Answer First
Risk‑neutral probabilities are not real probabilities. They are artificial weights that make the expected return of the stock equal to the risk‑free rate. Their purpose is to price derivatives, not to predict future stock movements.
Problem Setup
In a one‑period binomial model, the stock moves from \(S_0\) to either: \[ S_u = S_0 u, \quad S_d = S_0 d. \] The risk‑neutral probability is: \[ p^* = \frac{e^{r\Delta t} – d}{u – d}. \] This ensures: \[ S_0 = e^{-r\Delta t} \left( p^* S_u + (1 – p^*) S_d \right). \]
Step-by-Step Explanation
1. Real probabilities reflect the actual likelihood of up/down moves
Real‑world probabilities depend on market expectations, volatility, news, and investor behavior. They describe how the stock is expected to behave.
2. Risk‑neutral probabilities are mathematical tools
Risk‑neutral probabilities are chosen so that the expected return of the stock equals the risk‑free rate: \[ E^{*}[S_1] = S_0 e^{r\Delta t}. \] This eliminates risk premia and makes pricing arbitrage‑free.
3. Why they are not real probabilities
- They ignore investor risk preferences.
- They do not reflect expected returns.
- They may be greater than 0.5 even if the stock is more likely to fall.
- They are derived from no‑arbitrage, not from forecasting.
4. Why risk‑neutral probabilities matter
They allow us to price derivatives by discounting expected payoffs at the risk‑free rate: \[ \text{Option Price} = e^{-r\Delta t} \left( p^* V_u + (1 – p^*) V_d \right). \] This works because arbitrage forces all assets to earn the risk‑free rate in the risk‑neutral world.
5. Managerial interpretation
Risk‑neutral pricing is not about predicting the future. It is about ensuring that option prices are consistent with no‑arbitrage conditions in financial markets.
Intuition
Think of risk‑neutral probabilities as “pricing weights,” not “forecasting weights.” They are chosen to make the math work, not to describe reality.
Common Exam Mistakes
- Using real probabilities instead of risk‑neutral ones.
- Thinking \(p^*\) predicts the stock’s direction.
- Believing risk‑neutral pricing assumes investors are risk‑neutral.
- Confusing expected return with risk‑free return.
Final Summary
Risk‑neutral probabilities are not real probabilities. They are mathematical constructs that ensure the stock grows at the risk‑free rate in the pricing model, allowing arbitrage‑free valuation of options.
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