PARAMETER ESTIMATION IN A BLACK SCHOLES

Transcription

1 PARAMETER ESTIMATIO I A BLACK SCHOLES Musafa BAYRAM *, Gulsen ORUCOVA BUYUKOZ, Tugcem PARTAL * Gelisim Universiy Deparmen of Compuer Engineering, 3435 Isanbul, Turkey Yildiz Technical Universiy Deparmen of Mahemaics, 340, Isanbul, Turkey 3 Yildiz Technical Universiy Deparmen of Mahemaical Engineering, 340, Isanbul,Turkey * Corresponding auhor; mbayram@gelisim.edu.r. Inroducion In his paper we discuss parameer esimaion in Black Scholes model. A nonparameric esimaion mehod and well known maximum likelihood esimaor are considered. Our aim is o esimae he unknown parameers for sochasic differenial equaion wih discree ime observaion daa. In simulaion sudy we compare he nonparameric mehod wih maximum likelihood mehod using sochasic numerical scheme named wih Euler Maruyama. Key words: nonparameric esimaion mehod, maximum likelihood esimaion mehod, sochasic differenial equaions, Black Scholes model. When he sochasic differenial equaion is given wih cerain coeffciens, i is easy o solve his equaion wih numerical mehods, Euler Maruyama mehod, Milsein mehod, Runge Kua mehod, ec. [],[7]. However if we have only observed discree daa over any ime inerval i has difficuly. To reach diffusion and drif coefficiens are required in his case, so we need esimaion mehods. [3],[],[],[8] menion abou esimaion mehods for sochasic differenial equaions. A number of researchers have used nonparameric echniques [],[]. onparameric mehod is simple for o implemen and esimae he coefficiens, bu i is reasonably if we have frequen daa. Maximum likelihood mehod [0] is oher esimaion mehod which is more efficien han nonparameric mehod. In many pracical cases where a diffusion process has been observed a discree ime poins an explici likelihood funcion is rarely available. These ypes of daa have recenly received grea ineres. A simple mehod o obain an esimaor for discree ime is consruc from he sale daa, an approximaion o he esimaor found in he heory for coninuous observaions which is includes discree ime approximaions o sochasic inegrals. [9],[4] ec. sudy such esimaors analyically, in general i is difficul bu i can be done easily by numerical simulaion. In his paper our aim is discuss he differences beween nonparameric esimaion mehod and maximum likelihood esimaion mehod wih numerical applicaion in finance. This aricle is organized as follows. In Secion and Secion 3 we discuss he nonparemeric and maximum likelihood esimaion mehods respecively. Coefficiens of Black Scholes model [], are obained in Secion 4 using monhly YHOO sock, from o which model is very handy in finance. Then, we use he MATLAB package program o solve he sochasic differenial equaion numerically wih is obained parameers and compare original daa wih our approximae soluions. Our resuls are suppored via graphs a he end of he paper.. onparameric Esimaion Mehod We consider () diffusion process, which saisfies he sochasic differenial equaion

2 d ( ) ( ( )) d ( ( )) dw ( ) () Under proper resricions on, and arbirary funcion, from [5] condiional expecaion E [ (, )] in he Taylor series form can be wrien; E[ (, )] (, ) (, ) (, )( )... () where from Eq.(); n! n n n (, )( ) O(( ) ) E( (, ) x) ( x, ) ( x, ) lim (3) x x ( x, ) ( x, ) ( x, ) ( x) ( x) (, ) E [ (, ) (, )] (, )... (4) is obained. Taking he firs erm on he righ side of Eq.(4) and ignoring oher erms as an error we obain firs order approximae for. So we can wrie; (, ) E[ (, ) (, )] O( ) (5) If we wan approximae a specific ( x, ) funcion, we need only o specify funcion which provide (, ) ( x, ). To find ( ) coefficien we ake ( x, ) x and o find ( ) coefficien, is deermied in []. Then from Eq.(3); (, ) ( ) x x (, ) ( ) (6) (7) (, ) ( ) are obained. Accordingly ( ) E[ )] O( ) (8) are found. ( ) E [( ) ] O( ) (9)

3 3 If we wan o find parameer vecor via nonparameric esimaion mehod in d ( ) ( ( ); ) d ( ( ); ) dw ( ) (0) and we have x0, x,..., x observed daa of a he respecively uniformly disribued imes i for i 0,,..., where T, from Eq.(8) and Eq.(9) we can find using following equaions; (, x, ) ( x x ) () i i i i i0 i0 (,, ) ( ). i xi xi xi i0 i0 () i 3. Maximum Likelihood Esimaion Mehod We consider Eq.(0). Assume ha for 0,,,...,, x are known, densiy of he iniial case is g0( x0 ) and ransiion probabiliy densiy of (, x ) begining from (, x ) is g(, x, x ; ). Then from [6], he maximum likelihood esimaion (MLE) of is value of which maximizes following oin densiy equaion; ( ) g ( x ) g(, x, x ; ) (3) 0 0 wih L( ) ln( ( )) ransformaion we can rewrie Eq. (3) as following; L( ) ln( g ( x )) ln( g(, x, x ; )) (4) 0 0 Our aim is find he minimum value of which make L( ) funcion minimum. This proper value demonsrae wih. From Euler Maruyama approximaion schema we can wrie; x x (, x ; ) g(, x ; ) (5) where (0,) and ransiion probabiliy densiy is following; i where ( x ) g(, x, x ; ) exp[ ] (6) x (, x ; ) g(, x ; ).

4 4 We can approximae densiy funcion as in [6], for arbirary value of, we can find soluion of (0) numerically. This process is repeaed M imes and we show his esimaed values wih ransiion densiy g(, x, x ; ) can be esimaed by M ( M ) g x x K Mh i x i. Then he x xi (7) h (,, ; ) [ ] formula where K is a non-negaive kernel funcion which demonsrae following; and h is bandwidh equal o K( ) exp [ ] h /5 0.9 M. where M M xi xi M i M i ( ( ) ). 4. Analysis of Experimenal Daa In his secion we ackle he YHOO sock. The sock daa se is graphed for every monh in Fig. over he years 005 o 05. Figure : Monhly he YHOO sock, January 005 o January 05 We sui his daa o sochasic differenial equaion named Black Scholes model ( ) ( ) d ( ) dw ( ) (0) 3.49 where () is he sock price a ime and, is o be deermined. We need o esimae he parameer and using MLE and nonparameric parameer esimaion mehods. Firsly we esimae he and wih MLE mehod. If we use MLE procedure we obain he opimal values and approximaely. Therefore, a reasonable sochasic differenial equaion based on he daa for he YHOO sock for 3 monhs is T (8)

5 d d dw ow we esimae he and wih nonparameric esimaion mehod. If we use non-parameric esimaion mehod procedure for = 3 and, we obain he opimal values and approximaely. Therefore, a reasonable sochasic differenial equaion based on he daa for he YHOO sock for 3 monhs is d d dw The mean of sock price of acual daa is and (9) (0) 3.9, gives is 95% confidence inerval. Solving Eq.(9) and Eq.(0) wih Euler Maruyama approximaion mehod we obain he mean of sock price forecasing wih MLE is , gives and is 95% confidence inerval and he mean of sock price forecasing wih non-parameric parameer esimaion is , gives is 95% confidence inerval respecively. and In Fig., he simulaed daa, evaluaed wih MLE using he SDE model, and he monhly acual daa are ploed for every monh over he year "Acual" holds real daa for he YHOO sock, which is ploed as red sraigh lines. "Forecas" keeps MLE esimaion using Euler Maruyama approximaions, which is ploed as blue sraigh lines. Figure : Esimaed daa using MLE and acual daa of he YHOO sock, January 005 o January 05 In Fig. 3, he simulaed daa, evaluaed wih nonparameric parameer esimaion using he SDE model, and he monhly acual daa are ploed for every monh over he year "Acual" holds real daa for he YHOO sock, which is ploed as red sraigh lines. "Forecas" keeps nonparameric esimaion using Euler Maruyama approximaions, which is ploed as blue sraigh lines. All of he graphs indicae ha he Sochasic differenial equaion model supplies sensible fi o he daa.

6 6 Figure 3: Esimaed daa using nonparameric esimaion and acual daa of he YHOO sock January 005 o January Conclusion In his paper we have been concerned wih he esimaion of he and in he drif coefficien and in he difusion coefficien respecively of a Black Scholes model, when he observaion daa known. Using he YHOO sock daa monhly beween and maximum likelihood esimaion parameers, and nonparameric esimaion parameers, are obained.this obained parameers wrien in he Black Scholes model. Then applying Euler Maruyama mehod his sochasic differenial equaion wih is iniial value, he simulaed soluion is obained for each esimaion mehod a each ime. Afer ha, we compared acual daa wih numerical soluions for each esimaion mehod. According o our resuls we can say ha maximum likelihood esimaion mehod is have a good approximaion o observaion daa via nonparemeric esimaion mehod Acknowledgmen The auhors are graeful for he helpful commens and suggesions by he Ediors of he ournal and anonymous referees. References [] Allen, Edward. Modeling wih Iô sochasic differenial equaions. Vol.. Springer Science & Business Media, 007. [] Black, Fischer, and Myron Scholes. "The pricing of opions and corporae liabiliies." Journal of poliical economy 8.3 (973): [3] Chan, Kalok C., e al. "An empirical comparison of alernaive models of he shor erm ineres rae." The ournal of finance47.3 (99): [4] Florens-Zmirou, Danielle. "Approximae discree-ime schemes for saisics of diffusion processes." Saisics: A Journal of Theoreical and Applied Saisics 0.4 (989): [5] Hille, Einar, and Ralph Saul Phillips. Funcional analysis and semi-groups. Vol. 3. American Mahemaical Soc., 996. [6] Hurn, A. San, Kenneh A. Lindsay, and Vance L. Marin. "On he efficacy of simulaed maximum likelihood for esimaing he parameers of sochasic differenial equaions." Journal of Time Series Analysis 4. (003):

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