# Chapter II - Numerical& Simulation Techniques in Finance...

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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 [email protected] OUTLINE Chapter IIA – Everything PDE’s, 1 Dim • Types of PDE’s, 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 PDE’S • Linear vs Nonlinear • IVP’s, boundary value, free boundary problems • Types of linear PDE’s • 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 φ T + rS φ S + ½ σ 2 S 2 φ SS = rφ • Set S = e x (Logarithmic vs arithmetic RW) φ T + ( r - ½ σ 2 ) φ x + ½ σ 2 φ xx = rφ • Replace φ by e rT v (reverse discounting) v T + ( r - ½ σ 2 ) v x + ½ σ 2 v xx = • Replace v ( x, T ) by u ( x + ( r - ½ σ 2 ) T, T ) (no drift) u T + ½ σ 2 v xx = • Replace T by – t/σ 2 THE HEAT EQUATION: Analytic Solutions • Separation of variables Assume u ( x, t ) = X ( x ) T ( t ) X’’ / X = λ = 2 T ’/ T Decaying sine waves • Solution via Fourier Transform, F u ( ω, t ) d F u ( ω, t )/ dt = - ½ ω 2 F u ( ω, t ) , F u ( ω, 0) = F f ( ω ) F u ( ω, t ) = F u ( ω, 0) exp(- ω 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 π π- ∞ ∞--- = = ∫ 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 δ δ δ δ δ δ δ δ δ δ δ +- + +- + + ≈ ∂ ∂- +- + ≈ ∂ ∂- + ≈ ∂ ∂ 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 δ δ δ δ δ δ δ δ δ δ- +- + + = +- +- + ≈- + EXPLICIT (FTCS) FINITE DIFFERENCE METHOD u ( x- δx , t ) u n j- 1 u ( x , t ) u n j u ( x + δx , t ) u n j+ 1 u ( x , t + δt ) = u n j FULLY IMPLICIT FINITE DIFFERENCE METHOD u ( x- δx , t + δt ) u ( x , t + δt ) u ( x + δx , t + δt ) 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 in Finance...

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