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\$100 is invested at an annual interest rate of 6%6\%, it grows to $106\$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%6\%, $100\$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 rr (in decimals, so 6%6\% means r=0.06r = 0.06, and there are mm compounding periods in a year, the principal grows to ((1+r)m)m(\frac{(1+r)}{m})^m times the principal in a year. If the interest is continuously compounded, that is, we let mm become very large, and the expression ((1+r)m)m(\frac{(1+r)}{m})^m approaches the exponential ere^r, where ee is a special mathematical constant that is roughly 2.718282.71828. In our previous example, exp(0.06)=1.0618\text{exp}(0.06) = 1.0618. In general, if continuous compounding at the constant rate of rr occurs until time tt, we have exp(rt)\text{exp}(rt).

Zero-Coupon Bonds

A zero-coupon bond (ZCB) with face value (par value) KKand maturity TT is an interest-bearing instrument in which an investor can pay a discounted price B0B_0 relative to par value today to purchase the right to earn KKat time TT. This is also sometimes called a money market account. Assuming a constant continuously compounded risk-free rate rr, the price of a ZCB today is B0=Kexp(rT)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 100100 and a short put option with a strike price of 100100 (shown in the left column), contrasted with a portfolio consisting of that stock with a short bond with a future liability of 100100(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\text{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\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+KS(T)=0S(T) - K + K - S(T) = 0 , regardless of whether S(T)S(T) is greater than or less than KK. So, the portfolio must be worth 00 initially (at time t=0t= 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)KC(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=0t= 0. For example, let this be $112.5. We represent its future price at time t=Tt = T as S(T)S(T).

  • Let the current compounded interest rate be 5 percent a year, and let TT be one year so that exp(0.051)=0.95\text{exp}(-0.05*1) = 0.95. Note that the shorted bond needs to be repaid with the amount K=100K = 100 after one year. Short selling the bond today (i.e., borrowing it to sell) gives us 100exp(0.051)100*\text{exp}(-0.05*1).

  • The put costs p today. (If S(T)<KS(T) < K at expiration, then the put pays KS(T)K-S(T)).

  • The call costs c today. (If S(T)>KS(T) > K at expiration, the short call is worth KS(T)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)exp(rT)K+p=112.50.95100+2.5=20.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.

Last updated