Put–Call Parity

Builds onPresent Value of a Single Sum · Forward Price — if this page feels steep, start there.

S_0 + p_0 = c_0 + \frac{X}{(1+r)^T}

$S_0$: today's stock price
$p_0, c_0$: today's prices of the put and the call (same strike, same expiry)
$X$: the strike price both options share
$X/(1+r)^T$: the present value of the strike — a riskless bond paying X at expiry
$S_0 + p_0$: the protective put: stock plus downside insurance
$c_0 + PV(X)$: the fiduciary call: upside ticket plus the cash floor

Why it must be true

Two portfolios, one destiny. Protective put: own the stock, own a put — at expiry you hold the stock if it did well, or the strike $X$ in cash if it didn't. Fiduciary call: own a call, park the strike's present value in a riskless bond — at expiry you hold the stock (exercise) if it did well, or $X$ in cash if it didn't. Identical payoffs in every state of the world.

Identical futures must have identical prices today — that is the whole theorem. The practical magic: knowing any three of {stock, put, call, bond}, the fourth price is FORCED. Puts and calls are not independent instruments; they are the same insurance contract wired up differently, and this equation is the wiring diagram.

The derivation

Compare terminal values at expiry $T$ in the two possible worlds.

If the stock finishes above the strike ( $S_T > X$ ): the protective put holds $S_T$ (put expires worthless); the fiduciary call exercises, paying the bond's $X$ for the stock — also $S_T$ .

If it finishes at or below ( $S_T \le X$ ): the put delivers $X - S_T$ on top of the stock, totalling $X$ ; the call expires worthless and the bond pays $X$ — also $X$ .

Same payoff in every state, so the prices must match today:

S_0 + p_0 = c_0 + \frac{X}{(1+r)^T}

Rearrangements are free: $p = c - S + PV(X)$ prices a put from a call; $c - p = S - PV(X)$ says call-minus-put is a synthetic forward.

When to reach for it

Pricing one European option from its twin (same strike, same expiry) plus the stock and rate — or spotting an arbitrage when the two sides disagree.

Listen for

“put–call parity”“European options with the same strike and expiration”“price of the call given the put (or vice versa)”“synthetic position / protective put / fiduciary call”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Solve for the missing letter before plugging numbers: p = c − S + PV(X). One rearrangement, then arithmetic.
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At-the-money anchor: with S ≈ PV(X), call and put prices should nearly match — the call's small premium over the put reflects the interest on the strike.
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Sanity floor: no option price can be negative. A negative answer means two terms swapped sides.

Traps in applying it

✗Forgetting to discount the strike — X enters at PRESENT value, not face value.
✗Applying it to American options — early exercise breaks the equality (it becomes an inequality).
✗Mismatched strikes or expiries — parity binds only twins.
✗Ignoring dividends: an expected payout before expiry lowers the effective S₀.

Limits & criticisms

Exact only for European options on non-dividend assets with frictionless markets: American early exercise, dividends, borrowing costs and short-sale constraints all bend the equality into a band. In stressed markets (hard-to-borrow stocks, 2008-style funding) the "violations" persist because the arbitrage is genuinely expensive to execute — parity gaps are then measuring frictions, not free money.

Where it came from

Traders knew rough versions in 19th-century option markets (Russell Sage allegedly used parity to dodge usury laws with synthetic loans), but Hans Stoll's 1969 paper formalized it — four years before Black–Scholes. Unlike Black–Scholes, parity needs NO model of how prices move: only that arbitrage gets eaten. It is the arbitrage desk's daily bread — violations are scanned for by every market maker — and the conversion/reversal trades that enforce it are among the oldest systematic strategies in options.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Put from its twin

p_0 = c_0 - S_0 + \frac{X}{(1+r)^T}

The synthetic face: a put IS a call minus the stock plus a bond — build it or price it, same equation.

Drill this face →

Call from its twin

c_0 = p_0 + S_0 - \frac{X}{(1+r)^T}

The mirror: a call is a put plus stock minus the bond — the other rearrangement of the same wiring diagram.

Drill this face →

On the BA II Plus

Worked example: European options, strike $90.00, expiry 1 year(s): the call trades at $11.00 with the stock at $90.00 and rates at 3%. What must the put cost to rule out arbitrage?

1.90 [÷] [(] 1 [+] 0.03 [)] [y^x] 1 [=]PV of the strike
2.[+] 11 [−] 90 [=]add call, subtract stock → put

→ $8.38

Drill this formula Back to the library