Time value is not linear

A one-year option is not valued at twice the value of a six-month option. An “at the money” option has greater time value than an “in-the-money” or “out of the money” option. Accordingly, options unlike forwards and futures have convexity; that is, the change in the value of an option for a given change in the price of the underlying asset does not remain constant.

This makes pricing options much more complex than pricing other foreign exchange instruments.
A major advance in the general theory of options pricing was introduced by Professors Black and Scholes in 1973. Their work, which was subsequently adapted for foreign exchange options, showed that under certain restrictive assumptions, the value of a European option on an underlying currency depends on six factors:

1. The spot exchange rate
2. The interest rate on the base (or underlying) currency
3. The interest rate on the terms currency
4. The strike price at which the option can be exercised
5. The time to expiration
6. The volatility of the exchange rate.

Volatility, a statistical measure of the tendency of a market price—in this case, the spot exchange rate—to vary over time, is the only one of these variables that is not known in advance, and is critically important in valuing and pricing options.

Volatility is the annualized percentage change in an exchange rate, in terms of standard deviation (which is the most widely used statistical measurement of variation about a mean). The greater the forecast volatility, the greater the expected future movement potential in the exchange rate during the life of the option—i.e., the higher the likelihood the option will move “in-the-money,” and so, the greater the value (and the cost) of the option, be it a put or a call. (With zero volatility, the option should cost nothing.)

If the one-year forward dollar-Swiss franc exchange rate is CHF 1.6000 = $1, and the volatility of a one-year European option price is forecast at 10 percent, there is implied the expectation, with a 68 percent probability, that one year hence, the exchange rate will be within CHF 1.6000 per dollar plus or minus 10 percent—that is, between CHF 1.4400 and CHF 1.7600 per dollar.