Risk Neutral Pricing Explained From First Principles

Risk neutral pricing follows from replication and no arbitrage, not from any belief about real world probabilities or investor risk preferences.

Risk neutral pricing values a derivative as the discounted expectation of its payoff under a measure constructed from replication and no arbitrage, not from any assumption about what investors believe the probability of an outcome to be. The name is a source of confusion precisely because it sounds like a statement about risk preferences when it is actually a statement about a mathematical device used to compute a price consistently with the absence of arbitrage.

Suppose a derivative's payoff at maturity can be exactly reproduced by trading a portfolio of the underlying asset and a risk-free bond, adjusting the holdings continuously as prices move. If such a replicating portfolio exists, its cost today must equal the derivative's price today, because if the derivative were cheaper an arbitrageur could buy it and sell the replicating portfolio for a certain profit, and if it were more expensive the reverse trade would work. This argument requires no assumption about the probability of the stock going up or down. It only requires that the replicating strategy is achievable.

In the classic binomial model, the replicating portfolio's holdings are pinned down by matching the derivative's payoff in the up state and the down state exactly, and solving those two equations for the number of shares and the bond position gives a price that does not reference the actual, real world probability of the up move at all.

Once the replicating portfolio is known, its cost can be rewritten as a discounted expectation of the payoff under a particular set of probability weights, called risk neutral probabilities, chosen so that the discounted stock price itself is a martingale under those weights. These weights are derived from the no-arbitrage condition and the risk-free rate, not from any survey of investor sentiment or any historical estimate of how often the stock actually moves up. In the binomial model, the risk neutral probability of an up move depends only on the up factor, the down factor and the risk-free rate.

This is why the same derivative, priced by two people with completely different opinions about the real world probability of a rally, must still arrive at the same price, provided both agree on the replication argument. The martingale property that makes this work is developed fully in Martingales and Why Pricing Theory Depends on Them.