Put-call-forward parity creates a relationship between a forward contract, put and call options on an underlying. It is derived from the put-call parity relationship by modifying the protective put strategy.

A protective put is a combination of a long position in the underlying and associated put option. Instead of obtaining a long position in the underlying directly, we can create by going long a forward contract and a risk-free asset. For example, if we buy a forward contract on an asset with forward price F0(T) and invest the present value of the forward contract in a risk-free asset, it is equivalent to a long position in the underlying.

## Put-call parity vs put-call-forward parity

The put-call parity relationship is given as follows:

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

A long position in the underlying is represented by S_{0}, we substitute it with the present value of the forward price invested in the risk-free asset to get the following put-call-forward parity relationship:

p_0+\frac{F_0(T)}{{(1+r)}^T}=c_0+\frac{X}{{(1+r)}^T}

If we arrange it, we get:

p_0-c_0=\frac{X-F_0(T)}{{(1+r)}^T}

## Test

You want to create a put- call-parity portfolio using a 1-year forward contract. If the excess of put value over the call value is $3, the associated exercise price is $50 and the risk-free rate is 4%, the forward price must be:

- Lower than the exercise price
- Higher than the exercise price
- No such relationship can exist

## Show answer

A is correct.

The modified expression for the put-call-forward parity is given by the following equation:

$$ p_0-c_0=\frac{X-F_0(T)}{{(1+r)}^T} $$

If the left-hand side of the equation is positive, the forward price must be lower than the exercise price.