Put-call parity

Chapter 1 gave inequalities. This chapter gives an equality, and it is the single most useful result in the subject.

The equation

For a European call and put on the same underlying, with the same strike and the same expiry:

$C - P = S - Ke^{-rT}$

Rearranged the way it is usually remembered:

$C + Ke^{-rT} = P + S$

No volatility appears. No probability appears. No model appears. It holds whatever the underlying does, whatever anyone believes, and whatever pricing formula anyone uses.

Why it holds

Build the two sides as portfolios and look at expiry.

Left side: buy the call, and lend $Ke^{-rT}$ so that the loan matures at exactly $K$.

Right side: buy the put, and buy the underlying.

At expiry	Left: call + cash	Right: put + underlying
$S_T > K$	exercise the call with the $K$ → $S_T$	put lapses, hold the asset → $S_T$
$S_T = K$	both worthless, hold $K$ → $K$	both at zero value, hold asset → $K$
$S_T < K$	call lapses, hold $K$ → $K$	exercise the put, selling at $K$ → $K$

Identical in every state. By the static hedging principle of chapter 1, they must cost the same today. That is the whole proof, and it is why parity is the firmest thing in option pricing.

What it immediately buys you

Every position has a synthetic twin. Rearranging the same equation four ways:

You want	Build it from
A call	put + underlying − borrowing
A put	call − underlying + lending
The underlying	call − put + lending
Lending	put − call + underlying

That is the practical meaning of parity. If the market quotes any three of the four, the fourth's price is already determined. You cannot have a view on the call that is not also a view on the put.

And one correction it forces. People describe buying a put as bearish and selling a call as bearish, and treat them as near-equivalents. Parity says a bought put equals a bought call plus a short in the underlying plus lending — so it is a different position from a sold call in every way that matters, as the Derivatives chapter on selling options and tail risk argues from the payoff side.

The forward form, which is what you actually use

For index options the cash index is not directly tradeable, but the future is. Writing $F$ for the futures price of the same expiry, parity becomes

$C - P = e^{-rT}\left(F - K\right)$

This is the version to use on a real chain, because every quantity in it is quoted on a screen: two option prices, a futures price, a rate and a date. Nothing has to be estimated.

Where parity bends in practice

Parity is exact in the argument and approximate in the world, for reasons worth naming rather than hiding.

Dividends. For a single stock, the holder of the share receives dividends and the holder of a call does not, so the underlying's price in the equation must be net of dividends due before expiry. Index options avoid this by referencing the future, which already prices them in. Chapter 12 covers how Indian contracts treat dividends.

Early exercise. Parity as stated is a European result. For American options the relation becomes an inequality, because the early-exercise right is worth something on one side. Chapter 5 shows when that matters.

Bid-ask spreads. You cannot trade at the midpoint. A parity discrepancy smaller than the cost of crossing four spreads is not a trading opportunity, though it may still be a data problem.

Rates. Which rate? The one at which you can lend and borrow for that term, which is not the one a clearing corporation uses. NSE's own volatility index takes the 30-day or 90-day MIBOR rate as the risk-free rate for this purpose.

Working the problem

NSE's white paper publishes a worked example chain, and it makes a better exercise than a clean one would.

The quotes. Call 5,100: bid ₹144.50, ask ₹145.00, so mid ₹144.75. Put 5,100: bid ₹74.40, ask ₹74.50, so mid ₹74.45.

$C - P = 144.75 - 74.45 = ₹70.30$

The other side. The near-month future is 5,129, the at-the-money strike is 5,100, time to expiry is 0.02466 years and the rate 3.90%:

$e^{-rT}(F - K) = e^{-0.039 \times 0.02466} \times (5{,}129 - 5{,}100) = 0.99904 \times 29 = ₹28.97$

The discrepancy is ₹41.33 — on quotes whose bid-ask spreads are ₹0.50 and ₹0.10. It is nearly a hundred times the spread.

Restate it as a forward price, which is more informative. Invert parity to ask what forward the option quotes imply:

$F_{\text{implied}} = K + (C - P)\,e^{rT} = 5{,}100 + 70.30 \times 1.00096 = 5{,}170.37$

The options imply a forward of 5,170 while the futures market quotes 5,129 — a gap of 41 points.

What that means. In a live market, a 41-point gap between the options-implied forward and the quoted future is an arbitrage: sell the dear synthetic (short call, long put, long future) and the position locks in the difference. It would not survive to be screenshotted.

So the correct reading is that this chain is not a market snapshot — and the white paper says as much, describing it as a hypothetical example, offered to show the volatility index calculation rather than to be internally consistent. Parity detects that in two lines of arithmetic.

And that is the real lesson of the exercise. Parity's first use is not finding free money; professionals have already taken any that existed. Its first use is as a consistency check — on a data feed, on a quote you have been given, on a spreadsheet you have built. A chain that fails parity by far more than its spreads has a problem in the data, the rate, the expiry convention or the dividend treatment, and you should find out which before you price anything off it.

Which of the two quotes is wrong here? Parity cannot say — it constrains the pair, not either one. That is a limit of the result and worth holding onto: an equation between two prices tells you they disagree, never which one to trust.

The point

For European options on the same strike and expiry, $C - P = S - Ke^{-rT}$, or $C - P = e^{-rT}(F-K)$ in the form you can actually compute from a screen. It follows from two portfolios that are identical in every state at expiry, so it needs no model, no volatility and no probabilities — which makes it the most reliable statement in the subject. It means any three of call, put, underlying and loan determine the fourth, and it gives you a two-line test of whether a chain of quotes is coherent. It constrains the pair of prices without telling you which of them is wrong.