Black-Scholes-Merton

Company XYZ has issued equity S and 1 year zero-coupon debt D with a face value of 90 Assume that the value of the assets of the firm V are initially at 100. According to the Merton model (Merton JF 1974), debt and equity are contingent claims on the assets of the firm.

(a) Draw the payoff diagram for both the equity and debt of the firm as a function of the firm's value one year from now. Draw the individual payoffs of debt and equity, without the financing costs.

(b) What is the value of Company XYZ's equity at t 0. Assume an annual continuously compounding interest rate of 0% (zero), a time horizon of T 1 year and asset volatility of v 20%?

(c) What is the value of Company XYZ's debt at t 0, assuming an annual continuously compounded interest rate of 0%, a time horizon of T 1 year and assets volatility of ov 20%? Indicate two ways how you can calculate the value of debt.

(d) What is the risk-meutral probability of default?

(e) What is the delta, gamma, and vega of XYZ's equity?

(f) You, as a shareholder, would like to hedge yourself against market risk by making your holdings delta-, gamma- and vega-meutral. How can you do this by using only the company's stock and the two options indicated in the Table 2? Hint: Assume that each call (put) option contract gives the right to buy (sell) 1 unit of the underlying asset

Table 2: Options and Greeks

			vega
Option 1	0.6	0.5	2.0
Option 2	0.5	0.8	1.2