The study of option greeks and option valuation can be overwhelming. Pick up any textbook and it’s not long before you’re presented with something this:
Although this may be intuitive to some, it certainly isn’t to many. Having said that, an understanding of options should be a core part of any Product Controller’s knowledge and indeed is a regular topic for questions in the technical section of product control interviews.
This post will demystify options greeks and explain the concepts in plain English, giving you a solid platform to build on with more technical / mathematical examples if so desired.
The complexity behind option valuation stems from the fact that it is non-linear. You can buy an option, hedge the delta yet still lose money as the market moves. Why? Because there a number of individual components to the option valuation. These components, and their first level greeks are as follows:
|The rate of change in the price of an option with respect to the price of the underlying.||The change in the price of an option with respect to the time to expiry.||The change in the price of an option with respect to a change in the volatility of the underlying.||The change in the price of an option with respect to a change in interest rates.|
Second level option greeks are those which impact the first level greeks. The only one we will discuss in this article is Gamma.
Gamma is the rate of change of the delta of an option. All will be revealed below.
Simply, delta is the rate of change in the option price relative to the price of the underlying.
Delta can be any number between 0-100 but practically is usually referred to as being between 0-1. An option that is deeply out of the money has a delta close to zero, an ATM option has a delta of around 50 and an option deep in the money has a delta of 1. In other words a $1 move in the underlying would produce a $1 move in the option price.
You can think of delta as being the rough percentage chance that an option will finish in the money. If the option is deeply OTM then the chance of it finishing ITM will be close to 0%, deeply in the money and it’s close to 100% and an ATM option could go either way, hence a delta of around 0.5 or 50%.
Delta is crucial for hedging as the delta number will tell you how many units of the underlying you need to buy or sell to delta hedge the option.
Ie. Your position delta will = delta x contract size x number of option contracts.
The delta for a long call with a strike of 100 looks like this:
As you can see the delta tends closer to zero as the underlying stock price decreases and moves towards 100 as the underlying price increases. When the underlying price is 100 and the option is ATM then the delta is close to 50.
The crucial point to note from this graph is that delta is not linear! And this is where we introduce gamma.
As we mentioned previously, gamma is the rate of change of delta. Hopefully, with the above graph in mind, this now makes more sense.
Let’s focus firstly on the left hand side of the graph, where the underlying moves from 80 to 85. At this level the option is deeply out of the money and even a $5 move in the underlying has little impact on the option’s delta. It remains vey close to zero.
Now let’s move right and look at another $5 move in the underlying; this time between 95 and 100.
Here the delta move is much more significant – from 24.6 to 51.2. FOR THE SAME $5 move in the underlying!
This is gamma and is a crucial consideration when delta hedging options.
If you buy an option, you’re long gamma. So an increase in the underlying price will increase your delta ie. You become longer and would have to sell more of the underlying to remain delta neutral.
Conversely if the market falls then your delta decreases and you become shorter and have to buy more of the underlying to remain delta neutral.
Theta represents the decay in the option’s value over time. If you’re long a call option and the underlying price doesn’t move from one day to the next the option will still reduce in value. This is Theta.
Consider an option which is currently OTM; the more time passes and the closer we get to expiry then the less chance there is of the option ending up in the money. As such, the option reduces in value.
Vega represents the change in the price of the option for a 1% change in the volatility of the underlying. The more volatile the underlying, the more likely the option is to finish in the money and hence the more expensive the option will be.
Rho represents the impact to an option’s price of changes in the risk free interest rate. Of all the first order option greeks rho will impact the option’s price the least.
To explore this topic in more detail we highly recommend taking a look at Option Volatility and Pricing: Advanced Trading Strategies and Techniques, 2nd Edition
. This is one of the most comprehensive and, more importantly, readable books we’ve come across in this area. A great addition to the library for new and seasoned controllers alike. Also take a look a look at the Product Control HQ Amazon Store for other titles we recommend. You can access this via our RESOURCES page.
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