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Black and Scholes (1973) and Merton (1974) (hereafter referred to as BSM) introduced the contingent claim approach (CCA) to the valuation of corporate debt and equity. The BSM modeling framework is also named the 'structural' approach to risky debt valuation. The CCA approach considers all stakeholders of the corporation as holding contingent claims on the assets of the corporation. Each claim holder has different priorities, maturities and conditions for payouts. It is based on the principle that all the assets belong to all the liability holders.In the structural approach the arrival of the default event relies on economic arguments for why firms default as it is explicitly related to the dynamics of the economic value of the firm. A standard structural model of default timing assumes that a corporation defaults when its assets drop to a sufficiently low level relative to its liabilities.The BSM modeling framework gives the basic fundamental version of the structural model where default is assumed to occur when the net asset value of the firm at the maturity of the pure-discount debt becomes negative, i.e., market value of the assets of the firm falls below the market value of the firm's liabilities. In a regime of limited liability, the shareholders of the firm have the option to default on the firm's debt. Equity can be viewed as a European call option on the firm's assets with a strike price equal to the face value of the firm's debt. Actually, CCA can be used to value all the components of the firm's liabilities. Option pricing models are used to value stocks, bonds, and many other types of corporate claims.Different versions of the model correspond to different assumptions about the conditions when a firm defaults. Merton (1974) assumes that the firm only defaults at the maturity date of the firm's outstanding debt when the net asset value of the firm, in market value terms, is negative. Others introduce other conditions for default. Also, different authors introduce more complicated capital structure with different kinds of bonds (e.g. senior and junior), warrants, corporate taxes, ESOP, and more. Volume 1: Foundations of CCA and Equity ValuationVolume 1 presents the seminal papers of Black and Scholes (1973) and Merton (1973, 1974). This volume also includes papers that specifically price equity as a call option on the corporation. It introduces warrants, convertible bonds and taxation as contingent claims on the corporation. It highlights the strong relationship between the CCA and the Modigliani-Miller (M&M) Theorems, and the relation to the Capital Assets Pricing Model (CAPM). Volume 2: CCA Approach to Corporate Debt ValuationVolume 2 concentrates on corporate bond valuation by introducing various types of bonds with different covenants as well as introducing various conditions that trigger default. While empirical evidence indicates that the simple Merton's model underestimates the credit spreads, additional risk factors like jumps can be used to resolve it. Volume 3: Issues in Corporate Finance with CCA ApproachVolume 3 includes papers that look at issues in corporate finance that can be explained with the CCA approach. These issues include the effect of dividend policy on the valuation of debt and equity, the pricing of employee stock options and many other issues of corporate governance. Volume 4: CCA Approach to Banking and Financial IntermediationVolume 4 focuses on the application of the contingent claim approach to banks and other financial intermediaries. Regulation of the banking industry led to the creation of new financial securities (e.g., CoCos) and new types of stakeholders (e.g., deposit insurers).

The bulk of the option pricing properties established in Merton's Classic Theory when the option price is homogeneous of degree one in the underlying's value and the exercise price, are shown to extend to any Markovian diffusion world. The most important result is that calls are increasing convex functions of the value of the underlying. Still, some caveats are in order: Although an upward shift in the term structure of interest rates will increase a call's value, a decline in the present value of the exercise price can be associated with a decline in the call price; and a call's elasticity need not be everywhere increasing with the passage of time, or everywhere decreasing in the level of the stock price. As a direct implication of convexity, we are able to undertake a comparative static analysis of the effects of shifts in the term structure, in dividend policy, and in the underlying asset's instantaneous volatility function. We provide a new bound on the relative values of calls on otherwise equivalent dividend- and non-dividend- paying assets. With respect to volatility, we present two fascinating results. First, an equivalence between (i) a comparison of two different functional forms for the relation between instantaneous volatility and the contemporaneous stock price and time and (ii) increasing risk in the Rothschild-Stiglitz sense. Second, when the instantaneous volatility is bounded above (below), the call price is bounded above (below) by its Black-Scholes value evaluated at the bounding volatility level, and we can place upper and lower bounds on the stock positions necessary to hedge a given option position. We also show that if we relax either the continuity assumption or the Markovian assumption, then call options can be 'bloating' (not 'wasting') assets, whose value over some range is a decreasing, concave function of the value of the underlying. We argue that when considering the valuation of long-dated options on the stock of a firm, it is both intuitive and correct to view the dynamics of the underlying stock price as Non-Markovian.

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