Delta: Breaking Down the First of the Greeks

Hey guys, I've seen a few threads on options trading and how important the Greeks are to your overall options trade. I found a great read courtesy of OptionsHouse that breaks down Delta for those that are interested.

Delta is a numerical value that describes an option in several interesting and useful ways, such as how much the option's price should move for a $1 move in the underlying stock, how much the option is almost exactly like its stock, and what the probability is of the option being in-the-money by expiration.

A short definition of delta would be the sensitivity of an option's price to changes in the underlying price. A shorter definition would be the speed of the option. If an option has a delta of 0.500, you can think of this as 50% and it means that if the underlying stock moves $1 (higher or lower), the option price will move only 50 cents (again, higher or lower).

The option pricing model assigns a probability to every option that quantifies the likelihood of that option finishing in-the-money (ITM). This number is the delta, and it changes dynamically as underlying prices change and time passes. The number produced by the pricing model is between 0.000 and 0.999, and traders speak of 0.103 as a "10-delta" and 0.591 as a "60-delta."

How to Use Delta to Gage Option Buying vs. Selling

Some traders may use changes in delta to figure out if options are being bought or sold. Since option markets are so liquid and competitive, option prices tend to move in lock-step during the trading day with their stocks. So you would expect all "30-delta" options to move about 30 cents for every $1 move in the underlying stock. A $1 up move would have the calls up 30 cents and the puts down 30 cents, and vice versa for a $1 move lower in the stock.

What if the stock moved $1 higher and the 30-delta calls only moved 15 cents higher? Someone was probably selling those calls (trading them on or near the bid) and the market makers were perfectly willing to let them do so. What if the stock moved $1 lower and the 30-delta puts moved 50 cents higher? It's likely that someone was buying those puts (at or near the ask price) because they moved far higher in price than their delta would dictate was fair value.

Three Ways to Think About Delta That Make It Such a Great Tool

Here's a handy reference guide to delta to help you remember its power...

  1. Delta is the rate of change of the option relative to its underlying stock--in essence, the speed of the option.
  2. Delta is the probability the option will finish in-the-money--i.e., how likely is it the option will end up trading for parity because it is virtually equal to a position in the stock from the strike price?
  3. Delta is the "hedge ratio" or equivalence to the underlying stock--i.e., it tells you how much stock you need to buy or sell to exactly hedge an option position.

Comments (40)

Feb 8, 2011

Awesome, thanks for the read.

Feb 8, 2011

Take out point 2 in the interpretation of delta. Other than that its a good read.

Feb 8, 2011

^I believe that from a purely theoretical perspective, point 2 is correct.

Feb 8, 2011
alexpasch:

^I believe that from a purely theoretical perspective, point 2 is correct.

No it is not. There was a whole discussion on this before.

I'm looking forward to discussing GoV and Vanna.

Feb 8, 2011
Revsly:
alexpasch:

^I believe that from a purely theoretical perspective, point 2 is correct.

No it is not. There was a whole discussion on this before.

I'm looking forward to discussing GoV and Vanna.

I said "from a purely theoretical perspective", so a plain vanilla B-S option (since someone else mentioned exotics). I did not state that that was the case in reality. I believe that, in theory, it is the probability of finishing in the money. If you could link to the discussion you refer to that would be most helpful.

B-S is very theoretical and beautiful from a mathematical perspective. Since delta is synonymous with how many shares of the underlying you own, it makes sense to me, intuitively, that the number of shares you own is equivalent to the contract size (i.e. 100 for equity options) times the probability of it finishing in the money as of that precise moment.

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Feb 8, 2011
alexpasch:
Revsly:
alexpasch:

^I believe that from a purely theoretical perspective, point 2 is correct.

No it is not. There was a whole discussion on this before.

I'm looking forward to discussing GoV and Vanna.

I said "from a purely theoretical perspective", so a plain vanilla B-S option (since someone else mentioned exotics). I did not state that that was the case in reality. I believe that, in theory, it is the probability of finishing in the money. If you could link to the discussion you refer to that would be most helpful.

B-S is very theoretical and beautiful from a mathematical perspective. Since delta is synonymous with how many shares of the underlying you own, it makes sense to me, intuitively, that the number of shares you own is equivalent to the contract size (i.e. 100 for equity options) times the probability of it finishing in the money as of that precise moment.

Revsly is correct. Go search dervistrading's old post...

From a purely theoretical perspective, Delta a.k.a N(d1) is NOT the likelihood that option finishs in the money under risk neutral measure. N(d1) is, only under stock-numeraire measure, which is used in building efficient lattice-based option pricing models where the risk-free rate and volatility of the stock price are not constant.

In reality, no one knows the probability that option finishes in the money. The volatility is not constant, the interest rate may not be constant during the life of the option. The distribution of the stock return is not log normal. there are transaction cost... the list goes on...

Feb 8, 2011
alexpasch][quote=Revsly:

I said "from a purely theoretical perspective", so a plain vanilla B-S option (since someone else mentioned exotics). I did not state that that was the case in reality. I believe that, in theory, it is the probability of finishing in the money. If you could link to the discussion you refer to that would be most helpful.

B-S is very theoretical and beautiful from a mathematical perspective. Since delta is synonymous with how many shares of the underlying you own, it makes sense to me, intuitively, that the number of shares you own is equivalent to the contract size (i.e. 100 for equity options) times the probability of it finishing in the money as of that precise moment.

Not even from a theoretical perspective, I posted a thread about this a week ago.

Feb 8, 2011

---------------double post

Feb 8, 2011

Great post - looking forward to the series. As a PE investor, the more technical aspects of option trading have flown a bit under my radar, but I've always regretted not taking that derivatives class in college. SB for you, with more promised for the upcoming posts. I also front paged it.

Feb 8, 2011

Nice. I was thinking of doing a series explaining the Greeks, but you beat me too it. For that, I owe you a debt of gratitude, lol.

Feb 8, 2011

BSB... This is an excelent piece. I'm looking forward to the rest. Good stuff... I wish they taught me the Greeks like this when I started instead of trial by fire.

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Feb 8, 2011

delta is not the probability of the option finishing in the money. it's often close enough, but that is not correct. there are exotics which can have deltas far higher than 100%

Feb 8, 2011

Good post, makes me want to whip out my John Hull text.

SB for you :)

Feb 8, 2011

You have a really clear style of writing which makes this easy to understand. Thanks, look forward to the rest of the series.

Feb 8, 2011

Besides giving Silver Bananas, please remember to rank the original post by giving it (in this case, 5-bananas) a Banana Ranking :-) - you can see the 5 bananas (I thinkin IE they still might be stars) on the bottom left side of the post. All you have to do is click.

Thanks,
Patrick

Feb 8, 2011

Good stuff, let the Greeks keep on cominnn!

Feb 8, 2011

As an options amateur, I had never understood why the delta on calls cannot exceed 1.0, while the delta on puts cannot get any lower that -1.0. If a contract is knee-deep ITM, why can't an option value increase by $2, whilst the underlying stock increases by $1? Well, this thread just clarified it.

"Delta is the probability the option will finish in-the-money--i.e., how likely is it the option will end up trading for parity because it is virtually equal to a position in the stock from the strike price?"

Thanks. I'm looking forward to the other greeks.

I win here, I win there...

Feb 9, 2011
ArbitRAGE.:

As an options amateur, I had never understood why the delta on calls cannot exceed 1.0, while the delta on puts cannot get any lower that -1.0. If a contract is knee-deep ITM, why can't an option value increase by $2, whilst the underlying stock increases by $1? Well, this thread just clarified it.

"Delta is the probability the option will finish in-the-money--i.e., how likely is it the option will end up trading for parity because it is virtually equal to a position in the stock from the strike price?"

Thanks. I'm looking forward to the other greeks.

Say you have a 5 strike call and the spot price is 50. Assuming expiration is near the price will be 45.00. if the spot moves to 51, the option price will move to 46.00 which is a dollar for dollar move, hence delta one

Feb 8, 2011

http://www.wallstreetoasis.com/forums/option-delta...
N(d1) = p (S(T)>Xe^-(sigma^2)*(T-t)) & N(d2)= p(S(T)>X)

Feb 9, 2011

Two things:

  1. Since delta's the first derivative of option value with respect to underlying price and gamma is almost always nonzero for most strike price/maturity combos, delta is just an approximation and how good of an approximation it is depends on how big gamma is near the strike price of interest and how big of a move you are looking at. But if you have analytics that computes delta instantly at every point as the underlying changes, then you're probably good, unless the price of the underlying moves so fast that you can can't react properly to changes in delta.
  2. Delta CAN get larger than 1 on plain vanilla options when all hell breaks loose. Prices of options are still dictated by supply/demand, but the fact that 99% of option volume is traded on pricing formulas dictates that prices are quickly arbitraged back to "fair value" as determined by these models. However, in times of insanity, spreads widen like crazy and prices don't move in relation to models. E.g 1987 and Post-Lehman. Someone please correct me if I'm wrong on this.
Feb 9, 2011

Delta is a shitty American airline with non-stop service to nowhere.

The Greeks are socialists with a rotting corpse of an economy.

Oddly enough, OP analysis is on point.

Feb 9, 2011

Nevermind the relevance about how closely BS approximates reality (pretty amazing the crap that goes on about interpreting the salience of some of the finer points is neverending); good post by Ben Shalom Bernanke nonetheless.

Feb 8, 2011

Black-Scholes: the wrong model, you put the wrong input into to get the right price.

Feb 9, 2011

edtkh, please just go away b/c I didn't bring up any "finer" points. If something is nonlinear like option value vs underlying price, clearly you can't use the differential for huge jumps and assume you'll be anywhere close to the right value. That's obvious to anyone who has taking first semester calculus.

I'm actually not sure about the second point but it is clearly damn important because panics are the times where you can make or lose tons of money.

Feb 9, 2011
laudrup:

edtkh, please just go away b/c I didn't bring up any "finer" points. If something is nonlinear like option value vs underlying price, clearly you can't use the differential for huge jumps and assume you'll be anywhere close to the right value. That's obvious to anyone who has taking first semester calculus.

I'm actually not sure about the second point but it is clearly damn important because panics are the times where you can make or lose tons of money.

Was I addressing my post specifically at you?

It never ceases to amaze me some of the drivel people would go into such fine details over. Right, suppose we had spot at 50 and you have got 2 calls at say, 30 and 70 (with same tenor and deltas at say, 30 and 70 respectively). Is it intuitive that the latter has a higher chance of expiring in the money as things stand? Maybe that's what separates those who end up in academia and those who end up in the markets.

Invariably, you'd have some self-proclaimed "experts" trying to sell the notion as to how deltas aren't the probability of an option expiring in the money (right, theoretically, vols ain't constant too, so why don't we go get the Nobel committee to rescind the economics prize dished out to the folks who engineered such an idealistic and unrealistic model?).

Feb 8, 2011
edtkh:

Was I addressing my post specifically at you?

It never ceases to amaze me some of the drivel people would go into such fine details over. Right, suppose we had spot at 50 and you have got 2 calls at say, 30 and 70 (with same tenor and deltas at say, 30 and 70 respectively). Is it intuitive that the latter has a higher chance of expiring in the money as things stand? Maybe that's what separates those who end up in academia and those who end up in the markets.

Invariably, you'd have some self-proclaimed "experts" trying to sell the notion as to how deltas aren't the probability of an option expiring in the money (right, theoretically, vols ain't constant too, so why don't we go get the Nobel committee to rescind the economics prize dished out to the folks who engineered such an idealistic and unrealistic model?).

Actually constant vol is not that far of an assumption because volatility is a quantity measured over a period of time. So a 3 month period can have a single volatility if it is calculated over that time period. It may not necessarily move the exact amount everyday, but for those 3 months the volatility can be a single vol.

And you dont need to think of delta as probability it expires in the money (which is wrong) to know that a 70 delta call has a higher probability of ending up in the money than a 30 delta call.

Feb 9, 2011
derivstrading:

Actually constant vol is not that far of an assumption because volatility is a quantity measured over a period of time. So a 3 month period can have a single volatility if it is calculated over that time period. It may not necessarily move the exact amount everyday, but for those 3 months the volatility can be a single vol.

And you dont need to think of delta as probability it expires in the money (which is wrong) to know that a 70 delta call has a higher probability of ending up in the money than a 30 delta call.

Actually, you just effectively acknowledged that vols are theoretically still not constant (or at least, in practice, it seldom is). Effectively, that would still render the BS model as nothing more than an idealistic assumption.

Clearly, it's also not illogical to suggest delta, in a way, serves to predict or indicate the likelihood of an option expiring in the money.

So, with all said and done, why don't you commence a motion to get the Nobel Prize in Economics dished out to folks who built such an idealistic model rescinded?

Feb 8, 2011
edtkh:

Actually, you just effectively acknowledged that vols are theoretically still not constant (or at least, in practice, it seldom is). Effectively, that would still render the BS model as nothing more than an idealistic assumption.

Clearly, it's also not illogical to suggest delta, in a way, serves to predict or indicate the likelihood of an option expiring in the money.

So, with all said and done, why don't you commence a motion to get the Nobel Prize in Economics dished out to folks who built such an idealistic model rescinded?

Wow you just completely didnt read what I wrote did you? Daily movement is not constant, but vol is a quantity measured over time, so over a 3 month period, the daily variance will not be constant, but the 3m volatility in essence is constant, because theres only one 3 month volatility.

The black scholes model is a great model because it simplifies reality to a workable model, and allows for easy pricing. This doesnt mean its correct, but its simple enough that adjustments can be made to correct for some of the assumptions (ie skew).

The idea that delta is the probability of an option expiring in the money does not simplify anything, the concept is already intuitive, and yet its wrong.

The black scholes model is wrong but useful, your simplification is wrong and useless.

Feb 9, 2011
derivstrading:

Wow you just completely didnt read what I wrote did you? Daily movement is not constant, but vol is a quantity measured over time, so over a 3 month period, the daily variance will not be constant, but the 3m volatility in essence is constant, because theres only one 3 month volatility.

The black scholes model is a great model because it simplifies reality to a workable model, and allows for easy pricing. This doesnt mean its correct, but its simple enough that adjustments can be made to correct for some of the assumptions (ie skew).

The idea that delta is the probability of an option expiring in the money does not simplify anything, the concept is already intuitive, and yet its wrong.

The black scholes model is wrong but useful, your simplification is wrong and useless.

You don't understand basic English, do you? Or are you simply that dense you can't actually comprehend the fallacy in your statements?

The 3m volatility is constant simply because the market assumes a figure pretty much close to what your explanation purports it to be. If the markets didn't discount the variance (which you agree to be non-constant) in observing the vols in question and religiously factored that in, are you for a moment suggesting you could actually get vols that are constant? So much for precision! (Feel free to start an argument on the merits of using GARCH to model volatility changes to take into consideration the non-constant nature of the variance)

The BS model is great because it simplifies a workable model which works on assumptions that are widely used and generally agreed on - the model isn't wrong, mind! The model simply doesn't cater for the dynamics we see in markets in the real world!

If you understood English (and the ensuing assumptions that BS encapsulates including fundamentals like the assumption of constant vols and the absence of bid-ask spreads), it isn't rocket science to come to terms with the fact that, in your overt definition of being precise (at the expense of exercising some streetsmarts), the model is simply unworkable in reality if we interpreted everything in the strictest definition of the confines within which the model should operate. The fact that it works - and having your counterparty understand you when quoting a price and vice versa - boils down to the assumption that the BS assumptions hold and, to a large and reasonable extent, understood by market players in pricing an option.

Equally, if you can't identify how the BS model is actually "wrong", yet "useful", is it any wonder how you can't understand the concept why people have a knack of saying delta estimates the probability of an option expiring in the money?

The mathematical "fact" that deltas don't estimate the probability of an option expiring in the money is every bit as salient as vols simply isn't constant - throw out some common sense and we get enough technical jargons/arguments better suited for academia than the financial markets. Nice try in being contradictorily precise though...

Feb 8, 2011

^^^ This doesn't have anything to do with Delta, but it's interesting to note. That was one of the things we were forbidden by the CFTC to ever say to a client, even though everyone with a brain in their head knows it's true. For some reason the regulators were hung up on describing a dollar for dollar correlation between an option and the underlying.

Feb 8, 2011

How do you not understand what Im saying?

Black Scholes is an approximation/model that has a huge benefit on options trading.

Your probability approximation doesnt have any benefit. The intuitiveness that a higher delta option has a higher probability of ending up in the money than a lower delta call without thinking that delta IS the probability of the option ending up in the money.

Are you the same kid that was trying to argue with everyone about change of delta due to change in vol in an earlier thread?

Feb 9, 2011
derivstrading:

How do you not understand what Im saying?

Black Scholes is an approximation/model that has a huge benefit on options trading.

Your probability approximation doesnt have any benefit. The intuitiveness that a higher delta option has a higher probability of ending up in the money than a lower delta call without thinking that delta IS the probability of the option ending up in the money.

Are you the same kid that was trying to argue with everyone about change of delta due to change in vol in an earlier thread?

You're getting more amusing by the minute it seems. I see your English hasn't come on much either.

BS is a huge benefit on options trading because it remains a widely-accepted tool in the quoting of prices used in the industry - do you for a minute think everyone getting his/her hands on a BS pricer actually doesn't know the vols entered as an input in a BS pricer is assumed to be constant and primarily derived in the way you explained earlier?

By the same token, unless you're telling me options with vastly different deltas are priced similarly, I don't see how for the life of me one is intuitive to you, but not the other.

Jun 13, 2011

This was a really clearly written post... thanks very much!

Nov 13, 2016

.

Feb 8, 2011

Delta just refers to how the value of your portfolio changes when you bump the underlying. Basically, what position in the underlying you are running. The underlying has a delta of one.

If you: Bot 100 of a 25 Delta Call, your delta would be +25
If you: Sld 100 of a 25 Delta Call, your delta would be -25
If you: Bot 100 of a 25 Delta Put, your delta would be -25
If you: Sld 100 of a 25 Delta Put, your delta would be +25

If someone said they wanted to maintain 20% of notional, then they want to keep the portfolio +20d.

Let's say you initially Bot 100 of a 20d Call. If spot moves higher, you get long delta. Let's say Spot went such that the delta of your option was 30. Obviously you wanted to maintain a core position of +20 and you are actually +30. In this case you can Sell 10 of the underlying. Now your new Delta is +20 (+30 OPT - 10 Spot).
Let's imagine now that Spot goes back to where it originally was, so the delta of your option is +20, but your portfolio is +10 (+20 OPT -10 Spot). You can now Buy 10 of the underlying to bring your portfolio delta back to +20 (+20 OPT -10 Spot + 10 Spot).

Make sense?

Nov 13, 2016

Thanks, yes it does.

So in real simple terms:

You buy and OTM call and the underlying increases and you're long delta. If you're buying an OTM put and the underlying decreases are you by definition short delta?

I guess why I'm asking this is because the strategy that I'm looking at does the following:

1. Buy OTM puts to profit when the underlying decreases
2. Sells straddles to fund their long positions in the OTM puts

They say they're trying to assume a 20% notional long delta but they're not trading the underlying, just the options so I'm not sure how they're doing this or what they mean.

Nov 13, 2016
Gman31:

Thanks, yes it does.

So in real simple terms:

You buy and OTM call and the underlying increases and you're long delta. If you're buying an OTM put and the underlying decreases are you by definition short delta?

I guess why I'm asking this is because the strategy that I'm looking at does the following:

1. Buy OTM puts to profit when the underlying decreases
2. Sells straddles to fund their long positions in the OTM puts

They say they're trying to assume a 20% notional long delta but they're not trading the underlying, just the options so I'm not sure how they're doing this or what they mean.

It doesn't matter if it is OTM or ITM or puts or calls, options all behave the same way w.r.t delta.

A call is positive delta. A put is negative delta.
So if you are buying call, you are long delta.
If you are buying put you are short delta.
If you are selling call, you are short delta.
If you are selling put, you are long delta.

Makes sense so far? Now, from what the previous post says you might be getting it confused with gamma.
Gamma refers to how your delta changes as the underlying moves.
No matter what your delta currently is (+ or -), if you BUY options and your underlying moves up, then your net delta position is going to increase (+). And if the underlying moves down your net delta is going to decrease (-). The reverse is true if you SELL options.

Now, in your strategy, when you buy the OTM puts you are short delta. With your straddles if you are trying for a +20d position, you would have to select the strike so that by selling you achieve the delta you want. For example, you could buy the -10d put and sell 2 -15d straddles. If you do not trade the underlying, your delta is going to keep changing depending upon the gamma of the position, and you would have to buy/sell more options based on what you want to do.

Feb 8, 2011
Nov 13, 2016