"Numerical Methods for PDE" by Rituparna Sen and Sourish Das
 

Numerical Methods for PDE

Document Type

Book Chapter

Publication Title

Indian Statistical Institute Series

Abstract

The time evolution of prices of different financial quantities is often represented as a partial differential equation (PDE) with independent variables being time and prices of some other, often underlying, assets. Let V(St, t) be the price of an option at time t when the share price of the underlying stock is St. See Appendix A.1 for background on mathematical finance that is used in what follows. Under the Black–Scholes set-up, we have a risk-less asset bond Bt and a risky asset stock St. They evolve as where r is the interest rate, μ is the drift, σ is the volatility and W is a standard Brownian motion(BM). We apply Ito’s formula to the option price to get Consider the discounted option price Bt-1V(St,t). By the Fundamental Theorem of Arbitrage Pricing (see Appendix), the discounted option price must be a martingale under the risk-neutral measure. Also, under the risk-neutral measure μ= r. We have, from the above, For a martingale, the coefficient of the dt term has to be zero, otherwise there is a systematic drift.

First Page

53

Last Page

64

DOI

10.1007/978-981-19-2008-0_6

Publication Date

1-1-2023

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