Pricing Currency Options Under Stochastic Volatility

Japanese yen and Swiss franc as likely riskless interest rates to derive the impliedvolatility from two alternative option models. To match the maturity of currencyoptions, the 30-, 60- and 90-day interest rates are used to be the rates whose maturityare very close to the option expiration. Table 1 summarizes the sample sizes of fourcurrency options. There are a total of 394,940 records in the PHLX database includingthe British pound, Deutsche mark, Japanese yen, Swiss franc and other currencies (suchas, Australian dollar, Canadian dollar and EURO) over the period from 1994 to 2001.From Table 1, we can find that most options traded in the PHLX are American-style,especially on the British pound and Deutsche mark.To investigate the existence of volatility smile of implied volatility, we first dividethe option data into several categories in terms of either moneyness or time to maturity.We classify the money of a call (put) option to three categories, at-the-money (ATM)option, in which its S/K (K/S) is from 0.99 to 1.01, out-of-the-money option (OTM), inwhich its S/K (K/S) is below 0.99, and in-the-money option (ITM), in which its S/K (K/S)is above 1.01. A finer partition resulted in six moneyness categories. According to theterm of expiration, options are classified into (i) short term maturity (5~30 days); (ii)medium term maturity (30~60 days); and (iii) long term maturity (60~90 days). Table2.1 to 2.4 offers the sample properties of PHLX currency options in our studies. Thereported numbers are the average quoted prices, the standard deviations shown in theparentheses, and the total observed options listed in braces, for eachmoneyness-maturity category. The sample period extends from January, 1994 toDecember, 1998 including a total of 30410 options for British pound and 80779 optionsfor Deutsche mark, respectively; and from January, 1994 to December, 2001 includinga total of 63016 for Japanese yen, and a total of 50990 for Swiss franc, respectively.Note that for British pound, OTM, ITM and ATM options take up 45.3 percent, 10.812