The Merton model, developed by Robert C. Merton in 1974, is a widely used "structural" credit risk model. Analysts and investors utilize the Merton model to understand how capable a company is at meeting financial obligations, servicing its debt, and weighing the general possibility that it will go into credit default.
Approach Under this model, the value of stock equity is modeled as a call option on the value of the whole company – i.e. including the liabilities – struck at the nominal value of the liabilities; and the equity market value thus depends on the volatility of the market value of the company assets. The idea applied is that, in general, equity may be viewed as a call option on the firm: since the principle of limited liability protects equity investors, shareholders would choose not to repay the firm's debt where the value of the firm is less than the value of the outstanding debt; where firm value is greater than debt value, the shareholders would choose to repay – i.e. exercise their option – and not to liquidate. See Business valuation § Option pricing approaches and Valuation (finance) § Valuation of a suffering company. This is the first example of a "structural model", where bankruptcy is modeled using a microeconomic model of the firm's capital structure. Structural models are distinct from "reduced form models" – such as Jarrow–Turnbull – where bankruptcy is modeled as a statistical process. By contrast, the Merton model treats bankruptcy as a continuous probability of default, where, on the random occurrence of default, the stock price of the defaulting company is assumed to go to zero. This microeconomic approach, to some extent, allows us to answer the question "what are the economic causes of default?" Large financial institutions employ default models of both the structural and reduced-form types.
Mathematical formulation In the basic Merton model, the firm has assets with current market value V 0 {\displaystyle V_{0}} and constant volatility σ V {\displaystyle \sigma _{V}} , and has issued a single zero-coupon bond requiring a payment of D {\displaystyle D} at time T {\displaystyle T} . The model assumes that no payments are made to shareholders before the debt matures. Under the risk-neutral measure, the value of the firm's assets follows the process
d V t = r V t d t + σ V V t d W t , {\displaystyle dV_{t}=rV_{t}\,dt+\sigma _{V}V_{t}\,dW_{t},}
where r {\displaystyle r} is the continuously compounded risk-free interest rate and W t {\displaystyle W_{t}} is a Wiener process. In the basic model, default can occur only at maturity. If V T < D {\displaystyle V_{T}<D} , the firm defaults, the bondholders receive the firm's assets and the shareholders receive nothing. If V T ≥ D {\displaystyle V_{T}\geq D} , the debt is repaid and the shareholders receive the residual value. Thus, if E T {\displaystyle E_{T}} and B T {\displaystyle B_{T}} denote the values of equity and debt at maturity,
E T = max ( V T − D , 0 ) , {\displaystyle E_{T}=\max(V_{T}-D,0),}
and
B T = min ( V T , D ) = D − max ( D − V T , 0 ) . {\displaystyle B_{T}=\min(V_{T},D)=D-\max(D-V_{T},0).}
Equity is therefore equivalent to a European call option on the firm's assets with exercise price D {\displaystyle D} , while risky debt is equivalent to default-free debt minus a European put option on the firm's assets. Assuming no payouts from the firm's assets before maturity, the Black–Scholes–Merton formula gives the current market value of equity as
E 0 = V 0 N ( d 1 ) − D e − r T N ( d 2 ) , {\displaystyle E_{0}=V_{0}N(d_{1})-De^{-rT}N(d_{2}),}
where
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