Self-Financing, Replicating Hedging Strategies, an incomplete thermodynamic analogy

Abstract

In the theory of riskfree hedges in continuous time finance, one can start with the delta-hedge and derive the option pricing equation, or one can start with the replicating, self-financing hedging strategy and derive both the delta-hedge and the option pricing partial differential equation. Approximately reversible trading is implicitly assumed in both cases. The option pricing equation is not restricted to the standard Black-Scholes equation when nontrivial volatility is assumed, but produces option pricing in agreement with the empirical distribution for the right choice of volatility in a stochastic description of fluctuations. The replicating, self-financing hedging strategy provides us with an incomplete analogy with thermodynamics where liquidity plays the role of the heat bath, the absence of arbitrage is analgous to thermal equilibrium, but there is no role played by the entropy of the returns distribution, which cannot reach a maximum/equilibrium. We emphasize strongly that the no-arbitrage assumption is not an equilibrium assumption, as is taught in economics, but provides only an incomplete, very limited analogy with the idea of thermal equilibrium.