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  • Compounding Interest Rates
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  • Put-Call Parity

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Put-Call-Parity

PreviousBull Call SpreadNextOptions Math

Last updated 1 year ago

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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$100 is invested at an annual interest rate of 6%6\%6%, it grows to $106\$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\%6%, $100\$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 rrr (in decimals, so 6%6\%6% means r=0.06r = 0.06r=0.06, and there are mmm compounding periods in a year, the principal grows to ((1+r)m)m(\frac{(1+r)}{m})^m(m(1+r)​)m times the principal in a year. If the interest is continuously compounded, that is, we let mmm become very large, and the expression ((1+r)m)m(\frac{(1+r)}{m})^m(m(1+r)​)m approaches the exponential ere^rer, where eee is a special mathematical constant that is roughly 2.718282.718282.71828. In our previous example, exp(0.06)=1.0618\text{exp}(0.06) = 1.0618exp(0.06)=1.0618. In general, if continuous compounding at the constant rate of rrr occurs until time ttt, we have exp(rt)\text{exp}(rt)exp(rt).

Zero-Coupon Bonds

A zero-coupon bond (ZCB) with face value (par value) KKKand maturity TTT is an interest-bearing instrument in which an investor can pay a discounted price B0B_0B0​ relative to par value today to purchase the right to earn KKKat time TTT. This is also sometimes called a money market account. Assuming a constant continuously compounded risk-free rate rrr, the price of a ZCB today is B0=K∗exp(−rT)B_0 = K*\text{exp}(-rT)B0​=K∗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 100100100 and a short put option with a strike price of 100100100 (shown in the left column), contrasted with a portfolio consisting of that stock with a short bond with a future liability of 100100100(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}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}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)=0S(T) - K + K - S(T) = 0 S(T)−K+K−S(T)=0, regardless of whether S(T)S(T)S(T) is greater than or less than KKK. So, the portfolio must be worth 000 initially (at time t=0t= 0 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)KC(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= 0t=0. For example, let this be $112.5. We represent its future price at time t=Tt = Tt=T as S(T)S(T)S(T).

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

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

  • The call costs c today. (If S(T)>KS(T) > K S(T)>K at expiration, the short call is worth K−S(T)K-S(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=112.5–0.95∗100+2.5=20.c = S(0) - \text{exp}(-rT)K + p = 112.5 – 0.95*100 + 2.5 = 20.c=S(0)−exp(−rT)K+p=112.5–0.95∗100+2.5=20.

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