# Put-call parity

In this article, we explore a relationship between the price of a European call option and a European put option that both have the same underlier, expiry, and strike price. This knowledge can help you identify arbitrage opportunities in an options market if the put-call parity isn’t maintained, as well as pin down prices for a put given a call price or vice-versa.

We’ll first discuss some aspects of interest rates that we will need in this and future articles.

**Compounding Interest Rates**

Simple interest rates do not involve the compounding of money. For example, if $100 is invested at an annual interest rate of 6 percent, it grows to $106 a year from now. Compound interest is when interest is earned on the principal plus any accrued interest. For example, if the compounding interval is 1 month, with an annual rate of 6 percent, $100 grows to 100 + 6/12 = $100.5 in a month. In a year, there are 12 such compounding periods, so it grows to

$100*(1+6/12)¹² = $106.16.

In general, if the interest rate is *r* (in decimals, so 6 percent means *r = *0.06), and there are *m* compounding periods in a year, the principal grows to (1+*r/m*)ᵐ times the principal in a year. If the interest is continuously compounded, that is, we let *m* become very large, the expression (1+*r/m*)ᵐ approaches the exponential *eʳ*, where *e* is a special mathematical constant that is roughly 2.71828. In our previous example, exp(0.06)= 1.0618. In general, if continuous compounding at the constant rate of r occurs until time t, we have exp(*rt).*

**Zero Coupon Bonds**

A zero coupon bond(ZCB) with face value (par value) K and maturity T is an interest bearing instrument in which an investor can pay a discounted price B₀ relative to par value today to purchase the right to earn *K *at time* T*. This is also sometimes called a money market account. Assuming a constant continuously compounded risk-free rate r, the price of a ZCB today is B₀ = K*exp*(-rT).*

**Put-Call Parity**

Now to return to the previously scheduled option strategy content. Something magical happens when we buy (long) a call option, and write (short) a put option, with the same strike price and expiration:

In this case, we have a portfolio consisting of a long call option with a strike price of 100 and a short put option with a strike price of 100 (shown in the left column), contrasted with a portfolio consisting of that stock with a short bond with a future liability of 100 (shown on the right column). Notice that when added, both portfolios give the same payoff. We can write this as:

Long call + short put = long stock + short bond

Rearranging the terms in this equation also yields another interesting relationship:

Long stock + short call = short put + long bond

In words, a covered call = cash secured put, where the put and call have the same strike and expiration.

We can also represent this using a table:

In the above portfolio, we’ve moved the long call and short put on the left hand side of the equation above to the right hand side. Notice that at time *t = T*, the portfolio is always worth *S*(*T*)-*K + K -S*(*T*) = 0, regardless of whether *S*(*T*) is greater than or less than *K*. So, the portfolio must be worth 0 initially (at time* t=0*), otherwise we have an arbitrage opportunity. This allows us to write put-call parity in the following form, which is the same as the relationship obtained from the payoff diagram above (note that a long call has the negative payoff of a short call):

Let’s consider an example to see how this relationship can be used:

- A stock costs
*S*(0)*t*= 0, for example, let this be $112.5. We represent its future price at time*t= T*as*S*(*T*) - Let the current compounded interest rate to be 5 percent a year, and let
*T*be one year, so that exp(-0.05*1) = 0.95. Note that the shorted bond needs to be repaid with amount*K*= 100 - The put costs
*p*today. (If*S*(*T*) < K at expiration, then the put pays*K-S*(*T*)). - The call costs
*c*today. (If*S*(*T*) > K at expiration, the short call is worth*K-S*(*T*)) - Suppose the price of a put today is $2.5, now we can determine the arbitrage-free price of the call.

Set *c* = *S*(0) - *K**exp(-*rT*) + *p* = 112.5-0.95*100 + 2.5 = 20, which implies the arbitrage free price for the call option is $20.

**Next Time**

In our next article, we will look into Greeks, which capture rates of change for options prices with respect to various factors , and which will be necessary for us to explain the Arrow protocol in more detail.