Options and volatility · O3 · 8 min
Put–call parity and synthetics
Use the link between call and put prices to price one from the other, and build one position out of others.
In the news: When a company changes its dividend, its calls and puts reprice together: put–call parity shows by how much.
The essentials
- A call minus a put is a forward
- A bought call and a sold put at the same strike and expiry buy the share at the strike whatever happens. So the pair behaves like owning the share, except that the strike is paid later instead of the price now.
- Put–call parity
- The call minus the put equals the share price less the strike’s . A 100 strike six months away at 4% is worth 98.04 today, so with the share at 100 the call costs 1.96 more than the put.
- Rates and dividends
- Higher rates shrink the strike’s present value, so calls get dearer against puts. A dividend before expiry goes to the shareholder, not the option holder, so it makes calls cheaper and puts dearer.
- Synthetics
- Any one of call, put, share and cash can be built from the other three: a . A bought call plus a sold put is long the share; the share plus a bought put is a call; the share plus a sold call is a sold put.
- Conversions and reversals
- When the call is dear against parity, a sells it, buys the put and buys the share, locking in the gap. A reversal does the opposite for a cheap call. Traders run them all day, so gaps rarely last.
Put–call parity
C − P = S − PV(dividends) − PV(K); PV(K) = K ÷ (1 + r × T)
- C, P
- call and put prices, same strike and expiry
- S
- spot price of the stock
- K
- the strike
- r
- interest rate a year, simple
- T
- years to expiry
How it works
1. A bought call and a sold put
Buy a 100-strike call and sell a 100-strike put with the same expiry. If the share ends at 120, you use the call and buy at 100. If it ends at 80, the put’s buyer makes you buy at 100. Either way, you end up buying the share for 100.
2. That is a forward
Agreeing today to buy at a fixed price on a later date is a forward. So a call minus a put at the same strike behaves just like owning the share, except that you pay the strike later instead of the share price now. Two positions with the same result must cost the same.
3. The parity equation
Owning the share costs S today. The pair costs C − P today, plus the strike paid at expiry, which is worth PV(K) now: its present value, the strike less the interest you can earn until then. Same result, same cost: C − P = S − PV(K). This is .
4. Present value of the strike
With simple interest, PV(K) = K ÷ (1 + r × T), where r is the yearly rate and T the years to expiry. A 100 strike in six months with rates at 4%: 100 ÷ (1 + 0.04 × 0.5) = 100 ÷ 1.02 = 98.04. Higher rates make PV(K) smaller, so calls get dearer against puts.
5. Dividends
If the share pays a dividend before expiry, the shareholder gets it but the option holder doesn’t. So take its present value off the share: C − P = S − PV(dividends) − PV(K). A bigger dividend makes calls cheaper and puts dearer.
6. Synthetics
Rearrange parity and any one of call, put, share and cash can be built from the other three: a . Bought call + sold put = owning the share. Owned share + bought put = a call. Owned share + sold call = a sold put. Traders use whichever version is cheaper or easier to fund.
7. When parity breaks
If the call costs more than parity says, sell it, buy the put and buy the share with borrowed money. At expiry you sell the share at the strike whatever happens and repay the loan, keeping the gap. That trade is a ; the opposite, for a cheap call, is a reversal. Traders run these all day, so gaps rarely last.
8. The forward hidden in option prices
Turn parity round and it gives the forward price the options imply: F = K + (C − P) × (1 + r × T). If the 100 call costs 3.10 more than the 100 put and one-year rates are 4%, F = 100 + 3.10 × 1.04 = 103.22. Traders read the market’s dividend and interest assumptions from it.
Worked example
A share trades at 102. Six-month options struck at 100: the put costs 4.20 and the call 6.00. Rates are 4% (simple), and a dividend worth 1.00 today is due before expiry. Is the call fairly priced?
- PV(K) = 100 ÷ (1 + 0.04 × 0.5) = 100 ÷ 1.02 = 98.04.
- C − P should be 102 − 1.00 − 98.04 = 2.96.
- So the call should cost 4.20 + 2.96 = 7.16.
- It trades at 6.00, so it is cheap by 7.16 − 6.00 = 1.16.
- Reversal: buy the call, sell the put, short the share and lend the proceeds, keeping 1.00 aside for the dividend you owe as a short seller.
- At expiry you buy the share back at 100 through one option or the other, and keep about 1.16.
The call should cost about 7.16; at 6.00 a reversal locks in roughly 1.16 a share, before costs.
Check yourself
8 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
State put–call parity and show me a trade if it breaks.
How do you create a synthetic short stock?
Remember
C − P = S − PV(K)
A call minus a put at the same strike is a forward, so any three of call, put, stock and cash price the fourth.