomial option pricing model is:

C = e^(-rT)[pmax(U*S0-K,0) + (1-p)max(D*S0-K,0)]

where:

- C = call option price
- S0 = current spot exchange rate (in this case, £1 = $2.00)
- K = exercise price (£1 = $2.00/2 = $1.00)
- r = risk-free rate (5% in both countries)
- T = time to expiration (1 year)
- p = probability of the spot rate increasing (calculated using H = 2/3 as per the hint)
- U = factor by which the spot rate increases (2)
- D = factor by which the spot rate decreases (1/2)

Using these inputs, we get:

p = (e^(rT) - d) / (U - d) = (e^(0.05*1) - 0.5) / (2 - 0.5) = 0.6055

C = e^(-0.05*1)[0.6055*max(2*$10,000-$2.00*$10,000,0) + (1-0.6055)*max(0.5*$10,000-$2.00*$10,000,0)] = $6,666.67

Therefore, the best choice is B.

Value a 1-Year Call Option Written on £10,000 with an Exercise