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).
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) at 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 after one year. Short selling the bond today (i.e. borrowing it to sell), gives us 100*exp(-0.05*1).
- 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.
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.