Put-Call-Parity

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.

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 $6\%$, 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 $6\%$, $\$100$ grows to 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 $6\%$ means $r = 0.06$, and there are $m$ compounding periods in a year, the principal grows to $(\frac{(1+r)}{m})^m$ times the principal in a year. If the interest is continuously compounded, that is, we let $m$ become very large, and the expression $(\frac{(1+r)}{m})^m$ approaches the exponential $e^r$, where $e$ is a special mathematical constant that is roughly $2.71828$. In our previous example, $\text{exp}(0.06) = 1.0618$. In general, if continuous compounding at the constant rate of $r$ occurs until time $t$, we have $\text{exp}(rt)$.

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 $B_0$ 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 $B_0 = K*\text{exp}(-rT)$

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:

$$\text{Long call + short put = long stock + short bond}$$

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

$$\text{Long stock + short call = short put + long bond}$$

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:

$$C(K, T) — P(K, T) = S(0)-exp(-rT)K$$

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 be 5 percent a year, and let $T$ be one year so that $\text{exp}(-0.05*1) = 0.95$. Note that the shorted bond needs to be repaid with the amount $K = 100$ after one year. Short selling the bond today (i.e., borrowing it to sell) gives us $100*\text{exp}(-0.05*1)$.

• 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 . Now we can determine the arbitrage-free price of the call.

Set

$$c = S(0) - \text{exp}(-rT)K + p = 112.5 – 0.95*100 + 2.5 = 20.$$

This implies the arbitrage-free price for the call option is $20.