留まる所知らない
留まる所知らない
伊藤の補助定理の導出と使い方を相変わらず考えています。
It seems that the path of derivation of Black Scholes is
Brownian Motion->Geometric Brownian Motion->Ito's Lemma->Black-Scholes PDE
Geometric Brownian Motion is premised on: dS/S = mu*dt + sigma*dB
Determining the meaning of dB is not so simple, though.
1. Can the phi/F expression involving the error function be simplified?
Given that phi(x) = (1/2)(1+erf*1Integral(0,x)(e^(-t^2))dt, where the first parenthetical clause is the boundaries of integration,
how do we deduce F^-1 in terms of phi^-1?
We know phi^-1(p) = sqrt(2)*erf^-1(2*p-1), as then phi^-1(phi(x))= x.
By similar reasoning, F^-1(p;mu,sigma^2) = sqrt(2*sigma^2)*erf^-1(2*p-1)+mu.
Thus, F^-1(p;mu,sigma^2) = sigma*phi^-1(p) + mu
So, if the mean is 0 (as it is for Brownian motion), then a quantile of a normal distribution with volatility of sqrt(dt) will equal the quantile in a standard normal distribution times sqrt(dt). Thus, dB can be taken to equal a drawing from N(0,1)*sqrt(dt). Thus, dB provides a weighting of the volatility's effect on the percent change in S in Geometric Brownian Motion.
2. How is the chain rule related to Ito?
