paper19

Black-Scholes

---

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

The Black-Scholes model is a mathematical model used to calculate the theoretical price of European style call or put options.

If you're not familiar with what options are, options are a type of contract that gives the buyer the right, but not the obligation, to buy or sell an underlying asset at a specific price on or before a certain date.

The specific price is known as the strike price, the date is known as the expiration date, options historically were used to hedge risk, but are now used for speculation as well, certainly by retail investors.

Below is the Black-Scholes formula applied to both European style call and put contracts.

This isn't really an article, I originally constructed this to play around and understand for myself a bit of how options are priced.

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

In the above formula $N(d_1)$ and $N(d_2)$ is the cumulative distribution 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% $(\sigma)$