Black-Scholes option prices depend critically on the estimate of volatility (-) being used. In fact, traders typically use the implied volatility—the volatility that, when substituted in Equation 8A.2, yields the market price of the option—as an indication of the market’s opinion of future exchange rate volatility. Implied volatilities function for options in the same way as yields to maturity do for bonds. They succinctly summarize a great deal of economically relevant information about the price of the asset, and they can be used to compare assets with different contractual terms without having to provide a great deal of detail about the asset.
Application Pricing a Six-Month Swiss Franc European Call Option
What is the price of a six-month Swiss franc European call option having the following characteristics?
Solution. In order to apply Equation 8A.2, we need to estimate B(t,0.5) and B*(t,0.5) since T = 0.5 (6 months equal 0.5 years). Given the annualized interest rates on six-month bonds of 5.8% and 6.5%, the six-month U.S. and Swiss interest rates are 2.9% (5.8/2) and 3.25% (6.5/2), respectively. The associated bond prices are
Substituting in the values for B and B* along with those for S (0.68), X (0.70), and σ (0.2873) in Equation 8A.2, we can calculate
The easiest way to compute the values of N(—0.05786) and N(—0.26101) is to use a spreadsheet function such as NORMDIST in Excel. This Excel function yields computed values of N(—0.05786) = 0.47693 and N(—0.26101) = 0.39704. Using Equation 8A.2, we can now calculate the value of the six-month Swiss franc call option:
In other words, the value of the six-month option to acquire Swiss francs at an exercise price of $0.70 when the spot rate is $0.68 is 4.400¢/SFr. The relatively high volatility of the spot franc has contributed to the significant value of this out-of-the-money call option.
Indeed, option prices are increasingly being quoted as implied volatilities, which traders by agreement substitute into the Garman-Kohlhagen model (Equation 8A.2) to determine the option premium. This is not to say that traders believe that Equation 8A.2 and its underlying assumptions are correct. Indeed, they quote different implied volatilities for different strike prices at the same maturity. However, Equation 8A.2 by convention is used to map implied volatility quotes to option prices.