4.3
Structural Form Models
Structural form models employ modern option pricing theory in corporate debt valu- ation. The original framework developed on 1974 by Robert Merton Merton (1974). The theory was based on the principles of option pricing formulae introduced by Black and Scholes in 1973 (Black and Scholes, 1973). The Merton Model laid the cornerstone for all other structural models.
The basis of the structural model approach asses the probability of default of a com- pany by characterizing the companys equity as a call option on its assets. That is, debt and equity are contingent claims on the firms assets. The model assumes that a company has a certain amount of zero-coupon debt that expires at a timeT. Default occurs when the firm value falls to a low level such that the issuer cannot meet the par payment at maturity timeT. Hence if the value of its assets is less than the value of the debt at time T, the company will default.
The equity of the company is a European call option on the assets of the company with maturity T and a strike price equal to the value of the debt. The model can be used to estimate either the risk-neutral probability that the company will default (Hull et al., 2003).
4.3.1
The Merton Model
The theory behind the Merton Model was introduced in Merton (1974). At time t, for a company define:
Vt Value of assets (Value of company) Et - Value of Equity
Dt - Zero coupon dept. It might be necessary to transform the debt structure of the firm into a zero-coupon bond with maturity T and face value K
K - Face value of Zero Coupon Bond
T - The maturity T of the zero-coupon bond can be chosen either to represent the maturity structure of the debt, for T > t
The value of the companys assets at time t can be measured by the price at which the total of the firms liabilities can be bought or sold (Wang, 2009).
36 CHAPTER 4. OVERVIEW OF BANKRUPTCY PREDICTION MODELS
It them follows that:
Market Value of Assets = Market Value of Equity + Market Value of Bonds (4.12) or
Vt=Et+Dt (4.13)
Now if:
• VT > K the companys debtholders can be paid the full amount K, and share- holders equity still has value VT −K.
• VT < K debtholders have the first claim on the assets VT, take hold of the company and shareholders are left with nothing and the company defaults.
Therefor at time T, equity value Et can be written as ET =max(VT −K; 0)
This is the payoff of a European call option written on underlying assetVtwith strike price K maturing at time T. The Black and Scholes option price formulae can be applied under the Black and Scholes assumptions (Black and Scholes, 1973).
Applying the Black and Scholes option pricing formulae at any time t the market value of the companys equity is:
Et =Vtφ(d+)−Ke−r(T−t)φ(d−) (4.14)
Whereφ(.) denotes the N(0,1) cumulative distribution function withd+andd−given
by: d+ = ln(VtK) + (r+ 12σ2 V)(T −t) σV √ T −t (4.15) d− = ln( Vt K) + (r− 1 2σ 2 V)(T −t) σV√T −t (4.16) where:
σV is the standard deviation of Vt r is the standard riskfree rate
4.3. STRUCTURAL FORM MODELS 37
4.3.2
Assumptions of the Merton Model
Structural Models (including the Merton Model) have many advantages; firstly they model default on the realistic assumption that it is a result of the firms assets falling below the value of the debt. Disadvantages are that they are difficult to calibrate and are computationally intense (Focardi and Fabozzi, 2004)
The following key assumptions are set in the Merton Model (Sundaresan, 2013) (Mria Miankov, 1977):
• There are no transactions costs, taxes, or problems with indivisibilities of assets.
• The liabilities of the company consist of one zero-coupon bond.
• Debt structure is static, it doesnt change.
• There are a sufficient number of investors with comparable wealth levels such that each investor believes that he can buy and sell as much. of an asset as he wants at the market price.
• The riskiness of the investment will not be influenced by how close the company is to default.
• There exists an exchange market for borrowing and lending at the same rate of interest.
• Short sales of all assets, with full use of the proceeds, are allowed.
• Trading in assets takes place continuously in time.
• The value of the company in not dependent on its capital structure.
• The debt structure of the firm is modelled as a zero-coupon bond.
• The company assets has lognormal distribution, it cannot be negative.
• The dynamics for the value of the firm,Vtcan be described through a stochastic process.
Ever since the introduction of the Merton Model there have been many researchers who have criticized the underlying assumptions. There have been many proposed extensions to the original Merton model to relax some of the assumptions stated above. A description of these extensions will be explored in Chapter 5 Section 5.3.2.
38 CHAPTER 4. OVERVIEW OF BANKRUPTCY PREDICTION MODELS