One-Period Binomial Models for Stock Prices and Options

A One-Period Up-or-Down Model

A one-period binomial model describes a stock whose price, over one specified period, can move to one of two values. It is a deliberately small model: it makes the connection between a stock-price outcome and an option’s maturity payoff explicit. The calculations here concern payoffs at the end of the period and probability-weighted expected payoffs. An expected payoff under a stated probability model is not, by itself, a market price for the option.

Use the probability space $(\Omega,\mathcal{F},\mathbb{P})$ with $\Omega=\{U,D\}$, $\mathcal{F}$ the set of all subsets of $\Omega$, $\mathbb{P}(U)=p$, and $\mathbb{P}(D)=1-p$, where $0<p<1$. The event $U$ is the up outcome and $D$ is the down outcome. Let $S_0>0$ be the stock price now. At the end of the period its price is $S_1=S_0u$ in outcome $U$ and $S_1=S_0d$ in outcome $D$, where $0<d<1<u$. Thus $S_1>S_0$ in the up outcome and $S_1<S_0$ in the down outcome.

The stock-price process $(S_n)_{n=0,1}$ has state space $\mathcal{S}=\{S_0,S_0d,S_0u\}$ and time index $n=0,1$. Its initial value is $S_0$, and its terminal value is determined by the outcome. The information available at time 0 is $\mathcal{F}_0=\{\varnothing,\Omega\}$; at time 1 it is $\mathcal{F}_1=\mathcal{F}$. The probability $p$ describes the model’s probability of an up move. It is not specified as a pricing probability.

Definition: For a real number $x$, write $x^+=\max(x,0)$. A European call option with strike $K\geq0$ gives its holder the right to buy the stock for $K$ at the end of the period. Its maturity payoff is $C_1=(S_1-K)^+$. A European put with the same strike gives the right to sell the stock for $K$, with maturity payoff $P_1=(K-S_1)^+$. These payoffs are nonnegative: exercising an option is not required when doing so would produce a negative payoff.

The call payoff is positive exactly when the terminal stock price is above the strike; the put payoff is positive exactly when it is below the strike. If the stock price equals the strike, both payoffs are zero. These statements use strict inequalities for positive payoff, while equality at the strike gives zero.

Payoff Calculations and Their Basic Identities

For a fixed strike $K$, the two possible call payoffs are $(S_0u-K)^+$ and $(S_0d-K)^+$. The two possible put payoffs are $(K-S_0u)^+$ and $(K-S_0d)^+$. The model’s expected maturity payoffs follow by weighting the up outcome by $p$ and the down outcome by $1-p$.

Result (Expected maturity payoff): For any real-valued payoff $g(S_1)$ defined at both possible terminal stock prices, $$ \mathbb{E}[g(S_1)]=p\,g(S_0u)+(1-p)\,g(S_0d). $$ In particular, this formula applies to calls and puts. It gives an expectation under the specified outcome probabilities, not a rule for converting a payoff into a current option price.

Proof. The random variable $g(S_1)$ equals $g(S_0u)$ on $U$ and $g(S_0d)$ on $D$. Since these two events are disjoint, have union $\Omega$, and have probabilities $p$ and $1-p$, respectively, the definition of expectation for a finite probability space gives $\mathbb{E}[g(S_1)]=p\,g(S_0u)+(1-p)\,g(S_0d)$. The argument also holds if $p=0$ or $p=1$, although the standing model assumes $0<p<1$ so both outcomes have positive probability. $\square$

Result (Payoff relation): For every terminal stock price $S_1$ and every strike $K\geq0$, $$ C_1-P_1=S_1-K. $$ Consequently, the expected maturity payoffs satisfy $\mathbb{E}[C_1]-\mathbb{E}[P_1]=\mathbb{E}[S_1]-K$.

Proof. Fix any possible value $s=S_1$. If $s\geq K$, then $(s-K)^+=s-K$ and $(K-s)^+=0$, so the difference of the call and put payoffs is $s-K$. This includes $s=K$, where both sides are zero. If $s<K$, then $(s-K)^+=0$ and $(K-s)^+=K-s$, so their difference is $0-(K-s)=s-K$. These cases cover every $s$, proving the pointwise identity. Taking expectations of both sides gives $\mathbb{E}[C_1]-\mathbb{E}[P_1]=\mathbb{E}[S_1-K]$. Since $K$ is constant and the probabilities sum to one, $\mathbb{E}[S_1-K]=\mathbb{E}[S_1]-K$. $\square$

This identity concerns payoffs at maturity. It does not assert a relation between current market prices: that requires additional pricing assumptions and analysis. The examples below keep the distinction explicit.

Worked Example: A Call and Put at the Same Strike

Suppose $S_0=100$, $u=\tfrac65$, $d=\tfrac9{10}$, and $p=\tfrac13$. For strike $K=105$, calculate each terminal payoff and each expected maturity payoff.

First, the model has $0<\tfrac9{10}<1<\tfrac65$, and the outcome probabilities sum to $\tfrac13+\tfrac23=1$. The up stock price is $S_0u=100\cdot\tfrac65=120$. The down stock price is $S_0d=100\cdot\tfrac9{10}=90$.

For the call, the up payoff is $(120-105)^+=15$, and the down payoff is $(90-105)^+=0$. For the put, the up payoff is $(105-120)^+=0$, and the down payoff is $(105-90)^+=15$. Therefore, $$ \mathbb{E}[C_1]=\tfrac13(15)+\tfrac23(0)=5,\qquad \mathbb{E}[P_1]=\tfrac13(0)+\tfrac23(15)=10. $$

Check: The expected stock price is $\tfrac13(120)+\tfrac23(90)=40+60=100$, so $\mathbb{E}[S_1]-K=100-105=-5$. The expected payoff difference is $5-10=-5$, as required by the payoff relation.

Worked Example: An At-the-Money Strike

Let $S_0=80$, $u=\tfrac54$, $d=\tfrac34$, and $p=\tfrac25$. Take $K=80$, the current stock price. Find the terminal prices, call and put payoffs, and their expectations.

The multipliers satisfy $0<\tfrac34<1<\tfrac54$, and $\tfrac25+\tfrac35=1$. The up price is $80\cdot\tfrac54=100$; the down price is $80\cdot\tfrac34=60$. In the up outcome, the call pays $(100-80)^+=20$ and the put pays $(80-100)^+=0$. In the down outcome, the call pays $(60-80)^+=0$ and the put pays $(80-60)^+=20$.

Weighting by the outcome probabilities gives $$ \mathbb{E}[C_1]=\tfrac25(20)+\tfrac35(0)=8,\qquad \mathbb{E}[P_1]=\tfrac25(0)+\tfrac35(20)=12. $$

Check: $\mathbb{E}[S_1]=\tfrac25(100)+\tfrac35(60)=40+36=76$, so $\mathbb{E}[S_1]-K=76-80=-4$. Also, $\mathbb{E}[C_1]-\mathbb{E}[P_1]=8-12=-4$. At-the-money at time 0 does not mean that both maturity payoffs are positive; each payoff depends on the terminal price.

How the Strike Changes the Payoff

The strike determines where the call and put payoffs switch between zero and a positive amount. Raising the strike cannot increase a call payoff: the stock must finish higher for the call to pay, and its payoff at any fixed terminal stock price cannot grow. Raising the strike cannot decrease a put payoff. The pointwise statements also apply to expected payoffs, because expectations preserve inequalities for a finite probability space.

Result (Monotonicity in the strike): If $K_1\leq K_2$, then for every terminal price $S_1$, $$ (S_1-K_2)^+\leq(S_1-K_1)^+,\qquad (K_1-S_1)^+\leq(K_2-S_1)^+. $$ Thus the call payoff is nonincreasing in the strike and the put payoff is nondecreasing in the strike. Their expected maturity payoffs have the same respective inequalities.

Proof. For the call, $S_1-K_2\leq S_1-K_1$. The function $x\mapsto x^+=\max(x,0)$ is nondecreasing: if $x\leq y$, then either $y\leq0$, in which case both positive parts are zero; or $y>0$, in which case $x^+\leq y$. Applying this fact gives $(S_1-K_2)^+\leq(S_1-K_1)^+$. For the put, $K_1-S_1\leq K_2-S_1$, so the same property gives $(K_1-S_1)^+\leq(K_2-S_1)^+$. Each inequality holds in both outcomes. Multiplying the up-outcome inequality by $p\geq0$, the down-outcome inequality by $1-p\geq0$, and adding proves the corresponding expectation inequalities. $\square$

Worked Example: Comparing Two Strikes

Suppose $S_0=50$, $u=\tfrac32$, $d=\tfrac45$, and $p=\tfrac14$. Compare call and put payoffs at strikes $K_1=60$ and $K_2=70$.

The multipliers satisfy $0<\tfrac45<1<\tfrac32$, and the probabilities sum to $\tfrac14+\tfrac34=1$. The terminal stock prices are $50\cdot\tfrac32=75$ in the up outcome and $50\cdot\tfrac45=40$ in the down outcome.

At strike 60, the call pays $(75-60)^+=15$ up and $(40-60)^+=0$ down; the put pays $(60-75)^+=0$ up and $(60-40)^+=20$ down. Hence $\mathbb{E}[C_1]=\tfrac14(15)+\tfrac34(0)=\tfrac{15}{4}$, and $\mathbb{E}[P_1]=\tfrac14(0)+\tfrac34(20)=15$.

At strike 70, the call pays $(75-70)^+=5$ up and $(40-70)^+=0$ down; the put pays $(70-75)^+=0$ up and $(70-40)^+=30$ down. Hence $\mathbb{E}[C_1]=\tfrac14(5)+\tfrac34(0)=\tfrac54$, and $\mathbb{E}[P_1]=\tfrac14(0)+\tfrac34(30)=\tfrac{90}{4}=\tfrac{45}{2}$.

Check: Raising the strike from 60 to 70 lowered the call’s expected payoff from $\tfrac{15}{4}$ to $\tfrac54$, and raised the put’s from $15$ to $\tfrac{45}{2}$, consistent with the monotonicity result. At strike 70, the expected payoff difference is $\tfrac54-\tfrac{45}{2}=-\tfrac{85}{4}$. The expected stock price is $\tfrac14(75)+\tfrac34(40)=\tfrac{75}{4}+30=\tfrac{195}{4}$; subtracting 70 gives $\tfrac{195}{4}-\tfrac{280}{4}=-\tfrac{85}{4}$, which checks the payoff relation.

Expected Payoff Is Not the Option Price

The probability-weighted calculations answer a precise question: what is the expected payoff at maturity under the stated probabilities? They do not, on their own, answer what a buyer should pay now. A current price concerns a value at time 0, whereas $C_1$ and $P_1$ are random payoffs at time 1. Relating the two requires a pricing framework, which is separate from calculating the payoff in each outcome.

Common mistake. In the first example, it would be wrong to calculate the call’s down-outcome payoff as $105-90=15$. That calculation uses the put’s payoff formula. A call holder may buy for 105 when the stock is worth 90, so exercising would not help; the call payoff is $(90-105)^+=0$. Conversely, the put’s down-outcome payoff is $(105-90)^+=15$. Keeping the definitions $(S_1-K)^+$ for calls and $(K-S_1)^+$ for puts visible during the calculation prevents this reversal.

A reliable workflow is to calculate the two terminal stock prices first, then apply the relevant positive-part formula separately in each outcome, and only then take a probability-weighted expectation if requested. As a final check, the call-minus-put payoff in each outcome must equal that outcome’s stock price minus the strike.

Check Your Understanding

Use the payoff definitions and verify numerical answers by substituting the two outcomes.

If $S_0=40$, $u=\tfrac32$, $d=\tfrac12$, and $K=50$, what are the call and put payoffs in each outcome?
For the model in Question 1, take $p=\tfrac14$. Calculate the expected maturity payoff of the call and the put.
State the condition on $S_1$ for a call payoff to be strictly positive, and the condition for a put payoff to be strictly positive. What happens when $S_1=K$?
Suppose $K_1\leq K_2$. Which direction does each payoff move when the strike increases, and why?
Why does a probability-weighted expected maturity payoff not, without additional assumptions, determine an option’s current market price?

One-Period Binomial Models for Stock Prices and Options

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