Reading 74

The derivation of put-call parity for European options is based on the payoffs of two portfolio combinations: a fiduciary call and a protective put.

A fiduciary call is a combination of a call with exercise price $X$ and a pure-discount, riskless bond that pays $X$ at maturity (option expiration). The payoff for a fiduciary call at expiration is $X$ when the call is out of the money, and $X + (S - X) = S$ when the call is in the money.

A protective put is a share of stock together with a put option on the stock. The expiration date payoff for a protective put is $(X - S) + S = X$ when the put is in the money, and $S$ when the put is out of the money.

If at expiration $S \geq X$:

• The protective put pays $S$ on the stock while the put expires worthless, so the payoff is $S$.
• The fiduciary call pays $X$ on the bond portion while the call pays $(S - X)$, so the payoff is $X + (S - X) = S$.

If at expiration $X > S$:

• The protective put pays $S$ on the stock while the put pays $(X - S)$, so the payoff is $S + (X - S) = X$.
• The fiduciary call pays $X$ on the bond portion while the call expires worthless, so the payoff is $X$.

In either case, the payoff on a protective put is the same as the payoff on a fiduciary call. Our no-arbitrage condition holds that portfolios with identical payoffs regardless of future conditions must sell for the same price. We can express the put-call parity relationship as:

$c + X(1 + R_f)^{-T} = S + p$

Equivalencies for each of the individual securities can be expressed as:

• $S = c - p + X(1 + R_f)^{-T}$
• $p = c - S + X(1 + R_f)^{-T}$
• $c = S + p - X(1 + R_f)^{-T}$
• $X(1 + R_f)^{-T} = S + p - c$

Note that the options must be European style and the puts and calls must have the same exercise price and time to expiration for these relations to hold.

The single securities on the left-hand side of the equations all have exactly the same payoffs as the portfolios on the right-hand side. The portfolios on the right-hand side are the synthetic equivalents of the securities on the left. For example, to synthetically produce the payoff for a long position in a share of stock:

$S = c - p + X(1 + R_f)^{-T}$

This means that the payoff on a long stock can be synthetically created with a long call, a short put, and a long position in a risk-free discount bond.

Put-call-forward parity is derived with a forward contract rather than the underlying asset itself. Consider a forward contract on an asset at time $T$ with a contract price of $F_0(T)$. At contract initiation the forward contract has zero value. At time $T$, when the forward contract settles, the long must purchase the asset for $F_0(T)$. The purchase (at time = 0) of a pure discount bond that will pay $F_0(T)$ at maturity (time = $T$) will cost $F_0(T)(1 + R_f)^{-T}$.

By purchasing such a pure discount bond and simultaneously taking a long position in the forward contract, an investor has created a synthetic asset. At time = $T$ the proceeds of the bond are just sufficient to purchase the asset as required by the long forward position. Because there is no cost to enter into the forward contract, the total cost of the synthetic asset is the present value of the forward price, $F_0(T)(1 + R_f)^{-T}$.

The put-call-forward parity relationship is derived by substituting the synthetic asset for the underlying asset in put-call parity. Substituting $F_0(T)(1 + R_f)^{-T}$ for the asset price $S_0$ in $S + p = c + X(1 + R_f)^{-T}$ gives:

$F_0(T)(1 + R_f)^{-T} + p_0 = c_0 + X(1 + R_f)^{-T}$

which is put-call-forward parity at time 0, the initiation of the forward contract, based on the principle of no arbitrage. By rearranging, put-call-forward parity can also be expressed as:

$p_0 - c_0 = [X - F_0(T)](1 + R_f)^{-T}$

APPLICATION OF OPTIONS THEORY TO CORPORATE FINANCE

We can use options theory to describe the claims of a firm's shareholders and debtholders. Consider a hypothetical company with assets that are financed by equity and a single issue of zero-coupon debt. The value of its assets at any point in time is the sum of the value of its equity and the value of its debt.

Shareholders' position as a long call. Due to the limited liability nature of corporate equity, the price of a company's shares cannot fall below zero. Shareholders effectively have a European call option, with the company's assets as the underlying, and an exercise price equal to the face value of debt $D$.

• When the debt matures, if the value of the company's assets is greater than $D$, shareholders will "exercise their call option" to acquire the assets (i.e., pay off the debt and keep the residual).
• If the value of the company's assets is less than $D$, the shareholders will "let the option expire worthless" (i.e., default on the debt and let the debtholders claim the company's assets).

Shareholders' position as a long put and long the underlying. Alternatively, we can think of the shareholders' position as consisting of the firm's assets, plus a long put option that gives them the right to sell the firm's assets to the debtholders at an exercise price of $D$.

Debtholders' position. Debtholders may be viewed as having a risk-free claim to $D$ at maturity, plus a short put option exercisable by the shareholders. This put option effectively represents the firm's default risk, or its credit spread, because the shareholders will only exercise the put if the firm is insolvent (i.e., if the value of its assets is less than $D$). A default is synonymous with shareholders exercising the put option.

Using put-call parity, we can set up the following equations for the value of the firm's assets, $V_0$:

$V_0 + p_0 = c_0 + \text{PV}(D)$, and therefore:

$V_0 = c_0 + \text{PV}(D) - p_0$

This shows that we can view the value of the firm, $V_0$, as a portfolio of a long call option on the firm's assets, the risk-free present value of $D$, and a short put option on the firm's assets.

The last two components, $\text{PV}(D) - p_0$, make up the debtholders' position. Therefore, the long call option, $c_0$, must represent the shareholders' position. This makes sense because the payoffs of the shareholders' position as we have modeled it (combining a long put with a long position in the underlying assets) are equivalent to the payoffs of a call option. For this reason, these two ways of viewing the shareholders' position (long a call, or long the assets and long a put) are consistent with each other.