Risk-Neutral Distribution Explained

Worked Solution

How to Think About It: This is a "explain the concept" question, and the interviewer is testing two things at once. First, do you actually understand risk-neutral pricing deeply enough to explain it simply? Second, can you connect the intuition to the math? The best approach is to start from a concrete example that makes the idea feel obvious, then layer on the formalism. Do not start with measure theory -- start with a coin flip.

Key Insight: The risk-neutral distribution is not a belief about what will happen. It is a computational device: a re-weighting of probabilities so that every asset earns the risk-free rate in expectation, which lets you price derivatives by simple discounted expectation.

The Explanation:

*Level 1 -- No quant background:*

Suppose you want to figure out the fair price of a bet on whether a stock goes up or down. You might think you need to know the "real" probability of the stock going up. But here is the trick: the stock price today already reflects the market's views on risk and return. So instead of trying to estimate real probabilities and then adjust for risk (which is messy and subjective), we invent a set of fictional probabilities -- called risk-neutral probabilities -- where every investment earns exactly the safe bank rate.

In this fictional world, nobody demands extra return for taking risk. A stock, a bond, and a wild startup all earn the same rate. These fake probabilities are chosen so that market prices come out right. Once you have them, pricing any derivative is just: compute the average payoff under these fake probabilities, then discount at the risk-free rate. That is the whole idea.

*Level 2 -- The math:*

The risk-neutral measure $\mathbb{Q}$ is a probability measure equivalent to the real-world measure $\mathbb{P}$ (they agree on which events are possible) under which discounted asset prices are martingales:

$E^{\mathbb{Q}}\left[\frac{S_T}{B_T} \,\Big|\, \mathcal{F}_t\right] = \frac{S_t}{B_t}$

where $B_t = e^{rt}$ is the money market account. This is equivalent to saying that under $\mathbb{Q}$, every traded asset has expected return equal to the risk-free rate:

$E^{\mathbb{Q}}[S_T \,|\, \mathcal{F}_t] = S_t \, e^{r(T-t)}$

The First Fundamental Theorem of Asset Pricing says: a market is arbitrage-free if and only if at least one such measure $\mathbb{Q}$ exists. The Second Fundamental Theorem says: the market is complete (every contingent claim can be replicated) if and only if $\mathbb{Q}$ is unique.

Given $\mathbb{Q}$, the fair (no-arbitrage) price of a derivative with payoff $h(S_T)$ at maturity $T$ is:

$V_t = e^{-r(T-t)} \, E^{\mathbb{Q}}[h(S_T) \,|\, \mathcal{F}_t]$

The connection between $\mathbb{Q}$ and $\mathbb{P}$ is captured by the Radon-Nikodym derivative $dQ/dP$, which re-weights the real-world probabilities. In the Black-Scholes model, this amounts to shifting the drift of the stock from its real expected return $\mu$ to the risk-free rate $r$, while keeping the volatility $\sigma$ unchanged.

Practical Considerations:

• The risk-neutral distribution is not "what the market thinks will happen." It is a pricing tool. Under $\mathbb{Q}$, risky assets look less attractive than they do in reality (the equity risk premium disappears), which is why $\mathbb{Q}$ puts more weight on bad outcomes than $\mathbb{P}$ does.
• In practice, you can back out the risk-neutral distribution from option prices. The Breeden-Litzenberger result says the second derivative of the call price with respect to strike gives you the risk-neutral density: $f^{\mathbb{Q}}(K) = e^{r(T-t)} \frac{\partial^2 C}{\partial K^2}$.
• A common interview trap: someone says "the risk-neutral probability of the stock going up is 60%." That does not mean anyone believes the stock goes up 60% of the time. It means that if you price using 60% as the up-probability and discount at the risk-free rate, you get the correct market price.

Answer: The risk-neutral distribution $\mathbb{Q}$ is a re-weighting of real-world probabilities such that all assets earn the risk-free rate in expectation. It exists if and only if the market is arbitrage-free (FTAP). Derivative prices equal discounted expected payoffs under $\mathbb{Q}$: $V_t = e^{-r(T-t)} E^{\mathbb{Q}}[h(S_T)]$. The intuition is that market prices already embed risk preferences, so we can extract a "risk-adjusted" probability measure and use it to price anything by simple expectation.