Delta Hedging and Risk-Neutral Option Valuation

Summary

The note explains why the Black-Scholes option pricing equation does not depend on the stock’s expected return. Delta hedging combines an option position with the underlying asset to remove instantaneous exposure to stock-price movements. In the model, the resulting hedged portfolio is locally riskless, so no risk premium is required; its return must match the risk-free rate. This leads to the same pricing equation as if the stock’s expected return equaled the risk-free rate, which motivates risk-neutral valuation.

The evidence is a brief conceptual explanation rather than a derivation or worked example. The claim relies on Black-Scholes assumptions, including the ability to hedge continuously and the framework’s complete-market structure. It should not be read as saying investors are actually risk-neutral, or that the stock’s real-world expected return is the risk-free rate. The method concerns derivative pricing under the stated model; practical hedging frictions and model violations can limit its application.

Key ideas

• Delta hedging offsets an option’s local exposure to movements in the underlying stock.
• In the Black-Scholes framework, the hedged portfolio earns the risk-free rate because it is locally riskless.
• The pricing equation does not require the stock’s real-world expected return.
• Risk-neutral valuation is a pricing technique, not a claim that investors are truly risk-neutral.

Tags

# In the derivation of the Black-Scholes PDE, using delta hedging, how is this linked to the risk neutral valuation?


# In the derivation of the Black-Scholes PDE, using delta hedging, how is this linked to the risk neutral valuation?












I was reading this paper: http://www.columbia.edu/~mh2078/FoundationsFE/BlackScholes.pdf

I don't understand the paragraph here: "The most interesting feature of the Black-Scholes PDE (8) is that µ does not appear1 anywhere. Note that the Black-Scholes PDE would also hold if we had assumed that µ = r. However, if µ = r then investors would not demand a premium for holding the stock. Since this would generally only hold if investors were risk-neutral, this method of derivatives pricing came to be known as risk-neutral pricing."

## Answer by siou0107 (score 3)

https://quant.stackexchange.com/a/71813

It means that in the Black-Scholes framework (or more generally in a complete markets framework), since you can hedge all the risk away (by delta-hedging), you can price everything assuming that no risk premium is required, i.e. $\mu = r$.

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