Put-Call-Parity
Last updated
Last updated
We'll 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.
Simple interest rates do not involve the compounding of money. For example, if is invested at an annual interest rate of , it grows to 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 , grows to 100 + \frac{6}{12} = $100.05 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 (in decimals, so means , and there are compounding periods in a year, the principal grows to times the principal in a year. If the interest is continuously compounded, that is, we let become very large, and the expression approaches the exponential , where is a special mathematical constant that is roughly . In our previous example, . In general, if continuous compounding at the constant rate of occurs until time , we have .
A zero-coupon bond (ZCB) with face value (par value) and maturity is an interest-bearing instrument in which an investor can pay a discounted price relative to par value today to purchase the right to earn at time . This is also sometimes called a money market account. Assuming a constant continuously compounded risk-free rate , the price of a ZCB today is
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 and a short put option with a strike price of (shown in the left column), contrasted with a portfolio consisting of that stock with a short bond with a future liability of (shown on the right column). Notice that when added, both portfolios give the same payoff. We can write this as:
Rearranging the terms in this equation also yields another interesting relationship:
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 , regardless of whether is greater than or less than . So, the portfolio must be worth initially (at time ). 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) at . For example, let this be $112.5. We represent its future price at time as .
Let the current compounded interest rate be 5 percent a year, and let be one year so that . Note that the shorted bond needs to be repaid with the amount after one year. Short selling the bond today (i.e., borrowing it to sell) gives us .
The put costs p today. (If at expiration, then the put pays ).
The call costs c today. (If at expiration, the short call is worth )
Suppose the price of a put today is $2.5. Now we can determine the arbitrage-free price of the call.
Set
This implies the arbitrage-free price for the call option is $20.