SDE_Aug28FSU

SDE_Aug28FSU - Slides for MAD 5932-01 SCS-MATH FSU...

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Unformatted text preview: Slides for MAD 5932-01, SCS-MATH FSU Tallahassee * Prof. R. Tempone Version: Thursday August 31, 2006 * Based on the lecture notes Stochastic and Partial Differential Equations with Adapted Numerics, by J. Goodman, K.-S. Moon, A. Szepessy, R. Tempone, G. Zouraris. Aug 29, 2006 - Class contents: 1. Course Introduction, Admin details 2. Motivating examples (Chapter 1) 3. Brief Probability review 1 Admin details- syllabus- class location- student groups, email list, order of groups for assignments- HMW presentations by groups on Thurs- days. /Hand in days are Tuesdays for the group that makes the presentation/ 2- Matlab access, SCS computer facilities, per- sonal accounts.- course webpage Course goal: to understand numerical meth- ods for problems formulated by stochastic or partial differential equations models in sci- ence, engineering and mathematical finance. 3 Motivating examples (Chapter 1) 4 Example 1 (Noisy Evolution of Stock Values) Denote stock value by S ( t ) . Assume that S ( t ) satisfies the differential equation dS dt = a ( t ) S ( t ) , which has the solution S ( t ) = e R t a ( u ) du S (0) . Since we do not know precisely how S ( t ) evolves we would like to generalize the model to a stochastic setting a ( t ) = r ( t ) + ” noise ” . 5 For instance, we will consider dS ( t ) = r ( t ) S ( t ) dt + σS ( t ) dW ( t ) , (1) where dW ( t ) will introduce noise in the evo- lution. What is the meaning of ( 1 ) ? The answer is not as direct as in the deterministic ode case. One way to give meaning to ( 1 ) is to use the Forward Euler discretization, S n +1- S n = r n S n Δ t n + σ n S n Δ W n . (2) Here Δ W n are independent normally distributed random variables with zero mean and vari- ance Δ t n , i.e. E [Δ W n ] = 0 and V ar [Δ W n ] = Δ t n = t n +1- t n . Then ( 1 ) is understood as a limit of ( 2 ) when maxΔ t → . Applications to Option pricing European call option: is a contract signed at time t which gives the right, but not the obligation, to buy a stock (or other financial instrument) for a fixed price K at a fixed future time T > t . At time t the buyer pays the seller the amount f ( s,t ; T ) for the option contract. What is a fair price for f ( s,t ; T )? 6 The Black-Scholes model for the value f : (0 ,T ) × (0 , ∞ ) → R of a European call option is the partial differential equation ∂ t f + rs∂ s f + σ 2 s 2 2 ∂ 2 s f = rf, < t < T, f ( s,T ) = max( s- K, 0) , (3) where the constants r and σ denote the risk- less interest rate and the volatility, respec- tively. 7 Stochastic representation of f ( s,t ) The Feynmann-Kaˇ c formula gives the alternative probability representation of the option price f ( s,t ) = E [ e- r ( T- t ) max( S ( T )- K, 0)) | S ( t ) = s ] , (4) where the underlying stock value S is mod- eled by the stochastic differential equation ( 1 ) satisfying S ( t ) = s ....
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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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SDE_Aug28FSU - Slides for MAD 5932-01 SCS-MATH FSU...

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