Pricing Currency Options Under Stochastic Volatility

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∞ −iϕln( K)1 1 ⎡e fj( x, v, t; ϕ)⎤Pj( x, v, t;ln[ K]) = + Redϕ2 π∫ ⎢ ⎥iϕ0 ⎢⎣⎥, (8)⎦where j=1, 2, Re means the real part of the square bracket, f ( xvtϕ , , ; ) represents thecharacteristic functions of the conditional probability Pj.( ϕ )1 1 ∞ ⎡exp −i ln( K)⋅ f ⎤Pj= + ⎢ ⎥2 π ⎣ iϕ⎦j = 1, 2jj∫ Redϕ0, (9)where fj, j = 1,2 is the characteristic function of the parameters set,Φ= . 4 Given no-arbitrage condition and the price of a call option, C , we* *{ k , θ , σv, ρ}can use the put-call parity to obtain the European put option price as,−rfτ−rdτ= − + . (10)P C Se KeSince the American style currency options traded in the PHLX are used in this study,we employ the quadratic approximation method of Barone-Adesi and Whaley (1987) toadjust the early exercise premium to derive a more accurate empirical study. 53. Data Description and Sample PropertiesWe use the “Foreign Currency Options Pricing History (FCOPH)” database of thePHLX. 6 The FCOPH data of the PHLX contain options on the Australian dollar, Britishpound, Canadian dollar, Deutsche mark, Euro, Japanese yen, Swiss franc and otherrelated customized option contracts. The trade-by-trade option prices cover the periodfrom January 1994 to December 2001. The major advantage of using PHLX databaseis that the daily closing option prices and the simultaneous spot exchange rate quotesare recorded on the same transaction report.The trading hours is from 2:30 a.m. to2:30 p.m., Eastern Time, Monday to Friday. We use the mid-month option, whichexpires in March, June, September, December and the two near-term months and10
month-end option, which expires at three nearest months call and put options. Theprecise expiration date is the Friday preceding the third Wednesday of the month formid-month options and the last Friday of the month for month-end options.Daily closing prices of options and the simultaneous spot exchange rate quotes onBritish pound, Deutsche mark, Japanese yen and Swiss franc against US dollar areselected as our samples. Two reasons for choosing these four currencies are that theyare highly volatile and are most frequently traded in the PHLX in terms of the tradingvolumes. Since the European Union started using EURO in 1999, the currency optionsof British pound and Deutsche mark were traded less over the period from 1999 to 2001than that from 1994 to 1998. Thus, the selected sample periods are from January, 1994to December, 1998 for the British pound and Deutsche mark and those are from January,1994 to December, 2001 for Japanese yen and Swiss franc. Since the database ofPHLX contains the null values of simultaneous spot rates on Japanese yen over theperiod from October, 1994 to March, 1996, we fill the blank with daily spot rate fromthe DataStream.In this study we use the American-style call and put options and discard theuninformative options records from samples in the following categories. First, since forthe implied volatility of options, the maturity day is less than 5 calendar days that iserratic, we use options with the period from 5 to 90 calendar days to expiration.Second, options violate the European-style boundary conditions, for example, thecondition isC−r F τ −rDτ< Se − Ke for currency call options. And finally, the daily optionswith the strike/spot price ratio (moneyness, denoted as S/K for call options, K/S for putoptions) are from 0.8 to 1.2.We use the daily closing quotes of London Euro-Dollars rates collected fromDataStream, including the U.S. dollar against the British pound, Deutsche mark,11
Page 1 and 2: Pricing Currency Options Under Stoc
Page 3 and 4: 1. IntroductionIt has been shown th
Page 5 and 6: varying, and of sufficient magnitud
Page 7 and 8: The remainder of this paper is orga
Page 9: instantaneous variance Vt () follow
Page 13 and 14: percentage and 43.9 percent of the
Page 15 and 16: of the risk-neutralized stochastic
Page 17 and 18: contributed to the more resourceful
Page 19 and 20: actual return distribution better t
Page 21 and 22: Footnotes1. Although the SV option
Page 23 and 24: ReferencesArnold (1990), Mathematic
Page 25 and 26: Volatility: An Analytic Approach,
Page 27 and 28: Table 2.1 Sample Properties of PHLX
Page 29 and 30: Table 2.3 Sample Properties of PHLX
Page 31 and 32: Table 3. Implied Volatility from th
Page 33 and 34: Table 5. Return and Volatility Proc
Page 35 and 36: Figure 1. Implied Volatility of Gar
Page 37: Figure 2. (continued) Implied Volat

Pricing Currency Options Under Stochastic Volatility

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