Now, what it means is that competitive market equilibrium models in general and the Black-Scholes-Morten in particular, describe a state of thermodynamic equilibrium as a state in which the market is assumed to be. But this is a limit where entropy is already maximized and it doesn't fluctuate anymore. In physics, it's called thermodynamic death of the universe. In such world, everything moves and nothing changes at the same time. Both the Ptolemies and Newton's cosmologists would be good examples of systems in the heat death state. Here, comes not even a million dollar question, but rather a $50 billion question, by the size of the option market is such a limit a good starting point or a good first approximation to actual financial markets? The actual markets are very far from such heat death state. In fact, [inaudible] exactly opposite to such scenario isn't working in real markets. They continuously digest new information and adjust prices to reflect this new information. This information is used by the market players to make profits by mitigating risks and doing informed trades. This picture looks entirely different from a stationary state of market equilibrium and can be better thought of as an equilibrium, disequilibrium using an expression from a paper of Amihud and others that I will discuss in the next lesson. So, to bring more realism to option pricing, we need a few things. Most importantly, we need to bring back risk, namely get rid of the idea that these can be completely eliminated. The simplest and most natural way to do it is to give up the continuous-time approximation. That is commonly done in mathematical finance. This is a limit where non-existing perfect hedging becomes possible and all risk is instantaneously eliminated. A more general approach should start with a discrete-time setting. Then, it makes poor what happens in the limit when time steps go to zero but not vice versa. Such analysis, which we did in the previous course, shows exactly what is wrong with the continuous-time limit. This limit only leaves us with a mean of a replicating portfolio, which becomes in this limit a single number, a delta function at the value which is equal to the Black-Scholes price. This is the limit that makes perfect sense mathematically, but not financially as it appears in this limit that options are risk-free instruments contrary to reality. So, this directly contradicts the very existence of markets in options. Therefore, this problem has to be fixed if we want our models to be more useful and capture market dynamics better. In a previous course, we already discussed how to make first steps in this direction, and we formulated option pricing as a discrete-time MDP model. In the next lesson, we will talk more about markets for options and stocks, and what kind of models people use for option pricing can trade in more realistic scenarios.