Chapter II - Numerical & Simulation Techniques...

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Unformatted text preview: Numerical & Simulation Techniques in Finance FRE 6251 Chapter II Everything PDE Edward D. Weinberger, Ph.D, F.R.M. Adjunct Assoc. Professor Polytechnic University edw@panix.com OUTLINE Chapter IIA Everything PDEs, 1 Dim Types of PDEs, boundary conditions Analytic Solutions to Heat Equation Finite Difference Methods for H. Eqn. Von Neumann Stability Sparse Matricies Trees, Binary and Trinary (next week) Discrete Dividends Computing Greeks TYPES OF PDES Linear vs Nonlinear IVPs, boundary value, free boundary problems Types of linear PDEs First order Second order Elliptic Hyperbolic Parabolic (Black Scholes Equation) THE HEAT EQUATION Interpretations: Flow of heat as a function of time Probability density of diffusing particle as a function of time Prototypical parabolic PDE Initial temperature is f ( x ) for a x b Initial temperature is f ( x ) for - x 2 2 2 1 x u t u = FROM BLACK SCHOLES TO THE HEAT EQUATION T + rS S + 2 S 2 SS = r Set S = e x (Logarithmic vs arithmetic RW) T + ( r - 2 ) x + 2 xx = r Replace by e rT v (reverse discounting) v T + ( r - 2 ) v x + 2 v xx = Replace v ( x, T ) by u ( x + ( r - 2 ) T, T ) (no drift) u T + 2 v xx = Replace T by t/ 2 THE HEAT EQUATION: Analytic Solutions Separation of variables Assume u ( x, t ) = X ( x ) T ( t ) X / X = = 2 T / T Decaying sine waves Solution via Fourier Transform, F u ( , t ) d F u ( , t )/ dt = - 2 F u ( , t ) , F u ( , 0) = F f ( ) F u ( , t ) = F u ( , 0) exp(- 2 t/ 2) FOURIER TRANSFORM SOLUTION ( 29 ( 29 ( 29 source point a for 2 2 1 , 2 / 2 / 2 2 t e dy e y f t t x u t x t y x - --- = = FINITE DIFFERENCE METHODS ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 , , 2 , OR , , 2 , , , x t t x x u t t x u t t x x u x u x t x x u t x u t x x u x u t t x u t t x u t u +- + +- + + - +- + - + EXPLICIT (FTCS) FINITE DIFFERENCE METHOD ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] t x x u t x u t x x u x t t x u t t x u x t x x u t x u t x x u t t x u t t x u , , 2 , 2 , , , , 2 , 2 1 , , 2 2 - +- + + = +- +- + - + EXPLICIT (FTCS) FINITE DIFFERENCE METHOD u ( x- x , t ) u n j- 1 u ( x , t ) u n j u ( x + x , t ) u n j+ 1 u ( x , t + t ) = u n j FULLY IMPLICIT FINITE DIFFERENCE METHOD u ( x- x , t + t ) u ( x , t + t ) u ( x + x , t + t ) u ( x , t ) FULLY IMPLICIT FINITE DIFFERENCE METHOD ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 t x u t t x x u x t t t x u x t t t x x u x t x t t x x u t t x u t t x x u t t x u t t x u , , 2 , 1 , 2 o solution t the requires step each that implies , , 2 , 2 1 , , 2 2 2 2 = +-- + + + + +- +- + +...
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Chapter II - Numerical & Simulation Techniques...

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