## The use of Black-Scholes to value currency options

##### Published: Mar 2005

Over the last few issues, we have examined the role of Black-Scholes in valuing stock options. For the corporate treasurer, it is more likely to be necessary to value the currency and interest rate options used to hedge financial exposures, particularly if hedge accounting is not used.

In October 2004, we wrote that the value of a European-style call option (C) can be written as:

$$C=P*Nd_1–K*e\:^{-rT}*Nd_2$$

Where:

• $$P\:= current\: stock\: price$$
• $$N\:=\: cumulative\: standard\: normal\: distribution$$
• $$K\:=\: option\: strike\: price$$
• $$r\:=\: risk-free\: interest\: rate$$
• $$T\:=\: time\: to\: option\: expiry \:(in\: years)$$

And:

• $$d_1\:=\:\frac{ln\frac{P}{K}+r+\frac{\sigma^2}{2}\:*\:T}{\sigma\: *\:\sqrt{T}}$$
• $$d_2\:=\:d_1–\sigma\:*\:\sqrt{T}$$

Where:

• In = the natural logarithm
• σ = the standard deviation of stock returns (σ2 is the variance from the mean)

This can be adapted to be used to value currency options. The value of a European-style currency option can be calculated using the following formula:

C = P * e – FT * Nd3 – K * e – DT * Nd2

Where N, K, r and T are as above and:

• P = current spot foreign exchange rate
• D = risk-free domestic interest rate
• F = risk-free foreign currency interest rate

And:

• $$d_3\:=\:\frac{ln\frac{P}{K}+D-F+\frac{\sigma^2}{2}\:*\:T}{\sigma\: *\:\sqrt{T}}$$
• $$σ\:=\:the \:standard\: deviation \:of \:exchange\: rate$$

The value of the option is the difference between the spot foreign exchange rate and the strike price, discounted to a net present value. The spot rate is discounted using the foreign currency interest rate. This calculates the net present value of buying foreign currency at the current spot rate on expiration day. The strike price is discounted using the domestic interest rate. This is because the strike price is fixed in local currency when the option contract is agreed.

The principle of the currency option valuation can be seen by looking at the value of the option on expiration day. If the strike price is above the spot exchange rate, the transaction will be processed at the spot rate, meaning the value of the option is zero. If the spot price is above the strike price, the option will be exercised. The value of the option is the difference between what was paid (by exercising the option) and the cost of exercising the transaction at the spot rate.