Exploring Black-Scholes

This is a work-in-progress visualization for the Black-Scholes model, which is used by some in the financial industry to estimate the value of European-style options.

Some notes about this visualization are in this blog post. This is beta software without warranty of any kind - see the Disclaimer for more details.

The current version includes:

• a basic calculator of the Black-Scholes option values (based primarily on the Black-Scholes Wikipedia page)
• some utilities for playing with various option strategies (select the Options Strategies tab)
• calculation of the first order Greeks Delta, Theta, Vega, and Rho. The change in the value of the option with respect to Strike Price K is also calculated, but there doesn't seem to be a name used for it. I assume that is because it is generally not useful. It is referred to here as "Bromma", based on seeing this word for the company name of shipyard heavy equipment in Mobile, Alabama.
• calculation of the second order Greeks Gamma, DdeltaDvol (Vanna), DdeltaDtime* (Charm), DvegaDvol (Vomma), and DVegaDtime* (Veta)

  *Note: I am a bit confused about the sign for Charm and Veta. I am currently using the negative of the values calculated from the formulas on the Wikipedia page, because that seemed to be what was consistent with the underlying curves when overlaying the relevant tangent line.

• some interactive charts illustrating the dependence of the calculated values on various inputs and how the Greeks relate to these dependencies
• some other experimental charts highlighting intermediate aspects of the calculations

On mobile devices, this visualization will currently (probably) have visual hiccups and suboptimal performance, so use on desktop is highly recommended for now (with a width of at least 1024px). I have been testing using Chrome on a Mac, and am still poking at some quirks.

	Stock Price S	Strike Price K	Expiry t (years)	Volatility σ	Risk Free Rate r
Value of Call Option:  $ MathJax not available $$ = N(d_1) \, \class{eq-s}{S} - N(d_2) \, \class{eq-k}{K} e^{-\class{eq-r}{r} \class{eq-expiry}{t}} $$ $$\begin{eqnarray} N(d) &=& \text{Cdf of Normal Distribution} = {1 \over {\sqrt{2 \pi}}} \int_{-\infty}^d e^{-x^2 / 2} dx \\ d_1 &=& {1 \over { \class{eq-sigma}{\sigma} \sqrt{\class{eq-expiry}{t}} } } \left[ { \ln \left( { \class{eq-s}{S} \over \class{eq-k}{K} } \right) + \left( { \class{eq-r}{r} + { {\class{eq-sigma}{\sigma}^2 } \over 2} } \right) \class{eq-expiry}{t} } \right] = z_{\cssId{d1-z}{0.5}} \\ d_2 &=& d_1 - \class{eq-sigma}{\sigma} \sqrt{\class{eq-expiry}{t}} = z_{\cssId{d2-z}{0.5}} \end{eqnarray}$$ Value of Put Option:  $ MathJax not available $$ = N(-d_2) \class{eq-k}{K} e^{-\class{eq-r}{r} \class{eq-expiry}{t}} - N(-d_1) \class{eq-s}{S} $$
Call Option Value Curve (black) Slope of tangent is the first order Greek	Value Delta=	Value Bromma=	Value Theta=	Value Vega=	Value Rho=
	S →	K →	t →	σ →	r →
Second Order Greeks Slope of tangent is the second order Greek	Delta Gamma=	Delta DdeltaDvol =	Delta DdeltaDtime *=	Vega DVegaDvol =	Vega DvegaDtime *=
	S →	σ →	t →	σ →	t →
Put Option Value Curve (black) Slope of tangent is the first order Greek	Value Delta=	Value Bromma=	Value Theta=	Value Vega=	Value Rho=
	S →	K →	t →	σ →	r →
Second Order Greeks Slope of tangent is the second order Greek	Delta Gamma=	Delta DdeltaDvol =	Delta DdeltaDtime *=	Vega DVegaDvol =	Vega DvegaDtime *=
	S →	σ →	t →	σ →	t →
Value of Call Option = $ = N(d1) S — N(d2) K e- rt N(d1) S = $ N(d2) K e- rt = $
Value of Put Option = $ = N(-d2) K e- rt — N(-d1) S N(-d1) S = $ N(-d2) K e- rt = $

Note: using a desktop for this app is currently highly recommended.

Investors implement some option strategies that consist of the purchasing or selling of Call and/or Put options, each with potentially different strike prices. The Profit/Loss curve displays the profit or loss for the strategy as a function of the final stock price. Some of these strategies have colorful names that are (roughly) based on the shape of the Profit/Loss curve for the strategy.

Here, the following components are incorporated into the calculated Profit/Loss curve shown below:

• if the option is purchased, the initial cost of the option is incorporated as a debit, with the value based on the Black-Scholes formula
• if the option is being sold, the initial cost of the option is incorporated as a credit, with the value based on the Black-Scholes formula

Not included are commission charges, and "slippage" that might occur when trying to exit the positions. For the crazy random strategies, there may in fact be no practical way to even exit out all of the positions and capture the profit: that is an area for later research.

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Initial Stock Price S Expiry t (years) Initial Volatility σ Initial Risk Free Rate r
Clear Strategy Fit Strategy to Points (Experimental!) Clear Points Fit It # Buy Calls # Sell Calls # Buy Puts # Sell Puts Option Strategies (click to pick)	Processing... Processing... Processing... Under standard assumptions this is pdf of stock price at expiration (beta calcs!).	Contracts Purchased: , Contracts Sold: +Buy Call +Sell Call +Buy Put +Sell Put