OPTION PRICING WITH PADÉ APPROXIMATIONS

C om m unfacsciu niva nkseries A 1 Volum e 61, N um b er, Pages 45 50 (01) ISSN 1303 5991 OPTION PRICING WITH PADÉ APPROXIMATIONS CANAN KÖROĞLU A In this paper, Padé approximations are applied Black-Scholes model which reduces to heat equation This paper shows various Padé approximaitons to obtain an effective and accurate solution to the Black-Scholes equation for a European put/call option pricing problem At the end of the paper, results of closed-form solution of Black-Scholes problem, solution of Crank- Nicolson approach and the solution of (1, 1), (1, ), (, 0), (, 1), (, ) Padé approximations are given at a table 1 Introduction In recent years, studies of solutions of Black-Scholes partial differential equations have increased Although in 1970 s Merton [1,] and Black and Scholes [3] has formulated Black-Scholes model according to stochastic differential equations, nowadays the model has been solved both stochastic and numerical solutions Especially, in books of Seydel [6], Ugur [7] and Brandimarte [5] the results of examples solved by applying finite differences In this paper, we will give a new approach for solving the Black-Scholes model to reduced heat equation Firstly, as implementing the finite difference algorithms to the diffusion equation, the equation will be transformed the system of ordinary differential equation Then the system will be solved with Padé approximations ((1, 1), (1, ), (, 0), (, 1), (, )) and the results obtained will be compared with results of Crank-Nicolson solution, Closed-Form solution and Matlab solution Padé Approximations Black-Scholes equation for European option V(S,t): V t + 1 σ S V V + (r δ)s rv = 0 (1) S S Received by the editors Nov 16, 01, Accepted: Dec 05, 01 000 Mathematics Subject Classification 91B5, 41A1 Key words and phrases Option pricing, Padé Approximations 45 c 01 A nkara U niversity

46 CANAN KÖROĞLU is parabolic partial differential equation in domain D V = {(S, t) : S > 0, 0 t T } Black-Scholes equation with appropriate variable transformation is equating to heat equation: u τ = u x () Thus, domain of Black-Scholes equation is change with domain D u = {(x, τ) : < x <, 0 τ σ } [6] If x derivative of eq() is changed with following finite difference formula 1 h {u(x h, τ) u(x, τ) + u(x + h, τ)} + O(h ), then eq() can be written as du(τ) = 1 dτ h {u(x h, τ) u(x, τ) + u(x + h, τ)} + O(h ) (3) Heat equation can be reduced ordinary differential equation system as form: d dt V 1 V V N V N 1 = 1 h 1 1 1 1 1 1 That is, above matrix form can be shown as dv(τ) dτ V 1 V V N V N 1 + 1 h V 0 0 0 V N (4) = AV(τ) + b (5) where V (τ) = [V 1, V,, V N 1 ] T is approximation of u, b is a column vector which has zeros and known boundary values and 1 A = 1 1 1 h 1 1 1 is (N 1) order matrix Solution of ordinary differential equation dv dτ such that V (0) = [g 1, g,, g N 1 ] T = g initial condition is = AV + b V (τ) = A 1 b + exp(τa)(g + A 1 b) (6) In step (τ + k), eq(6) can be written as V (τ + k) = A 1 b + exp(ka)(v (τ + A 1 b)

OPTION PRICING WITH PADÉ APPROXIMATIONS 47 [4] In this paper, we have made approximation to exp(ka) with Padé approximations (1, 1) Padé approximation as matrix form: (I 1 ka)v (τ + k) = (I + 1 ka)v (τ) + kb (1, ) Padé approximation as matrix form: (I 1 3 ka)v (τ + k) = (I + 3 ka + 1 6 k A )V (τ) + (I + 1 6 ka)b (, 0) Padé approximation as matrix form: (I ka + 1 k A )(V (τ + k) = V (τ) + (kb 1 k Ab) (, 1) Padé approximation as matrix form: (I 3 ka + 1 6 k A )V (τ + k) = (I + 1 3 ka)v (τ) + (I 1 6 ka)kb (, ) Padé approximation as matrix form: (I 1 ka + 1 1 k A )V (τ + k) = (I + 1 ka + 1 1 k A )V (τ) + kb Padé approximations given above form following systems of linear equations: CV (j+1) = BV (j) + b (j) (7) The matrix C can be written as LU decomposition C = LU, where L is a lower and U is an upper triangular matrix The solution to the system of linear equations (7) can be written, V (j+1) = U 1 L 1 (BV j + b j ) For each Padé approximations, the solution of Black-Scholes Model reduced to heat equation is given The results can be seen at Table 1 Also, in this table the results of Crank-Nicolson solution of Black-Scholes Model reduced to heat equation is illustrated For put option and call option, it has been taken S0 = 10, K = 10, r = 05, sigma = 06, div = 0 and maturity time T = 1 Values at Table 1 and Table has been found respectively put and call options

48 CANAN KÖROĞLU S C-N (1,1) (1,) (,0) 10 168873 168873-8591455509448166 1688514 0 033834 033834-1914057658597 0333149 30 0084809 0084809 048896 008508 40 004170 004170 004057 004190 50 0009358 0009358 0009330 000983 60 0003313 0003313 000378 000316 70 0001435 0001435 0001407 0001351 80 000059 000059 000057 0000530 90 000030 000030 0000305 000073 100 0000169 0000169 0000159 0000135 S (,1) (,) Closed-Form Sol Matlab Sol 10 168888 168884 1593673 1690363 0 033945 033948 084594 034044 30 008490 008490 006680 008533 40 004174 004173 0017639 00559 50 0009330 000939 0006431 0008799 60 000380 000379 00013 0003366 70 0001407 0001407 0000866 0001401 80 000057 000057 0000333 000065 90 0000304 0000305 0000171 000095 100 0000158 0000158 0000085 0000147 Table 1

OPTION PRICING WITH PADÉ APPROXIMATIONS 49 S C-N (1,1) (1,) (,0) 10 088070 088070-85914555090493 087858 0 8983799 8983799 19140567883981 903449 30 1689984 1689984 17055754 1689349 40 5437537 5437537 5439717 5437547 50 377360 377360 3776947 3773515 60 41745939 41745939 4175139 4174587 70 49759815 49759814 49766811 49759713 80 55811386 55811386 558353 5581197 90 66103337 66103334 66099704 6610367 100 73874374 73874366 7387456 73874313 S (,1) (,) Closed-Form Sol Matlab Sol 10 088174 088170 199973 0897 0 90348 90351 8983799 8970 30 16893074 16893074 1687485 16859 40 5437538 5437536 5430801 49868 50 3773571 3773569 377043 331573 60 41745900 41745900 41744440 41339 70 49759780 49759780 49758884 49545 80 55811356 55811356 55810648 577111 90 6610331 6610331 661074 658981 100 73874349 73874349 73873777 74085 Table 3 Conclusion In this study Black-Scholes equation for European put/ call options model solved by using Padé approximations which applied to heat equation which is the classical reduced form of Black-Scholes model Tables 1 and show various estimations of Padé approximations along with Crank-Nicholson (C-N), closed form solutions and Matlab solution The maturity times shown in Tables 1 and illustrate how the closed form (exact) solutions as well as the approximate solutions obtained via Crank-Nicholson solution, various Padé approximations and Matlab solution behave Although the discretizations used for spatial variables are uniform, the discretization for asset price is non-uniform, this generate the slight differences on the maturity times Although, these slight differences insignificant, I think that it s due to the transformation S = e x Hence, a shortcoming of these transformations perhaps that the resulting grid is not uniform for the asset price but it is inevitable in our case Of course, one can get a uniform grid in the asset price by choosing constant step size in the asset price However, the construction of non-uniform grids for the finite difference methods for the heat equation may not be as easy as the one over a uniform grid Another diffi culty is that after solving the equation

50 CANAN KÖROĞLU numerically, a back transformation must be used to interpret the solution in terms of option values which is also a tricky task which we will confront, but this is our prospect study in the future Acknowledgment: This paper was built up on using results in the author s PhD dissertation Özet: Bu makalede Padé yaklaşımları ısı denklemine indirgenen Black-Scholes modeline uygulanıyor Makale Avrupa put ve call opsiyon problemi için Black-Scholes denkleminin etkili ve doğru çözümünü elde etmek için çeşitli Padéyaklaşımlarını gösteriyor Makalenin sonunda tablo halinde Black-Scholes probleminin kapalıçözümü, Crank-Nicolson çözümü ve (1, 1), (1, ),(, 0), (, 1), (, ) Padé yaklaşımlarının çözümleri verilmektedir R [1] R C Merton, Optimum consumption and portfolio rules in a continuous time model, Journal of Economic Theory, Vol3, No4(1971), 373-413 [] R C Merton, Theory of rational option pricing, Bell Journal of Economics and Management Sciences, Vol4, No1(1973), 141-183 [3] F Black and M Scholes, The pricing of options and corporate liabilities, Journal of Political Economiy, Vol81, No3(1973), 637-654 [4] G D Smith, Numerical Solution of Partial Differential Equations, Clarendon Press, 198 [5] P Brandimarte, Numerical Methods in Finance, Wiley Series, 00 [6] R U Seydel, Tools for Computational Finance, Springer, 009 [7] Ö Uğur, An Introduction to Computational Finance, Imperial College Press, 009 Current address: Department of Mathematics, Hacettepe University, 06800, Ankara, TURKEY E-mail address: ckoroglu@hacettepeedutr URL: http://communicationsscienceankaraedutr/indexphp?series=a1