Put-call parity: the price that needs no model
A call minus a put at the same strike is the stock minus the discounted strike. No volatility, no view, no model. When it breaks, it is a stale quote, a dividend nobody mentioned, or free money.
- Compute a put from a call, or the other way round, with the discounted strike
- Back out the stock price, or the interest rate, the options imply
- Size and describe the arbitrage when a put is quoted away from parity
- Say what rates do to calls against puts, and why volatility never appears
The intuition
Own a call and sell a put at the same strike and expiry. Above the strike the call pays; below it the put costs you; either way you end up buying the share for K at expiry. That is the same as owning the share today and owing K at expiry. Two positions with identical payoffs must cost the same now, or someone buys the cheap one, sells the dear one and keeps the difference with no risk.
So C − P = S − PV(K). A six-month call at $7.00 on a $100 stock struck at $100, with rates at 5%, forces the put to $4.53: the strike discounted is $97.53, and 7 − 100 + 97.53 is 4.53. Volatility raises both prices equally, which is why it is nowhere in the formula. Rates are in it: a higher rate shrinks PV(K), so calls gain on puts, because a call defers paying for the stock and a put defers receiving the strike.
C − P = S − K e^(−rT) for European options on a stock with no dividends. Rearranged: P = C − S + PV(K); S = C − P + PV(K); r = −ln((S − C + P) ÷ K) ÷ T. A put quoted above parity: sell it, buy the call, short the stock, lend PV(K). Below: the reverse. The profit is the gap, locked in today.
Why it works
- The conventions here: European options, same strike and expiry, a stock that pays no dividends, continuous compounding with T in years. The call is priced with Black-Scholes and rounded to cents; the put is then set exactly by parity.
- Replication, not valuation. Long call plus short put pays S_T − K at expiry whatever happens. So does long stock plus a loan that repays K. Same payoff, same price today, or there is an arbitrage. Nothing about the distribution of S_T was used.
- The arbitrage package. Sell what is rich and buy the synthetic that replicates it. A rich put: sell it, buy the call, short the stock, lend PV(K). At expiry every leg cancels and only the premium collected up front remains. A naked short put is not an arbitrage; it has market risk.
- Rates favour calls. Only PV(K) depends on r. A higher rate lowers it, so S − PV(K) rises: C − P rises, the call gets dearer and the put cheaper.
- Dividends and borrow break the clean version. With dividends, C − P = S − PV(dividends) − PV(K): the options imply a stock price below the screen. A hard-to-borrow stock does the same. Box spreads, parity run on two strikes, are how traders extract the rate, because the stock drops out.
- Implied rate is fragile. A one-cent error in either option is a one-cent error in PV(K), which over a few months on a large strike is several basis points of rate.
| Discount factor: e^(−0.05 × 0.5) | 0.97531 |
| PV of strike: $100 × 0.97531 | $97.53 |
| Put: $7.00 − $100 + $97.53 | $4.53 |
| C − P = S − PV(K) | $2.47 |
| Implied stock from the options: $7.00 − $4.53 + $97.53 | $100.00 |
| Rates to 6%: PV(K) = $97.04; C − P | $2.96, up $0.49 |
If the put were quoted at $5.03 instead, sell it, buy the call, short the stock and lend $97.53: $0.50 a share, $50 a contract, locked in today.
The formulas
European, no dividends: a call minus a put is the stock minus the discounted strike.
Rearranged for the put, or for the stock the options imply.
PV(K) from parity, then the rate that discounts K to it.
Sell the rich thing, buy its synthetic; every leg cancels at expiry.
Worked example
Discount the strike, then rearrange parity for the put. The follow-up explains which of the two costs more and why.
See it move
Same stock and options. Change the rate, the time to expiry, the volatility and where the strike sits against the stock.
- Raise the rate. PV(K) falls, C − P rises, the call gets dearer and the put cheaper.
- Raise the volatility. The call and the put both rise; C − P does not move at all.
- Lengthen the expiry. PV(K) falls and C − P rises; both options are worth more with more time.
- Raise the strike. The call falls, the put rises, and C − P falls with PV(K) rising.
Run it backwards
The stock screen is down but the options still trade. What stock price do they imply?
S = C − P + PV(K). The two option prices and the discounted strike are enough; no model is needed.
The follow-up: if the screen came back showing a lower price, suspect a dividend or a borrow cost, both of which lower the call and raise the put.
Traps
Say it in the interview
“A dealer quotes the put fifty cents above where parity puts it. What do you do?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- C − P = S − PV(K). No volatility, no model.
- Rearrange for whichever piece is missing: the put, the stock, the rate.
- Rich put: sell it, buy call, short stock, lend PV(K). Every leg cancels; the gap is the profit.
- Higher rates favour calls. Dividends and borrow break the clean version.