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 S0, 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.