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Lesson 1 of 6 · 11 min

Why options need a price model: the one-period binomial setup

Because an option's payoff is one-sided, valuing it requires a model of where the underlying can go; the binomial model assumes it can only move up to one price or down to another.

In short

Forward commitments can be valued by replication alone, without assuming anything about the underlying's future price, because their payoff is symmetric.
Options and other contingent claims have asymmetric payoffs, so their value depends on which future prices are possible: we need a model.
In the one-period binomial model the underlying moves from $S_0$ either up to $S_1^u = R^u S_0$ or down to $S_1^d = R^d S_0$ , with $R^u > 1 > R^d$ .
q is the actual probability of the up move and 1 − q of the down move. Surprisingly, q is not needed to value the option.
The gap between $S_1^u$ and $S_1^d$ stands for the underlying's volatility, a key driver of option value.

Forwards versus contingent claims

The law of one price says that if two assets or portfolios produce identical payoffs in every possible future state, they must cost the same today. Replication and no-arbitrage pricing both rest on this idea.

A forward commitment is easy to replicate. Its payoff, $S_T - F_0(T)$ for the long side, rises and falls one-for-one with the underlying, gains and losses alike. Buying the underlying and borrowing copies it whatever the price turns out to be, so no assumption about future prices is needed.

An option is different. A call pays $\max(0, S_T - X)$ : it gains above X but is floored at zero below it. That kink means the option behaves like the underlying in some states and like nothing at all in others. To know how much of the underlying copies it, we must say which future prices are possible. Option valuation therefore needs a model of the underlying's future (random) price.

	Forward commitment	Contingent claim (option)
Payoff shape	Symmetric, linear	Asymmetric, kinked at X
Holder's position	Obligation to transact	Right, not obligation
Price model needed?	No: replicate with spot and borrowing	Yes: payoff depends on the path of S

The one-period binomial model

The binomial model is the simplest useful price model. Over one period, which we set equal to the option's life, the underlying either goes up to $S_1^u$ or down to $S_1^d$ . Nothing else can happen. Which outcome occurs is random, like a coin toss (a Bernoulli trial).

It is handy to describe the moves as gross returns: $R^u = S_1^u / S_0$ , which is above 1, and $R^d = S_1^d / S_0$ , which is below 1. A share at EUR 40 that can rise 30% or fall 20% has $R^u = 1.30$ and $R^d = 0.80$ , so $S_1^u = 52$ and $S_1^d = 32$ .

Let q be the actual probability of the up move, so 1 − q is the probability of the down move. It seems obvious that q must matter for the option's value. The next lessons show it does not: once $S_1^u$ and $S_1^d$ are fixed, the option's no-arbitrage value is fixed too.

What does matter is how far apart the two outcomes are. The spread between $S_1^u$ and $S_1^d$ (equivalently between $R^u$ and $R^d$ ) captures the underlying's volatility, so the model's up and down sizes should be chosen to match how volatile the asset really is.

A one-period binomial tree: from EUR 40 the share ends at either 52 or 32. With a EUR 44 exercise price, a call finishes in the money only in the up state and a put only in the down state.

Option payoffs at the two end points

Once the two end prices are set, the option's value at expiration is known in each state, because it is simply the payoff. Call the call's two possible values $c_1^u$ and $c_1^d$ , and the put's $p_1^u$ and $p_1^d$ . The only unknown left is today's value, $c_0$ or $p_0$ .

The interesting case is when the exercise price lies between the two outcomes, $S_1^d < X < S_1^u$ . Then the call pays off only after an up move and the put only after a down move. In the example, with X = 44: $c_1^u = \max(0, 52 - 44) = 8$ , $c_1^d = 0$ , $p_1^u = 0$ and $p_1^d = \max(0, 44 - 32) = 12$ .

An at-the-money option (X = $S_0$ ) fits the same pattern: since $S_1^u > S_0 > S_1^d$ , an at-the-money call ends in the money after an up move and out of the money after a down move, and a put the other way round.

With X = 44 each option pays in exactly one state. Knowing these four numbers and the risk-free rate is all the binomial model needs to value both options.

The one-period model looks crude, but it is the building block for binomial trees: chain many short periods together, each with its own up or down move, and the end prices spread out into a realistic distribution. The same pricing logic then values more complex contingent claims.

Formulas

Up and down gross returns: $R^u = \frac{S_1^u}{S_0} > 1 \qquad R^d = \frac{S_1^d}{S_0} < 1$
$S_0$ = price today; $S_1^u$ , $S_1^d$ = price at the end of the period after an up or a down move.
Option values at the end of the period: $c_1^u = \max(0,\, S_1^u - X) \quad c_1^d = \max(0,\, S_1^d - X) \quad p_1^u = \max(0,\, X - S_1^u) \quad p_1^d = \max(0,\, X - S_1^d)$
The period ends at the option's expiration, so each value is just the payoff in that state.

Worked examples

Example 1 Building the tree

A share trades at GBP 80. Over the next year it will either rise by 25% or fall by 25%. What are $R^u$ , $R^d$ , $S_1^u$ and $S_1^d$ ?

Example 2 Option payoffs in each state

Using the GBP 80 share above, find the end-of-period values of a call and a put, both with an exercise price of GBP 90.

Example 3 Why the forward needs no model

An analyst says: 'I can value a one-year forward on this share without any view on where the share will go, but I cannot do the same for a one-year call.' Is she right?

On the exam

Expect conceptual items on why contingent claims, unlike forwards, need a model of future prices, plus quick computations of $R^u$ , $R^d$ , end prices and option payoffs at each node.

Writing the gross return as the percentage move (0.25) instead of one plus the move (1.25).
Thinking the actual probability q is an input to the option value. Only the two end prices (and the risk-free rate) are needed.
Using today's price instead of the end-of-period price when computing $c_1^u$ , $p_1^d$ and the rest.

Check yourself

Q1A share trades at NOK 250 and will either rise to NOK 300 or fall to NOK 220 over the period. The up and down gross returns are:
Q2Pricing an option, unlike pricing a forward, most likely requires a model of the underlying's future price because the option:
Q3In a one-period binomial model, the underlying's volatility is represented by:
Q4A put has an exercise price of EUR 75. The underlying trades at EUR 70 and will rise 20% or fall 10% by expiration. The put's value in the down state is:
Q5In a one-period binomial model, an at-the-money call on the underlying will be:

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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