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I'm studying the risk-neutral derivation of Black-Scholes formula and feel confused about the requirement for the volatility of the underlying asset and the risk-free rate to be constant. It seems that in all stages of the derivation:

  1. Change probability measure to make discounted stock price a martingale
  2. Show that stock price is a Geometric Brownian Motion with drift equal to the risk-free rate
  3. Prove that the discounted wealth of any replicating portfolio consisting of stock and bond is a martingale
  4. Derive the formula using the above conclusion

It doesn't really matter weather the risk-free rate or volatility are constants or change with time. I wonder if the assumptions are removed, what stages of the derivation will not hold?

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  • $\begingroup$ You should not confuse the BS pricing formula and the BS pricing PDE. You are right in that the pricing PDE easily generalises to a term structure of risk-free rates or a term structure of volatilities and so does the pricing formula. Actually, the PDE still holds when volatility is a deterministic function of $(t,S_t)$ e.g. local volatility models. But there are no closed form formulae for the price of vanilla options in that case. Incomplete models (e.g. Heston) verify a different PDE. Deriving a closed form pricing formula for vanillas is however not possible (semi-closed form) etc. $\endgroup$ – Quantuple Jan 19 '18 at 13:59

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