 # Option Greeks: A Practical Guide

The Option Greeks give us all the information we need to set up a trade and profit and loss expectations. I’m talking about Delta, Gamma, Theta, Vega and Rho. There are many more, but we’ll start with the first order derivatives and the most important second order one: Gamma.

A lot of articles, books and videos discuss the Option Greeks, therefore, I will try to write something practical.

Options can be a bit confusing at first so if you’re not familiar with them check out how they work before you carry on reading.

## What are the Option Greeks

An option’s price is derived from the stock price, time, volatility, strike and the interest rate.

First thing’s first: we can’t discuss the Option Greeks before you understand what is a derivative.

### What is a derivative in mathematics

I assume that you have never solved a differential equation (if you have skip this) so we’ll start with a train.

When you board the London to Manchester service you expect to travel the distance over a period of time.

#### Derivatives of position

You can calculate the average speed of the train by dividing the distance by the number of hours it takes to get to Manchester. Speed is a first order derivative of your position, measured in miles per hour.

Speed = ds/dt or the change in distance divided by the change in time.

Velocity is also a first order derivative of position as it gives us the speed relative to a direction, for example 70 mph North.

Acceleration however, is a second order derivative of position because it represents the rate of change of the velocity, a first order derivative. It is measured in m/s² because you have to divide the change in velocity by time or dv/dt = (m/s)/s = m/s². Then you add the direction.

It doesn’t end there, we have jerk or lurch which is a third order derivative: the rate of change of the acceleration relative to time. The formula looks like this j = da/dt = d²v/dt² = d³r/dt³ where a is acceleration, v is velocity and r is position. You can go on and on to the 8th or 9th order.

In summary a derivative is a function of functions like f(x) = y². Don’t worry you won’t calculate anything but you have to understand how it works.

#### Practical use

The maps app on your phone uses derivatives because of the curvature of the Earth’s surface. It is minor when we’re talking about a 100 mile trip but enough to render the software useless.

The GPS software flattens the curvature by using derivatives.

## Option Greeks explained

All of the Option Greeks are important but there is no way to start with anything else than the Delta.

### Delta

Delta is the change in the price of the option relative to a \$1 change in the price of the underlying stock. Each strike has a different Delta depending on how close or far it is from the current share price. Therefore, Delta is a first order derivative.

I’ll only use examples with actual stocks because I feel that they are easier to understand compared to stock XYZ. We’ll start with XLV, an ETF, by using the mid-price of the options.

There is no stock trading in \$1 increments which makes it harder to calculate the Delta on the fly. Nevertheless, we can say that XLV is close to \$108 which should be ok.

Note that the Delta is quoted per single option and it is negative for put options. You need to multiply it by 100 to get the contract’s Delta, e.g. \$110 strike = 0.42 x 100 options = 42.

42 Delta means that the option contract’s price will increase by \$42 if the price of XLV increases by \$1. It also means that it will decrease by \$42 if XLV’s price decreases by \$1.

You will also notice the put-call parity. If we subtract the put Delta from the call Delta we always get 1. For example, \$107 strike = 0.55 – (-0.45) = 1. The \$110 strike values add up to 0.99 probably due to a rounding error.

#### Call Option Delta

Based on the information above we expect that if we buy the \$110 call for \$271 the value of the contract will increase by a bit over \$42 if XLV’s price increases to \$109.

However, if the price decreases to \$107 the contract will lose a bit less than \$42 as XLV is trading for \$107.82 and not \$108. The exact difference can be calculated but I can assure you that you won’t be doing that for a live position.

The Delta is positive when you buy the option because you have a positive directional exposure and benefit from a share price increase:

##### Negative Delta

If you sell the \$110 call option you get the reverse effect as you will benefit from a decrease in the price of XLV.

Consequently, you have negative Delta and an inverse directional exposure to the underlying asset like when you short sell stock.

An increase in the price of the underlying stock will increase the price of the option and the Delta will become more negative.

However, you already received a payment for the option when you sold it. Therefore, if you have to buy it back you will incur a loss:

You can see that the profit and loss of this position is negative. However, if the Delta increases it will turn positive. When I say increase I mean that the short call option benefits when the Delta moves towards 0. It will never be zero unless the company goes bankrupt.

There is always a chance, however improbable, that a \$5 stock can go up to a theoretical limit of infinity or simply \$1,000 in a manner of days.

When we sell a call we want the share price to go down which drags our Delta closer and closer to 0. This is what happened 3 days later:

Note that this position was part of a strategy and should not be considered in isolation. Nevertheless, it works for illustration purposes.

##### Strike, Delta and Call Option price

Below you can see a chart of the \$105 – \$115 strike call options in XLV:

The Delta increases the lower the strike because the \$100 call option will become more and more in the money if the share price of XLV keeps going up. Conversely, if the share price starts to decrease it will be less and less in the money.

It will end up out of the money if XLV trades below \$100 at expiration.

Inevitably this relationship affects the price of the option: The left vertical axis shows the strike price, the Delta is on the horizontal and the price of the option is on the right axis

The option price expansion or contraction is not linear. Part of this is because of Gamma, one of the Option Greeks, which we’ll discuss after Delta.

###### Call Option Prices and Delta

You need to have this non-linearity in mind if you are to have reasonable expectations of a trade. You probably noticed that the option price line becomes almost straight beyond 60 Delta.

This is because when the Delta is very high the option starts to behave more and more like stock. Think about it for a moment, a 20 Delta option will gain \$0.20 for each \$1 up-move of a stock, while an 80 Delta option will gain \$0.80.

Consequently, when the stock price goes down the option will behave less like stock due to the loss of Delta. If our 80 Delta option loses 40 Delta it will only gain \$0.40 from a \$1 up-move of the underlying.

#### Put Option Delta

The Delta of the put options is negative because their price increases when the underlying stock goes down. Therefore, you have an inverse directional exposure because you lose money when the underlying’s price goes up. Here is XLV again:

This time if you buy the \$107 strike put you get a negative Delta of -45. You expect to make \$45 if XLV’s price drops a bit below \$107. In contrast it will become more positive if XLV goes up to \$109 and you will lose \$45.

If you bought the put you will lose money the less negative the Delta becomes. An imaginary put option in XLV with a strike price of \$1 would have a Delta very close to 0.

Negative Delta looks like this:

As the share price goes down the delta will become more and more negative. Therefore, the it will decrease and your option’s price will appreciate:

##### Positive Delta

You get positive Delta when you sell a put because this time your directional exposure matches the stock’s price movements. It is the same as buying a call, in both cases you will have profits if the underlying’s price goes up.

However, the Delta implication is different compared to a long call. When you sell a put you need the positive Delta to start moving towards 0 to make a profit. This is because you are short negative Delta or -1 contract x 100 options x (-0.2286 Delta) = 22.86 in the example below:

By now you would have figured out that the stock moved against me, therefore the P&L is in the red. This is because I sold an option with a negative Delta of -18. As the share price went down the Delta became more negative, -22.86.

However, my position’s Delta became more positive as I am short the negative Delta.

As time went by and the stock cooperated, the Delta started to move towards 0. The seller’s positive Delta decreased while the buyer’s negative Delta increased:

##### Strike, Delta and Put Option price

Below is a graph of the XLV puts Delta values respective to strike price:

The Delta decreases the higher the strike price as the option becomes more and more in the money. Despite the negative Delta the relationship to a 1\$ share price movement is the same. Let’s see how that looks on the put side: The strike price is on the left vertical axis, the price on the right and the Delta is on the horizontal

Similarly to the calls the relationship between the option price and the Delta is not linear. A put option will behave more and more like short stock the closer the Delta moves to -1. This is again because of the greater increase in option premium per \$1 decrease in the underlying stock price.

###### Delta driven P&L example

If you buy the -37 Delta \$105 strike put for \$279 and XLV’s share price drops by \$1 you can expect to make \$0.37 per option or \$37 per contract. You could be short the same put which will cause your P&L to show a paper loss of \$37.

In case that the XLV’s share price keeps going up the -22 Delta of the \$100 strike put will keep increasing towards a value of 0. Therefore, the option will be further and further away from the money until it expires worthless.

#### Calls and Puts Delta Summary

To summarize, calls and puts lose value as their delta moves towards 0. If you are the seller you make money and if you are the buyer you lose money.

The price of calls increases as the Delta increases bringing profit to the buyer. Delta decreases for the seller by becoming more negative and causes losses.

When the Delta of a put option decreases and becomes more negative an option buyer will make money. The seller will see a loss and an increase in the Delta which becomes more positive.

#### Static Delta vs Dynamic Delta

Each share of the underlying stock has a Delta or 1 or -1 depending on whether you are long or short respectively. This is another way to explain why the long puts have negative Delta.

The stock Delta is static. Whatever happens, no mater if the share price surges or drops 1 share still has 1 Delta. This is normal because the option is a perishable derivative financial instrument based on the performance of the underlying stock.

You are aware that the Delta of the option changes as the stock price fluctuates. Therefore, the option has dynamic Delta. This can be an advantage or a handicap depending on whether a position is moving in your favour or not.

##### Implications for practice

Both static and dynamic delta can be used to reduce or increase the directional exposure of a trade.

There are many ways to neutralise directional exposure which can result in direction neutral or direction indifferent trades with varying risk profiles.

###### Probability proxy

Most brokerage platforms have a %ITM (in the money) indicator which shows the likelihood of an option ending up ITM.

It is only valid at the time when your order is filled because it will change throughout the life of the trade. The other disadvantage is that it calculates the chance of making a profit of \$1 which is not our target.

Therefore, I prefer to use Delta as a proxy for probability of profit. Buying static delta (stock) has a 50% probability of profit because it will either go up or down.

Consequently, it is not unreasonable to assume that a 50 Delta option has about 50% probability of profit because it is right at the money or ITM. It follows that if I buy a 20 Delta option I can expect about 20% probability of profit.

###### Another way to look at Delta

I could say that I ‘control’ 20 shares when I buy a 20 Delta option. This is not true as I won’t control anything if it expires worthless.

However, it gives us another way to look at Delta. If I’m long options I want my the Delta to move away from 0 and I will control more shares. In contrast I’d like it to decrease if I’m short options.

### Gamma

Gamma is next on our Option Greeks list. It is actually a second order derivative with respect to the stock price but it is very important so it got its rightful place as number two.

Gamma is the rate of change of an option’s Delta relative to a \$1 move of the underlying stock. Therefore, it measures the acceleration of our profit or loss.

Let’s include it to the XLV example:

This seems quite clear, it is 0.04 or 4 per contract for all near the money strikes. However, Gamma is not always symmetrical. Have a look at XLF, another ETF:

#### Long Gamma

All long option positions have positive Gamma as the Delta will increase for calls or decrease for puts if the underlying stock moves in the appropriate direction.

##### Calls

Let’s say I buy the \$110 call for \$271, how much Delta am I going have if XLV goes to \$111? I’ve got a \$3 move and I started off with 42 Delta + \$3 x 4 Gamma = 42 + 12 = 54.

I’ll get \$42 + \$46 + \$50 = \$138 from the Delta increase adding up to a total of \$409. You can check above that the mid-price of the 55 Delta option is \$421, on par with the calculation.

It seems that if I’m buying an option, all things being equal, I’d prefer to get high Gamma because it has a compounding effect.

###### Probability of profit

Unfortunately a low Delta option has a much lower probability of profit than a high Delta option regardless of the underlying stock or the Gamma.

The 17 Delta option in XLV costs \$74, much less than the 42 Delta. We don’t need any math to figure out that investors won’t be keen on paying a lot for an option which has a slim chance of making any money.

Therefore, high Gamma alone is not a predisposing factor for success. A high priced stock like Amazon will show Gamma values near 0 but you can still make a ton of money from the options. That is if you are willing to fork out \$20K for a 55 day option:

##### Puts

A long put also has positive Gamma, however, it still works to your advantage as it renders the Delta more negative if the underlying cooperates.

If you bought the \$24 strike put in XLF and it drops by approximately \$1 your delta will decrease to -31 – 12 = -43. Another \$1 down move will bring it to -43 – 14 = -57.

##### In the money

Your long options, puts or calls, will start to pick up intrinsic value whenever they move in the money.

This is another reason for them to start behaving more and more like stock as the Delta expands. You probably noticed that the XLF options have smaller and smaller Gamma the more in the money they are.

The 93 Delta \$20 strike call in XLF costs around \$535, you have an intrinsic value = (\$25.25 price – \$20 strike) x 100 options = \$5.25 x 100 options = \$525. Therefore, if you buy it you can look at it as if you bought stock. It has only \$10 of extrinsic value and there’s not a lot of scope for a Delta increase, hence the 3 Gamma.

##### Out the money

The same is true the further out of the money an option goes, this time because it behaves less and less like stock.

You’d notice that the 3 Delta option in XLF also has 3 Gamma. Granted, it will go up a lot if XLF prints \$30 but this is unlikely to happen within the 55 day period.

#### Short Gamma

Whenever you sell an option you will be short Gamma. This is because a movement of the underlying in you desired direction chips away the Delta bringing it closer to 0.

##### Put options

I sold the EWZ \$24 strike put for \$69 which I thought at the time was a reasonable price. Two days later EWZ kept going down and my Delta increased from 18ish to 23 causing a \$14 (20%) paper loss:

Perhaps I could have done better but if you sell options you have to accept the fact that it may show a loss at one point or another so I left it alone for a bit:

You see that EWZ increased by \$1, therefore, the Delta result is 22.86 – 5.43 Gamma = 17.43 Delta. I suppose I can present it more accurately like this:

-1 contract x 100 options x (-0.2286 Delta + 0.0543 Gamma) = 17.43 Delta

There are 2 Deltas missing somewhere and the gain should have been \$22.86 while we can clearly see it is \$38.

This is why I support the view that none of the Option Greeks can be taken in isolation. The missing 2 Deltas can be explained by the effects of time and volatility on the option’s price.

##### Call options

Short Gamma works exactly the same for call options, however, their Delta is negative. If you sold the 17 Delta call in XLF and it moved up by \$1, your delta would be -17 – 10 = -27 and you’d be down \$17.

Consequently, a \$1 down move will result in -17 + 10 = -7 Delta but the gain is likely to be less than \$17. Even a 2 Delta option has some value because it caries risk. Therefore, there is an element of censorship as the Delta gets close to 0.

Your short option has the inverse directional exposure compared to the corresponding long option. Therefore, the effect of Gamma on the option price remains the same. It just affects you in a different way when you are the option seller.

##### Short Gamma risk

The main risk is that whenever you are short options the payoff is fixed while the loss potential is undefined.

It is not impossible to sell a \$5 contract and lose \$1,000 and when it comes to calls the sky is the limit.

If you are long options you have an integrated stop loss. For example, if I buy the \$110 call in XLV I can only lose \$271. I don’t care what’s going on with the Gamma.

However, had I sold the \$110 call and XLV goes up to \$130, I’d be looking at \$20 x 100 = \$2,000 loss in intrinsic value alone. Even if it just goes up by \$2 to \$110, the loss is \$42 + \$46 = \$88.

We see that the loss increases with each \$1 increment. If XLV trades for \$111 it will be \$42 + \$46 + \$50 = \$138. Now, I’m sitting at 50% loss because the short Gamma increases the loss with each additional \$1 move of the underlying stock.

Another reason to be careful with short options is that Gamma increases around 20 days prior to expiration. Therefore, the magnitude of a loss will be greater should the underlying move against you. The easiest way to manage this is to close the position a few weeks before expiration.

### Theta

Time decay is probably the easiest to explain out of all the Option Greeks. Every option has an expiry date. In the XLV example we are looking at 55 days to expiration. Consequently, options lose value as each day goes by, some say every second even.

#### Theory and Practice

Theta is a theoretical construct which helps us to spread time decay in equal chunks of dollar value.

I said theoretical because the option may not lose value for a few days and then erode rapidly. It’s difficult to isolate the effect of Theta in real world examples. However, we know that it starts to accelerate around 40-50 days to expiration.

One of the reasons for this ambiguity is that nothing is ever static. Share price, Delta, Gamma and volatility fluctuate all the time so you can never pin down the effect of individual Option Greeks to the cent.

#### How does it work

If an option is out of the money it has to have a value of 0 at the time of expiration. There is no other way.

What will happen if I buy an in the money option in an underlying which doesn’t move a penny, the volatility remains constant and the interest rate is 0%? The option’s extrinsic value cannot change based on Delta, Gamma, Vega or Rho.

However, no one can stop the arrow of time. It always goes forward and it will chip away a dollar or two each day until our frozen option ends up trading for intrinsic value at expiration.

##### Long options

Here is XLV again:

You see the dollar value of Theta per contract so we don’t have to multiply by 100. Depending on the brokerage or website it can be in dollar value (3.16) or per option (0.0316).

The result is the same, a long \$110 strike call will lose \$3.16 per day all things being equal. If you multiply that by 55 days you’ll get \$173.80 while the price of the contract is \$271.

This happens because time decay accelerates the closer you get to expiration, therefore, the Theta will increase if everything else remains the same.

It is exactly the same for put options.

##### Short Options

Theta works in your favour when you are selling options. We just want everything to remain the same and to clock in a sure profit. Sadly, it doesn’t work like that but time decay helps.

I argue that time is always negative for the option itself. Some talk about positive or negative Theta but this refers to your position not the option.

If you are buyer you’ll experience negative theta, however, the seller will see positive theta:

Long \$110 strike call = +1 contract x -3.16 = -3.16 Theta

Short \$110 strike call = -1 contract x -3.16 = 3.16 Theta

This is how time decay becomes positive. I think that you should be looking at it any way you like, there’s no rule.

#### Theta risk

When it comes to time decay it is clear that the risk is present if you are long options.

##### Worthless at expiration

One of the main issues investors have with options is that they are time limited. I see it as an advantage or a disadvantage depending on the circumstances.

However, they are not wrong. It is possible to buy an option and see the underlying moving in your favour only to lose your cash especially if the option stays out of the money.

You need to choose the right length of time if you are buying to get the most out of it. And even then the stock can hit your strike 3 days after expiration.

##### Theta and Short Gamma risk

Some traders prefer to sell options close to expiry to benefit from the increased time decay.

The problem is that Gamma risk increases a lot too, there’s no free lunch. Check out the Theta and Gamma of the \$25 put and the \$26 call, they have pretty much doubled:

Intuitively, this is normal as XLF is more likely to end up trading for \$24 or \$27 within 54 days than 12. Furthermore, if it hits either a few days before expiration there is less time to wait and see if it will go back to \$25.

You benefit from time decay but you risk higher losses. I prefer to sell the longer duration options but others disagree. There are no right or wrong choices as long as you are happy with the risk profile of the trade.

### Vega

Interestingly Vega is not an actual Greek letter but never mind it’s still one of the Option Greeks. It is the change in the option’s price for each 1% increase in implied volatility. Option premium increases when implied volatility increases.

Volatility is extremely important when it comes to options. The average SIPP or ISA investor may not even know what it is. However, if you are looking into options trading it’s your new best friend.

There is no way you can trade options without looking at the volatility of the underlying stock and the Vega of the option. Maybe you could but you’d be missing out.

#### Example

I’ve seen the following question pop up: I bought an option before an earnings report and the stock moved in my direction but I still lost money, what happened?

Volatility happened is what any options trader would reply. It was higher in the anticipation of the earnings and then it got crushed once the report was released.

The option must have been out of the money, like a 10-20 Delta. It remained there because as volatility went down the expected move of the stock contracted. This pushed the option further out of the money, it lost Delta and subsequently the buyer’s money.

#### How does it work.

We are going back to XLV. For the purpose of this I have to share that the IVx is 25.2%, the IV Rank is 27 and the IV Percentile is 47.1. These are not Option Greeks but we need them for the Vega discussion.

##### IVx, IVR and IV Percentile

The IVx (Implied Volatility index) is like an individual stock intraday VIX index. It shows how much we can expect to pay or get paid for the options. You can compare companies within an industry or check it against historical values.

IV Rank compares the current volatility with past volatility within the past 12 months. If your stock’s lowest volatility was 40 and the highest was 80 a current value of 60 will give you an IV Rank of 50 as it is right in the middle.

The IV Rank got a bit skewed due to the explosion in volatility during the coronavirus crash. Therefore, the ranges are wider than usual resulting in lower IV ranks. Ideally you want an IV Rank over 50 if you are selling options.

IV Percentile gives you a more refined value of current volatility against historic. It calculates the percentage of days during the past year when volatility was lower than today’s value. This means that XLV traded at lower volatility 47.1% of all days during the last year.

#### Option Vega

Here is the table of XLV with the added Vega and implied volatility:

You can see that each option has it’s own implied volatility depending on it’s position in the option chain.

Note that it starts increasing the further down you go in terms of strike price. This is because of tail risk, everyone is worried about a crash.

The same is the case when we move to higher strikes as the distribution has two tails. However, in the case of XLV we have to go all the way up to the \$140 strike to get to 25% implied volatility.

This is how Vega and implied volatility look graphically in the 20 – 80 Delta call range: Implied volatility is on the left vertical axis, Vega on the right and Delta on the horizontal Axis

We can see that the Vega risk is the highest in the 40-60 Delta range.

Intuitively this is normal because the expected move of the stock will expand if volatility increases. Therefore, these options become more lucrative due to a higher profit potential.

In the opposite case if volatility decreases the expected move will shrink. This causes the near the money options to become less likely to enter the money and their price will decrease.

##### Long Vega

Vega resembles Gamma a bit because whenever you are long any options you are long Vega too. Therefore, all long options are long Gamma, long Vega and short Theta:

You may have bought a call or a put but you’ll make money either way if volatility goes up. It’s part of the options pricing model and you’ll see it on your P&L.

The general recommendation is to buy options or long option strategies when volatility is low. This way you could profit from directional moves and an increase in volatility.

But it’s not just that, you’ll pay less for the option and you will experience less time decay.

###### Puts vs Calls

Puts benefit more from volatility because it tends to expand during a down move. Therefore, a put option will appreciate due to the gain in Delta and the increase in the option’s implied volatility. If the down move exceeds \$1 Gamma will compound the profit.

What will happen if I buy the \$107 put for \$349 and XLV goes down by 1\$ and the implied volatility of the option goes up by 1%? I’ll gain \$45 from the Delta and \$17 from the Vega adding up to \$62. The opposite scenario will result in a loss of \$62.

Buying the \$108 call for \$367 under the same conditions will deliver a Delta gain of \$51 and a Vega gain of \$17 to a total of \$68. The reverse will result in a corresponding loss of \$68.

It is important to note that volatility often decreases during up moves. Therefore, call options benefit less often from an increase in volatility compared to put options.

However, while the put gains a lot on the way down it tends to lose a lot on the way up too. This gives you a symmetry in terms of expectations.

This effect of volatility on long options is normal because it measures uncertainty or fear. It is understandable that a sharp down move will result in fear while a sharp up move will make investors happy. Sometimes volatility goes up during an up move, therefore, you shouldn’t consider this a causality.

##### Short Vega

All short options are short Gamma, short Vega and long Theta. Moreover, if you are short the option you get the inverse of its directional exposure: long Delta for a put and short Delta for a call.

Here is EWZ again:

Since we have put some of the pieces together now we have a better understanding because we can use more of the Option Greeks. First we have the directional move which chipped away some of the Delta and Gamma of the option. Then a few days went by and we expect some time decay to kick in, although not exactly \$1.58 per day.

Luckily the IV rank went down too which reduced the implied volatility of the option and depressed the Vega. This also added a few dollars to the P&L.

It sounds logical enough but it is still a very simplistic explanation. This is a live position, therefore, I can’t explain everything based on first order Option Greeks and Gamma.

There are higher order Option Greeks which affect Delta, Gamma, Theta and Vega. However, from a practical standpoint we have to stop somewhere. Therefore, I never calculate them I just imagine how they influence the position.

#### Vega risk

The risk for long positions is that you may overpay for volatility and end up out of pocket when it contracts as in the earnings example.

Short Vega is risky for put options because you are Short Gamma too. Therefore, you lose money on both during a down move.

It is important to sell options when volatility is high. You may not get it right, it may continue to expand but at least you know you collected decent premium.

The only difference when selling calls is the directional exposure during a down move. Your P&L may get hit by short Vega but you will make money from the Delta. It’s not set in stone but it is more likely than shrinking volatility during a down move.

Overall, it is difficult to determine whether you sold an option at the right time. Volatility may expand or contract after you sell it. Consequently, you have to get used to seeing a bit of red in your P&L from time to time.

That’s not so different from buying stock. We’ve all bought stock and seen it go in the red then in the green or the other way around, it’s just part of the process.

### Rho

Rho is the last of the Option Greeks we’ll talk about and I promise this will be really short.

It is the dollar change of the option’s price for a 1% increase in the interest rate. Nowadays, with rock bottom interest rates and Quantitative easing ad infinitum it’s not even on the agenda.

The coronavirus pandemic assured all of us that these interest rates are here to stay. Therefore, I will not cover Rho because it has no practical implications at the moment.

I promise that I will either edit this or white a new article should things change.