Real-World and Risk-Neutral Probabilities in Black-Scholes
Summary
The answer distinguishes the real-world stock-price process, whose drift is the expected return, from the risk-neutral process used to price options, whose drift is the risk-free rate. It presents the geometric Brownian motion solution under each measure and explains that the standard call price is the discounted expected payoff under the risk-neutral measure. This connects the Black-Scholes pricing formula to the no-arbitrage framework rather than to a direct forecast of the stock’s actual return.
For a call option, the answer identifies the probability that the terminal stock price exceeds the strike as the exercise probability under the chosen measure. It advises using the risk-neutral process for the risk-neutral probability and the real-world drift for the real-world probability. The excerpt does not carry out the probability calculation or specify numerical inputs, and it directs readers to other references for further detail. Its explanation is limited to the Black-Scholes assumptions and should not be read as a claim that risk-neutral probabilities are real-world forecasts.
Key ideas
- The real-world stock process uses the expected-return drift, while the risk-neutral process uses the risk-free rate.
- Black-Scholes prices a call as a discounted expected payoff under the risk-neutral measure.
- The probability that the terminal price exceeds the strike depends on which probability measure is used.
- Risk-neutral exercise probabilities are pricing quantities and need not match real-world probabilities.
- The explanation assumes the stated geometric Brownian motion framework and gives no numerical worked calculation.
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# Answer by lukada (score 1)
# Can someone please help me answer this question about Black-Scholes model? (risk-neutral & true probability of the call option)
I don't even know where to get started with this question...can someone please help me? How do I answer it?
## Answer by lukada (score 1)
https://quant.stackexchange.com/a/78496
The unique solution of the stochastic differential equation that is mentioned: $$dS_t = S_t\mu dt + S_t \sigma dW_t$$ is given by: $$S_T = Se^{(\mu - \frac{\sigma^2}{2})(T-t) + \sigma (W_T-W_t)}=Se^{(\mu - \frac{\sigma^2}{2})(T-t) + \sigma \sqrt{(T-t)} X}$$ where $X\sim\mathcal{N}(0,1)$. It gives the "real-world" dynamic of the risky asset $S$.
On the other hand, the risk-neutral dynamic is almost the same but with $r$ instead of $\mu$, therefore: $$dS_t = S_trdt + S_t \sigma dW_t^Q$$ and thus $$S_T = Se^{(r - \frac{\sigma^2}{2})(T-t) + \sigma (W_T^Q-W_t^Q)} = Se^{(r - \frac{\sigma^2}{2})(T-t) + \sigma \sqrt{(T-t)} X}$$
where $W^Q$ is the Standard Brownian Motion under measure $Q$ (you can refer to Girsanov Theorem for details) and thus, $X \overset{Q}{\sim}\mathcal{N}(0,1)$.
The option with payoff $(S_T-K)_+$ is priced using the risk-neutral probability $Q$ which is induced by the no-arbitrage assumption. The option price is given by:
$$e^{-r(T-t)}\mathbb{E}_Q\left[(S_T-K)_+ | \mathcal{F}_t \right]$$
which is basically the discounted expected payoff at time $t$ under the risk-neutral measure. The formula that is given is the result of this conditional expectation within the framework of Black-Sholes model.
In short, for a) it is enough to calculate the risk-neutral probability that $S_T>K$ which means that the option will be exercised (you can purchase the asset $S$ for $K$ instead of $S_T$). Therefore, using the aforementioned dynamic (just substitute) calculate:
$$P^Q(S_T>K)$$
For b) the reasoning is exactly the same, but with $\mu$ exchanged for $r$.
For details I would refer you to any textbook or online resources on Black-Sholes model.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.