HW4 - Numerical Methods of SDEs-Dr Ra´ul Tempone...

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Unformatted text preview: Numerical Methods of SDEs-Dr. Ra´ul 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 ) =- α (1 + Y ( t )) + 0 . 4 √ α p 1- ρ 2 dt + 0 . 4 √ αd b Z ( t ) , (2) where W and Z are independent Wiener process and b Z ( t ) ≡ ρW ( t ) + p 1- ρ 2 Z ( t ) (3) Note: The correlation coefficient is ρ =- . 3. To solve the SDE (1), define the following: f ( x ) = log( x ) (4) Solving the SDE (1) by Itˆ 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 =- α (1 + Y t ) + 0 . 4 √ α p 1- ρ 2 dt + 0 . 4 √ αd b Z t = - αY t + 0 . 4 √ α p 1- ρ 2- α | {z } = M dt + 0 . 4 √ α | {z } σ d b Z t = (...
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This note was uploaded on 12/14/2011 for the course MAD 5932 taught by Professor Gallivan during the Fall '06 term at FSU.

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HW4 - Numerical Methods of SDEs-Dr Ra´ul Tempone...

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