What Are The Greeks?

Learn how Market Risk Managers and Traders manage their Risk

The Greeks, as they are known in the finance world, are certain market risk measures which allow you to understand how your portfolio would behave under certain conditions.

Market Risk is the risk a market participant is exposed to, due to market factors that cannot be diversified away. Examples of such market factors include Interest Rates, FX, Stock and Commodity Prices.

In essence, the Greeks allow you to know how much exposure (how much money you stand to lose or make) if the market was to move in a particular direction and magnitude.

In this article, we are going to cover the most commonly used Greeks. There are, however, more which we may cover in the future.

So, what do the Greeks mean? What do they tell you? Let’s find out!

Delta

Delta is a Greek letter (Δ δ) that is used across in mathematics to mean change. In very much the same way, in finance, Delta signifies the value change of your portfolio per unit change in the underlying asset.

In other words, you can think of Delta as:

Delta = d(Value of Portfolio) / d(Price of Market Variable)

To assist in understanding this further, imagine that you have a portfolio of positions relating to Brent Crude Oil. The total value of the portfolio is $1000, and an underlying price change to Brent Crude Oil of $1, causes your portfolio to appreciate by $300. Then, your portfolio’s Delta is:

Portfolio's Delta: 300/1 = 300

Delta, however, is generally only useful for small changes in price. That is because a linear relationship is assumed between the portfolio value and price change. Bring in optionality, however, and the relationship quickly changes.

In other words, imagine you had a deep out of the money option in your portfolio whose strike price is $5 away from the current price. This would mean that the option contributed $0 to the value of our portfolio in the previous calculation.

Following a $10 price move; however, one would expect the portfolio value to change significantly and hence have a significantly different delta.

Gamma

Gamma is yet another Greek letter (Γ γ), and it is used to supplement Delta. Gamma is the rate of change of Delta.

In other words, it is the second derivative of the value of the portfolio per unit change of the underlying market variable. This is why Gamma is known as a second-order Greek.

Gamma = d(Gamma) / d(Price of Market Variable)
      = d2(Value Of Portfolio) / d2(Price of Market Variable)

A low Gamma number means that our portfolio is relatively linear, and small delta changes are expected day on day. A large number, however, means that our portfolio is likely to have an ever-changing delta!

Rho

Now that we got the hang of Delta and Gamma, and we understand how they work and how they can be used, Rho (Ρ ρ) is very easy to explain. Rho helps to identify the impact a unit change in interest rates will have on your portfolio.

Rho is essentially Delta, where the market variable in question is Interest Rates. In other words:

Rho = d(Value of Portfolio) / d(Value of Interest Rate)

An easy way to remember this is to focus your attention on the sound the Greek letter Rho makes. It is the same sound as the English letter R, which we use in finance to signify interest rates.

Theta

Theta (Θ θ) is a Greek letter that makes the sound ‘th’ (think Thesaurus), and it captures the rate of change of portfolio value in regards to time, assuming everything else remains constant.

Theta = d(Value of Portfolio) / d(Time)

Given the assumption of everything else remaining constant, Theta is only really useful when it comes to portfolios involving optionality. That is because as time decays (goes by), there is a decrease in the value of an option whereas other asset-classes have constant Theta.

For most cases, the Theta of an option is negative (value decreases over time), all else equal.

Vega

Interestingly, Vega is not a Greek letter. To add to the story, the letter used to symbolise Vega is actually the Greek letter Nu (Ν ν).

Vega is probably the least intuitive of the Greeks we will cover, although very useful. It captures the rate of change of the portfolio’s value in regards to volatility (of the price of the underlying).

In other words, Vega highlights how our portfolio’s value will change per unit (%) change in the volatility of the underlying asset.

Vega = d(Value of Portfolio) / d(Volatility)

Hull defines volatility as:

A variable’s volatility, σ, is defined as the standard deviation of the return provided by the variable per unit of time when the return is expressed using continuous compounding.

Closing Thoughts

In this blog, we have covered some of the most commonly used Greeks. There are, however, many more which we have not covered. Specifically, there are four categories.

1. First-order: Delta, Vega, Theta, Rho, Lambda, Epsilon
2. Second-order: Gamma, Vanna, Charm, Vomma, Veta, Vera
3. Third Order: Speed, Zomma, Color, Ultima
4. Multi-Asset: Cega (Correlation Delta), Cross Gamma, Cross Vanna, Cross Volga

If you want to read more about them, I suggest you start from the Wikipedia entry, as it is very informative.

References:

• Risk Management and Financial Institutions — John C Hull
• Wikipedia

If you liked this article, you might also like: