A Quick Review – the Math Behind the Black Scholes Model

 

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 dS_{t} follows a geometric Brownian motion process:

dS_{t} = \mu S_{t}dt + \sigma S_{t}dW_{t},        (1)

Where \mu is the drift constant and \sigma is the volatility constant.

According to Ito’s Lemma (Newtonian calculus doesn’t work here because of the existance of stochastic term W_{t}), the price of an option of this stock, which is a function C of S and t, must satisfy:

dC(S,t) = (\mu S_{t} \frac{\partial C}{\partial S} + \frac{\partial C}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}) dt + \sigma S_{t} \frac{\partial C}{\partial S} dW_{t},        (2)

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 B_{t} and the stock underlying said option. Given the no-arbitrage assumption, the value of this replicating portfolio, P_{t}, should be exactly the same as the option price. Therefore we have:

P_{t} = a_{t}B_{t} + b_{t}S_{t},        (3)

dP_{t} = a_{t}dB_{t} + b_{t}dS_{t}

= ra_{t}B_{t}dt + b_{t}(\mu S_{t}dt + \sigma S_{t}dW_{t})        replace dS_{t} with (1)

= (ra_{t}B_{t} + b_{t}\mu S_{t})dt + b_{t}\sigma S_{t}dW_{t}, and        (4)

dP_{t} = dC_{t},        (5)

where a and b 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

b_{t} = \frac{\partial C}{\partial S}, and        (6)

ra_{t}B{t} = \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2_{t}\frac{\partial^2 C}{\partial S^2}        (7)

Feed (6) and (7) into (3), we get the Black Scholes partial differential equation:

rC_{t} = rS_{t}\frac{\partial C}{\partial S} + \frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2_{t}\frac{\partial^2 C}{\partial S^2}        (8)

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 K is the strike price, C(S,T) = max(S-K, 0), C(0, t) = 0 for all t and C(S, t) approaches S as S approaches infinity.

C(S,t) = S_{t}\Phi(d_{1}) - e^{-r(T-t)}K\Phi(d_{2})        (9)

where

d_{1} = \frac{\ln{\frac{S_{t}}{K}} + (\frac{r-\sigma^2}{2}) (T-t)}{\sigma\sqrt{T-t}}

d_{2} = d_{1} - \sigma\sqrt{T-t}

\Phi{(.)} 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.

Roy

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