Risk-Neutral Pricing & No-Arbitrage

You know how to model a price as it wanders. What you cannot yet do is attach a fair number to a claim written on it — an option, a forward, anything whose payoff depends on where the price ends up. The surprising fact, and the heart of this chapter, is that you can do it without ever predicting where the price will actually go. That sounds impossible. It is the deepest idea in derivatives pricing, and once you see it you will never read a fair value the same way.

No-arbitrage and the law of one price

Start with a principle so simple it feels trivial: no arbitrage. An arbitrage is a free lunch — a trade that costs nothing today, can never lose, and sometimes wins. In a functioning market such trades do not persist: the moment one appears, someone exploits it, and the exploiting moves prices until the free lunch closes. So we assume, as a modeling axiom, that no free lunch exists. This is not a claim about psychology or efficiency — it is a much weaker, much more reliable thing: a constraint on which prices are mutually consistent.

Watch what that single constraint forces. Suppose you can build two portfolios that pay out exactly the same amount in every future state of the world — one an option, the other a recipe of stock and cash that mimics the option's payoff. Then they must cost the same today. Because if one were even a penny cheaper, you'd buy the cheap one, sell the dear one, pocket the difference, and hold a position that nets to zero forever — an arbitrage, which we forbade. This is the law of one price, and it is the engine under every fair value the platform shows.

Replication

The trick of pricing, then, is replication: if you can copy a claim's payoff with simpler instruments whose prices you know, the claim's price is fixed by the cost of the recipe. You never forecast the asset — you only copy the payoff. Doing this by hand for every claim would be brutal, so there is a far more powerful way to say the same thing, through a change of probability.

The martingale, and the measure Q

Consider the discounted price of a tradable — its price divided by the growth of a risk-free bank account, $e^{-rt}S_t$. Under the real-world probabilities $P$, this tends to drift, because risky assets earn more than cash to compensate for risk (that excess is the risk premium — the VRP you met in 201). Now ask a strange question: is there some other set of probabilities — a re-weighting of the same future states — under which the discounted price has no drift, i.e. is a martingale, its expected future value equal to today's?

$\mathbb{E}^{Q}\!\big[e^{-rT}S_T\big] = S_0.$

The fundamental theorem of asset pricing says yes, and sharply: a market is free of arbitrage if and only if there exists at least one such measure. We call it an equivalent martingale measure $Q$ — the risk-neutral measure. Two words carry the weight. Martingale: under $Q$ the discounted tradables have zero drift — the risk premium is engineered away. Equivalent: $Q$ agrees with the real world about which events are possible (any state $P$ gives positive probability, $Q$ does too, and vice versa) — it may disagree wildly about likelihood but never about possibility, which is exactly what blocks conjuring an arbitrage through an impossible-event loophole.

Risk-neutral pricing: one line

With $Q$ in hand, pricing collapses to a single line. The fair value today is the discounted expected payoff under $Q$:

$V_0 = e^{-rT}\,\mathbb{E}^{Q}\!\big[\text{payoff}(S_T)\big].$

Compute the average payoff using the risk-neutral probabilities, discount at the risk-free rate, done. No risk premium appears anywhere. The asset's true expected return — the thing you spent a career trying to forecast — has vanished from the pricing formula. It got absorbed into $Q$ the moment the drift was removed.

The risk-neutral SDE and the market price of risk

Here is the operational content of "the drift vanished." Under the real world the stock follows $dS = \mu S\,dt + \sigma S\,dW$. Switching to $Q$ changes the drift — and only the drift — from $\mu$ to the risk-free rate $r$:

$dS = rS\,dt + \sigma S\,dW^{Q}.$

The volatility $\sigma$ is untouched (it is a real, observable property of the paths); the drift is rewritten. The bookkeeping that does this is Girsanov's theorem: define the market price of risk

$\lambda = \frac{\mu - r}{\sigma} \qquad\text{and}\qquad W^{Q}_t = W_t + \lambda t,$

and $W^Q$ is a Brownian motion under $Q$ under which the drift becomes $r$. That $\lambda$ — excess return per unit of volatility, a Sharpe-ratio-like quantity — is precisely the risk premium, and re-weighting the probabilities zeroes it out.

Two routes to the same price

You have now seen pricing from two directions, and they are the same mountain. Route 1 (last chapter): apply Itô to the option, delta-hedge away the $dW$, and the risk-free portfolio must earn $r$ — giving the deterministic Black-Scholes PDE for $V(S,t)$. Route 2 (this chapter): change to $Q$ and write the discounted expectation $V_0 = e^{-rT}\mathbb{E}^Q[\text{payoff}]$. The Feynman-Kac theorem proves these give the identical number: the solution of the PDE is that expectation. And when the expectation has no closed form (path-dependent, exotic), you compute it by simulation — the Monte Carlo of Module 202-6, run under $dS = rS\,dt + \sigma S\,dW^Q$.

This is why risk-neutral pricing is preference-free: two investors who violently disagree about how much the asset will rise, and how much risk they can stomach, must still agree on the option's price — provided both refuse a free lunch. The disagreement lives in the real measure; the price lives in $Q$. And a warning that matters for the rest of the institute: the risk-neutral probabilities are a pricing device, not a forecast. The $Q$-probability of an up-move is not your honest belief the asset will rise. Pricing a thing and having an edge on it are different questions — keep them apart.