The Nikkei index is 22,550,K = 21,000,σ = 0.19,rf = 0.04,S = 0.10,r = 0.08 and ?div = 0.01.The yen to dollar spot rate is 104 and the correlation coefficient is 0.30.What would be the dollar price of a 2-year equity-linked foreign exchange call?

Question

Quizplus · Accepted Answer

First, we need to calculate the drift rate:

drift = rf - ?div + correlation * σ * S
drift = 0.04 - 0.01 + 0.30 * 0.19 * 0.10
drift = 0.0533

Next, we can calculate the expected return on the Nikkei index:

expected return = drift + r
expected return = 0.0533 + 0.08
expected return = 0.1333

Then, we can calculate the expected index price in 2 years:

expected index price = K * exp(expected return * time)
expected index price = 21,000 * exp(0.1333 * 2)
expected index price = 28,796.96

Now we can calculate the payout of the equity-linked foreign exchange call option:

payout = (final stock price - initial stock price) / initial stock price * exchange rate + 1

Assuming a strike price of K = 21,000, the final stock price is expected to be 28,796.96, so:

payout = (28,796.96 - 22,550) / 22,550 * 104 + 1
payout = 1.4502

Finally, we can calculate the option price using the Black-Scholes formula:

d1 = [ln(S/K) + (r - ?div + 0.5 * σ^2) * time] / (σ * sqrt(time))
d1 = [ln(22,550/21,000) + (0.08 - 0.01 + 0.5 * 0.19^2) * 2] / (0.19 * sqrt(2))
d1 = 1.158

d2 = d1 - σ * sqrt(time)
d2 = 1.158 - 0.19 * sqrt(2)
d2 = 0.5481

N(d1) = 0.8757
N(d2) = 0.2916

option price = S * N(d1) - K * exp(-r * time) * N(d2) * payout
option price = 22,550 * 0.8757 - 21,000 * exp(-0.08 * 2) * 0.2916 * 1.4502
option price = $45.02

The Nikkei Index Is 22,550,K = 21,000,σ = 0