Black-Scholes

Edit: The youtuber Veritasium produced a great video regarding the history/background of the Black-Scholes equations here.

This isn't really an article, I originally constructed this to play around with some reactivity, Katex math stuff, and to understand for myself a bit of how options are priced, as such there's no long explanation aside from what's below.

Below is the Black-Scholes formula applied to both European style call and put contracts, it's derived from the Black-Scholes equation which is a partial differential equation and goes largely over my head. It involves a branch of mathematics called stochastic calculus.

$$Call = N(d_1)S_t^{-yt} - N(d_2)Ke^{-rt}$$ $$Put = N(-d_2)Ke^{-rt} - N(-d_1)S_t^{-yt}$$ $$\text{ where } d_1 = \frac {\ln \frac{S_t}{K} + (r + \frac{\sigma^2}{2})t} {\sigma \sqrt{t}}$$ $$\text{ and } d_2 = d_1 - \sigma \sqrt{t}$$

In the above formula $N(d_1)$ and $N(d_2)$ is the cumulative normal probability of $d_1$ and $d_2$.

Additionally a contracts delta is defined as $N(d_1)$ for call contracts, and $N(d_1) - 1$ for put contracts.

The defaults in the form below represent:

• A contract with an underlying stock price of $50 $(S_t)$
• A strike price of $52 $(K)$
• One year till expiration $(t)$
• An annualized dividend yeild of 5% $(y)$
• A risk-free rate of 4.5% $(r)$
• And an implied volatility of 20% $(\sigma)$