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Erschienen in:
2023 | OriginalPaper | Buchkapitel
6. Numerical Methods for PDE
verfasst von : Rituparna Sen, Sourish Das
Erschienen in: Computational Finance with R
Verlag: Springer Nature Singapore
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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 \(V(S_t, t)\) be the price of an option at time t when the share price of the underlying stock is \(S_t\). 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 \(B_t\) and a risky asset stock \(S_t\). They evolve as where r is the interest rate, \(\mu \) is the drift, \(\sigma \) 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 \(B_t^{-1}V(S_t,t)\). 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 \(\mu =r\). We have, from the above, For a martingale, the coefficient of the dt term has to be zero, otherwise there is a systematic drift.