I recently met an options trader that said to me that the price of an option is the expected present value of the payoffs of an option (present value as in discount by the risk free rate and expected value as in real world not risk neutral probabilities).

Anyway I tried to demonstrate to him why this approach is flawed. So typically most books eg Mark Joshi's Concepts and Practice of Mathematical Finance (which is a fantastic book in my opinion) demonstrate why this is wrong by constructing an arbitrage opportunity.

Unfortunately the options trader just dismissed it and started telling me about what Black, Scholes and Merton did...

So my question is there a reference where Black, Scholes or Merton actually say that this is not true in their original framework?

EDIT A more recent survey paper written by them or something along those lines would also be extremely helpful.

  • $\begingroup$ The trader made a specific claim - so he has the burden of proof to provide something supporting it. I don't think you'll find either of the terms "risk neutral" or "real world" in the two 1973 papers though. They both develop the pricing formulae via dynamic hedging and the fundamental valuation PDE, not via the martingale approach. $\endgroup$ Jun 19, 2017 at 22:22
  • $\begingroup$ Thanks for that. Do you happen to know if they write about the martingale approach being equivalent to the PDE approach later on? Or perhaps something that they wrote for a more general audience? $\endgroup$
    – RNvsRW
    Jun 19, 2017 at 22:33
  • $\begingroup$ The first papers that made a clear use of the martingale approach were Harrison & Kreps Martingales and arbitrage in multiperiod securities markets 1979 for the discrete time case and Harrison & Pliska Martingales and stochastic integrals in the theory of continuous trading 1981 for the continuous time case. $\endgroup$ Jun 20, 2017 at 6:45

1 Answer 1


Paul Wilmott's book Frequently Asked Questions in Quantitative Finance shows 10 different ways of proving the Black-Scholes equation.

One of them is certainly using the risk-neutral pricing approach, so just showing that this approach is equivalent to the B-S result should be sufficient to prove it is sound.

If the trader said that B-S used discounted expected payoff to yield their formula, he understood strictly nothing to what they did.

Also, remember that their approach is based on strong assumptions, one of which being the GBM dynamics and the other being that you can trade the underlyings (to dynamically hedge).


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