Risk-Neutral Expected Returns in the Black–Scholes Model

Summary

The document considers why the Black–Scholes stock-price expression includes volatility while its expected price growth appears independent of volatility. It asks whether this means investors should accept a risky asset whose expected return equals the risk-free rate, and whether options prices imply a risk premium.

The key clarification is that Black–Scholes pricing uses a risk-neutral probability measure. Under that measure, the expected growth rate of the underlying, after accounting for dividends where applicable, is tied to the risk-free rate for valuation. This is a pricing device, not a claim that investors in the real world expect every risky asset to earn the risk-free rate. The question is conceptual and the brief answer supplies no derivation or empirical evidence. It does not discuss how real-world expected returns or risk premia are estimated.

Key ideas

• The Black–Scholes price process is evaluated under a risk-neutral measure for pricing.
• Under that measure, expected asset growth is linked to the risk-free rate rather than volatility.
• This pricing assumption does not imply that real-world investors expect risky assets to earn only the risk-free rate.
• The document clarifies the distinction but gives no empirical estimate of risk premia.

Tags

# Does Black-Scholes imply that the expected return for an asset is fixed as the volatility increases?


# Does Black-Scholes imply that the expected return for an asset is fixed as the volatility increases?












I'm new to this, and just trying to understand what options prices imply about asset growth. I'm looking at the following expression for the underlying asset price in the Black-Scholes model, in particular:

`S(t) = S(0)*exp((r - 0.5*\sigma^2)*dt + \sigma*dW(t)`

Where `S(t)` is the asset price at time `t` in years, `r` is the risk-free rate, `\sigma` is the volatility.

Which, implies that that expected price of the asset at time `t` can be computed as `exp(rt)` -- independent of `\sigma`.

What I'm trying to understand is why a rational investor would purchase a risky asset if the expected growth rate was the same as the risk-free rate. Wouldn't it be expected that if an asset is more risky then investors would only purchase it if it offered higher returns than a risk-free asset?

## Answer by fairidox (score 1, accepted)

https://quant.stackexchange.com/a/74437

As per the comments above, Black-Scholes assumes a risk-neutral measure.

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