Now, let me start with one of the central themes in Physics. Namely, an idea that a right physical theory should be simple and beautiful. Albert Einstein famously said that everything should be made as simple as possible, but not simpler. Paul Dirac insisted on mathematical beauty. Lev Landau was famous for his ability to trivialize highly complex physical problems. Now catch-22 here, is that for someone trained in theoretical physics, PDEs are generally synonymous with simplicity and beauty. Newtonian mechanics and its Lagrangian Hamilton Jacobi formulation, a diffusion equation, the Schrodinger equation, and now the Black-Scholes equation, are all great examples of both theoretical clarity and greater stability of a physical or financial model. Therefore, we physicists are almost conditioned on taking the continuous-time limit, leading to a PDE, almost universally for nearly all physical systems. Now, let's move from physics to finance. Donald Duffie gave a very good account of contribution of Black Scholes and Merton to economics, as well as the role played by the notion of competitive market equilibrium, in modern finance. While he mentioned that important alternatives exist, Duffie said in 1997, 20 years ago, that the competitive market equilibrium is still a dominant paradigm in both the industry and academia. Things did not change much, in this sense since 1997. Duffie talks about three Nobel Prize works in economics that are all based on these paradigm. The first one is Modigliani-Miller, theory of 1958 or irrelevance of capital structure for market evaluation of a company. The second one is the Capital Asset Pricing Model or CAPM, of William Sharpe of 1974. We briefly mentioned CAPM when we talked about models of stock returns and PCA analogies. And the third one is the Black-Scholes theory of 1973, that used no-arbitrage pricing as a weak form of market equilibrium. Now, these are all models or theories, that in Mr. Soros' words, are modeled on Newtonian physics, and something that he declared absurd. To be more accurate, these models follow equilibrium statistical mechanics of Ludwig Boltzmann. But this is relatively a small remark that doesn't change the essence of Soros's statement. So, who is right, Mr. Soros or the academics? To answer this, let's try to conceptualize what the Black-Scholes model does. It does in essence two things. First, it uses the idea of option replication by a portfolio of stock and cash. Second, it takes the problem of evaluation of such portfolio to the continuous time limit. Together, these two steps produce the celebrated Black-Scholes PDE, and it does look both simple and beautiful. Now, my understanding, given this quote of Black, is that he took the continuous-time limit in his analysis from the very beginning. Black says, ''I applied the Capital Asset Pricing Model to every moment in a warrant's life for every possible stock price and warrant value. I stared at the differential equation for many many months.'' And a very interesting question is indeed, why didn't Black analyze the discrete-time case tours? Now, the summary of this model can be formulated as follows, we only need two features to price an option, the current stock price and the stock volatility. The option price is unique, and is given by a solution of the Black-Scholes equation. The optimal hedge, that is the number of stocks in the replicating portfolio, is computed after the option price is found. And the most striking conclusion though, is that options have zero risk as they can be perfectly replicated by a stock and cash, at all points in time. But then it also means that options according to these very theory are entirely redundant. But this didn't seem to bother Black, who said that when people are seeking profits, equilibrium will prevail. So, Black basically says that as long as an option market exists, the fair option price obtained in the equilibrium, should be the true option price.
