Maths: Greek

In the course of my reading what I consider to be fairly technical financial reports and analyses, I often come across references to Greek. Having covered them briefly in the finance courses of my commerce degree years ago, I thought that it would be a good time for a slight refresher on what I already know in addition to learning things whilst providing a reference guide for the future. 

The first two Greeks that I'll discuss relate to the broader notions of investing and have a fairly wide range of applicability while the remaining deal specifically with options.

Risk Ratios

Alpha

Alpha is defined as excess return on investment when compared to an indexed benchmark. Therefore alpha is calculated as actual rate of return less benchmark rate of return. The resulting figure gives an indicator of whether the investment over-performed (if the number is positive) or if it under-performed (if the number is negative), essentially providing a measure of relative return.
A positive alpha indicates an investment had a good return given it's underlying risk whilst a negative alpha indicates the opposite.
When people say they have a high alpha, it means they have a tendency to outperform the market.

Beta

Beta is defined as the measure of relative volatility compared to the market as a whole. It is used to measure systemic risk of a portfolio when compared to the benchmark.
The formula for beta: 
Although it looks fairly complex, it can easily be calculated with excel by using variance and covariance formulas.
By definition, the market as a whole has a beta of 1. A resulting number of less than 1 indicates that it is less volatile than the market which can often be found in lower risk investments such as bonds or gold. A higher beta indicates that it has more volatility than the market. For those who dabble in inverse indices, a negative beta demonstrates that the investment moves in the opposite direction of the market.
High beta vehicles often offer higher returns whereas low beta investments offer lower returns. By utilising the Capital Asset Pricing Model, one can then derive what a fair return on investment ought to be given any beta.
In combining the two, it can therefore be concluded that the most attractive investments are those that offer the highest alpha with a low beta.

Options Greeks

Delta

For those who remember from their high school days, Delta is a measure of change. In the finance context it measures the rate of change of an option with the change in underlying asset's price.
The formula for delta:
The resulting number ranges between -1 to 1 with negative numbers for puts and positive for calls. Delta represents the change in the price of the option for every dollar movement of the share.
Options which are deep In The Money (i.e. calls where the market price > strike price and puts where market price < strike price) will have a delta closer to one whereas options which are deep Out of The Money will have delta closer to zero. In practically applying delta, it is often used as a rough indicator of the probability that the option will expire in the money. A delta of 50 would mean that it has roughly half a chance of profiting by expiry. The higher the delta the more chance you have of profiting, but usually this would correlate to higher premiums for the contracts.

Gamma

Again one from high school mathematics, Gamma represents the derivative, that is rate of change of Delta with respect to the change in the underlying asset's price.
The formula for gamma: 
Simply put, gamma is to delta as acceleration is to speed. Gamma is always positive and highest when the option is closest to At The Money (market price = strike price). It is sometimes called the "Uncertainty Factor" because of this, since when options are closest to  ATM the higher the chances that it could end up either way.
Practical example, if a stock had a value of $100 with the correlating call having a delta of 0.45 and a gamma of 0.05, when the stock goes to $101, the delta becomes 0.50, so the value of the premium goes up correspondingly.

Theta

Theta is a function of measuring an option's time sensitivity. It gives an indication of the change in the value of the contract given a one day change in time.
The formula for theta:
Short options, that is selling contracts, has positive theta whereas long options have negative theta because the closer you get to expiration date, the less the underlying contract is worth because of the lower probability of market value making the required strike price. By way of example, a Theta of -0.10 means that every day that the stock price does not move, the value of each contract will reduce by $0.10. The lower the theta, the slower the rate of decay is on the contract.
For those who browse forums as much as I do, you may have come across the term Theta Gang. Theta gang represents those who sell options to profit from the decay in value when it expires OTM.

Vega

To clarify, Vega is not actually a Greek letter, rather it uses the letter nu, but as the character bore similarities to the letter v, it has been named vega in line with beta and theta. It measures the change in the option's value with every percent change in implied volatility.
The formula for vega: 
All options have a positive vega. When implied volatility is higher, options are worth more as people think that there is a higher likelihood of meeting the required strike price. As such, with the most recent volatility in the market, options would have sold for a far higher premium and as volatility subsides they will be worth considerably less, hence the term IV crush.

Though I have little intention to dabble in options, having a good understanding of the above is always useful when assessing other's option purchases to consider their viability. It's also very interesting to learn on the side anyway.

by 小福