On the White Noise of the Price of Stocks related to the Option Prices from the Black-Scholes Equation

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1 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 4 On the White Noise of the Price of Stocks related to the Option Prices from the Black-Scholes Equation A Kananthai, Kraiwiradechachai Abstract In this paper, we study the white noise from the stock model and obtained some interesting properties Moreover such white noise can be applied to the black-scholes equation in the form of white noise and obtained the option price of such equation We also found the kernel which has interesting properties Keywords: Black-Scholes equation, white noise, kernel Introduction We know that the white noise is the cause of the fluctuation of the price of stock In the past, the white noise has not been computed properly from the stock model Fortunately, we can compute such white noise by using the idea of generalized function or distribution theory When we get the valued of white noise we can understand how much the fluctuation of any kind of stock Moreover we know the interesting properties such as the tempered distribution and a generalized stochastic process and also the Gaussian normal distribution Moreover, we have applied such white noise to the Black-Scholes equation in the form of white noise It is well known that the Black-Scholes equation plays an important role in financial mathematics, particularly in finding the option price of the stock market In this work we start with the stock model ds µsdt + σsdb where s is the price of stock at the time t, µ is the drift of stock, σ is the volatility of stock and B is the Brownian motion From we define the white noise ξ db dt Ac- Manuscript received August 4, 7; revised November 6, 7 his work was supported in part by Naresuan University under Grant R559C68 he authors are very grateful to the referee for their valuable suggestions and comments that improved the paperhis project was partially completed while the first author visit to Department of Mathematics, Naresuan University A Kananthai malamnka@gmailcom is with the Department of Mathematics, Faculty of science, Chiangmai University, Chiangmai 5, hailand Kraiwiradechachai corresponding author tanyongk@nuacth is with the Research center for Academic Excellence in Mathematics, Department of Mathematics, Faculty of science, Naresuan University, Phitsanulok 65, hailand tually, db does not exist in the classical sense or Newtonian sense But it has meaning in the distributional dt sense that is in the space of tempered distribution By applying the Ito s formula and the tempered distribution to we obtained the white noise ξ in the form ξ s tσ ln µ s σ + σ where s is the price of stock at t We can relate to the Black-Scholes equation which is given by us, t t us, t + rs + σ s s us, t s rus, t 3 with the terminal condition us, t s p + 4 see ], pp for t where us, t is the option price at time t, σ is the volatility of stock and p is the strike price Let us, t V ξ, t where ξ is given by hen 3 is transformed to the equation V ξ, t + V ξ, t r t t ξ + tσ σ V ξ, t rv ξ, t t ξ 5 with the condition V ξ, t t fξ, t 6 where fξ is the given generalized function We obtain the solution V ξ, t of 5 with 6 in the convolution form V ξ, t Kξ, t fξ 7 where t tξ r Kξ, t π t exp tr exp σ σ ln t ] t 8 is the kernel see ], pp Preliminary Notes Recall the stock model ds µsdt + σsdb 9 Advance online publication: 8 May 8

2 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 4 or ds µsdt + σsḃtdt db where Ḃt is the white noise denoted by ξt dt Ḃt Apply the Ito s formula to 9, we obtain dln sτ where τ t hus ln st ln s µ σ dτ + σ µ σ t + σ Ḃτdτ ξτdτ where ξτ Ḃτ and s s db Since Ḃt dt does not exist in classical sense or Newtonian sense But it can be show that ξt Ḃt is a tempered distribution, that is ξ ŚR- the space of tempered distribution see 3], pp 6-8 hus for any testing function φ SR-the Schwartz space, define ξ, φ ξτφτdτ hus, from ln st s µ σ φτ t + σ φτ ξτdτ for φτ Now ξτ ŚR, also ξτ ŚR φτ Let F τ ξτ thus φτ ξτ φτ F τφτdτ Since F τφτ is a smooth function of τ By mean value theorem, there exist τ for τ t such that F τφτdτ F τ φτ F τ φτ t ξτ φτ φτ t ξτ t dτ for t By changing the variable s to ξ from and let us, t V ξ, t we have and u s V s ξ u s V s V ξ ξ s tσs tσs hus 3 is transformed to V ξ, V ξ ξ s t σ s V ξ tσs V ξ V ξ, t + V ξ, t r t t ξ + tσ σ V ξ, t rv ξ, t t ξ where < t with the terminal condition of 4 and V ξ, Let fξ ] + s exp µ σ + σ ξ p 3 ] + s exp µ σ + σ ξ p thus V ξ, fξ 4 Definition Let fx is a locally integrable function he Fourier transform fω of fxis definition by fω e iωx fxdx 5 and the inverse Fourier transform of fω also defined by fx F fω π 3 Main Results e iωx fωdω 6 hus from hus ln st s µ σ t + σξτ t ξ s tσ ln µ s σ + σ heorem 3 he equation given by with the terminal condition given by 4 has a solution V ξ, t Kξ, t fξ in the convolution form, where r Kξ, t π t e tr exp σ σ ln t + ξ ] t is the kernel of Advance online publication: 8 May 8

3 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 4 Proof ake the Fourier transform defined by 5 to, we obtain is bounded hus the inversion V ω, t ω t t V ω, t+ r t σ σ iω V ω, t r V ω, t hus and from 4, and hus r V ω, t Cωe rt e ω t iω σ σ lnt V ω, t fω V ω, Cωe r e ω r iω σ σ ln Cω fω ω r e iω r σ σ ln fωe r + ω + iω r σ σ ln V ξ, t π π exp π exp dydω e iωξ V ω, tdω e iωξ tr e t ω ln r t iω σ σ ] fωdω tr e e ξ yiω tr e π t ω ln t iω r σ σ t exp ω fydydω tr e π exp t ω ] fy ln t r σ σ + ξ y t ln t r σ σ + ξ y t ln t r σ σ + ξ y ] fydydω t ]] iω i hus Put u V ω, t e tr exp t ω + ln ln tiω r σ σ then du fω, t ω t dω or dω i ln t r σ σ + ξ y t du hus t, as a solution of for < t Now V ω, t e tr e ω t fω Let M max fω Now e tr and e t ω are bounded, thus V ω, t e tr e t ω M K V ξ, t tr e e u du π t exp ln t r σ ] σ ξ y fydy t tr π t e π exp ln t r σ ] σ ξ y fydy, t Advance online publication: 8 May 8

4 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 4 since e u du π hus tr V ξ, t e π t ln exp t r σ σ + ξ y ] fydy t Actually Kξ, t is the kernel of hus V ξ, t Kξ, t fξ in the convolution form Now,Kξ, t tr e t π ln r t σ σ ] + ξ exp t hus Kξ, t is a Gaussian function or normal distribution with mean e tr σ r σ ln t tr t and variance e hus Kξ, t fξ we need to show that as t, V ξ, t δξ fξ fξ that 4 holds hat means lim t Kξ, t δξ where δξ is the Dirac-delta distribution Moreover, we see that the kernel Kξ, t involving the white noise ξ which causes the fluctuation of the price of stock as mentioned before Actually, such kernel plays the significant role for find the particular solution of the nonhomogeneous differential equation For example, given the nonhomogeneous differential equation Lux fx where L is the partial differential operator, then we can find the particular solution ux Kx fx where Kx is the kernel of such equation Corollary 3 From V ξ, t Kξ, t fξ, < t We obtain the following conditions hus lim Kξ, t lim t t e t lim t π t ln exp t r σ σ + ξ ] t δξ δξ hus V ξ, t δξ fξ fξ ii We have lim t + V ξ, t lim t o + Kt, ξ, t fξ By applying L Hospital rule we obtain lim t + Kξ, t hus Now, we have lim V ξ, t fξ t + ] V ξ, t us, t u s exp µ σ t + σtξ, t but from 3, V ξ, it follows that V ξ, us, Now consider the option price at t, s s and we have us, which is not really appear in the real world Actually when t, s s the option price us, need not be zero his can be conclude that the initial condition V ξ, of mat be different from the initial condition us, of 3 heorem 33 Properties of Kξ, t i Kξ, satisfies equation ii Kξ, is a tempered distribution, that is Kξ, S R iii Kξ, > for < t iv e tr Kξ, dξ v lim t Kξ, δξ i V ξ, fξ that is the terminal condition ii lim t + V ξ, t Proof i We need to show that lim t Kξ, t δξ Now, vi Kξ, is Guassian distribution with mean e tr σ r σ ln t tr t and variance e e tr σ r σ ln t, e hat is Kξ, is tr t Proof i By computing directly,kξ, satisfies ii Since Kξ, is a Gaussian function and ln Kξ, t π t e t exp t r σ σ + ξ ] Kξ, LR- the space of integrable function on the real R It follows that Kξ, is N a tempered distribution t Advance online publication: 8 May 8

5 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 4 iii Kξ, > for < t is obvious iv e tr Kξ, tdξ e tr tr e π t r exp σ σ ln t + ξ ] dξ t r exp σ σ ln t + ξ ] dξ π t t Let u π t ξ + r σ σ ln t, t then du dξ or dξ du π t hus e tr Kξ, tdξ t e u du π t t π π t v lim t Kξ, δξ by Corollary 3 vi Since Kξ, t r π t e tr exp σ σ ln t + ξ ] t σ e tr π t ξ exp r ln ] σ t t hus Kξ, t is a Gaussian distribution with mean E exp e tr π ξ σ r σ t e tr σ r σ t ln t ln t, ] where E is expectation And variance V where V is variance hus Kξ, is e tr π t σ ξ exp r σ ln ] t t e tr V π t σ ξ exp r σ ln ] t t e tr t e tr σ r ln, e σ t tr t Note : he solution V ξt of is called the option price in the white noise form where the white noise ξ can be computed from, now tr V ξ, t e π t ln exp t r σ σ + ξ ] y fydy t or V ξ, te tr π t ln exp t r σ σ + ξ ] y fξ t he left hand side of the above equation is the value of money that the option price V ξ, t put in the Bank with the riskless interest r at the time t t 4 Conclusion It is well known that the volatility σ causes the fluctuation of the price of stock But there is another factor that Advance online publication: 8 May 8

6 IAENG International Journal of Applied Mathematics, 48:, IJAM_48 4 called the white noise ξ which is not really well known Such white noise ξ also cause the fluctuation of the price of stock We are succeeded in formulating the white noise ξ given in Such white noise ξ is helpful for the investor to estimate the expected return of the price of stock for trading Moreover, we can relate such white noise ξ to the Black- Scholes equation which is the important area of studying the option prices of the stock market Acknowledgment he authors would like to thank Naresuan University for financial support References ] F Black and M Scholes, he pricing of options and Corporate Liabilities, Journal of Political Economy, Vol 8, No3973, ] A Kananthai, On the Kernel of the Black-Scholes Equation in form of White Noise, Applied Mathematical Science, Vol 4, ] H M Kuo, White Noise Distribution heory, CRC Press, Boca Raton, 996 4] I M Gel fand and G E Shilov, Generalized Functions, Generalized Functions, Vol 964 Advance online publication: 8 May 8