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

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = +-- + + + + +- +- + +...
View Full Document

{[ snackBarMessage ]}

Page1 / 63

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

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online