SDE_HW4

SDE_HW4 - Numerical Methods of SDEs-Dr. Raul Tempone...

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Unformatted text preview: Numerical Methods of SDEs-Dr. Raul Tempone Homework #4 Exercise 1 The following stochastic volatility model [FPSS00] generalizes the well known Black-Scholes geometric Brownian motion model [BS73], improving some aspects of option pricing. A simplified version of the model reads dS ( t ) = rS ( t ) dt + e Y ( t ) S ( t ) dW ( t ) (1) dY ( t ) =- (1 + Y ( t )) + 0 . 4 p 1- 2 dt + 0 . 4 d b Z ( t ) , (2) where W and Z are independent Wiener process and b Z ( t ) W ( t ) + p 1- 2 Z ( t ) (3) Note: The correlation coefficient is =- . 3. To solve the SDE (1), define the following: f ( x ) = log( x ) (4) Solving the SDE (1) by It o formula: d log S t = 1 S t dS t- 1 2 S 2 t ( dS t ) 2 = r dt + e Y t dW t- 1 2 e 2 Y t d t = r- 1 2 e 2 Y t dt + e Y t dW t (5) Integrating (5), obtain the following: log S t = log S + Z t ( r- 1 2 e 2 Y s ) ds + Z t e Y s dW s S t = S e R t ( r- 1 2 e 2 Ys ) ds + R t o e Ys d W s (6) Solving the SDE (2): dY t =- (1 + Y t ) + 0 . 4 p 1- 2 dt + 0 . 4 d b Z t = - Y t + 0 . 4 p 1- 2- | {z } = M dt + 0 . 4 | {z } d b Z t = (...
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SDE_HW4 - Numerical Methods of SDEs-Dr. Raul Tempone...

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