Put-Call Forward Parity

The put-call forward parity extends the put-call parity to include the forward contracts. To get the put-call forward parity, we substitute the present value of the forward price, \(F_0(T)\), for the underlying price:
$$F_0(T)\left(1+r\right)^{-T}+p_0=c_0+X\left(1+r\right)^{-T}$$

Deriving Put-Call Forward Parity

Consider an investor whose main objective is to benefit from an increase in underlying value and hedge against a decrease in underlying value. Consider the following portfolios:

Portfolio A

At time \(t=0\), an investor buys a forward contract and a risk-free bond whose face value is equal to the forward price \(F_0\left(T\right)\). The investor then puts an option on the underlying at a price of \(p_0\) whose exercise price is \(X\) at \(t=T\). The cost of this strategy is:

$$F_0(T)\left(1+r\right)^{-T}+p_0$$

Portfolio B

At time \(t=0\), an investor buys a call option at a price of \(c_0\) on the same underlying exercise price of \(X\) and a risk-free bond redeemable at a price of \(X\) at time \(t=T\). The cost of this transaction is:

$$c_0+X\left(1+r\right)^{-T}$$

Portfolio A is called synthetic protective put. Compared to a synthetic put, a synthetic protective put replaces the underlying cash position with a synthetic position using forward purchase and a risk-free bond.

Portfolio B is the same fiduciary call as in the put-call parity seen previously.

Cash flows at time \(t=T\) for the synthetic protective put and the fiduciary call are shown in the following table:

$$
\begin{array}{l|c|c|c}
 \textbf { Portfolio Position } & \begin{array}{l}
\textbf { Put exercised } \\
\boldsymbol{S}_{\boldsymbol{T}}<\boldsymbol{X}
\end{array} & \begin{array}{l}
\textbf { No Exercise } \\
\boldsymbol{S}_{\boldsymbol{T}}=\boldsymbol{X}
\end{array} & \begin{array}{l}
\textbf { Call Exercised } \\
\boldsymbol{S}_{\boldsymbol{T}}>\boldsymbol{X}
\end{array} \\
\hline \textbf { Fiduciary Call: } \\
\hline \begin{array}{l}
\text { Purchased Call } \\
\text { Option }
\end{array} & 0 & 0 & S_T-X \\
\hline \text { Risk-free Asset } & X & X & X \\
\hline \text { Total: } & X & X\left(=S_T\right) & S_T \\
\hline \textbf { Synthetic Protective Put: } \\
\hline \text { Purchased Put at } p_0 & X-S_T & 0 & 0 \\
\hline \begin{array}{l}
\text { Purchased Forward } \\
\text { Contract }
\end{array} & S_T-F_0(T) & S_T-F_0(T) & S_T-F_0(T) \\
\hline \begin{array}{l}
\text { Risk-free bond are } \\
\text { currently priced as } \\
F_0(T)(1+r)^{-T}
\end{array} & F_0(T) & F_0(T) & F_0(T) \\
\hline \text { Total: } & X & S_T(=X) & S_T \\
\end{array}
$$

Since portfolios A and B have identical payoffs at time \(t=T\), the costs of these portfolios must be identical at time \(t=0\). Therefore, based on no-arbitrage conditions, the put-call forward parity is given by:

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

Example: Put-Call Forward Parity

Capital Investments would like to buy a 6-month put option on a company’s shares, whose current price is $195 per share. The exercise price of the put options is $190.00 per share.

The 6-month call option on the same shares trades at $64 per share with the same exercise price of $190.00. Using the put-call forward parity and assuming a 1.5% risk-free rate, the price of the put option is closest to:

Solution

Using the put-call forward parity

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

Making \(p_0\) the subject of the formula, we get:

$$p_0=c_0+\left(X-F_0\left(T\right)\right)^{-T}$$

We need to solve for \(F_0\left(T\right)\), which, if you recall, is given by:

$$\begin{align}F_0\left(T\right)&=S_0\left(1+r\right)^T\\&=195\left(1.015\right)^{0.5}\\&=\$196.4571\end{align}$$

As such,

$$\begin{align}p_0&=c_0+{\left(X-F_0\left(T\right)\right)(1.015)}^{-T}\\&=64+{\left(190-196.4571\right)(1.015)}^{-0.5}\\&=\$57.59\end{align}$$

Option Put-Call Parity Applications: Firm Value

The put-call parity relationship can be used to define a firm’s value based on equity holders’ and debt holders’ interests.

As a rule of thumb, at time \(t=0\), a company’s market value, \(V_0\), is equivalent to the present value of its outstanding debt obligations, \(PV(D)\), and equity, \(E_0\), where the borrowed funds are in zero-coupon debt with a face value of \(D\).

In an equation, we can express this relationship as:

$$V_0=E_0+PV(D)$$

In the event of debt maturity at \(t=T\), the assets and debts of the company will be split between debtholders and shareholders, with two possible outcomes based on the company’s value at that given time:

Solvency (\(V_T>D\))

Recall that solvency refers to a company’s ability to meet its financial obligations and long-term debt. If at time \(T\), a firm’s value (\(V_T\)) is greater than the face value of the debt, ( \(\mathbf{V}_\mathbf{T}>\mathbf{D})\) the firm is solvent, and thus able to return capital to both the shareholders and the debtholders.

Debt holders come first when distributing capital returns. As such, they receive the debt repayments (D) in full. On the other hand, the shareholders receive what remains. That is \(E_T=V_T-D\).

In summary, we’ve established that shareholders benefit if a company can meet its debt obligation and maintain solvency. On the other hand, debt holders benefit when a company is solvent and hence meets its debt obligations.

Insolvency (\(V_T<D\))

Insolvency refers to a company’s inability to meet its financial obligations and long-term debts. This occurs if, at debt maturity (\(T\)), a company’s value is less than the debt’s face value, \(V_T<D\).

When a firm is insolvent, the shareholders receive the residual, which is equal to zero \(\left(E_T=0\right)\), and the debtholders are owed more than the firm’s total assets. As such, the debtholders receive \(V_T<D\) to cover the debt of \(D\) at time \(T\).

Payoff in Terms of Options

Note that shareholders retain the unlimited upside potential in solvency and limited downside potential in insolvency. On the other hand, the debtholders are limited to receiving debt repayment in the case of solvency and principal and interest in the case of insolvency.

Intuitively, the payoff profiles can be mathematically represented as follows:

• At time \(t=T\), the payoff of the shareholders can be expressed as \(max(0,D-V_T)\).
• On the other hand, the debt holder’s payoff is expressed as \(min(V_T,D)\).

In terms of options, the payoff profiles can be expressed as follows:

• Shareholders’ payoff: They hold a long position in the underlying firm’s assets value. For instance, assume that we have bought a put option on the firm value, \(V_{T}\), with the exercise price of \(D\). The payoff is \(max(0, D-V_T)\).
• Debtholders’ payoff: The hold a long position in the risk-free bond, valued at \(D\), and have sold a put option to the shareholders on the firm value, \(V_T\), with an exercise price of \(D\).

Remember the put-call parity relationship:

$$ S_0+p_0=c_0+PV\left(X\right)$$

If we replace the underlying asset, \(\left(S_0\right)\), for the company’s value at time 0, \(\left(V_0\right)\), and further replace the risk-free bond, \(\left(X\right)\), with debt, \(\left(D\right)), the equation becomes:

$$V_0+p_0=c_0+PV(D)$$

We can also rearrange the formula to solve for the value of the company, \(\left(V_0\right)\):

$$V_0=c_0+PV\left(D\right)-p_0$$

From the above results, the shareholders have a payoff equivalent to that of a call option (\(c_0\)) on the firm’s value. On the other hand, the debtholders hold a position of \(\left(D\right)-p_0\), which is the risk-free debt plus a short position in a put option.

This put option may be seen as a credit spread on a company’s debt or the premium above the risk-free rate a company must pay debtholders to bear insolvency risk. The value of the put option to shareholders rises as the probability of insolvency grows.

Question

Which of the following best describes the replication of a risk-free bond under the put-call parity?

A. Long underlying, short call option, and long put option.

B. Long underlying, short risk-free bond, and long put option.

C. Short underlying, long risk-free bond, and long call option

Solution

The correct answer is A.

Recall that the put-call parity relationship may be expressed as:

$$\begin{align}c_0+X\left(1+r\right)^{-T}&=p_0+s_0\\\Rightarrow X\left(1+r\right)^{-T}&=p_0+s_0-c_0\end{align}$$

The risk-free bond replicating individual positions under put call parity is a long underlying, short call option and long put option.

B is incorrect: Call option individual replication position equals long underlying, short risk-free bond, and long put option.

$$c_0=p_0+s_0-X\left(1+r\right)^{-T}$$

C is incorrect: Put option position equals short underlying, long risk-free bond, and long call option.

$$p_0=c_0+X\left(1+r\right)^{-T}-s_0$$