4. Put-call parity: the equation that ties every piece together

Derivatives 2: Options, the mechanics

The three previous modules have treated calls and puts as separate instruments. They are not. There is a relationship between them so tight that, given the price of one, the other's price is determined. It is called put-call parity, and it is not a valuation formula or a model with arguable assumptions: it is an identity about what two portfolios end up being worth. If it does not hold, there is free money on the table.

Two ways of building the same profile

An investor wants to share in the rise of a share trading at 100 without risking more than a certain limit. He has two routes, both expiring in a year with a strike of 100.

• Route A. He buys the share for 100 and buys a put with a strike of 100 for 6. Outlay: 106. This is called a protective put.
• Route B. He buys a call with a strike of 100 for 10 and sets aside whatever it takes to have 100 euros in a year's time. If that is 96 today, his outlay is also 106.

A year later:

• The share closes at 80. In A the share is worth 80, but the put sells it at 100: total 100. In B the call lapses and the money set aside has turned into 100. Both, 100.
• The share closes at 100. In A, share at 100 and put worthless. In B, call worthless and 100 in cash. Both, 100.
• The share closes at 130. In A the share is worth 130. In B the call gives 30 plus the 100 in cash. Both, 130.

There is no scenario in which they differ. And two portfolios worth the same in every future state have to cost the same today: otherwise anybody would buy the cheap one and sell the dear one with no risk at all.

The identity, put into words

Put-call parity says exactly this: holding the share alongside a protective put is equivalent to holding a call plus the money needed to pay the strike price at expiry.

Put another way: the difference between what a call costs and what a put of the same strike and expiry costs equals the difference between the share's price today and the present value of the strike. In our numbers, 10 minus 6 is 4, exactly what separates the share at 100 from the present value of 96.

Those four euros are neither a coincidence nor the intermediary's margin: they are interest. Whoever buys the call instead of the share does not advance the 100 euros, and that implicit funding has a price. If rates were zero, call and put would cost the same with the share at the strike.

Three conditions make it clean: same underlying, strike and expiry; European style; and no dividends. If there is a dividend, it enters the calculation as an outflow of value for the shareholder and the relationship adjusts.

What parity lets you do: synthetic positions

From an identity with four elements — share, cash, call and put — you can solve for any one of them. That is: any position can be manufactured out of the other three. Some that come straight out:

• Synthetic call. Long share plus long put of the same strike.
• Synthetic put. Long call plus short share.
• Synthetic share. Long call plus short put of the same strike and expiry: it rises and falls like the share without owning it.

This is not an intellectual game: it explains why somebody who needs a particular profile can build it by several routes and pick the cheapest, and why the premiums of calls and puts of the same strike move together.

If the relationship breaks, there is arbitrage

Suppose the 100 call trades at 10, as it should, but the 100 put can be bought at 5.40 instead of 6. Parity is broken by 60 cents. A trader does this all on the same day:

• Buys the share for 100.
• Buys the put for 5.40.
• Sells the call for 10.
• Borrows 96, committing to repay 100 in a year's time.

The inflows add up to 106: the 10 from the call sold and the 96 borrowed. The outflows add up to 105.40: 100 for the share and 5.40 for the put. He pockets 0.60 today without putting up a euro of his own.

A year later, that combination is worth exactly 100 in any imaginable scenario. If the share is at 40, the put sells it at 100. If it is at 250, the call takes it away at 100. With that 100 he repays the loan and comes out level. The initial 0.60 was riskless profit.

That is why parity holds with remarkable precision in liquid markets: as soon as the deviation covers commissions and spreads, somebody corrects it. Persistent deviations usually signal real frictions, not money being given away.

What parity does not say

It is worth drawing its boundaries, because it is easily over-interpreted. Parity does not say what a call ought to be worth: it says what the put ought to be worth given the price of the call. It is a relationship between prices, not a theory of the fair price: if the market inflates both premiums, parity will hold between the two just the same. And it says nothing about whether any position is a good idea.

Worked case: the same protection by two routes

An investor has 106 euros and wants exposure to the share with the loss capped at 6 for a year. He buys the share and the 100 put: if the share falls to 55, his portfolio is worth 100 and he has lost 6. His neighbor buys the 100 call for 10 and leaves 96 in a one-year bill: the call lapses and he is left with the 100 from the bill, the same loss of 6.

If the share rises to 140, the first has 140 and the second has 40 from the call plus 100 from the bill, also 140. Same outlay, same result in every outcome. The difference is one of form, not of substance.