# 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 <a href="#c247" id="c247"></a>

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 $$100 + \frac{6}{12} = $100.05$$ 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 $$r$$ (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 href="#b39d" id="b39d"></a>

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 <a href="#ccdb" id="ccdb"></a>

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:

<figure><img src="https://25047179-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F2ezG7k2AmbmPLJDUAiOv%2Fuploads%2FzWrf9kJJ1flK6eWcPIeh%2FScreen%20Shot%202022-12-22%20at%205.40.51%20PM.png?alt=media&#x26;token=05a3bf18-25dc-4e99-bbcc-5e764c3fd9d1" alt=""><figcaption></figcaption></figure>

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:

<figure><img src="https://miro.medium.com/max/1400/0*ztaY-uXClO5tRZzz" alt=""><figcaption></figcaption></figure>

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 $$ $2.5 $$. 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.