Positivity Preserving Schemes for Black-Scholes Equation

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1 Research Joural of Fiace ad Accoutig IN -97 (Paper) IN -7 (Olie) Vol., No.7, 5 Positivity Preservig chemes for Black-choles Equatio Mohammad Mehdizadeh Khalsaraei (Correspodig author) Faculty of Mathematical ciece, Uiversity of Maragheh, Maragheh, Ira Muhammad.mehdizadeh@gmail.com Reza hokri Jahadizi Faculty of Mathematical ciece, Uiversity of Maragheh, Maragheh, Ira reza.shokri.@gmail.com Abstract Mathematical fiace is a field of applied mathematics, cocered with fiacial markets. I the market of fiacial derivatives the most importat problem is the so called optio valuatio problem, i.e. to compute a fair value for the optio. The solutio of the Black-choles equatio determies the optio price, respectively accordig to the used iitial coditios. I this paper, first we show that the positivity is ot esured with classical fiite differece schemes whe applied to the Black-choles equatio for very small time steps. Next, by reformig the discretizatio of the reactio term of equatio, a family of efficiet explicit schemes are derived that is free of spurious oscillatios aroud discotiuities ad preservig positivity. Keywords: Positivity, Nostadard discretizatio, Black-choles equatio. Itroductio I this work, we are iterested i the optio valuatio problem satisfies the Black-choles partial differetial equatio preseted i [5] as: V V V + r + σ rv =, () t V is the price of the optio ad edowed with iitial ad boudary coditios: where (, t ) V (,) = ( K, ) (), [ L, U ] V (, t) as or, with updatig of the iitial coditio at the moitorig dates t i, i =, L, F : V (, t ) V (, t ) (), where [, ]() is the idicator fuctio, i.e., L U i = i L U [, ] = t < t < L < t = T, F [ ] [ L, U ], if L, U [, ]() = L U if () here, the parameter r > is the iterest rate ad the referece volatility is σ >. To obtai the fiite differece approximatio for equatio (), let the computatioal domai [, ] [, T ] is discretized by a uiform mesh with steps, t i order to obtai grid poits (, t), =, L, M ad =,, L, N so that M = = M ad T = N t. By usig the forward differece for V ad cetered differece for discretizatio of V ad V ad approximatios V of V t at the grid poits, we have the followig explicit fiite differece method []: + V V V + V V V + V + + r + σ rv =. () t This method has lower accuracy ad ofte geerates umerical drawbacks such as spurious oscillatios ad

Research Joural of Fiace ad Accoutig IN -97 (Paper) IN -7 (Olie) Vol., No.7, 5 egative values i the solutio whe applied to ().

Aalytical solutio Explicit method Trucated call optio value (V) 95 5 5 5 Figure. Trucated call optio value for explicit method with =., t =. parameters: 9 K =, U =, r =.5, σ =., T =.

explicit ostadard fiite differece method as follows: L =, r ( a + b ) V = + ra V + r V t t r σ + + rb V +, r σ σ + () ad the matrix form of the () is t + r ( a + b ) V = AV, (5)

2 Research Joural of Fiace ad Accoutig IN -97 (Paper) IN -7 (Olie) Vol., No.7, 5 egative values i the solutio whe applied to (). Wheever the fiacial parameters of the Black-choles model σ ad r satisfy the relatioship σ r, see Figure. The parameters used i this simulatio is take from [5]. Aalytical solutio Explicit method Trucated call optio value (V) Figure. Trucated call optio value for explicit method with =., t =. parameters: 9 K =, U =, r =.5, σ =., T =., =.. Costructio of ew scheme Followig the suggested strategy i [,,,, 5], by reformig the discretizatio of the reactio term to V,t = av + V + V a + b V +, + we get a family of explicit ostadard fiite differece method as follows: L =, r ( a + b ) V = + ra V + r V t t r σ + + rb V +, r σ σ + () ad the matrix form of the () is t + r ( a + b ) V = AV, (5) where A is the followig tridiagoal matrix r σ σ r σ A = + ra r + rb t,,. () The costat a is chose accordig to the followig theorem: Theorem. ufficiet for scheme (5) to be positive is r r a, b, t <. σ σ σm + r (7) Proof. From (5) it is eough to show that r σ + ra, ()

3 Research Joural of Fiace ad Accoutig IN -97 (Paper) IN -7 (Olie) Vol., No.7, 5 r σ + rb, (9) t σ r, () r ( a b ), t + () from () we ca write σ r ra, σ r a r ( ), σ r r r a, r + σ σ σ () σ r r a, r σ σ σ r r a ( ), r σ σ ow, the last iequality i () shows sufficiecy of a r σ for (7), ad similarly, from (9) we derive σm + r r that b. I the other had () ad () leads to t < σ The proposed positive scheme is coverget due to followig theorem:, ad the proof is complete. Theorem. Uder coditios (7), the ew scheme is stable ad coverget with local trucatio error O t, for a b O t, otherwise. = ad t + for spectral radius, ρ of the iteratio matrix we derive Proof. Uder coditios (7) we have r ( a b ) A = r a b r, t + ad. I the other had we have r ( a + b ) r ρ A A A t, < r ( a + b ) r ( a + b ) r ( a + b ) r ( a + b ) t t t t therefor the scheme is stable ad coverget with local trucatio error: T V (, t + ) V (, t ) V ( +, t ) V (, t ) = + r t V (, t ) (, t ) (, t ) V + V + + σ ( + + ) r av (,t ) + V (,t ) + bv (,t ) ( a + b) V (,t ), ()

substitutio ito the expressio for T the gives V V V V V T = + r + σ rv t r( a b) + t t t t r V V r V ( a + b) + r( a + b) t + ( a + b) t + L.

. Numerical Results To illustrate the advatage of the desiged ew positive explicit scheme see Figure that shows the ew explicit scheme is positivity-preservig ad

4 Research Joural of Fiace ad Accoutig IN -97 (Paper) IN -7 (Olie) Vol., No.7, 5 by Taylor s expasio, we have V V V + t t t V (, t ) = V (, t ) + t + t + t + L, V V V + V (, t ) = V (, t ) L, V V V + V (, t ) = V (, t ) + + L, substitutio ito the expressio for T the gives V V V V V T = + r + σ rv t r( a b) + t t t t r V V r V ( a + b) + r( a + b) t + ( a + b) t + L. But V is the solutio of the Black-choles equatio so (5) Therefore, if a b = the T O ( t, ) V V V + r + σ rv. = t =, otherwise T O ( t, ) =.. Numerical Results To illustrate the advatage of the desiged ew positive explicit scheme see Figure that shows the ew explicit scheme is positivity-preservig ad spurious oscillatios are avoided. 9 Aalytical solutio New method Trucated call optio value (V) Figure. Trucated call optio value for ew explicit method with =., t =. parameters: L = 9, K =, U =, r =.5, σ =., T =., =. I the case of larger time step, we see the same behavior, see Figure. These umerical results are obtaied with σ r.

5 Research Joural of Fiace ad Accoutig IN -97 (Paper) IN -7 (Olie) Vol., No.7, 5 9 Aalytical solutio New method 9 Trucated call optio value (V) Trucated call optio value (V) Time(t) Figure. Trucated call optio value for ew explicit method with =., t =. parameters: L = 9, K =, U =, r =. 5, σ =., T =., =.. Coclusios ad discussio We costructed a family of explicit method based o a ostadard discretizatio scheme to solve optio valuatio problem with double barrier kock-out call optio. I particular, the proposed method uses a ostadard discretizatio i reactio term ad the spatial derivatives are approximated usig stadard fiite differece scheme. It has bee show that the proposed ew scheme preserve the positivity as well as stability ad cosistece. Future work will iclude extedig the method to oliear Black-choles equatio. Refereces [] Mehdizadeh Khalsaraei, M. & Khodadoosti, F. () A ew total variatio dimiishig implicit ostadard fiite differece scheme for coservatio laws. Computatioal Methods for Differetial Equatios,, 5 9 [] Mehdizadeh Khalsaraei, M. & Khodadoosti, F. () Nostadard fiite differece schemes for differetial equatios. ahad Commu. Math. Aal, Vol. No., 7 5 [] Mehdizadeh Khalsaraei, M. & Khodadoosti, F. Qualitatively stability of ostadard -stage explicit RugeKutta methods of order two, Computatioal Mathematics ad Mathematical physics. I press. [] Mickes, R. E. (99) Nostadard Fiite Differece Models of Differetial Equatios. World cietific, igapore. [5] Milev, M. & Tagliai, A. (). Efficiet implicit scheme with positivity preservig ad smoothig properties, J. Comput. Appl. Math.,, 9 [] Tagliai, A., Fusai, G. & afelici,. (). Practical Problems i the Numerical olutio of PDEs i Fiace. Redicoti per gli tudi Ecoomici Quatitativi,, 5 5

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Hopscotch and Explicit difference method for solving Black-Scholes PDE

Hopscotch and Explicit difference method for solving Black-Scholes PDE Mälardale iversity Fiacial Egieerig Program Aalytical Fiace Semiar Report Hopscotch ad Explicit differece method for solvig Blac-Scholes PDE Istructor: Ja Röma Team members: A Gog HaiLog Zhao Hog Cui 0