Recently I had an epi

Introduction

This is the first post in a series of articles that will seek to demystify stock options. In my line of work, it is common for companies to give stock options to employees to encourage them to stay. I was always curious about the underlying mechanism of options. What we will dive into in this article are stock options for publicly traded companies. Options are an hedging strategy used to offset risks, especially in volatile markets. It is also used by speculators to take advantage of said volatility. Buying an option gives you the right, but not the obligation (hence the name) to buy a stock at a certain price, before a certain date, after which the option expires.   On the other side, selling an option gives you an immediate return, the premium, the price paid by the buyer to purchase the option contract, but you have the obligation to fulfill your part of deal, should the buyer decide to exercise his/her right.   Options have a strike price. The strike defines the price at which you can buy or sell the underlying stock. Purchasing an option to buy is called Buying a Call.   Buying a call with a strike price higher than the current share price is also referred to as a long call You will profit if the stock rises above the strike price. Purchasing an option to sell is called Buying a Put. You will profit if the stock drops below the strike price. Selling an option to buy is called Writing a Call. You will profit if the stock drops below the strike price. Selling a call option a strike price higher than the current share price is also referred to as a short call Selling an option to sell is called Writing a Put. You will profit if the stock rises above the strike price. Important Note An option contract is typically for a 100 shares block, but the option price shown on exchanges is usually listed for a single share. Extrinsic/Extrinsic Value An important thing to understand with options is that options can end up worthless at the expiration date. This is why it's inherently riskier to trade options. For stocks, there is always some intrinsic value since you own a slice of the company. That is not the case for options. Extrinsic value is captured by time. That means an option that is set to expire in the future has some extrinsic value, since there is time for the share price to move and therefore capture some intrinsic value. In the Money (ITM) Describes an option that have intrinsic value. That means that for call options, the stock is trading higher than the strike price. For a put option, that means the stock is trading lower than the strike price. At the Money (ATM) This is the situation where the strike price is equal to the current stock price. At this point, the option has no intrinsic value. Out of the Money (OTM) An option that is out-of-the money means it has no intrinsic value. It is at risk of expiring worthless.

Introduction to the greeks

Here we introduce the Black-Sholes equation

[ {\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0 ]

Let us break it down and decipher what each part means.

Delta represents the variation of the option price with respect to the underlying stock value. It is has a value between 0 and 1 for call options and -1 to 0 for put options. [ \Delta = \frac{\partial V}{\partial S} ]

Gamma represents the variation of the option value with respect to the rate of change of the option with respect to the price (delta). This also expresses the convexity of the option price. The maximum value is when the option is at-the-money, since this marks an inflection point in the option price.

[ \Gamma = \frac{ \partial \Delta}{\partial S} = \frac{\partial^2 V }{\partial S^2} ]

[ rS \cdot \frac{\partial V}{\partial S}]

Represents a short position in the underlying stock [ \frac{\partial V}{\partial S}].

[ \nu = \frac{\partial V}{\partial \sigma} ]

Vega express the change on the option price relative to the volatility of the underlying asset. A vega of $1 would indicate that a 1% change in implied volatility corresponds with a $1 change in option price.

[ \theta = \frac{\partial V}{\partial \tau}]

Theta expresses the change in option price with respect to the time to maturity. Usually this value is negative, meaning the option price decays as the option gets closer to maturity (expiration date)

The important thing to notice here is that there can be an equilibrium between the left-hand side and the right-hand side of the equation. The riskless return of an option over Δt, can then be expressed as a sum of theta and gamma.