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Volatility is one of the parameters needed to calculate the price of a currency option between its trade date and expiration date. Other parameters provided by the markets are: the spot price of the underlying, interest rates of the two currencies involved.
The valuation model used for the European style currency options (vanilla) is Garman-Kohlhagen. This is an adaptation of the Black & Scholes pricing model that includes two interest rates. Pricing model also takes into account the remaining term to maturity and the strike price.
Historical volatility is that observed during a past period. In the context of currency options, it is the dispersion of the spot price around its average over a period of observation. More precisely, it is the standard deviation of changes in the spot price, a statistical term that is defined by the sum of squared deviations between the spot price and the average recorded over. The standard deviation (represented conventionally in financial math by the Greek letter Sigma is formulated as:
Implied volatility represents the expected changes in the spot price for a future period. It is represented by the standard deviation of the anticipated changes of the spot relative to futures prices and observed at the start of the period. When we want to model data sets in financial math, we’re using a specific model of distribution: the Normal distribution. This is because the mean and the standard deviation are indicators that characterize the Gaussian type distribution. Therefore, applying a model of a Normal distribution for these anticipated changes, we get a bell curve called “Gaussian” as follows:
With the normal distribution, the distribution of anticipated changes of the spot compared to the forward rate at the maturity of the option is given by the following distribution:
68.46% of the variation of the spot will be located in the interval [-1, 1] standard deviation.
95.44% of the variation of the spot will be located in the interval [-2, 2] standard deviations.
4.56% of the variation of the spot will be located outside of the interval [-2, +2] standard deviations.
Example
Let’s take a currency option on FX spot EUR / USD. The forward one month EUR / USD given by markets is 1.3481. The spot price is 1.3457. Implied volatility given by markets is 3.25%. In monthly equivalent, this corresponds to a monthly volatility of 0.9382% (3.25% / SQRT (12)). The normal distribution gives us the following distribution at maturity for the option:
68.46% of the variation of the spot EUR / USD will be located in the interval [+ / – 0.9382%] around 1.3481, so between 1.3355 and 1.3607.
95.44% Of the variation of the spot EUR / USD will be located in the interval [+ / – 2% x 0.9382] around 1.3481, so between 1.3228 and 1.3734.
4.56% Of the variation of the spot EUR / USD will be outside of the interval [+ / – 2% x 0.9382] around 1.3481, so spot <1.3228 and spot> 1.3734.
Graphically this will give:
Note that the pricing models are also used in opposite side: taking the price of an option listed on the market, we can determine the value of sigma that the model has taken to get this price. And thus we get the implied volatility from the model on the basis of the price.