# Put-call parity

Put-call parity refers to the relationship between the value of a put option and a call option on the same underlying. It is derived from the fact that two options strategies, i.e. protective put, and fiduciary call, have the same payoffs and hence their initial values must be equal.

## Protective put

A protective put is a strategy in which an investor pays an option premium p0 to get a long position in a put option with an exercise price of X on underlying he already owns whose value is S0. At expiration T, if the underlying price ST is greater than X, he would let the option expire and receive ST, but if the option is in-the-money (because the underlying’s price is lower than the exercise price) he would exercise the option and receive X. This creates a lower bound X on his portfolio value below X and an unlimited upside potential above X.

## Fiduciary call

A fiduciary call is a strategy in which a trader buys a call option for c0 with exercise price X and invest an amount in a zero-coupon bond paying X at maturity. The value of the portfolio at time 0 is c0 plus the present value of a zero-coupon bond which equals X/(1 + r)T. If the underlying price at expiration (ST) is above the exercise price, the option is in-the-money, the trader uses the proceeds from the zero-coupon bond to buy the underlying at X and receive a net payoff of ST. However, if the call option is worthless, he just received the maturity value of the zero-coupon bond X.

Since the payoff of both strategies is the same, we can create the following equivalence:

p_0+S_0=c_0+\frac{X}{{(1+r)}^T}

If we rearrange the above expression, we can derive the value of one variable from the other three, as shown below:

p_0=c_0+\frac{X}{{(1+r)}^T}-S_0
c_0=S_0+p_0-\frac{X}{{(1+r)}^T}

It follows that we can create a position synthetically by transacting in the other three assets. For example, we can create a long call position by going long the underlying asset, buying a put option and going short the zero-coupon bond, and so on.

## Test

Using the put-call parity strategies, a synthetic long position in a put option can be created by:

A) Going long the call option, a risk-free bond and short the underlying
B) Going short the call option, a risk-free bond and long the underlying
C) Going long the call option and short the risk-free bond and the underlying

A is correct. The relationship can be obtained by rearranging the put-call parity expression:

$$p_0+S_0=c_0+\frac{X}{{(1+r)}^T}$$

$$p_0=c_0+\frac{X}{{(1+r)}^T}-S_0$$

The right-hand side shows a long position in a put option, and the left side shows a position in the other three assets. A positive sign means a long position and a negative sign means a short position.