Question

•Find the value of 1-period call option on €10,000 with a strike of £7,700.

i£ = 14%, i€ = 7%, S0(£/€) = £0.8500/€

In the next year, there are two possibilities: S1(£/€) = £1.10/€ or S1(£/€) = £0.7

Answer 1

Pound interest rate = 14% | Euro interest rate = 7% | Spot Exchange rate = 0.85 Pound per Euro

Up-state Exchange rate, S1 = 1.10 Pounds per euro | Down-state Exchange rate, S2 = 0.7 Pounds per euro

Call option on 10,000 euros with a strike of 7,700 pounds. Strike Exchange rate = 7700 / 10,000 = 0.77 Pounds per euro

As we know the Up-state and Down-state exchange rate, Binomial 1-period tree model can be used and we can calculate the risk-neutral probabilities.

In risk-neutral environment, Forward exchange rate should equal expected exchange rate in 1 year.

Using Interest rate parity, we can find the Forward exchange rate. In this case, I am assuming continuous compounding where Foreign interest rate acts as asset dividend.

Forward Exchange rate = Spot Rate * e^{(Domestic interest rate - Foreign Interest rate)*T}

Forward Exchange rate (Pound/Euro) = 0.85 * e^{(14% - 7%) * 1} (In our case Domestic curreny is Pound sterling)

Forward Exchange rate (Pound/Euro) = 0.85 * 1.07251

Forward Exchange rate (Pound/Euro) = 0.91

Let probability of Up-state be p and probability of down-state be (1 - p)

Risk Neutral: Forward exchange rate = p * Upstate exchange rate + (1-p)*Downstate exchange rate

=> 0.85 * 1.07251 = p * 1.10 + (1 - p) * 0.7

=> 0.85 * 1.07251 = 1.10 p + 0.7 - 0.7 p

=> p = (0.85 * 1.07251 - 0.7) / (1.10 - 0.7)

Probability of Upstate = 0.52908

Probability of Downstate = 1 - 0.52908 = 0.47092

Now we can calculate the Payoffs for each state at Year 1 using the Strike exchange rate of 0.77 Pound/Euro (7,700 / 10,000)

For a call option, Payoff = Max(Current rate - Strike, 0)

Payoff (upstate) = Max(Upstate exchange rate - Strike exchange rate, 0) = Max(1.1 - 0.77, 0) = 0.33 Pound/Euro

Payoff (downstate) = Max(Downstate exchange rate - Strike, 0) = Max(0.7 - 0.77, 0) = 0

Now we can find the Value of the call option by discounting the expected Payoff from Year 1 to 0 using Domestic and Foreign interest rate.

Expected Payoff at Year 1 = Probability of Upstate * Payoff (upstate) + Probability of Downstate * Payoff(downstate)

Expected Payoff at Year 1 = 0.52908 * 0.33 - 0.47092 * 0

Expected Payoff at Year 1 = 0.17460 Pound/Euro

Value of the Call Option = Amount in Foreign currency * Expected Payoff at Year 1 * e^{-(Domestic Rate - Foreign rate)*T}

Value of the Call Option = 10,000 Euro * 0.17460 * e^{(-14% + 7%)*1}

Value of the Call Option = 10,000 Euro * 0.17460 * e^-7%

Value of the Call Option = 10,000 Euro * 0.17460 * 0.93239

Value of the Call Option = 1,627.93 Pound Sterling

Hence, The Value of 1-period call option on 10,000 euro with a strike of 7,700 Pound Sterling is 1,627.93 Pound Sterling