Risk-Neutral Pricing

Price any claim as its expected payoff under pretend probabilities, discounted at the risk-free rate.

The idea

Risk-neutral pricing is a general recipe for pricing any claim on a binomial tree as a discounted expectation. The replication argument of Binomial Option Pricing prices one claim by solving a pair of simultaneous equations — a fresh pair for every new contract. Solving those equations once, in general, shows that the claim's details survive only through its two payoffs.

Theorem (Risk-neutral pricing formula).

Consider a one-period binomial tree with up factor $u$, down factor $d$ and risk-free rate $r$, where $d < 1 + r < u$, and set $q = \frac{(1+r) - d}{u - d}.$ The arbitrage-free price today of a claim paying $V_{u}$ after an up move and $V_{d}$ after a down move is $V = \frac{1}{1+r}\,\big(q\,V_{u} + (1-q)\,V_{d}\big) = \frac{1}{1+r}\,E_{q}[\text{payoff}].$

The number $q$ is the risk-neutral probability of an up move. With $q$ in hand, we price every claim on the tree by averaging its payoffs with weights $q$ and $1-q$ and discounting the average one period. The same two lines of arithmetic price calls, puts, digitals, and any other claim on the tree.

The formula for $q$ contains only the tree's two factors and the interest rate. The real probability of an up move, $\mathbb{P}(\text{up})$, appears nowhere, so $q$ is not a forecast. It is the one probability under which the discounted stock price is a fair game.

Ways to work on it

• Walkthrough. Price a claim by replication, and see why the risk-neutral probability is bookkeeping rather than belief.
• Practice. Find the risk-neutral probability, then price a two-state claim with it.
• Hardest. Price a digital option and its complement over one tree, and compare the two prices.

Not sure where to start? Take the ten-question placement test.