Comutational and Alied Mathematics Journal 15; 1(1: 1-6 Published online January, 15 (htt://www.aascit.org/ournal/cam he fast Fourier transform method for the valuation of Euroean style otions in-the-money (IM, at-the-money (AM and out-of-the-money (OM Fadugba Sunday Emmanuel Deartment of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria Keywords Contingent Claim, Euroean Otion, Fast Fourier ransform Received: December 5, 14 Revised: December 31, 14 Acceted: January 1, 15 Email address emmasfad6@yahoo.com, sunday.fadugba@eksu.edu.ng Citation Fadugba Sunday Emmanuel. he Fast Fourier ransform Method for the Valuation of Euroean Style Otions in-the-money (IM, at-the-money (AM and out-of-the-money (OM. Comutational and Alied Mathematics Journal. Vol. 1, No. 1, 15,. 1-6. Abstract his aer resents the fast Fourier transform method for the valuation of Euroean style otions in-the-money, at-the-money and out-of-the-money. he fast Fourier transform method utilizes the characteristic function of the underlying instrument s rice rocess. A fast and accurate numerical solution by means of the Fourier transform method was develoed. he fast Fourier transform method is useful for emirical analysis of the underlying asset rice. his method can also be used for ricing contingent claims when the characteristic function of the return is known analytically. 1. Introduction he Black-Scholes model and its extensions constitute the maor develoments in modern finance. Much of the recent literature on otion valuation has successfully alied Fourier analysis to determine otion rices such as [1],[3], [5], ust to mention a few. hese authors numerically solved for the delta and the risk-neutral robability of finishing in-the-money, which can be combined easily with the underlying asset rice and the strike rice to generate the otion value. Unfortunately, this aroach is unable to harness the considerable comutational ower of the fast Fourier transform [7], which reresents one of the most fundamental advances in scientific comuting. Furthermore, though the decomosition of an otion rice into robability elements is theoretically attractive as exlained in [], it is numerically undesirable due to discontinuity of the ayoffs [4]. For more literature see ([8], [9], [1] ust to mention a few.. Fast Fourier ransform Method he fast Fourier transform is an efficient algorithm for comuting the discrete Fourier transform of the form; 1 N 1 πi m( ex x N (1 wheren is tyically a ower of two. (1 reduces the number of multilications in the required N summations from an order of N to that of N ln ( N, a very considerable reduction.
Fadugba Sunday Emmanuel: he Fast Fourier ransform Method for the Valuation of Euroean Style Otions in-the-money (IM, at-the-money (AM and out-of-the-money (OM Let and be written as binary numbers i.e. 1 + and 1 + with 1,, 1, {,1}, then (1 becomes 1 1 πi m( 1, ex ( 1 + ( 1 + x( 1, 1 N 1 1 1 1 ( 1 + ( 1 + 1 ( 1 + A x( 1, A A x( 1, 1 1 ( he fast Fourier transform can be described by the following three stes as 1 1 1 1 1 ( 1 + (, (, 1, (, (,, m x A m m A 1 1 m(, 1 m (, 1 (3 he basic idea of the fast Fourier transform is to develo an analytic exression for the Fourier transform of the otion rice and to get the rice by means of Fourier inversion..1. he Fast Fourier ransform Method for the Valuation of Euroean Call Otion he Fast Fourier transform method is a numerical aroach for ricing otions which utilizes the characteristic function of the underlying instrument s rice rocess. his aroach was introduced by [4]. he Fast Fourier transform method assumes that the characteristic function of the logrice is given analytically. Consider the valuation of Euroean call otion. Let the risk neutral density of S logs be f ( S. he characteristic function of this density is given by ϕ ( v ivs e π f ( S (4 he rice of a Euroean call otion with maturity and exercise rice K denoted by C ( k is given by where S C ( ex( r( e K f ( S (5 K e (6 Substituting (6 into (5 yields S C ( ex( r( e e f ( S (7 lim C ( lim ex( ( ( k k S r e e f S S (8 From (8, it is clearly seen that Euroean call rice given by (7 is not square integrable function. We consider a modified version of (7 derived by [4] given by α c ( e C (, α > (9 Equation (9 is square integrable in over the entire real line. Using the definitions of the Fourier transform and its inversion, we have that πik ( ( ( ( F c cɶ k e c d (1 1 1 π ik π F ( c( c ( e cɶ dk (11 Substituting (11 into (1 and solving further, then the new call value is obtained as α 1 1 π ik α ɶ π ik ɶ C ( e ( ( e c k dk e e c k dk π π (1 (13 is a direct Fourier transform and lends itself to an alication of fast Fourier transform method. (1 is comuted as follows:
Comutational and Alied Mathematics Journal 15; 1(1: 1-6 3 πik S ɶ c e ex( r ( e e f ( S d r S πik α ( e f ( S e ( e e e r ( α S πik ( ( e f S e e e e d S S ( α π e ( α π ( ( S S r e + ik + ik e f ( S e e + + + 1 α πik α πik d (13 aking the limit for ( ik lim e α + π withα >, then the last equation in (13 becomes ( α + + πik S ( α + + π r ex 1 ex 1 ik S cɶ e f ( S α + πik α + 1+ πik ( α + + π r ex 1 ik S e f ( S ( α + πik( α + 1+ πik r e ϕ ( πk ( α α + α 4π k + πki(α + 1 aking the Fourier transform for f ( Sex (( α + 1+ πik S f ( Sex i( ( k ( α + 1 + πk S we get the characteristic function for the risk neutral rice rocess ϕ ( k ( πk ( α. Finally we have r e ϕ ( πk ( α cɶ (14 α + α 4π k + πk(α Settingv πk, then (.57 becomes r e ϕ ( v ( α cɶ (15 α + α v + v(α he corresonding ut values can be obtained by defining α ( e P (, v πk, α > with Fourier transform r e ϕ ( v ( α ɶ (16 α α v + v( α where ϕ is the Fourier transform of f ( S. A sufficient condition for c ( to be square-integrable is given by cɶ ( 1 being finite. his is equivalent toe Q ( S α + <. Substituting (15 into (13 withv πk, we have r α 1 iv e ϕ ( v ( α C e e dv π α + α v + v(α (17 Similarly, for the rice of ut otion we have that: r α 1 iv e ϕ ( v ( α C e e dv π α α v + v( α (18 For the ut formula to be well defined, is suffices to choose an aroriate α > in a way that E Q ( S α < (see [6]. he Euroean call values are calculated using (17. Carr and Madan [4] established that if α the denominator of (17 vanishes when v, inducing a singularity in the integrand. Since the fast Fourier transform evaluates the integrand at v, the use of the factore α. Now, we get the desired otion rice in terms of cɶ using Fourier inversion of the form: α 1 iv C ( e e c dv π ɶ (19 Using basic traezoidal rule, (19 can be comuted numerically as: where 1 C ( e ( η N 1 α iv e c v π ɶ ( v η,,1,,3,... N 1, η >. (1 Since we are interested in (at the money call values C (, the case where the strikes near the underlying sot rice of the asset. his tye of otions is traded most frequently. he fast Fourier transforms method returns N values of and we then consider a uniform sacing of size ρ > for the log-strikes around the log-sot rice S of the form: k a + ρu, u,1,,..., N 1 ( u Equation ( gives us log-strike levels ranging from a to a, where
4 Fadugba Sunday Emmanuel: he Fast Fourier ransform Method for the Valuation of Euroean Style Otions in-the-money (IM, at-the-money (AM and out-of-the-money (OM N ρ a (3 Substituting ( and (3 into (, we have N 1 Nρ 1 Nρ C ( ku ex α ρu ex iv + ρu cɶ ( v η (4 π Now, the fast Fourier transforms method can be alied to x in (1 rovided that π ρ. Hence the integration (19 is η N an alication of the summation (1. For an accurate integration with larger values of η we aly basic Simson s 1 rule weightings into (4 with the 3 condition π ρ, then the accurate call rice which is the η N exact alication of the fast Fourier transform method is obtained as: N 1 Nρ 1 πi η C ( ku ex α ρu ex u iav c ( v ( 3 ( 1 δ 1 + ɶ + (5 π N 3 where ku a + ρu, u,1,,..., N 1 and δ is called the Kronecker delta function exressed as:, if δ 1, if (6.. he Fast Fourier ransform for the Valuation of Otions in-the-money (IM, at-the-money (AM and out-of-the-money (OM For in-the-money and at-the-money otions rices, call values are calculated by an exonential function to obtain square integrable function whose Fourier transform is an analytic function of the characteristic function of the logrice. Unfortunately, for very short maturities, the call value aroaches to its non-analytic intrinsic value causing the integrand in the inversion formula of Fourier transforms to vary above and below a mean value and therefore remains tedious to be integrated numerically. We introduce an alternative aroach called the ime Value Method roosed by [4] to mitigate this numerical inconvenience. his aroach works with time values only, which is quite similar to the revious aroach. But in this context the call rice is obtained by means of the Fourier transform of a modified time value, where the modification involves a hyerbolic sine function instead of exonential function. Let z reresent the time value of out-of -the-money (OM, that is, for S < k we have the call rice for z and for S > k we have the ut otion for z ( k. Scaling S 1 for simlicity, z is defined by ( >, < ( <, > (7 z e r e S e k Ι f ( S + e r e S e k Ι f ( S k S k k S k Where f ( S is the risk-neutral density of the log-rice S and Ι denotes the indicator function. Let the Fourier transform of z ( k be defined by ivk υ e z dk (8 he rices of OM otions can be obtained by the inversion formula of the Fourier transform of (8 of the form ivk z e υ dv (9 By substituting (7 into (8 and writing in terms of characteristic functions then (8 becomes ivk r S k r S k υ e e ( e e Ι k> S, k< f ( S + e ( e e Ιk< S, k > f ( S dk r r r e e ϕ ( i e ϕ ( v i 1 + iv iv iv(1 + iv (3
Comutational and Alied Mathematics Journal 15; 1(1: 1-6 5 here are no issues regarding the integral of this function in (9 ask or, the time value at k tends to zero can get rather stee as tends to zero and this can make the Fourier transform to be wide and oscillatory. By considering the daming function sinh( α k, the time value follows a Fourier inversion: αk αk ivk e e ζ e z dk υ ( v + iα υ ( v iα (33 where 1 ivk z e ζ dv π sinh( αk (31 ivk ζ e sinh( αk z dk (3 Solving (.7 further and relace sinh( α k by αk αk e e, he use of the fast Fourier transform for calculating outof-the-money otion rices is similar to (5. he only differences are that they relace the multilication by Nρ ex( αku ex( α( a + ρu ex α ρu with a division by sinh( αk and the function call to cɶ ( v be relaced by a function ζ ( v. Hence the formula for out-ofthe-money otion rice is given by then we have 1 πi η C ( k ex k ex u + iav ( v 3 + ( 1 N1 ( α ζ ( δ 1 u u π sinh( αku N 3 (34 where N ρ ku a + ρu ρu, u,1,,..., N 1 and called the Kronecker delta as given by (6. 3. Numerical Exeriment and Discussion of Results his section resents numerical exeriment and discussion of results as follows: Numerical Examle We consider the ricing of in-the-money (IM and atthe-money (AM and out-of-the-money (OM Euroean call otion with the following arameters given below: κ.8, θ.5, σ.5, v.8, ρ {.5,,.5}, S 6, r.8,.75, K {, 4, 6,8,1} δ is (35 We examine the effects of correlation coefficient, strike rice and volatility of volatility; on otion values using the fast Fourier transform method. he results generated are shown in the ables 1A and 1B for in-the-money (IM and at-the-money (AM and out-of-the-money (OM Euroean call otions resectively. able 1A. he Effects of Correlation ρ and Exercise PriceK on in-themoney (IM and at-the-money (AM Euroean Otions Values K ρ.5 ρ ρ.5 41.737 41.4985 41.16 4 7.796 7.676 7.417 6 18.136 18.4365 18.6684 8 11.9475 1.57 13.1965 1 7.8495 8.768 9.6557 able 1B. he Effects of Correlation ρ, and Exercise PriceKon out-ofthe-money (OM Euroean Otions Values K ρ.5 ρ ρ.5 41.9377 41.784 41.66 4 7.897 7.6654 7.4978 6 18.1999 18.4337 18.688 8 11.98 1.5585 13.1959 1 7.889 8.751 9.6468 3.1. Discussion of Results ables 1A and 1B show the effect of correlation coefficient ρ and strike ricek on IM, AM and OM otions values using fast Fourier transform. he rice of otion and the time value for Nine-month call otions associated with volatility of volatility σ.5 are relatively close as we can see from the ables 1A and 1B for IM, AM and OM resectively. he effect of correlation coefficient deends on the relationshi between the current underlying rice of the asset and strike rice. For a ositive correlation coefficient, the rice of out-the-money call otion becomes lower. For negative correlation coefficient, the rice of in-the-money and at-the-money call otions becomes higher. When the correlation coefficient is zero, the effect of volatility of
6 Fadugba Sunday Emmanuel: he Fast Fourier ransform Method for the Valuation of Euroean Style Otions in-the-money (IM, at-the-money (AM and out-of-the-money (OM volatility is negligible. he fast Fourier transform method is useful for emirical analysis of the underlying asset rice. his method can also be used for ricing contingent claims when the characteristic function of the return is known analytically. 4. Conclusion he fast Fourier transform method is used because of its advantages when comared to the analytic solution. Using the fast Fourier transform with risk neutral aroach rovides simlicity in calculations.an analysis of the fast Fourier transform method reveals that the volatility of volatility σ and the correlation coefficient ρ have significant imact on otion values, esecially long-time otion, stock returns and negatively correlated with volatility and these negative correlations are imortant for the valuation of Euroean style otions. References [1] Bakshi, G., and Chen, Z. (1997, An alternative valuation model for contingent claims.journal of Financial Economics, 44(1, 13 165 [] Bakshi, G., and Madan, D.B. (1999, Sanning and derivative security valuation.journal of Financial Economics, [3] Bates, D. (1996. Jums and stochastic volatility: Exchange rate rocesses imlicit indeutschemark otions, Review of Financial Studies, 9, 69--18. [4] Carr, P. and Madan, D. (1999, Otion valuation using the fast Fourier trans-form, Journal of Comutational Finance,, 61-73. [5] Chen, R.R., and Scott, L. (199, Pricing interest rate otions in a two-factor Cox-Ingersoll-Ross model of the term structure, Review of Financial Studies, 5, 613--636. [6] Lee, R., (4, Otion ricing by transform methods: extensions, unificationand error control, Journal of Comutational Finance, 7. [7] Walker, J.S. (1996. Fast Fourier ransforms, CRC Press, Boca Raton, Florida. [8] Wu, L. (8, Modeling financial security returns using Levy rocesses, In: Handbooks in oerations research and management science: Financial Engineering, Volume 15, Eds. J. Birge and V. Linetsky, Elsevier, North-Holland. [9] Yan, G. and Hanson F. B. (6, Otion ricing for a stochastic volatility um-diffusion Model with Log-Uniform um-amlitudes, Proceedings of the 6 American control conference, Minneaolis, Minnesota, June 14 16. [1] Zeliade systems (9. Heston 9, Zeliade systems white aer, Setember. Available at htt://www.zeliade.com/whiteaers/zw-4.df.