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Unformatted text preview: 1 Temple University Continuous Time Finance Discretization of Ito Process 2 Discretization of Ito Process Continuous Time Models 3 Formula for Ito Process ) )( , ( ) , ( ) , ( ) , ( (t)dW (t)dt (s)ds (s)dW(s) X(0) X(t) 2 2 1 t t t t t xx t t x t t t t dX X t f dX X t f dt X t f X t df dX + + = + = + + = Stochastic Calculus (in Finance) is little more than repeated use of this formula 4 Continuous Time Finance Discretization of Ito Process Discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is first step toward making them suitable for numerical evaluation. 5 Discretization of Ito Process Drift Coefficient 6 Discretized Stock Process t t S t S t t S dt t S t dS e S t S t + = + = = ) ( ) ( ) ( ) ( ) ( ) ( ) ( 7 Discretized Stock Process t t e t S t t S t t S t t S dt t S d t S t S dt t S t dS e S t S = + + = + = + = = = ) ( ) ( ) ( ln ) ( ln ) ( ln ) ( ln ) ( ln ) ( ) ( ) ( ) (...
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 Fall '09
 Soss
 Finance, Numerical Analysis, Brownian Motion, dt, Stochastic differential equation, GBM Paths

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