The price of a call option, denoted c, is a function of the stock price and time remaining: c = c(S,t). Assuming a stock price follows a geometric Brownian motion dS = Sdt + St^1/2 where ~N(0,1) we can use a Taylor series to estimate c(S,t) = c(S₀ ,t₀ ) + c_sdS + c_tdt + [c_ssdS² + c_stdSdt + c_ttdt² ]

Substituting in dS, eliminating, and collecting terms, we end up with:

dc = [c_sS + c_t + c_ss ²S²]dt + [c_sS]dt^1/2

If we then set-up a portfolio of a call and a short position in stock in the proportion S, we have = c - DS. The change in the portfolio value is d = dc - DdS

As we have defined dc and dS above, substitute these in.

Proceed from this point to where you derive the famous BS PDE: -c_rf + c_sSr_f + c_t + c_ss²S² = 0

Hint: you will need to recognise that d = r_fdt, where r_f is the risk free rate. explain as you go.