Understanding Option Greeks – Introducing Gamma
Gamma is the second derivative of the option price with respect to the price of an underlying asset. Alternatively, it is the rate of change in the option Delta due to a change in the underlying asset price.
Figure 1: Gamma against Spot
What does that mean in simpler language?
In the image […]
As part of our Sales & Trading technical interview guide series we have done a number of posts on Greeks, Delta hedging and estimation of Delta hedging Profit & Loss.
A recent client/student/interview request indicated a preference/need for a sheet dedicated to European Put Options Greek plots. Hence the Greeks Put Option suspects’ gallery. While some of these are making a second appearance, we think a Put only collection is indeed useful given our focus on European Call options.
The Analysis framework used for dissecting put option Greeks is simple. We break the contracts down by “money-ness”. The three categories are Deep In, At/Near, Deep Out of money options.
And the five option pricing variables – Spot, Strike, Vol, Interest Rates & Time. This analytical combination produces interesting results.
The options for which Greeks have been plotted below assume a volatility of 20%, a risk free rate of 5% and a time to maturity of 1 year. In addition Spot and Strike prices are:
Deep In
Spot = 50
Strike = 100
At/Near
Spot = 50
Strike = 100
Deep Out
Spot = 50
Strike = 100
Understanding Greeks – Introduction
Understanding Greeks – Analyzing Delta & Gamma
Understanding Greeks – The Guide to delta hedging using Monte Carlo Simulation
Understanding Greeks – Quick Reference Guide to Delta, Gamma, Vega, Theta & Rho
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While we have done a few posts earlier about option price sensitivities, here is a quick reference guide for the truly lost and confused. For convenience the reference guide has been broken down into the following sections
While there are many ways of dissecting Greeks a framework or frame of reference helps. Here are some basic ground rules.
1. Remember the first order Greeks and separate them from second order sensitivities. Delta, Theta & Rho are first order (linear) Greeks which means that they will be different for Call Options and Put Options. Gamma and Vega are second order (non linear) Greeks which means that they will be exactly the same for Calls and Puts.
2. Remember that in most cases Greeks will behave differently depending on the “in-the-money-ness” of the option. Greeks will behave and look differently between Deep Out, At, Near and Deep In the money options.
3. Think how the Greeks will change or move as you change the following parameters:
Rather than remember the formula try and remember behavior, shape and shifts. For example, see the following three panels that show the shift of the 5 Greek shapes across spot prices and “money-ness”. Starting off with a Deep out of money call option we plot the same curves for an At and near money option as well as a Deep in money option. Can you see the shift and the transition?
Figure 1 Delta, Gamma, Vega, Theta & Rho for a Deep out of money Call Option
Figure 2 Delta, Gamma, Vega, Theta & Rho for At and Near Money Call Option
Figure 3 Delta, Gamma, Vega, Theta & Rho for a Deep In Money Call Option
The five derivative pricing and sensitivities (aka Greeks) with their equations and definition reference
Figure 4 Option Greeks: Delta & Gamma formula reference
Figure 5 Option Greeks – Vega, Theta & Rho, formula reference
Greeks Against Spot Prices. Here is the short series for Deep out of Money Call Option and Deep In and Out of Money Put options.
Figure 6 Deep out of money call options – Greeks plot
Figure 7 Deep In money put options – Greeks plot
Figure 8 Deep out of money put option – Greek plot
The way to read the above graphical set is to take one Greek at a time. So starting with Delta you will see that while the shape is the same, the sign is different between the Call and the Put. For illustration we have also produced the Greek plot for a Deep out of money Put option and while there are some similarities between the Deep out of money Call and the Deep in money Put, they disappear completely when we look at the Deep out of money Put contract.
Figure 9 At money Call option – Greek Plot against changing volatilities
Figure 10 At money put option – Greek plot against changing volatilities
However the difference really crops up between Calls and Puts when you switch the frame of reference from changing spot prices to changing volatilities. With this new point of view Calls and Put are clearly different animals. Why is that? Or is that really the case? If you look closely you will see that as far as Vega, Delta and Rho are concerned the basic shape and shift is similar, it looks different because the LHS axis has shifted. Still Delta is different because of the sign change. But its Gamma and Theta that are really different when it comes to dissecting the behavior of Greeks across Calls and Puts. But would these differences stay if you plot the 5 Greeks across money-ness?
Figure 11 Plotting N(d1), N(d2) and Price against volatility
What do you think is the most common question most students have when they see figure 9 above? Do you see a contradiction? Need a hint? Take a look at Delta. Then think about how we calculate Delta for a European call option. We look at N(d1) as a conditional probability? Intuitively speaking what should we expect N(d1) to do as volatility rises? Rise or Fall? What is N(d1) doing in Figure 9 above?
Now take a look at figure 11 above? What are N(d1) and N(d2) doing as volatility rises? Is that intuitive or counter intuitive? Need a hint? Two words – volatility drag.
Think about the above question and tell us about your answers through the comment sections below in this post. Would love to hear more from you.
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Here is a short and sweet extract from the Sales & Trading Interview Guide series on Understanding Greeks (iBook and plain vanilla PDF version in the works). Rather than focus on formula and derivations, we have tried to focus on behavior. Our hope is the pretty pictures and colored graphs would help take some of the pain away from comprehending this topic.
There are five primary factor sensitivities that we are interested in when it comes to option pricing and derivative securities.
Figure 1 The Five Greeks. Plotted against changing spots
The image above presents a plot of the five factors for an At The Money (ATM) European call option.
Delta (Spot Price)- Measures the change in the value of the option price, based on a change in price of the underlying. Delta is the dark red line in the image above.
Vega (Volatility) – Measures the change in the value of the option price, based on a change in volatility of the underlying. Vega is the dark indigo line in the image above.
Rho (Interest Rates) – Measures the change in the value of the option price based on a change in interest rates.
Theta (Time to expiry) – Measures the change in the value of the option price based on a change in the time to expiry or maturity.
The first four sensitivities measure a change in the value of the option price based on a change in one of the determinants of option prices – spot price, volatility, interest rates and time to maturity. The fifth and final sensitivity is a little different. It doesn’t measure a change in option price, but measures a change in one of the sensitivities, based on a change in the price of the underlying.
Gamma – Measures a chance in the value of Delta, based on a change in the price of the underlying. If you are familiar with fixed income analytics, think of Gamma as Delta’s convexity.
As promised above, we won’t hit you with any equations. However a quick notation summary is still required to appreciate the shape of the curves you are about to see.
Delta, Vega, Theta and Rho are all first order changes, while Gamma is second order change. If you take a quick look at the plot of the five factors presented above, you will see that the shape of the curves are similar for Delta and Rho (the slanting S) and similar for Gamma, Vega and Theta (the hill or inverted U). We will revisit the shape debate later on in our discussion.
Delta has a handful of interpretations. Some common, some exotic.
The common interpretation is the one we have just covered above. Delta tracks option price sensitivity to changes in the price of the underlying. The second interpretation is as a conditional probability of terminal value (St) being greater than the Strike (X) given that St > X for a call option.
The third and the most relevant definition to our discussion comes from the option replicating and hedging portfolio example from the Black Scholes world.
Figure 3 Delta Hedging. Replicating portfolio for call option using option Delta
As a seller of a call option if you would like to hedge your exposure (short call option) so that when (or if) the call option is actually exercised your loss is ideally completely offset by the change in value of your replicating portfolio.
This replicating portfolio is defined as a combination of two positions. A long position in the underlying given by Delta x S, less a borrowed amount.
Figure 4 – The Delta Hedge Relationship
For a European call option Delta is defined as
If we adjust Delta and with it the borrowing amount at suitably discrete time intervals we will find that our replicating portfolio will actually shadow or match the value of the option position. When the option is finally exercised (or not exercised) the two positions will offset each other.
Figure 5 Delta Hedging. Replicating portfolio performance for hedging a short call option exposure
The two replications snap shots shown above show how closely the two portfolios move with changes in the underlying price over a one year time interval with fortnightly rebalancing (24 time steps). The tracking error will reduce if the rebalancing frequency is increased but it will also increase the cost of running the replicating portfolio.
Now that we have gotten the basic introduction out of the way, let’s spend some time on dissecting Delta by evaluating how this measure of option price sensitivity changes as you change:
a) Spot,
b) Strike,
c) Time to expiry and
d) Volatility.
Where relevant and important, we will add more context by also looking at how Delta’s behavior changes if the option is in, at, near or out of money.
So how does Delta behave across a range of spot prices.
If we assume that we have purchased or sold a call option on a non-dividend paying stock with a strike price of US$100. The underlying is currently trading at a spot price of US$100. The time to expiry or maturity is one year.
Figure 6 Delta plotted against changing spot prices
The graph above shows the change in the value of Delta as spot prices move higher or lower from the original US$100.
In this specific instance while we have moved spot prices we have held maturity constant. As a result while spot prices for the underlying change from 60 to 130, the option’s delta doesn’t touch zero or 1, since there is a chance that it may still switch direction and go the other way.
How does the behavior of Delta change if you move across At money options to options that are deep out of money or deep in the money? Think about this for a second before you move forward. Would you expect to see a different curve? Or a different shape? How different?
Let’s start with at, in and near money options.
For at money or near money options the shape remains the same. For options that are deep in the money, it becomes asymptotic before finally touching 1. From our hedging definition above, this means that the seller of the option should now own the exact numbers of shares of the underlying committed to the call option (Delta = 1) since the option will most certainly expire in the money. From a probability definition perspective, for a call option a conditional probability of 1 indicates that the option is certain to expire in the money.
Figure 7 Delta against Spot. At, In and near money options
But what about deep out of money options? What happens to Delta or for that matter to all the other Greeks discussed earlier when it comes to deep out of money options.
We answer this question by plotting the Greeks for a European call option written with strike price of US$200, while the current spot price is only at US$100. In the price ranging between US$60 and US$130, the value of Delta touches zero and then slowly rises to about 8% as the underlying spot price reaches US$130.
The overall shape remains the same, all we are doing now is just looking at a different pane of the option sensitivity window. Slide a little further or put the two images (figure 7 and figure 8) side by side and you should be able to see the complete picture.
Figure 8 The Greeks against Spot. Deep out of money options
Figure 9 The Greeks against spot. AT, In and near money options
The next natural question deals with the valid range of values that Delta is expected to take. For a call option the range is between 0 and 1, as we have seen demonstrated above. Zero for deep out of money options, one for deep in money options. In between for all other shades.
For put options, Delta ranges between 0 and -1. Deep in money put options touch a Delta of -1, deep out of money put options reach a Delta of zero. The negative sign corresponds to a short position. To hedge a put, unlike a call, we short the underlying and invest the proceeds, rather than buy the underlying by borrowing the difference.
Figure 10 Delta & Gamma against changing strike price.
The next graph plots Delta and Gamma against changing strike price. We use a plot of both Delta and Gamma to reinforce the relationship between the two variables. Once again before you proceed further think about why do you see the two curves behave the way they do?
As the strike price moves to the right, the option gets deeper and deeper out of money. As it gets deeper in the deep out territory, the probability of its exercise and the amount required to hedge the exposure fall. Hence the steady decline in Delta as the strike price moves beyond the current spot price.
As the rate of change of Delta increases, we see Gamma rise by a proportionate amount. Gamma will only flatten out once the rate of change of Delta flattens out in the image above.
The next three plots show how Delta and Gamma change as we vary time to expiry from a day to one year.
In the three snapshots that follow below, time moves from right to left (more to less). Once again we use both Delta and Gamma to reinforce the relationship between the two factors.
The only other variation from the options above is that we are now looking at three different options. An at money Call (Spot = 100, Strike = 100), an in money call (Spot = 110, Strike = 100) and a deep out of money call (Spot = 100, Strike = 200).
Notice how delta declines with time for an at money call, but rises to 1 for an in money call. Beyond a certain cut off point, it also rises for a deep out of money call but not as much as our first two pairs.
Figure 11 Delta & Gamm against Time for in, at and out of money options
Figure 12 Delta, Gamma against Volatility. For at and out of money options.
For our last act, we plot Delta and Gamma against volatility and see a result which some students find counter intuitive.
For in, near or at money option, Delta actually falls with rising volatility. For most students this is a surprising result. Once would expect that with rising volatility, the value of the option should go up (correct) because the range of values reachable by the underlying is higher (also correct) hence leading to a higher probability of exercise (incorrect).
For deep out of money options, Delta rises with rising volatility. Gamma keeps pace initially but then runs out of steam as the rate of increase in Delta begins to flatten out.
To appreciate this behavior you actually have to move away from the Greeks and look at exercise probabilities.
Our next three plots, show how the conditional probability of exercise N(d1), the unconditional probability of St > X, N(d1) and price behave and change for in, at and out of money European call options.
In the images beneath, Price is measured using the right hand scale, while the two probabilities are measured using the left hand scale.
For at, in and near money options, the two probabilities actually decline as volatility rise. Sounds counter intuitive when you consider that while the two probabilities are declining, the price of the option is actually rising.
Here is a hint, look up and think about volatility drag. For a deep out of money option the trend is reversed. Once again ask yourself why?
Figure 13 Vol, N(d1), N(d2) and Price for in, at, and deep out of money call options
If you are interested in a career on the floor or on a derivatives trading desk, you need to get very comfortable with the above graphs and behavior of Greeks across them. To the point of the lessons discussed becoming second nature – like riding a bike, breathing or drinking coffee. To remove the shock and awe caused by the partial differential equations behind the Greeks, we completely eliminated them from this post. In real life and when modeling them in excel you will have to get re-acquainted. Drop us a line with your questions or other dimensions that you would like us to address and if we can, we will do a few more posts on this topic.
Enjoy.
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Preparing for the quantitative portion of a sales and trading interview for a main street bank is a nightmare. Specially if the bank is an active derivative trader and wants it intake class of interns and full time analysts and associates to hit the Sales and Trading desk running.
While basic option concepts generally get covered quite well in the MBA curriculum, when it comes to option price sensitivities and Greeks, our understanding remains rudimentary and superficial. One reason is the focus on formulas and calculation rather than intuition and understanding. Most courses have run out of time when it comes to delta, gamma, vega, theta and rho and stop after a basic rudimentary coverage of the material.
There is a lot of good material available on basic quantitative and numerical techniques tested in a Sales and Trading interview. But when it comes to option price sensitivities or Greeks, available material generally looks like this.
As part of the work we do with customers and students, our Apple iPad iBooks team is working on two very interesting and exciting titles.
The first is the Sales and Trading Interview Guide – Understanding Greeks for Dummies. Using the interactive iBook template we will help you master your Greeks to such a level where future mention of delta, gamma, vega, theta and rho would no longer break you out in cold sweat and palpitations. The iBook will cover Greeks behavior across time, volatility, spot and strike prices using easy to understand language, graphs and self assessment quizes.
But it’s the second iPad iBook title that we are really excited about. Sales and Trading Interview Guide – Delta Hedging and other higher dimensions, will help you build your own delta hedging sheet in excel using Monte Carlo Simulation. Both iPads books will have options for purchasing supporting excel spread sheets that extend the concepts covered in the iBook.
Planned for release in mid September, the two books will increase our inventory of interactive iBook for iPad titles to 5. Please feel free to drop me a note if you would like to learn more about the release dates and table of content for both titles.
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