Transcription of Asian Options with Cost Of Carry Zero - www.espenhaug.com
1 Asian Options with cost Of Carry ZeroEspenGaarderHaug October22,2006 AbstractThe turnbull and Wakeman (1991) formula is a well known formulafor continuous Arithmetic average rate Options . turnbull and Wakemanoriginally only developed their formula for Asian Options when cost -of- Carry is different from zero. In many commodity and energy marketswhere Asian Options frequently trade the average is typically based onfutures or forward prices, that is cost -of- Carry on the underlying asset iszero1. Many people have contacted me over the years to ask me for howto extend the turnbull and Wakeman (1991) formula to also hold in thecase of cost -of- Carry zero. This quick note gives you the Zero SolutionIn the case of Asian Options when cost of Carry is zero the original Turnbulland Wakeman formulas do not hold and must be modified.
2 If we assume thearithmetic average is approximately lognormally distributed all we need to valuean Asian futures option is to adjust the volatility of the Black-76 formula. Thisentails replacing the futures volatility with the volatility of the average on thefutures A:cA e rT[FN(d1) XN(d2)],(1)pA e rT[XN( d2) FN( d1))],(2) This solution is based on some calculations I did on May 16 on stocks can naturally also have cost -of- Carry zero if the continuous dividendyield equal to the risk free rate, the extension given in this note can then naturally also the time to maturity,ris the risk-free rate,Fis the futures priceandXis the strike (F/X) +T 2A/2 A T,d2=d1 A T,where2 A= ln(M)T,M=2e 2T 2e 2 [1 + 2(T )] 4(T )2,where is the time to the beginning of the average period and is the volatilityof the futures contract.
3 If the option is into the average period the strike pricemust be replaced by Xand the option value must be multiplied byTT2, where X=XT2T FA(T2 T)T,whereT2is the original time in the average period andFAis the average futuresprice during the realized or observed time periodT2 Xshould be negative the call option will for sure be exercised at maturityand the value becomes the discounted value of the expected average at maturityEQ[A] minus the strike price:EQ[A] X. The expected average is equal toEQ[A] =FA(T2 T)T2+ a put the value will be 0 if Xshould be negative. This is basically theTurnbull-Wakeman formula extended to Asian Options on futures ( cost -of-carryzero).
4 ConclusionWe have here extended the turnbull and Wakeman formula to also hold foroptions in the case of zero- cost -of must however say it is a mystery to me why so many people in practice areinterested in continuous average rate Options formulas when all Asian optionsin practice are discrete type, for more on this see for example Haug (2006b) andalso the 2nd edition of The Complete Guide To Option Pricing Formulas .2It is only the second moment we give here, the first moment is 1 with cost -of- Carry , (2006a):The Complete Guide To Option Pricing Formulas, 2ndEdition. McGraw-Hill, New York.(2006b):Derivatives: Models on Models. New York: John Wiley & , , (1991): A Quick Algorithm forPricing European Average Options , Journal of Financial and QuantitativeAnalysis, 26, 377
