This is a high-level quick review on the derivation of the Black Scholes model, i.e., I will not spend time on putting down rigorous definitions or discussing the assumptions behind the equations.They are definitely important, but I’d rather focus on one thing at a time.
Suppose the change of a stock’s price follows a geometric Brownian motion process:
Where is the drift constant and is the volatility constant.
According to Ito’s Lemma (Newtonian calculus doesn’t work here because of the existance of stochastic term ), the price of an option of this stock, which is a function of and , must satisfy:
Now theoretically, instead of buying an option, we could replicate the payoff of an option by actively and seamlessly allocating our money between a risk-free asset and the stock underlying said option. Given the no-arbitrage assumption, the value of this replicating portfolio, , should be exactly the same as the option price. Therefore we have:
replace with (1)
, and (4)
where and represent the portions of money allocated in each asset; r is the risk-free rate.
Knowing (5), we can map some terms in (2) and (4) to get
, and (6)
Feed (6) and (7) into (3), we get the Black Scholes partial differential equation:
Apparently, a couple of Nobel Laureates solved this equation here, and now we have the Black Scholes pricing model for European options, which means if is the strike price, for all and approaches as approaches infinity.
is the CDF of the standard normal distribution.
Using this general approach, we should be able to model any derivatives as long as we have some basic assumptions about the underlying process. Of course, it’s pointless to use these models in a dogmatic way because almost all of the assumptions behind them are not true. An investment professional’s job is not to follow the textbook and blindly apply the formula in the real world and hope it sticks, instead it’s to investigate the discrepancies between the theoretical model and empirical evidences and figure out which assumptions are violated and if so, can we translate these violations into trading opportunities.