Question

Briefly explain the following statements(ie, are they true? Why and why not?)

The stock of a company with a single zero-coupon bond issue is a call option on the assets,

"The claim of the bondholders, which is subject to default, can be viewed as a default -free bond and a short put on the assets.

The bondholders have implicitly written the stockholders a put on

the assets.”

Answer 1

All the three sentnces are TRUE

Ina traditional framework, there is a put option, which we know is an option to sell the underlying at a fixed price. In fact, we know from put-call parity p0=c0-S0+X/(1+r)T-p0 .

The put is equivalent to a long call, a short position in the underlying stock and a long position in the risk-free bond with face value equal to the exercise price. In the current framework, the standard expression of put-call parity is p0+A0 (put plus underlying)= S0+F/(1+r)T (stock plus present value of bond principal). Turning this expression around and reversing the order of put and bond, we obtain

A0= S0+F/(1+r)T - p0

Noting however, that by the definition, the asset value A0 equals the stock's market value S0, plus the bond's market value B0

We see that the Bond's market value must be equivalent to

B0= F/(1+r)T - p0

The first term on the right hand side is equivalent to a default-free-zero-coupon bond paying F at maturity. The second term is a short put. The bondholders claim, which is subject to default, thus can be viewed as a default-free bond and a short put on the assets. In other words, The bondholders have implicitly written the stockholders a put on the assets.”

From the stockholders' perspective, this put is their right to fully discharge their liability by turning over the assets to the bondholders, even though those assets could be worth less than the bondholders' claim. In legal terminology, this put option is called the stockholders' right of limited liability.