# Put-Call Parity

Put and call values are mathematically bound together by an equation referred to as put-call parity. In its basic form, put-call parity states:The put-call parity assumes that options are not exercised before expiration (that is, that they are European style). This version of the putcall parity is for European options on non-dividend-paying stocks. Put-call parity can be modified to reflect the values of options on stocks that pay dividends. In practice, equity-option traders look at the equation in a slightly different way:

Traders serious about learning to trade options must know put-call parity backward and forward. Why? First, by algebraically rearranging this equation, it can be inferred that synthetically equivalent positions can be established by simply adding stock to an option. Again, a put is a call; a call is a put. For example, a long call is synthetically equal to a long stock position plus a long put on the same strike, once interest and dividends are figured in.

A synthetic long stock position is created by buying a call and selling a put of the same month and strike. Understanding synthetic relationships is intrinsic to understanding options. A more comprehensive discussion of synthetic relationships and tactical considerations for creating synthetic positions is offered in Chapter 6.Put-call parity also aids in valuing options.

If put-call parity shows a difference in the value of the call versus the value of the put with the same strike, there may be an arbitrage opportunity. Arbitrageurs tend to hold synthetic put and call prices pretty close together. Generally,only professional traders can capture these types of profit opportunities, by trading big enough positions to make very small profits (a penny or less per contract sometimes) matter.

Retail traders may be able to take advantage of a disparity in put and call values to some extent, however, by buying or selling the synthetic as a substitute for the actual option if the position can be established at a better price synthetically.Another reason that a working knowledge of put-call parity is essential is that it helps attain a better understanding of how changes in the interest rate affect option values. The greek rho measures this change.

Rho is the rate of change in an option’s value relative to a change in the interest rate.Although some modeling programs may display this number differently,most display a rho for the call and a rho for the put, both illustrating the sensitivity to a one-percentage-point change in the interest rate.When the interest rate rises by one percentage point, the value of the call increases by the amount of its rho and the put decreases by the amount of its rho.

Likewise, when the interest rate decrease by one percentage point, the value of the call decreases by its rho and the put increases by its rho. For example, a call with a rho of 0.12 will increase \$0.12 in value if the interest
rate used in the model is increased by one percentage point. Of course,interest rates usually don’t rise or fall one percentage point in one day. More commonly, rates will have incremental changes of 25 basis points. That means a call with a 0.12 rho will theoretically gain \$0.03 given an increase of0.25 percentage points.

Mathematically, this change in option value as a product of a change in the interest rate makes sense when looking at the formula for put-call parity.But the change makes sense intuitively, too, when a call is considered as a cheaper substitute for owning the stock. For example, compare a \$100 stock with a three-month 60-strike call on that same stock. Being so far ITM, there would likely be no time value in the call. If the call can be purchased at parity, which alternative would be a superior investment, the call for \$40 or the stock for \$100? Certainly, the call would be.

It costs less than half as much as the stock but has the same reward potential; and the \$60 not spent on the stock can be invested in an interest-bearing account.This interest advantage adds value to the call. Raising the interest rate increases this value, and lowering it decreases the interest component of the value of the call.A similar concept holds for puts.Professional traders often get a shortstock rebate on proceeds from a short-stock sale.

This is simply interest earned on the capital received when the stock is shorted. Is it better to pay interest on the price of a put for a position that gives short exposure or to receive interest on the credit from shorting the stock? There is an interest disadvantage to owning the put. Therefore, a rise in interest rates devalues puts.This interest effect becomes evident when comparing ATM call and put prices.