Pricing an Option in a Two-Period Risk-Neutral Tree

With spot

100

, per-period stock multipliers

u=1.15

and

d=0.90

, and a risk-free gross return

R=1.02

per period, the risk-neutral up probability is

0.48

at each node. A European call with strike

105

pays

27.25

only if the stock goes up twice. Its current value is about

6.03

Derive the one-step weight and enumerate the terminal nodes

At any node with stock price $S$ , require $S=R^{-1}[puS+(1-p)dS]$ . Cancelling $S$ gives

p=\frac{R-d}{u-d}=\frac{1.02-0.90}{1.15-0.90}=0.48.

The down probability is $0.52$ . The two-step terminal stock prices and state weights are

Terminal history	Stock	Risk-neutral weight
Up, up	$132.25$	$0.48^2=0.2304$
Up, down or down, up	$103.50$	$2(0.48)(0.52)=0.4992$
Down, down	$81.00$	$0.52^2=0.2704$

The weights sum to one. At a time-one node, price the payoff one step later by weighting its two values and dividing by

R

. Then repeat that procedure from the root using the time-one values. For a European payoff of terminal stock, this is equivalent to using the terminal state weights and dividing by

R^2

Price the call at each node

The payoff at maturity is $(S_2-105)^+$ . Only the all-up node is above the strike. That node pays $132.25-105=27.25$ , and the other two nodes pay zero.

At the time-one up node, the two successor payoffs are

27.25

and zero. The option is worth

\big(0.48(27.25)+0.52(0)\big)/1.02\approx12.8235

. At the down node, both successors pay zero. One more step gives

V_0=\frac{0.48(12.8235)+0.52(0)}{1.02}\approx6.0346.

At each node you apply the same two weights and discount one period. The middle terminal stock price occurs on two paths. The table adds their weights explicitly, whereas the rollback visits each path from its own parent. For this strike, the two middle paths pay zero anyway. A quick check from maturity is

27.25(0.48)^2/(1.02)^2

, since the only cash flow arrives two periods from now after two up moves.

You can also check the price with stock and cash. At time one the stock is $115$ or $90$ , while the call value is about $12.8235$ or zero. Holding $12.8235/(115-90)\approx0.513$ shares matches the difference between the two option values. Borrow enough to repay $90$ times the share quantity after one period, which makes the down-state value exactly zero. The time-zero cost of the stock minus the amount borrowed is again about $6.03$ .

Pricing an Option in a Two-Period Risk-Neutral Tree

Derive the one-step weight and enumerate the terminal nodes

Price the call at each node

Watch a binomial tree approach the Black–Scholes price

References and next steps