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Why Black–Scholes Pricing Uses Risk-Neutral Expectations

Article Quant Q&A · Author: RNvsRW
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Summary

The document challenges the claim that an option’s Black–Scholes price is the discounted expected payoff under real-world probabilities. The response recommends comparing this claim with risk-neutral valuation, which uses a pricing measure consistent with arbitrage-free valuation, and notes that this approach can be shown to produce the Black–Scholes result. Discounting a real-world expected payoff generally does not provide that result because expected returns depend on risk preferences and cannot substitute for a no-arbitrage pricing argument.

The response also stresses the model’s assumptions: geometric Brownian motion for the underlying and the ability to trade it dynamically for hedging. It cites a book presenting multiple derivations but gives no direct quotation or original-paper reference, despite the question asking for one. Thus, it conveys the core distinction and important limitations, but leaves the requested historical source unresolved.

Key ideas

  • Black–Scholes pricing is consistent with discounted expected payoffs under risk-neutral probabilities, not generally real-world probabilities.
  • Risk-neutral valuation is one route to the Black–Scholes result and rests on no-arbitrage reasoning.
  • The framework assumes geometric Brownian motion for the underlying and dynamic trading for hedging.
  • The response suggests a later reference but does not identify a direct statement by the original authors.

Tags

OptionsDerivatives pricingRisk management
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# Answer by SRKX (score 1)


# Reference that states that the price of an option is not the expected present value of the payoffs under Black and Scholes?












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.

## Answer by SRKX (score 1)

https://quant.stackexchange.com/a/34776

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).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.