  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.
