# Option Delta

The delta of an option is the sensitivity of an option price relative to changes in the price of the underlying asset. It tells option traders how fast the price of the option will change as the underlying stock/future moves.

The above graph illustrates the behaviour of both call and put option deltas as they shift from being out-of-the-money (OTM) to at-the-money (ATM) and finally in-the-money (ITM). Note that calls and puts have opposite deltas - call options are positive and put options are negative.

Option delta is represented as the price change given a 1 point move in the underlying asset and is usually displayed as a decimal value. Delta values range between 0 and 1 for call options and -1 to 0 for put options. Note - some traders refer to the delta as a whole number between 0 to 100 for call options and -100 to 0 for put options. However, I will use the decimal version of -1 to 0 (puts) and 0 to 1 (calls) throughout this site.

## Call Options

Whenever you are long a call option, your delta will always be a positive number between 0 and 1. When the underlying stock or futures contract increases in price, the value of your call option will also increase by the call options delta value. Conversely, when the underlying market price decreases the value of your call option will also decrease by the amount of the delta.

The above graph shows how the delta of a call option changes as the underlying price changes.

When the call option is deep in-the-money and has a delta of 1, then the call will move point for point in the same direction as movements in the underlying asset.

## Put Options

Put options have negative deltas, which will range between -1 and 0. When the underlying market price increases the value of your put option will decreases by the amount of the delta value. Conversely, when the price of the underlying asset decreases, the value of the put option will increase by the amount of the delta value.

The above graph shows how the delta of a put option changes as the underlying price changes. So, when the underlying price rallies, the price of the option will decrease by delta amount and the put delta will therefore increase (move from a negative to zero) as the option moves further out-of-the-money.

## Comments (76)

Peter

August 16th, 2015 at 10:30pm

Hi Kenan,

Mmm, tough question! Honestly, I've no idea sorry. But is sounds like it's asking for the VaR at the different confidence levels. Check out the method and graph in the following page;

Calculating VaR for Options and Futures

Does this help?

Kenan

August 15th, 2015 at 1:36pm

Hi Peter,

Hope you are doing well, I stuck one question can't figure out. I would really appreciate if you help about that. Here is the question:

Assume that we operate under the assumptions in BlackScholes. Also assume the following: S=200 (current stock price), K=200 (strike price) return stock per year (nu)=8% Volatility of stock per year=25%, r=3% per year and the time to maturity for the option is 126 trading day(=0.5 year).

a) calculate the true 10 day VaR and the 10 day Delta-Gamma-VaR at the 97.5% confidence level for a long standard european put option. (z0.025=-1.960(=-z0,975).

b) draw figure to illustrate the difference in VaR estimates (in question a)

Solution a)

d1=0.173

Critical stock price S=230.5 (prob=0,975)

S=174.9 (p=0,025)

VaR0,975=25

delta put=-04312

gamma put=0,0111

VaR(long put)0,975(delta)=-12,936

VaR(long put)0,975(gamma)=-7,98

I could not draw the figure in "b)". Thanks in advance.

Peter

June 10th, 2015 at 10:57pm

Hi Gags,

1) I would say OTM options are more attractive to option traders because they contain more "optionality". That is, they are more sensitive to option specific factors like volatility and time to expiration. As an option becomes more and more ITM they behave more like the underlying stock and less like options.

2) OTC market makers aren't making their quotes live on the exchange; they are being made via messaging services like Bloomberg chat. Because of this, a strike and price quote won't be valid when the underlying market moves. So they then peg their quote to a delta instead of the strike. If the market maker quotes a price for a call and the market rallies, s/he may still honour the quote on a higher strike provided the delta is still within range.

Gags

June 10th, 2015 at 7:43pm

Hi ,

I few basic questions :

1) Why 25 delta options are the most liquid option .

2) why otc markets trader quote in terms of deltas and implied vol. For a layman i would approach a trader to quote a call / put for a strike price.

Gags

Peter

January 26th, 2015 at 4:46am

Hi Raja,

You can enter that data in my option pricing spreadsheet to calculate the option delta and other greek values.

Raja

January 26th, 2015 at 3:11am

Underlying price = 20

Exercise price = 18

Today's date = 16 Apr 2013

Expiry date = 30 Jun 2014

Historical volatility = 22%

Risk free rate = 5%

Dividend yield = 0%

How to calculate Delta Gamma

Please explain step by step

Peter

November 30th, 2014 at 7:32pm

Hi sHag91,

Why do you say that? As the stock price declines the put will approach a delta of -1 and as it rises the delta approaches 0 as there is less and less chance of the option expiring in the money (ITM put is where stock < strike).

sHag91

November 29th, 2014 at 2:57pm

I think the second graph (put delta) is wrong. It should be graphed just like it is in the first graph

Peter

November 3rd, 2014 at 5:21pm

That's right BullDaddy. The contract delta of a put is negative but because you are short the put, your position delta is positive.

BullDaddy

November 1st, 2014 at 8:09am

Peter,

So with 1 short put @100 in zztop with a delta of .5 and a gamma of .04 what is the delta if zztop goes to 99?

I realize that a short has a positive delta, it would seem to me that the delta would go to .54 because the risk of the short put closing ITM would be greater? Is that correct?

thx!

BD.

Peter

October 10th, 2014 at 4:25pm

Hi SaulusPaulus,

The theoretical price for a call and put will be the same where the strike = ATM Forward price. Where the ATM Forward is the spot price + cost of carry - expected dividends. However, this is not the same as (Call Price * Delta) = (Put Price * Delta). I don't think there is a relationship between (Call Price * Delta) and (Put Price * Delta) that is easily observable.

Typically the ATM Forward price is slightly higher than the current spot price. But even at this price the deltas of the options won't be the same; the call delta will be approximately 52 and the put -48.

You're welcome to use my option pricing spreadsheet - it's a good way to familiarise yourself with the theoretical values by playing around with various scenarios and viewing the changes that take place after changing the inputs to the model.

SaulusPaulus

October 10th, 2014 at 11:04am

Hello Peter,

Thanks for your very informative website.

I have a few theoretical questions regarding the delta for European Calls/Puts in the Black Scholes Framework.

1. For what spot price is |delta*Call| = |delta*Put| ?

2. When |delta*Call| = |delta*Put|, what is the delta? Which Option is worth more?

Delta should be 0 and Call option should be worth more as its value is not capped through the stock price?

Thanks in advance!

Peter

March 27th, 2014 at 5:37am

Hi Anu,

Not sure what you mean by CE/PE - but you can either use my option spreadsheet or an online option calculator to simulate various option greek and pricing values.

anu

March 27th, 2014 at 1:58am

hi..

i started he option trading now a days.so please give me guidance.i know the basics.but is there any calculations for Eg:what give the market today(CE/PE) and how much points. or what will be the tomorrows status..Please help..

Thank you.

Veggies

June 2nd, 2013 at 1:18pm

I'm not sure how to solve this question. Can anybody help me please. ugently!

A delta-neutral position is a portfolio that is immune to changes in the stock price, the portfolio of options and stock has a position delta of 0.0.

∆p=n1∆1 + n2∆2 + ...=0

Example

Suppose you are 100 puts long with a delta of -0.3.

How many calls, delta of which is -0.83, should you buy or sell to create a delta-neutral position?

∆p=n1∆1 + n2∆2 =0

(100)(-0.3) + n2(-0.85) =0

n2 = -35.29

Negative sign means the call should be sold.

Peter

April 16th, 2013 at 6:31pm

Hi Johnny,

I see now - it's the definition of gamma that has caused confusion. "Gamma" measures the change in delta for a "1 point" move in the underlying i.e. from 25 to 26.

Your example has used "Gamma 1%", which will measure the change in delta from a 1% move i.e. 25.25. Hence the need to divide by 100.

johnny

April 16th, 2013 at 2:12am

Hi Peter, let's stimulate the below scenario with the free spreadsheet in your site.

Underlying price = 20

Exercise price = 18

Today's date = 16 Apr 2013

Expiry date = 30 Jun 2014

Historical volatility = 22%

Risk free rate = 5%

Dividend yield = 0%

We come up with below:

Theoretical price (call) = 3.7011

Delta = 0.79

Gamma = 0.0597

Let multiplier = 500 and quantity = 25

Total market value = 3.7011 * 500 * 25 = 46264

Cash delta = 0.79 * 20 * 500 * 25 = 197505

Cash gamma = 0.0597 * 20 * 20 * 500 * 25 / 100 = 2983

So assume underlying price moves up by 1% (0.2) to 20.2

New theoretical price (call) = 3.8603

Total market value = 3.8603 * 500 * 25 = 48254

Total PL impact = 48254 - 46264 = +1990

Delta PL impact = 197505 * 1% = +1975

Gamma PL impact = 2983 * 1% / 2 = +15

Delta and gamma PL impact = 1975 + 15 = +1990 which reconciles to total PL impact above

So cash gamma has to be divided by 100 to arrive the sensitivity PL impact - but why...?

Can you please advise and explain? Thanks!

Peter

April 16th, 2013 at 12:01am

Hi Johnny,

Yep, you're right about the multiplier - I missed that. I'll change the formula in my comment. However, I'm not sure why they have divided by 100.

If you simulate your position by moving the base price by 1 point does your cash delta of position change by the cash gamma amount?

johnny

April 15th, 2013 at 9:46pm

Thanks Peter for the cash greeks formula. I refer to the cash gamma forumla, from my company's risk system, the formula would be:

Cash Gamma of position = gamma of contract * position * underlying price * underlying price * multiplier / 100 (in which * multiplier / 100 are not found in your formula)

Could you please advise and explain?

Peter

March 25th, 2013 at 9:30pm

Hi Johnny,

To calculate cash greeks;

Cash Delta of position = delta of contact * multiplier * position * underlying price

Cash Gamma of position = gamma of contract * multiplier * position * underlying price * underlying price

Cash Vega of position = vega of contract * position * multiplier

Cash Theta of position = theta of contract * position * multiplier

Note: Vega and Theta are already expressed in dollars hence no need to multiply by the underlying price.

johnny

March 21st, 2013 at 10:00pm

Hi Peter,

I refer to the delta exposure in dollar term of an call option, say:

Spot = 20

Strike = 18

Delta = 0.79

Multiplier = 10

The delta exposure in dollar term = 0.79*10*18 = 142.2

So when spot increases by 1%, delta p&l will be roughly 142.2*1% = 1.42

My question is, is there any forumla which I can quickly calculate the other greek exposure in dollar term, say vega/ gamma/ theta and rho?

Mnay thanks!

SATISH GUPTA

June 27th, 2012 at 9:34am

Please help me for delta hedging or delta skew. How can i find them.

Peter

February 19th, 2012 at 7:01pm

Hi Eg,

Mmm...if you use a flat volatility (i.e. the same volatility for all strikes) you will see this and I think this is just one of the limitations of using a theoretical pricing model.

In the option markets, the volatility will be different for every strike price - for equity options, downside strikes generally have a higher volatility as stocks fall faster than they rise and hence will reach the strike faster than for upper strike prices.

I could be wrong though - there may well be a quantitative explanation for this, however, I had a quick look through Natenbergs' - Option Volatility and Pricing but couldn't see it explained.

If you find another reason for this, please let me know and I will document it here.

Eg

February 19th, 2012 at 1:49pm

Given lognormal prices it would be expected that, say, a 30 Call would have a higher time value than a 20 Put when the price is at 25 (both equally OTM) due to the slight skew to the positive. But why does a 30 Put have have a higher time value than a 20 Call when the price is 25? You would expect it to be the other way around! It seems to depend on the strike, but why?

Peter

February 15th, 2012 at 10:15pm

They will be very close to it, however, as soon as the market moves in either direction the position will accumulate/lose delta, which will need to be re-hedged to remain delta neutral.

Mike

February 15th, 2012 at 6:57am

Is a portfolio consisting of a Long Put and a Long Call delta-neutral if both options have the same Strike price and are trading at the money?

Peter

January 19th, 2012 at 3:46pm

Thanks Eric! I work in software sales and trade in my spare time ;-)

Eric

January 19th, 2012 at 10:50am

Thank you very much. Excellent site btw - what is your line of work?

Peter

January 18th, 2012 at 3:55pm

Yes, correct - Delta is calculated from a pricing model such as B&S so it represents the theoretical change in the option price given a one point move in the underlying asset.

Eric

January 18th, 2012 at 8:20am

I notice that on the vega page you write that the vega represents the THEORETICAL change in the option price/ change in volatility.

Does the same go for the delta? Is it only theoretical since the change in price is assuming hte market is using BS to price the option?

Thank you,

Peter

November 9th, 2011 at 8:27pm

If the underlying stock drops by 5pts then the option price (theoretically) will either rise or fall (depending on if it is a call or put option) by 0.75 (0.15 x 5).

Ty

November 9th, 2011 at 8:16pm

So what happens if the underlying stock price goes down 5pts, and the delta was .15 the day before....wouldn't the value of the delta also decrease?

Chris

November 2nd, 2011 at 5:55pm

Yes, I think the diagrams imply a normal distribution of share price movements, but I guess that's because of the erroneous assumption in black-scholes.

Peter

November 2nd, 2011 at 5:08pm

Hi Chris,

Yes, the skew affects the prices (and hence the greeks) of calls and puts differently. Generally, for equity options puts have higher volatilities than for call options with the same strike difference from ATM.

Is this what you mean?

Chris

November 2nd, 2011 at 4:05pm

Thanks this site is very helpful.

Could you clarify one thing - assuming equity movements are skewed to the downside, would skew alter the delta of a put option vs a call option (i.e. would the delta of an out-of-the money put option be further from zero than a similarly out-of-the money call option?)

Chris

Peter

September 26th, 2011 at 6:41pm

My deltas for AAPL look fine, see link below;

AAPL Options

Can you send me a screen shot of what you see?

Jose

September 26th, 2011 at 2:55pm

Today apple calls have been tradin with an inverted delta curve, meaning OTM calls have a higher delta than ATM calls. Is that common. Can someone explain this to me?

Peter

September 4th, 2011 at 6:39pm

No, the graphs are correct. You are not reading them correctly.

The first graph is plotting the delta values against OTM/ATM/ITM concepts - not market price. For an OTM put the delta is zero, which is what this graph shows.

A put delta is never +1 as you mention - a put delta can only be between -1 and 0.

Moha

September 4th, 2011 at 4:33pm

Hi,

Excuse me, but your Graph is WRONG: the delta of a put is -1 when the underlying is around zero (Out of The Money OTM) and around +1 when the put is In The Money ITM,Please amend

Peter

August 20th, 2011 at 1:37am

A call option delta is between 0 and 1, while a put option delta is between -1 and 0. But because the stock IS the underlying its delta is always 1.

kanchan

August 19th, 2011 at 9:46am

isn't it between o and 1 ??

Peter

August 16th, 2011 at 7:34am

That isn't possible: the delta of a stock is always 1.

kanchan

August 16th, 2011 at 7:19am

If a stock has a delta of 0.6 at $45 and 0.8 at $50... what does this mean?

Peter

June 25th, 2011 at 2:18am

Yes, exactly. The graphs above are for long call and put deltas.

Anita

June 24th, 2011 at 10:53pm

Hi ,

Will the graph of short call and short put be the inverse of the 2 graphs shown above .

Peter

March 1st, 2011 at 10:05pm

Hi Tom, you'll need some kind of option pricing software to do this. You can use my option pricing spreadsheet as a starting point. However, you might also want to check with your broker as many online brokers provide such functionality in client front ends.

What broker do you use?

TOM

March 1st, 2011 at 9:40pm

If i buy 10 calls and 10 puts ATM of a 50 dollar stock, and say the calls cost me 4 each and the puts cost 3 each and the expiration is 60 days out, when the stock moves up or down how do i know when and how to adjust to get back to delta neutral. As the stock goes to 53 or 47, how do i know what the delta is and how do i trade it.............

Peter

February 11th, 2011 at 3:15am

Yes.

Saravanan

February 11th, 2011 at 12:30am

I am from india. I am a basic learner of options. Is put delta nd put option value inversely proportional?

Peter

January 3rd, 2011 at 10:41pm

Delta values range between -1 and + 1, so -1,466.80 seems strange...unless there is some kind of multiplier being applied.

Anyway, it just means that if the base price (e.g. stock price) moves up 1 point then the value of the put option is expected to decrease 1,466.80 points.

YEO

January 3rd, 2011 at 9:46pm

Hi,

If the put option got -1466.80 (delta), what is this means ?

Peter

December 22nd, 2010 at 3:57pm

Yes, although it doesn't depend on the time to expiration as much as it does on the interest rate. As long as the strike is equal to (or as close as possible) to the forward price, then yes, ATM options will have deltas very close to 50%.

You can try it on this web based online option calculator. Make the interest rates and dividend yield = 0 so that the forward price will equal the strike you are after and just change around the days to expiration field.

Prasun

December 22nd, 2010 at 6:22am

Hi,

for an ATM Call Option, will the Delta always hover around 50%? doesnt maturity period have any impacts? In other words, will 2 ATM options, one with an expiry of 1m and another with 1 yr, have 50% deltas?

Peter

November 23rd, 2010 at 6:53pm

Yep, you're right. Thanks for the clarification!

K

November 23rd, 2010 at 2:11pm

Hey Peter,

Love your site. Good work, and thanks.

Your last comment on this page was, "the put delta will also decrease as the option moves further out-of-the-money."

However, won't the put option increase (e.g. move closer to zero from negative one) as the option moves further OTM?

Peter

October 10th, 2010 at 12:22am

No, but here's an online version;

http://www.sitmo.com/live/OptionBarrier.html

George

October 9th, 2010 at 2:38pm

I guess it can't calculate the Greeks of barrier options

any links?

Peter

August 28th, 2010 at 12:52am

How do you mean...because it's negative?

juan

August 27th, 2010 at 11:55pm

the put graph seems to be wrong ?

Peter

August 1st, 2010 at 9:01pm

It's the relationship between volatility (probability of option expiring in the money) and time being non-linear - asset volatility follows a log-normal distribution.

Option Theta is highest for strikes at (close to) the money and tapers off either side in a non-linear fashion.

sam

July 31st, 2010 at 2:23pm

what is the financial intuition behind time value of option decreasing convexly for strikes away from asset price?

Peter

June 3rd, 2010 at 10:04pm

You'll have to calculate the Greek values. You can use the spreadsheet found under the pricing link. Or, you can go to;

www.option-price.com

Sundraa

June 3rd, 2010 at 12:47pm

Forget continuous or discrete compounding.. just take it this way. Long Call option profit is virtually unlimited... whereas with a long put, your profits has a cap (because stock prices cannot go below 0). So call option can give you more returns than a put option and hence delta of ATM call is greater than a put.

Ray

June 2nd, 2010 at 1:38pm

Gentlemen, where do I go to get current option delta values?

Peter

December 23rd, 2009 at 4:33pm

I disagree. It is the compounding of those factors that causes the curve to skew to the upside, hence becoming log normal. Without compounding the curve is symmetrical as the returns to the upside have no bias over those to the downside. When you begin to compound the returns, you will notice that a compounded negative rate of return yields a lower absolute change than a return that is positive.

For example, if you take $100 at a 5% return and compound it for 10 years you end up with a profit of $62. If you take -5% you will lose only $40, hence the skew to the upside.

Marc

December 18th, 2009 at 2:35pm

Your explanation of the log normal distribution (LGD) is wrong. The LGD is not used over a normal because option models are "continuous". Both normal and lognormal are continuous. Lognormal is used for the simple fact that is a natural way to enforce positive asset prices. This in turn introduces a skew that does not exist in the normal distribution. Continuous compounding rates, dividends, and volatility, have absolutely nothing to do with it.

Alan

December 17th, 2009 at 11:53pm

Thank you very much Peter. Really appreciate your help.

Peter

December 15th, 2009 at 6:40pm

Hi Alan,

Yes, this is due to the Log Normal Distribution curve that is used by the Black and Scholes model to estimate the "rate of return" (interest and volatility). The Log Normal curve is used over a Normal Distribution because option models are considered continuous, where volatility, interest and dividends are taken to be continuously compounded and hence produce and upward bias in returns.

Alan

December 15th, 2009 at 8:19am

Hi Peter, i have a question regarding ATM call and put. ATM calls seems to be like 52 delta and ATM put seems to be around 48 delta. there were some comments made saying its due to Black Scholes model preference for puts over call. Would appreciate if you can help to explain.

Peter

November 10th, 2009 at 4:21am

Hi Ashi, a Box Spread is a combination of two opposing vertical spreads i.e. a long call spread and a short put spread. Both spreads would have the same strikes and expiration date.

The idea is that the credit received for the short spread is more than what is required to be paid for the long spread and hence a risk-free profit is locked in.

Regarding Collars vs Bull Spread...this depends on your capital requirements and the prices for the option components. A Collar consists of a long stock meaning a much greater burden on your trading account.

Ashi

November 9th, 2009 at 5:10pm

Hiya

I stumbled upon your page while preparing for an exam :) and I found your material really useful.

what is a BOX SPREAD by the way? And I am always confused between choosing a Collar options verus a call Bull spread...both profiles look the same... when do you choose one or the other?

Jo Jack

July 7th, 2009 at 2:04am

Peter,

Your graph is correct. Thank you for all the information on this site.

Peter

May 22nd, 2009 at 3:14am

Hi Steve,

Actually, I think it is correct. The graph is showing the delta of a 50 strike put option, which has a negative delta. As the stock price declines, the option becomes shorter hence the delta approaches -1. When the put option is deep in the money the delta will reach -1 and behave like a short underlying position.

As the stock price increases and becomes out of the money the delta will approach zero and eventually become worthless.

Let me know if you dissagree.

Steve

May 22nd, 2009 at 1:15am

Your put option graph is reversed. The red line in the bottom graph should has the wrong slope.

aranjan

April 9th, 2009 at 3:59am

Very good explanation

Pratap

January 21st, 2009 at 11:42pm

Very Useful......

Rating - 5 out of 5

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