At this point, I would like to take a short pause and ask a question. How would Mark Twain approach option-pricing? What are the wrong words in this list that should be crossed out, should be, No arbitrage, or maybe it should be the notion of Risk-less hedge, et cetera? To understand this a bit better, let's talk again about risk. Risk is a story about fluctuations, that in our case will be fluctuations of portfolio value. One of the simplest measure of risk is variance. For example, in the Markowitz portfolio theory, portfolio risk is proportional to the variance of the total portfolio return. One multiplied by risk aversion parameter lambda, if user risk penalty, that is subtracted from expected return to get a risk-adjusted return. There are other specifications of risk that can be defined in terms of higher moments of portfolio value distribution. In physics, fluctuations are known to play a key role in many physical effects, for example in phase transitions. There exists equilibrium fluctuations and non-equilibrium fluctuations. Equilibrium fluctuations describe small deviations of absorbable thermodynamic quantities such as pressure, volume or entropy from their constant equilibrium values. If our system is of infinite volume, all fluctuations die off, and thermodynamic absorbables become strictly constant in time, in this limit. This limit is called the thermodynamic limit, so fluctuations are Sonic that requires that we work with a finite system, not an infinite one. There also known equilibrium fluctuations, these are fluctuations around the non-equilibrium steady-state over system. An example of a non-equilibrium system would be an electric circuit, where electric current running in wires produces heat. Physics has developed various approaches to deal with fluctuations, both equilibrium and non-equilibrium ones. Now, the thing with the Black-Scholes model is that it throws away fluctuations in the replicating portfolio of an option. This happens because in the Black-Scholes equation, we work from the start in continuous time. Because of that instantaneously, risk-free hedging becomes mathematically possible. This means that by continuously hedging, the Black-Scholes model kills all fluctuations in the option replicating portfolio. So, by taking a non-physical or better say, maybe non-financial limit of continuous time, the Black-Scholes model gets a non-physical result, that option hedging and hence the option itself is riskless. This is entirely equivalent to taking the thermodynamic limit in physics. Instead of taking the volume of a system to infinity, here we take time steps to zero, but the net result is the same, all fluctuations die off in this limit.