📕 Finance students often encounter the Black Scholes Model (BSM) in their undergraduate degree. However, it's not always clear how the formula works or why it includes terms like $d_1$ and $d_2$. In this write-up, I'll explore the derivation of the Black Scholes equation, shedding light on the inner workings of this famous financial model.
This BSM derivation is split into the following sections
Under the Black Scholes model, we assume that the stock price at time $t$ follows the following equation, where the stock grows at the risk free rate $r$ under the risk neutral measure.
$$ dS_t = rS_t d_t + \sigma S_t dW_t $$
Solving the stochastic differential equation can be done using Ito’s Formula. Once solved, we arrive at the following process for Stock Price.
$$ S_t = S_0e^{(r - \frac{1}{2}\sigma^{2}T + \sigma W_T)} $$
We also introduce a risk less money market bond which grows at the risk free rate $r$.
$$ dB_t = rB_tdt $$
Solving the stochastic differential equation for the bond can also be done using Ito’s formula. Once solved, the following process is derived.
$$ B_t = B_0e^{rt} $$
In summary, we have the following processes and equations
Stock Price SDE and Process
Bond Price SDE and SDE Process
$$ dS_t = rS_t d_t + \sigma S_t dW_t $$
$$ dB_t = rBtdt $$
$$ S_t = S_0e^{(r - \frac{1}{2}\sigma^{2}T) + \sigma W_T} $$
$$ B_t = B_0e^{rt} $$