The Black-Scholes-Merton (BSM) Equation, a deterministic fundamental partial differential equation (PDE) used to characterize price-movement of broad derivative financial instruments based on a stochastic representation of price-movement of a primitive financial instrument, can be derived from a variety of assumption sets; hence, identifying a Watershed of BSM Equation Derivations that is even smaller than the Watershed of BSM Formula Derivations, where the resolution of the BSM Equation is referred to as the BSM Formula, as the fundamental PDE is solved for Boundary Conditions that, among other things, specify the maturity payout of a Call option—and, hence, result in a formula for “pricing” a Call option when such an option’s “price” deterministically adjusts (in a specifically restricted manner) to an underlying stock price movement. Yet, despite the existence of a Watershed of Derivations, different derivations require different model validation. In fact, some derivations have low-hanging fruit for model invalidation (e.g., the use of arbitrage arguments that readily fail in both derivation and application, when a hedge portfolio is constructed and implemented inappropriately). Hence, a micro-corporate-governance of model validation (micro-CGMV) process should choose wisely the derivation utilized when a corporation uses the BSM Formula to “price” an option held on the corporate books, even ensuring that the application is consistent with the derivation—to avoid ready invalidation according to “application use is inconsistent with derivation (AUID).” Accordingly, derivations matter; that is, the choice of derivation can matter in the model validation process, an observation that fulfills a “mathematics with economics (MWE)” construct that is counter to a common “mathematics without economics (MWoE)” construct, especially when mathematics devoid of proper economics is present in a validation argument. Interestingly, Malliaris (1983) presents what can be referred to as the Malliaris Single-Sentence (MSS) Derivation (of the BSM Equation), as a “third” derivation embedded in a presentation of a version of the Merton (1973) derivation. Further, Black and Scholes (1973) also presents what can be referred to as the Black-Scholes Single-Sentence (BSSS) Derivation, yet with additional argumentation baggage, and Merton (1973) introduces what can be referred to as the Merton Single-Sentence BSM Formula (MSSF) Derivation, which leads only to the BSM Formula and not the BSM Equation. And while all three Single-Sentence derivations eliminate low-hanging model invalidation fruit, the MSS Derivation offers a slight advantage. And if a corporation chooses to valuate a Call option based on the BSM Formula, while also not simultaneously holding the option in a BSM-specified hedge portfolio (with weights based on either a Hedge Ratio or a Sigma Ratio), the corporation should likely choose to use the MSS Derivation, to make invalidation more challenging. (Notwithstanding, the MSS Derivation still has low-hanging fruit for invalidation (LHFI) that also applies to other derivations.) Further, regulators should identify which derivation a corporation uses when reporting the corporation’s financial position for Sarbanes-Oxley certification, since there is a chance that the corporation may be applying a “pricing” formula in a manner that is inconsistent with corporate use
