Put–call parity

A call and a put with the same strike and expiry are two sides of the same object: hold one, sell the other, and you have manufactured the stock. Parity is the single most-used relationship on an options desk, and it is a replication argument rather than a model — it holds whatever you think volatility is.

Equation 2.1

The relationship

For European options on a non-dividend-paying asset, with the same strike and expiry.

C - P = S_0 - Ke^{-rT}

$C,\ P$: Call and put premiums.
$S_0$: Spot price of the underlying.
$Ke^{-rT}$: Present value of the strike.

Derivation 2.2

Why it must hold

Build two portfolios and compare their payoffs at expiry. If the payoffs match in every state, the prices must match today.

$\text{A: long call, short put} \Rightarrow \max(S_T - K, 0) - \max(K - S_T, 0)$
$= S_T - K \quad \text{in every state}$
Above the strike the call pays and the put expires; below it the put is exercised against you. Either way you end up buying the stock at $K$ .
$\text{B: long stock, borrow } Ke^{-rT} \Rightarrow S_T - K$
$\text{Same payoff in every state} \Rightarrow \text{same price today}$
Otherwise buy the cheap portfolio, sell the dear one and hold to expiry for a certain profit.

C - P = S_0 - Ke^{-rT}

Figure 2.3 · A call minus a put is a forward. The two dashed hockey sticks add to the straight line:

S_T - K

in every state, with no optionality left anywhere. That is the whole of parity — the prices must match because the payoffs do, and the only difference today is that the forward defers paying the strike, which is the

Ke^{-rT}

.

You want	Build it from
Long stock	Long call, short put, lend $Ke^{-rT}$
Long call	Long stock, long put, borrow $Ke^{-rT}$
Long put	Short stock, long call, lend $Ke^{-rT}$
Short stock	Short call, long put, borrow $Ke^{-rT}$

Table 2.4 · Every position has a synthetic. This is why a desk can quote a put it does not want to hold: it can manufacture the risk from the call and the underlying.

Proposition 2.5

The versions you need

With a known dividend stream of present value $D$ , the spot is reduced by it: $C - P = S_0 - D - Ke^{-rT}$ . On a futures contract there is nothing to carry, so $C - P = (F - K)e^{-rT}$ . For American options parity becomes a pair of inequalities rather than an equation, because early exercise breaks the replication.

Holds when

A borrow cost on a hard-to-borrow name acts exactly like a dividend and belongs in the same place.
The futures version is why "options on futures" questions look simpler: no carry term.

After this lesson you should be able to

The relationship

Why it must hold

The versions you need

The rest of this lesson is in Premium