Replication and risk-neutral pricing

An option is priced by building a portfolio of stock and cash that pays exactly what the option pays. Because the payoffs match in every state, the prices must match today — and the real-world probability of the stock going up never appears anywhere in the argument.

Definition 1.1

The one-period model

A binomial step, $S_0 \to \{uS_0,\ dS_0\}$ — Over one period the stock either rises by a factor $u$ or falls by a factor $d$ . An option on it pays $f_u$ in the up state and $f_d$ in the down state. That is the whole model, and it is enough to price the option exactly.

Figure 1.2 · The stock and call share the same two terminal states. For the worked example below, a half-share plus a borrowing of 40 pays 20 in the up state and 0 in the down state. The call must therefore cost the same 10 today. The 0.5 arrow labels are risk-neutral pricing weights derived from replication, not forecasts.

Derivation 1.3

Building the replicating portfolio

Hold $\Delta$ shares and $B$ in cash. Choose them so the portfolio pays the option’s payoff in both states.

$\Delta\, u S_0 + B e^{rT} = f_u, \qquad \Delta\, d S_0 + B e^{rT} = f_d$
$\Delta = \frac{f_u - f_d}{(u - d)S_0}$
Subtract the two equations. This is the delta — literally the hedge ratio.
$f_0 = \Delta S_0 + B$
Two portfolios with identical payoffs must cost the same today, or there is an arbitrage.

f_0 = e^{-rT}\big[q f_u + (1-q) f_d\big], \qquad q = \frac{e^{rT} - d}{u - d}

Proposition 1.4

What $q$ is, and what it is not

The quantity $q$ arrives out of the algebra, not out of any belief about the stock. It is the probability under which the stock’s expected return equals the risk-free rate — which is why it is called the risk-neutral probability. It is not anyone’s forecast, and it is not the real chance of an up-move.

Holds when

For $q$ to be a probability you need $d < e^{rT} < u$ — otherwise there is a riskless arbitrage in the stock itself.
The real probability $p$ cancels out of the replication entirely.

Why your view on the stock does not matter. Two traders who violently disagree about whether the stock will rise must still agree on the option price, because each can build the same hedge out of the same stock. Any disagreement would let the other one trade against them for a certain profit. Pricing by replication removes opinion from the answer — which is exactly why it works.

Replication and risk-neutral pricing

After this lesson you should be able to

The one-period model

Building the replicating portfolio

What $q$ is, and what it is not

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