# Black-Scholes Model question

Suppose a company has a zero coupon bond with face value of 200 million which matures in a year. Its assets has a market value of 300 million. The standard deviation is 30% and the risk free rate is 5%. It does not pay any dividends. Based on the Black-Scholes model, what is the value of debt and equity? How can we replicate this option?

I'm new to finance, but don't we need to know whether this is a call or a put option to replicate it? Also how do we go about solving for the value of debt and equity using the B-S model? Wouldn't the value of debt just 200M/(1+5%)? What does it have to do with the model?

## 1 Answer

For simplicity, numbers are in millions.

We assume that markets are efficient, and that the value of equity and liabilities equals that of assets. So debt + equity = 300. We want to solve for the value of the debt, and get equity as the residual.

The valuation debt $$=\frac{200}{1.05}$$ assumes no default. We model default by the probability of the value of the assets dropping below 200 (redemption value of debt) after one year.

We can value the default risk by nothing that it could be cancelled out by buying a put on the asset values, with a strike at 200. I leave to the reader the determination of this value. Buying the bond and the put creates a default risk free bond, which has the discounted value above.

The value of bond with default is therefore the value of the default free bond less the value of the put.

• I see! One more question: can we replicate this put option somehow? – Rainroad Nov 10 '18 at 21:22
• The only thing I can think of is look at delta hedging the bond; not sure whether it willwork. I just remembered that there is a quantitative finance stack exchange, and probably get more people over there for such questions. – Brian Romanchuk Nov 11 '18 at 13:11