Section 26.1

Binomial Trees and Risk-Neutral Pricing

One- and multi-period trees, risk-neutral probabilities, pricing non-standard payoffs.

Toolkit

No-arbitrage pricing replaces real-world probabilities with the unique risk-neutral measure under which discounted asset prices are martingales; any payoff is then priced as the discounted risk-neutral expectation.

For a single period with stock moving $S\to Su$ or $S\to Sd$ and gross return $R=e^{r\Delta t}$ (or $R=1$ at zero rates):

q=\frac{R-d}{u-d},\qquad V_0=\frac{1}{R}\big(qV_u+(1-q)V_d\big).

Over $n$ independent periods, the terminal stock price is $S_0u^kd^{n-k}$ with $k\sim\mathrm{Bin}(n,q)$ , so any European payoff $f(S_n)$ prices as

V_0=\frac{1}{R^n}\sum_{k=0}^n\binom{n}{k}q^k(1-q)^{n-k}f\big(S_0u^kd^{n-k}\big).

Replication (holding $\Delta$ shares plus cash) gives the same price and justifies using $q$ regardless of the true probability; path-dependent or nonlinear payoffs are handled by evaluating $f$ node-by-node before averaging.

Problems

26.1 Two-Step Tree for a Squared Payoff+3 variations Easy Free