Continuous-Time Models for Firm Valuation and Their Collateral Effect on Risk-Neutral Probabilities and No-Arbitraging Principle

Abstract

Extensions of Merton's model (EMM) considering the firm's payments and generating new types of firm value distribution are suggested. In the open log-value/time space, these distributions evolve from initially normal to negatively skewed ones, and their means are concave-down functions of time. When payments are set to zero or proportional to the firm value, EMM turns into the Geometric Brownian model (GBM). We show that risk-neutral probabilities (RNPs) and the no-arbitraging principle (NAP) follow from GBM. When firm's payments are considered, RNPs and NAP hold for the entire market for short times only, but for long-term investments, RNPs and NAP just temporarily hold for individual stocks as far as mean year returns of the firms issuing those stocks remain constant, and fail when the mean year returns decline. The developed method is applied to firm valuation to derive continuous-time equations for the firm present value and project NPV.