I'm currently learning Black-Scholes-Merton partial differential equation, and there are some confusions I can't work out.
Under the Black-Scholes assumption, we have: $$df=\left(\frac{\partial f}{\partial S}\mu S+\frac{\partial f}{\partial t}+\frac 12\frac{\partial^2f}{\partial S^2}\sigma^2S^2\right)dt+\frac{\partial f}{\partial S}\sigma S dB_t$$
To construct a portfolio: $$\Pi=-f+\frac{\partial f}{\partial S}S$$ We can eliminate the randomness, so we have: $$\Delta \Pi=\left(-\frac{\partial f}{\partial t}-\frac 12\frac{\partial^2f}{\partial S^2}\sigma^2S^2\right)\Delta t$$
Any riskless asset must satisfy $\Delta\Pi=r\Pi\Delta t$, then we can reach to the differential equation.
My question is:
- Is S in $\Delta \Pi=\left(-\frac{\partial f}{\partial t}-\frac 12\frac{\partial^2f}{\partial S^2}\sigma^2S^2\right)\Delta t$ still a stochastic process? or the stock price at the beginning, which is deterministic?
- If S is a stochastic process, then according to the Geometric Brownian Motion assumption, $S=S_0e^{\sigma B_t+(\mu-\frac12\sigma^2)t}$ , which contains $\mu$ as a parameter. Then how can we say it has no dependence on $\mu$ and replace $\mu$ with $r^f$ in the risk-neutral world?