SDE_Aug28FSU

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

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

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

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

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 ....
View Full Document

## This note was uploaded on 12/14/2011 for the course MAD 5932 taught by Professor Gallivan during the Fall '06 term at FSU.

### Page1 / 98

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

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

View Full Document
Ask a homework question - tutors are online