Question

Recall from your previous corporate finance studies. For a levered firm, firm’s assets are financed by equity and debt. That is, ?? = ?? + ??, where ??, ?? & ??
represents asset value, debt value and equity value at time ?. Suppose the firm makes no dividend payment and has a zero-coupon debt maturing at time ?. At maturity, if the value of the company asset is greater than the maturity value of the debt (?? > ??), the company will simply pay off the debt. Otherwise, the company will declare bankruptcy and debt holders will own the firm.

At maturity ? = ?:
(i) From an equity holder perspective, describe a company’s equity as a call option. In your
answers, identify the underlying asset, strike price, maturity, and the payoff function of the
call option.

(ii) From a debt holder perspective, describe a company’s debt using a call option. In your
answers, identify the positions that the debt holder takes on each asset (i.e. long or short?).

At time ? = 0:
(iii) Suppose ? and ? are option prices (i.e. premium) for call and put options. Using the result
from part (i) and put-call parity, describe a company’s debt using a put option from a debt
holder perspective. In your answers, identify the positions that the debt holder takes on each
asset (i.e. long or short?).

Answer 1

In this question we are talking about a levered firm i,e, there is a componenet of debt as well, other than just equity.

(i) Compnay's equity as a Call option

It is a call option on the assets of the firm where, strike price or the exercise price is the face value of the debt. to understand the comcept it is important to realise that when the firm takes out a debt, it pledges it's assets against the money for a certain duration with an option to buy back the assets by paying off the debt obligation.

In essence, a call option is exercised when the value>strike price, much like equity where there is positive pay-off i.e. intrinsic value if at the end of debt contract, the value of the asset>debt obligation. In a contrary situation, the option will not be exercised or the pay off on the equity will be zero (this is the insolvency or bankruptcy case).

(ii) Debt as a call option

In case of debt, the pay off is the same as the pay off on the same debt at risk freee yield plus a short put option on the assets of the firm where the strike price or the exercice price is the actual and full value of the debt.

There are two possible outcomes for the debt holder in this case:

1. Either the debt is payed-off in full here (the put option here is not exercised since the option expires out of the money here)

2. Or, the shreholders default and the put option is exercised where the contract is honoured through assets rather than the amount in the contract.

This is why the YTM on a risky debt is higher than a relatively risk free debt since a premium component is included in riskier debt. This is the price that the company pays its debtor's as a premium for the right to default which is captured in the interest rate that the company pays above the risk free rate.

(iii) Put Call Parity

Understand the put call parity here i.e.

C= Call Premium

P= Put Premium

S= Current price of underlying asset

PV(x) = Present value of the strike

The debt holder position has already been covered in part 2 above. Considering part 1, the option may expire in the money i.e. the put will not be exercised by the company and the debt contract will be settled in full to the debt holders. Hence, the debt holders will be going long on the contract and short on the put option.

Try to create a synchronization for this answer from the above two points.