A call and a put at the same strike and expiration are not two separate prices that happen to be quoted next to each other. They are locked together by one equation, and a market maker who sees them drift apart will trade until they snap back. The equation is called put-call parity, and it is the most useful piece of arithmetic on the chain.
Build stock out of options
Buy a call and sell a put, both at strike K, both for the same expiration. Follow it to expiration. If the stock finishes above K, you exercise the call and buy stock at K. If it finishes below K, the put you sold is exercised against you and you buy stock at K. Either way you end up owning the stock at K on that date. So the package of a long call and a short put is a promise to buy the stock at K, which is the same thing as a forward contract struck at K.
That promise has a value today: the forward price minus K, discounted back. Written out with the carry from the last lesson: C - P = S - K + carry, where carry is the interest on K until expiration minus any dividend paid before then. Rearrange it and you get the tool from lesson one: K + C - P = forward.
Notice what is missing from that equation: volatility. Parity compares two packages that pay exactly the same at expiration, so it holds whatever the stock does next. There is no model and no forecast in it. The only inputs are the stock price, the strike, the rate, the dividends and the time left.
The $100 call minus the $100 put with the stock at $100, against days to expiry. The 20% and 60% volatility lines sit exactly on top of each other. Drag the rate up and the gap widens; set the dividend above the rate and it turns negative.
So the call-put gap at the money is carry, not opinion. When volatility rises, both get dearer by the same amount and the gap stays put.
A worked row
Stock at $100, strike $100, three months to expiration, rates at 4%, no dividend. Interest on $100 for a quarter is $1, so C - P must be about $1. If the call is $5.00, the put should be near $4.00. Now suppose the put is quoted at $3.50. Sell the call at $5.00, buy the put at $3.50, buy the stock at $100. You have spent $98.50 for a position that is worth exactly $100 at expiration whatever happens: above $100 the call takes your stock at $100, below it the put sells it at $100. That is $1.50 of gain against a $1 interest bill, a risk-free 50 cents. Market makers run this trade in size the moment it appears, which is why parity holds to within the bid-ask spread all day long.
The trade in that example, long stock, short call, long put, is called a conversion. Its mirror, short stock, long call, short put, is a reversal. A conversion is worth exactly the strike at expiration wherever the stock ends, so it behaves like a loan: you pay about the strike, discounted, today and get the full strike back on expiration day.
Long 100 shares at $100, short the $100 call, long the $100 put, 90 days out at a 4% rate. It pays $100 a share wherever the stock ends, so the line is flat at about $98 per contract: the interest on the $99.02 it cost. Drag days down to watch that interest build; drag the rate to change it.
The flat line is the whole point. A desk holding conversions has no view on the stock at all. Its profit is the gap between the interest the package earns and the interest the desk pays to fund it.
What every row of the chain must satisfy
- At the money, the call mid minus the put mid equals the carry: a small positive number when rates exceed the dividend yield, and it grows with time to expiration.
- Move up one strike and C - P falls by almost exactly the strike spacing, because S - K fell by that much.
- The call and the put at the same strike carry the same implied volatility, or very nearly. If the chain shows the put at a much higher IV than the call at the same strike, that is usually a stale quote or borrow cost, not an opportunity.
Where it bends a little
US stock options are American style, so they can be exercised early. That adds a few cents to deep in-the-money puts, and to calls just before an ex-dividend date. Hard-to-borrow stock acts like a stock with an extra dividend, pushing the put up relative to the call. Bid-ask spreads blur everything by a few cents. None of these break the equation; they just set the width of the band it lives in.
The maths: parity from two portfolios
Build two portfolios today. Portfolio A: one call at strike K, plus cash of K × e^(-rT) on deposit, which grows to exactly K by expiration. Portfolio B: one put at strike K, plus e^(-qT) shares of stock, which grows to one full share by expiration if the dividends are reinvested in more stock.
At expiration, if the stock ends at S_T above K, A is worth (S_T - K) + K = S_T and B is worth 0 + S_T = S_T. If it ends below K, A is worth 0 + K = K and B is worth (K - S_T) + S_T = K. Both are worth the larger of S_T and K in every case, so they must cost the same today: C + K e^(-rT) = P + S e^(-qT), or C - P = S e^(-qT) - K e^(-rT).
Worked example, the row from the text: S = K = $100, T = 0.25, r = 4%, q = 0. K e^(-rT) = 100 × e^(-0.01) = $99.005, so C - P = 100 - 99.005 = $0.995, the "about $1" of carry. This is exact for European options. American options add a small early-exercise premium, which is why US stock chains obey it inside a narrow band rather than to the penny.
Your turn
Open the AAPL chain on the next monthly expiration. Read the at-the-money call mid and put mid, then work out C - P - (S - K). What is left should be small: the carry, straight off the chain.