Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

11.5 Static Structural Models

Static structural models trace back to Merton (1974).¹⁷ Merton observed that a risky bond from a company that issues both debt and equity is equivalent to a risk-free bond plus a put option on the company's assets. This may seem simple with the benefit of hindsight, but it is a profound insight that provides a powerful approach for thinking about determinants of default.

Merton's Model

Take a very simple one-year framework in which a firm has assets, V, which are random (will go up and down in value over the next year), and the company issues equity, S, and a one-year bond with a promise to pay a fixed amount, B. Default is very simple: When the value of the assets are below the bond payment (V < B), the bond is in default. Otherwise the bond is paid and shareholders receive the excess. From this, we can see that the equity is a call option on the value of the assets, with a strike equal to the promised bond payment, B.

Setting up notation:

Firm value (assets): V₀,: value at beginning, end of period. is the driving variable in this model, assumed to be random, generally log-normal.

Bond: B₀, , B: bond value at beginning, end of period. is random (since ...