How Risk-Neutral Pricing Relates Derivative Values to Risk Premia

Summary

The document addresses why risk-neutral pricing can value derivatives despite investors demanding compensation for risk. Its explanation distinguishes a derivative’s pricing relationship to the underlying asset from the underlying asset’s expected return. A derivative’s future return can depend on the underlying’s return, so changes in the underlying’s risk and price affect the derivative’s value and expected return as well.

The answer emphasizes that a pricing formula expresses a derivative’s price in relation to the underlying spot price, rather than independently specifying the underlying’s expected return. It offers an intuitive explanation, not a derivation of risk-neutral valuation or a discussion of how the risk-neutral measure incorporates risk premia. The claim that a derivative has no separate risk of its own is stated broadly; actual derivative exposures can depend on contract terms and other market factors.

Key ideas

• A derivative’s price is expressed in relation to the underlying asset’s price.
• The derivative’s return can depend on the underlying asset’s return and expected return.
• Risk-neutral pricing does not mean that the underlying’s risk premia have no effect on market prices.
• The explanation is conceptual and does not derive the risk-neutral measure or pricing formula.
• Derivative exposures can also depend on contract terms and other market factors.

Tags

# How can risk-neutral pricing find the right price for securities if it doesn't account for risk premia?


# How can risk-neutral pricing find the right price for securities if it doesn't account for risk premia?












I'm confused as to how a method that values securities purely on their expected return works in the real world if it doesn't take into account the fact that investors demand a higher return for greater variance in expected return.

## Answer by Kiwiakos (score 4)

https://quant.stackexchange.com/a/31898

The price of a derivative does not explicitly depend on the expected return of the underlying, however the price change or return of the derivative depends on the return of the underlying. Hence the expected return of the derivative depends on the expected return of the underlying, which is what matters.

Also remember that the price is a function of the spot of the underlying, and this is what a pricing formula does: establishes how the derivative is priced relative to the underlying price. A derivative inherits all risk from the underlying, it does not have its own risk, so to speak. If the underlying becomes very risky, then its price will fall to offer higher expected return in the future; the derivative price will also change to reflect that.

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