suppose that the exchange rate is 1 dollar for 120 Yen. The dollar interest rate

Question

suppose that the exchange rate is 1 dollar for 120 Yen. The dollar interest rate is 5%(continuously compounded) and the yen rate is 1%(continuously compounded). Consider an at the money American dollar call that is yen-denominated. The option has 1 year to expiration and the exchange rate volatility is 10%. Let n=3.

1) What is the price of a European call? An American call?

2)What is the price of a European put? An American put?

3)How do you account for the pattern of early exercise across the two options?

Explanation / Answer

The European Call option pricing is done basis Black-Scholes Options Pricing Model which uses 6 parameters for calculating Option Pricing i.e. Premium; all of which are provided to us in this problem At the Money Dollar Call Option spot means that Spot Exchange rate USD/JPY = Strike Exchange Rate USD/JPY = 120 JPY for each USD.. European Call Option mandates the trader to exercise the right to buy underlying asset only on expiration date of the Option . Let us solve this problem assuming standard normal distribution

Currency options are priced using a variation of the Black-Scholes formula for stock prices. The

main revision comes from the fact that the opportunity cost to invest in a foreign currency is not

the domestic risk-free rate, as for an ordinary asset, but rather the interest rate differential

(domestic minus foreign). The intuition behind this modification is very simple: an investment

in a foreign currency costs the domestic interest rate (to finance the purchase of currency) but

earns the foreign interest rate. The Black-Scholes formula for European currency call options is

given by:

C = e-if T S N(d1) - X e-id T N(d2),

where

d1 = [ln(S/X) + (id - if + .5 2) T]/( T1/2),

d2 = [ln(S/X) + (id - if - .5 2) T]/( T1/2).

The constants id, if, and are the continuously compounded riskless domestic interest rate per

unit time, the continuously compounded riskless foreign interest rate per unit time and the

standard deviation of the rate of return on the stock per unit time, respectively. N(d1) and N(d2)

represent the cumulative normal distribution evaluated at points d1 and d2, respectively.

The well-known put-call parity relationship implies that the price of a European currency put

option, P, is given by:

P = C - e-if T S + X e-id T

Note that there are six factors affecting the premium of a currency option:

i. the current exchange rate (S)

ii. the strike price (X)

iii. the time to expiration (T)

iv. the volatility of the exchange rate ()

v. the domestic interest rate (id)

vi. the foreign interest rate (if)

Applying the above formulae to data provided; Eurpean Call Option price= 7.340382

European Put Option price= 2.681933..

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