Amerian Journal of Applied Mathematis 2016; 4(4): 181-185 http://www.sienepublishinggroup.om/j/ajam doi: 10.11648/j.ajam.20160404.13 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) Effiient Priing of European Options with Stohasti Interest Rate Using Fourier Transform Method Mihael Chimezie Anyanwu, Roseline Ngozi Okereke Department of Mathematis, Mihael Okpara University of Agriulture, Umudike, Abia State, Nigeria Email address: manyanwu71@yahoo.om (M. C. Anyanwu), okerekeroseline@gmail.om (R. N. Okereke) To ite this artile: Mihael Chimezie Anyanwu, Roseline Ngozi Okereke. Effiient Priing of European Options with Stohasti Interest Rate Using Fourier Transform Method. Amerian Journal of Applied Mathematis. Vol. 4, No. 4, 2016, pp. 181-185. doi: 10.11648/j.ajam.20160404.13 Reeived: May 6, 2016; Aepted: May 20, 2016; Published: July 13, 2016 Abstrat: The inlusion of stohasti interest rate is an essential element of any realisti option priing method. Therefore, the purpose of this paper is to inorporate interest rates in the Fourier transform method for priing European options in exponential Levy models. With the assumption of stohasti independene between the underlying log asset prie and the stohasti interest rate, we obtain a priing of pure disount bond available in the market. Our method of valuation is to apply eigenfuntion expansion to the variable that desribes the evolution of the interest rate, and Fourier transform to the variable that desribes the log asset prie. Keywords: Fourier Transform, Eigenfuntion Expansion, Stohasti Interest Rate, Levy Proess, Simple Harmoni Osillator 1. Introdution The first aeptable ontribution to the field of quantitative modelling of derivative seurities is given by Blak and Sholes [1], in whih the prie of a European option on an underlying asset with prie proess given by a geometri Brownian motion was given in lose-form. However, in most exponential Levy models and stohasti volatility models, analyti formula for the option prie does not exist. Hene, alternative methods are sought. One of suh approahes is the Fourier transform and inversion methods, sine it has been observed that when harateristi funtion of the density funtion is mapped to the option payoff in Fourier spae, option pries are easily alulated muh easier for some of the muh omplex proesses. The idea behind Fourier transform is to take an integral of the payoff funtion over the probability distribution funtion, where the probability distribution funtion is obtained by inverting the orresponding Fourier transform. In mathematial finane, Fourier transform method was first used in Stein and Stein [3] volatility model, to obtain the distribution of the underlying asset prie. This was followed by Heston [2], where through harateristi funtion, obtains an analytial formula for the valuation of European options when the volatility of the underlying asset prie is stohasti. Following these two papers, quite a lot have being written on the appliation of Fourier transform and inversion methods in the valuation of more omplex ontingent laims. For example, Carr and Madan [4] introdues fast Fourier transform algorithm for the numerial valuation of the Fourier integral enountered in option priing. Sine then, fast Fourier transform algorithm has beome an effiient mathematial tool used in mathematial finane in the valuation problems, e.g. see [10, 7, 8, 12, 13]. A more general priing framework was developed in Lewis [9] by separating the underlying asset prie from the option payoff through the use of Planherel-Parseval theorem. Through his method, a variety of priing formula for different types of ontingent laims an be obtained. For other appliations of Fourier transform methods to option priing, see [5, 6, 8] and bibliographies therein. All the existing Fourier transform methods in option priing known in literature make a simplifying assumption that the interest rate in the market remain onstant throughout the life of the option. This assumption is not generally true, espeially when priing options of long maturity, as interest rates are bound to hange, and this level of hange is not known before time (that is, it is stohasti). Therefore, the ontribution of this paper to the existing literature on Fourier transform method for option priing is to inorporate
182 Mihael Chimezie Anyanwu and Roseline Ngozi Okereke: Effiient Priing of European Options with Stohasti Interest Rate Using Fourier Transform Method stohasti interest in priing European style ontingent laim under the assumption that the underlying log asset prie is an exponential Levy proess. Therefore, to prie the European option, we assume that the interest rate,, is a stohasti funtion of the Markov proess :, when state spae is D =. The method of valuation we adopt here is the joint appliation of the Fourier transform and eigenfuntion expansion. To the best of our knowledge, this is the first time the two methods are being applied together. The idea is that we use the Fourier transform for the variable that models the log-asset of the firm, and apply the eigenfuntion expansion with respet to the variable that desribes the evolution of interest rate. The study of eigenfuntion expansion of linear operators was motivated by the seminal work of Mkean [14], where the spetral representation for the transition density of one dimensional diffusion proess is obtained. Sine then, eigenfuntion expansion has been shown to be an important methodology for derivative priing. For example, Gorovoi and Linetsky [20] models the Japanese interest rate using eigenfuntion expansion method, to obtain analytial solutions for zero oupon bonds and bonds options under the Vasiek [19] and CIR [18] proesses for the shadow rate. Lewis [27] uses eigenfuntion expansion to prie generalized European style options on stoks that pay onstant dividends, and finds an estimate for the yield urve assoiated with the Merton [26] eonomy. Davydov and Linetsky [16] develops eigenfuntion expansion approah for valuing options on salar diffusion proesses. Boyarhenko and Levendorskii [17] applies eigenfuntion expansion to multifator models of interest rate. For more appliations of eigenfuntion expansion method to derivative priing, see in applying eigenfuntion expansion, the problem of derivative priing is redued to that of solving a single eigenvalue-eigenfuntion equation. One the equation is solved, the approximate value of the derivative an be omputed. In this paper, we do not do eigenfuntion expansions with the infinitesimal generator of the priing semigroup, rather we do expansion with infinitesimal generator, is the infinitesimal generator of the proess. This ultimately leads to solving an infinite system of disjoint first order ordinary differential equation for the bond prie, whih is easy to solve. In this ase, we obtain an infinite series that gives the bond prie. The remaining of this artile is organized as follows: in the remaining part of setion one, we disuss the preliminaries, whih inlude Fourier transform and harateristi funtion, Levy proess and its theory and eigenfuntion expansion. In setion two, our main method for priing European option with stohasti interest rate is presented. This is the joint appliation of eigenfuntion expansion and Fourier transform. In setion three, we proess our numerial result, disuss the result and onlude the artile. 1.1. Fourier Transform and Charateristi Funtion Let be a pieewise ontinuous real funtion whih satisfies the ondition. The Fourier transform of defined for any omplex variable is given by This transform is known to exist and is analyti for all in the strip "#:$ %&,'()* +,, $-.,,. The inverse of the transform through whih the funtion an be reovered is 023 (1) / 01, $ 4 %. (2) 02 1.2. Brief on Levy Proesses Levy proesses are beoming very popular in mathematial finane beause they have been disovered to desribe the observed reality of finanial markets more aurately than models with Brownian motion suh as geometri Brownian motion of Blak and Sholes [1]. In real world, it is known that asset prie paths do exhibit jump or spikes, and these have to be taken into aount in order to manage risks effiiently. Moreover, the empirial distribution of asset returns exhibits fat tails and skewness. These show that the distribution of asset returns are far from being normal, as predited in Brownian motion models. Levy proesses are known to possess the appropriate properties that aurately desribe all these properties of asset prie and its returns in the real world and risk neutral world. By definition, a Levy proess is stohasti proesses with stationary and independent inrements, and it is ontinuous in probability. A general 1-dimensional Levy proess an be represented as 56 7 4 7 8 7 9*: ;< = ; (3) Where 4 is a 1-dimensional Brownian motion, 5 ",8 is a ompound Poisson proess whih inludes the jumps of, and = ; is a ompensated ompound Poisson proess whih inludes the jumps of with ",, > 1. The harateristi funtion of the distribution of an be represented in the form The funtion @ is alled the harateristi exponent for. the general form of harateristi exponent of any Levy proess an be dedued from the Levy. Khinthine formula ψη BC 0 η0 ibη 7 F1 7 iη1 G/,/H e Jη KFd \ (4) where O P 0,% ",$- R is a measure on \0 satisfying S 0,1&R \ The triple %,O 0,R is alled the Levy triple for X t. The proof of this Levy Khinthine formula an be found in Cont and Tankov [21]. The known exponential Levy models in finanial modeling literature orrespond to assigning different values to O 0 $- R. For example, the ase
Amerian Journal of Applied Mathematis 2016; 4(4): 181-185 183 where F( is a finite measure and O 0, gives rise to jump-diffusion Levy proesses, e.g. Merton [22] and Kou [23] models. The ase where R() an infinite measure, and O=0, gives rise to pure jump model of infinite ativity, e.g. Variane Gamma (VG) model of Madan, et al [24], Normal Inverse Gaussian model of Barndorff-Nielson [25], et. As an example in this artile, we use a VG proess whose density of jumps is given by R()=U 3 V W / 1 (,3) ()+U V X / 1 (,) () Here, 1 (+..) is an indiator funtion of the interval (a, b). The oeffiients + (respetively, - ) gives the intensity of upwards jumps (respetively, downward jumps). The harateristi exponent @(η) of the VG proess an be obtained using (4): @(η)=u[ln( [ *η) ln( [ )+ln([ 3 *η) ln([ 3 )] 1.3. Eigenfuntion Expansion for the Bond Prie Consider a pure disount bond that pays the owner a ertain unit amount of urreny at maturity date T. The prie \(6,],) of this bond at time 6<] is given by \(6,],)=^_,` d e, " f (5) where f is the interior of f, and g is the risk-neutral measure hosen for priing. In the CIR (1985) model, the variable satisfies the stohasti differential equation = (h i)6+ojk (6) Where 0i,O>0 $- k is one dimensional Brownian motion. In (6), we assume that the feller ondition, 2h>O 0 is satisfied to ensure that the point 0 is not attained by the proess. Applying Feymann-Ka theorem to \(6,],), we obtain the bakward Kolmogorov equation s C op + ()q\(6,],)=0 (7) Subjet to \(],],)=1,kh =(h i)p + 0 p 0 is the infinitesimal generator of the proess. We assume that = () is diagonalizable so that we an expand 1 t () into the series 1 t = vx U v w v () (8) where w v are the orthogonal eigenfuntions of $- U v are the oeffiients of expansion given by U v = y/ z,{ }~ C (,t) y{,{ }~ C (,t), (9) : is the measure on f hosen so that the infinitesimal generator beomes unbounded in the Hilbert spae 0 (:,f), with disrete spetrum, and y1 t,w v }=S 1f()w v ():() t Is the inner produt in 0 (:,f). We an now expand the bond prie \(6,],) in the series \(6,],)= vx U v \ v ( )w v () (10) where =] 6. If we substitute (10) into (7), with p = p, we get vx U v w v ()(p +[ v )\ v ( )=0, n= 0,1,2 Hene (p +[ v )\ v ( )=0 (11) Subjet to \ v (0)=1. Then, (11) is an infinite system of disjoint salar first order ordinary differential equations for \ v ( ). This is solved to get \ v ( )= V for eah -. therefore, the eigenfuntion expansion (10) for the bond prie beomes \(6,],)= U v V vx w v () (12) Hene, if the onstant of expansion v onverges fast, and if the eigenvalues and eigenfuntions are known, then with moderately few terms, the series (12) an be trunated appropriately to find the approximation to the bond prie. Suppose the risk-free interest rate is quadrati funtion of the stohasti variable =()= / 0 $ 0 0 +$ / +$,$ 0 0,6h- an be transformed to the operator of the simple Harmoni osillator whose eigenvalues and eigenfuntions are well known. The transformed infinitesimal generator beomes ƒ = ã, where and ã are onstants to be determined in the transformation, = / 0 (0 p 0 ) is the operator of the simple harmoni osillator. The normalized eigenfuntions of an be generally written as @ ()=Ĥ () ()C /0 With @ 0 =2 Š!, Š=0,1,, and the orresponding eigenvalues are [ =Š+ 1 2, Š=0,1,2, /0 (/) (0) WC!( 0 ) Where Ĥ ()=Š! x are the Hermite polynomials, and =9(( () means the largest integer whih is not greater than. it is verifiable that @ ()= o / 1 q C C is the initial eigenfuntion that orresponds to the initial eigenvalue [ = / 0. We use the following theorem from [17] whih we state without proof. Theorem 1. The eigenfuntions of the transformed infinitesimal generator ƒ and that of are the same. 2. The eigenvalues of ƒ $ [ ã, where [ are the eigenvalues of for j = 0,1,2, With these, the eigenfuntion expansion for the bond prie beomes \(0,],)= x @ () V (13)
184 Mihael Chimezie Anyanwu and Roseline Ngozi Okereke: Effiient Priing of European Options with Stohasti Interest Rate Using Fourier Transform Method The detail of this transformation an be found in [17] The graph below shows how the bond pries from the eigenfuntion expansion method approximates losely, the bond pries from the analytial solution. bond prie 1.1 1.05 1 0.95 0.9 0.85 0.8 Figure 1. Approximate and Exat Bond Prie for T =.5 years. 2. Priing European Option with Stohasti Interest Rate approx Exat 0.75 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 time to maturity We begin by hoosing a risk neutral measure Q, suitable for priing options on the underlying stok or index with stohasti interest rate, where is a Levy proess of exponential type [,[ 3 ). In the presene of stohasti interest rate, the time 0 < T prie of a European option with payoff (, ) and expiry date ] an be written as š(6,],)=^_,, [ (, ) =, =] (14) With ()= / 01 ž(ÿ)ÿ, (17) ž(ÿ)= () R Putting (16) into the priing formula (14), we get (18) š(0,],)=\(0,],)^_,`/ 01 ž(ÿ)ÿ e (19) We an apply Fubinis theorem to hange the order of taking expetation and integration to obtain š(0,],)=\(0,],) / 01 ^_, ( ) ž(ÿ)ÿ (20) Using the harateristi funtion for the distribution of X, we an write (20) as š(0,],)=\(0,],) / 01 ( ) ž(ÿ)ÿ (21) Where @(Ÿ) is the so-alled harateristi exponent for the Levy proess X. the payoff for European all and put options with strike prie K are ( ) $- ( ) respetively. Hene, using (18), we have ž(ÿ)= W ª for all option and ž(ÿ)= W ª (3 ) (3 ) for put option. The major task in applying Fourier transform to priing European option is the evaluation of the integral in (21). To do this, we trunate the integral and disretize the integrand. This involves hoosing a onvenient grid Ÿ =*4 3 +Š«, to obtain = 01 F K žfÿ K ;Z (22) where Š " Z,4 3 >0, and «is the grid size Where (, )= ( ).*-U $- are independent, we an write (14) to be š(6,],)=^_, [ ( ) =)]^_, [ ( ) =] (15) Where ( )=^_,` e is the risk neutral value of a pure disount bond that pays one unit of urreny at maturity date ]. Define \(0,],):= ^_,` e. Then, the value of the option an be written as š(0,],)=\(0,],)^_, [ ( ) =] (16) Suppose there exists some 4 "([,[ 3 ) suh that () 2 () is integrable, we an deompose H into Fourier integral to get 3. Numerial Example Table 1. Call option values for K = 100, T = 1. In this setion, we implement the method disussed in We onsider this paper. We experiment with the following parameters: For the Levy proess, we use =0.45,O=0 (sine we are onsidering pure jump proess), [ / = 3,[ 0 = 2,U=1.5. For the CIR interest rate proess, we use h= 1.2,i=1,O=1.6428,$ 0 =1,$ / =0.9,$ =0.0005. We experiment with different stok pries with fixed strike prie, K = 100 for both European all and put options. The result of our alulations are presented in Table 1 and Table 2 for all and put options respetively. In this tables, N represent the number of simulations used in (22). S/N 20 30 40 50 60 70 80 90 120 1.3781 1.3745 1.3753 1.3750 1.3751 1.3751 1.3751 1.3751 150 6.2955 6.3002 6. 2996 6. 2996 6. 2996 6. 2996 6. 2996 6. 2996 180 11.3756 11.3751 11.3758 11.3759 11.3758 11.3758 11.3758 11.3758 200 14.7766 14.7728 14.7727 14.7728 14.7729 14.7729 14.7729 14.7729
Amerian Journal of Applied Mathematis 2016; 4(4): 181-185 185 Table 2. Put option values for K = 100, T = 1. S/N 20 30 40 50 60 70 80 90 20 2.9252 2.9255 2.9255 2.9255 2.9255 2.9255 2.9255 2.9255 40 5.2309 5.2310 5.2312 5.2311 5.2312 5.2312 5.2312 5.2312 60 5.6639 5.6667 5.6672 5.6672 5.6672 5.6672 5.6672 5.6672 80 4.2085 4.2084 4.2042 4.2042 4.2043 4.2043 4.2043 4.2043 4. Summary Priing European options with short maturity date on a given stok, may not require hange in interest rate. But when the maturity date is long, it is neessary to onsider the possibility of stohasti hange in the level of interest rate in the market. In this artile, we have sueeded in inorporating stohasti interest rate in the Fourier transform method for valuing European-style ontingent laim on a stok driven by an exponential Levy proess. Moreover, the tables above reveals that the method of simplified trapezoid rule adopted here for evaluating the Fourier integral onverges fast, thereby allowing the option value to be omputed fast. Referenes [1] Blak, F. and M. S. Sholes (1973). The Priing of Options and Corporate Liabilities, Journal of Politial Eonomy 81, 637-654. [2] Hoston, S. L. (1993) A losed-form Solution for Options with Stohasti Volatility with Appliation to Bond and Curreny Options. Review of Finanial Studies 6 (2), 327-343. [3] Stein, E. and Stein, J. (1991) Stok Prie Distributions with Stohasti Volatility: An Analyti Approah. Review of Finanial Studies 4, 727-752. [4] Carr, P. P. and Madan, D. B. (1999) Option valuation using the fast Fourier transform, Journal of Computational Finane 2 (4), 61-73. [5] Bailey, D. and Swarztrauber, P. (1991). The frational Fourier transform and appliations, SIAM Review 33, 389-404. [6] Attari, M. (2004) Option Priing using Fourier Transforms: A Numerially Effiient Simplifiation. Working Paper, Charles River Assoiates. Available at SSRN: http://ssrn.om/abstrat=520042. [7] Benhamou, E. (2000). Fast Fourier Transform for Disrete Asian Options, EFMA 2001 Lugano Meetings. Available at SSRN: http://ssrn.om/abstrat=269491 [8] Hurd, T. R. and Zhou, Z. (2009). A Fourier transform method for spread option priing, Preprint, arxiv: 0902.3643v1. [9] Lewis, A. (2001). A simple option Formula for General Jump- Diffusion and other Exponential Levy Proesses, Envision Finanial Systems and Option City. net, California. Available at http://optionity.net/pubs/explevy.pdf. [10] Borak, S., Detlefsen, K. and Hardle, W. (2005) FFT Based Option Priing. Disussion Paper, Sonderforshungsbereih (SFB) 649, Humboldt Universität Berlin. [11] Dempster, M. A. and Hong, S. S. (2002) Spread Option Valuation and the Fast Fourier Transform. Tehnial Report WP26/2000, University of Cambridge. [12] Itkin, A. (2005) Priing options with VG model using FFT. arxiv: physis/0503137v1 [13] O Sullivan, C. (2005) Path Dependent Option Priing under Levy Proesses. EFA 2005 Mosow Meetings Paper. Available at SSRN: http://ssrn.om/abstrat=673424 [14] Mkean, Henry P. J. (1956). Elementary solution for ertain paraboli differential equations. Transation of the Amerian mathematial soiety, 82 (2), 519-548. [15] Linetsky, V. (2004). The spetral deomposition of the option value. International Journal of Theoretial and Applied Finane, 7 (3), 337-384. [16] Davydov, V. and Linetsky, V. (2000). Priing Options on salar diffusions: An Eigenfuntion Expansion Approah, Operations Researh 512, 185-209. [17] Boyarhenko, N. and Levendorskii, S. Z. (2004). Eigenfuntion Expansion method in multi-fator quadrati term-struture models. Working paper. Available at SSRN: http://ssrn.om/abstrat=627642 [18] Cox, J. C., Ingersol, J. E and Ross, S. A. (1985). A theory of the term struture of interest rates. Eonometria 55, 349-383. [19] Vasiek, O. A. (1977). An equilibrium haraterization of the term struture. Journal of Finanial Eonomis 5, 177-188. [20] Gorovoi, V. and Linetsky, V. (2004) Blak s model of interest rates as options, eigenfuntion expansion and Japanese interest rates. Mathematial Finane, 14 (1), 48-78. [21] Cont, R. and Tankov, P. (2004). Finanial Modeling with Jump Proesses. (Finanial Mathematis Series), Chapman and Hall/CRC, New York. [22] Merton, R. C. (1976). Option priing when the underlying stok returns are disontinuous. Journal of Finanial Eonomis 3, 125-144. [23] Kou, S. G. (2002). A jump diffusion model for option priing. Management Siene 48, 1086-1101. [24] Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The Variane Gamma Proess and Option Priing. European Finane Review 2, 79-105. [25] Barndorff-Nielson, O. Proesses of Normal Inverse Gaussian Type. Finane and Stohastis, 2, (1998), 41-68.