Why Black–Scholes Uses the Risk-Free Drift When Price Uncertainty Vanishes

Summary

The answer resolves a proposed Black–Scholes paradox by considering a stock with no uncertainty. Under the objective probability measure, the stock’s drift must equal the risk-free rate in this deterministic case; otherwise, investors could choose between two certain investments with different growth rates and obtain an arbitrage opportunity.

The explanation compares investing cash in the stock with placing the same amount in a risk-free account over the same horizon. Since both outcomes are certain, their growth rates must match in an arbitrage-free setting. The answer also distinguishes this objective-measure argument from the risk-neutral pricing setup: the risk-free rate is the stock drift used under the risk-neutral measure when deriving the Black–Scholes formula. The reasoning is limited to the stated absence of uncertainty and does not discuss extensions where risks, trading constraints, or other market features affect the argument.

Key ideas

• With no uncertainty, a stock and a risk-free account must have the same drift to avoid arbitrage.
• Different certain growth rates would let investors choose the investment with the higher payoff.
• The risk-neutral stock drift in the Black–Scholes setup is the risk-free rate.
• The argument addresses the deterministic case rather than broader market settings.

Tags

# Black Scholes paradox exercise


# Black Scholes paradox exercise












Any idea where lies the problem? Thank you for suggestions.

## Answer by Quantuple (score 5, accepted)

https://quant.stackexchange.com/a/28327

In the absence of uncertainty, the drift of the stock $\mu$ under the objective measure $\mathbb {P} $ should be equal to the risk-free rate $r$ (*) to preclude arbitrage opportunities. Hence, there is no paradox, since the two formulas coincide under such circumstances.

To convince yourself, consider the following thought experiment. At time zero, suppose that you possess a cash amount $S_0$, which you would like to invest. You come up with 2 different investment ideas:

- Buy the stock at time zero. With certainty your long position will be worth $S_0e^{\mu T}$ at time $T$.

- Invest your cash in the risk-free money market account at time zero. This will get you $S_0e^{r T} $ with certainty at time $T$.

It is clear that you should have $r=\mu$ to preclude arbitrage opportunities in that case.

(*) $r$ happens to be the drift required under the risk-neutral measure $\mathbb {Q} $, i.e. the drift assumed when deriving the BS formula

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