# Put-Call-Forward Parity for European Options

Another important concept in the pricing of options has to do with put-call-forward parity for European options. This involves buying a call and bond (fiduciary call) and a synthetic protective put, which requires buying a put option and a forward contract on the underlying that expires at the same time as the put option.

## Put-Call-Forward Parity

An alternative structure for a protective put is to buy a forward contract and a risk-free bond in which the face value is the forward price rather than purchasing the underlying asset. As we have established that a fiduciary call is equivalent to a “regular” protective put, it holds that a fiduciary call must also be equivalent to a protective put with a forward contract.

The fiduciary call consists of a long call and a long position in a zero-coupon bond:

$$\text{Value at inception} = c_0 + \frac{X}{(1+r)^T}$$

The synthetic protective put is made up of a long put and a long forward:

$$\text{Value at inception} = p_0 + \frac{F_0(T)}{(1+r)^T}$$

As the two portfolios have precisely the same payoff, their original investments should be the same as well. By setting the fiduciary call equal to the synthetic protective put, we establish the put-call parity for options on forward contracts.

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

Solving for $$F_0(T)$$, we acquire the equation for the forward price in terms of the call, put, and riskless bond.

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

Where $$\frac{F_0(T)}{(1+r)^T}$$ is the value of the forward today multiplied by $$(1+r)^T$$ to get its value at expiration.

Therefore, a synthetic forward combines a long call, a short put, and a zero-coupon bond with a face value of $$X – F_0(T)$$.

## Question

A European put has an exercise price of $58 that expires in 120 days. The long forward is priced at$55 (also expires in 120 days) and makes no cash payments during the life of the options. The risk-free rate is 4.5% and the put is selling for $3.00. According to the put-call-forward parity, what is the price of a call option with the same strike price and expiration date as the put option? A.$50.43

B. $3.31 C.$0.83

Solution

c0 = p0 + Fo/(1 + r)T – X/(1 + r)T

c0 = 3.00 + 55/(1.045)120/365 – 58/(1.045)120/365

c0 = 0.043

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