# Call option and put option delta different

#### hairamki

##### New Member
Iam using Zerodha B&S calculator to calculate greeks of options below for Nifty now..Why is that for the same Spot and strike price the Call option delta is more than the put option delta ? Ideally should'nt it be 0.5 both sides as probability for the ATM to go in the money is same both sides mathematically .. Can someone pls help .. Thanks

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#### Rusty_Banks

##### Member
Iam using Zerodha B&S calculator to calculate greeks of options below for Nifty now..Why is that for the same Spot and strike price the Call option delta is more than the put option delta ? Ideally should'nt it be 0.5 both sides as probability for the ATM to go in the money is same both sides mathematically .. Can someone pls help .. Thanks

View attachment 34001
The reasons due to which the deltas of the ATM call and put are different, as shown in your screenshot is because:
• The calculator that Zerodha uses is based on the BSM model. It means that the interest rate that you input is used by the model to calculate a forward price for the stock. The model then uses that forward price (and not the stock price that you input) as the underlying price. Going by this, the higher the interest rate and more time to remaining, the higher would be the call price and lower the put price and different deltas. This should be simple.
On a side note, the theoretical premium (in the screenshot) for the 11500 call and put is different. Hence, once can easily tell that it is the Black-Scholes-Merton (BSM) model.
• In addition to this, even if the model did not calculate the forward price (such as the Black76 model) and took the forward price as the actual input, it would show the deltas of the ATM calls and put different.
Why?

Because if you look at what the delta calculation formula for a call, it is: Exp(-1 * riskfree_rate * time_to_expiration) * N(d1) and for a put it is: : Exp(-1 * riskfree_rate * time_to_expiration) * N(d1-1)

Here N(x) is the standard cumulative normal distribution function. It returns the probability of getting an occurrence less than x by calculating the area under the standard normal distribution curve between the values of –∞ and x

d1 = tells us, in standard deviations, how far the exercise price is from the mean. This means that if in case, d1 = 0.5, then the absolute delta would be same for both the ATM call and put. But is d1 equal to 0.5 for ATM strike?

Thought it may seem that for an ATM option, d1 should be exactly half, but the caveat here is that stock prices do not follow normal distribution, but rather the log-normal distribution. There is a slight upward drift in Geometric Brownian motion.

To elaborate more, the actual formula for d1 that we use in option modelling is (ln(S/K)+(r+0.5σ* σ)(t))/σ√t
This is already adjusted for log-normal distribution ( by adding (0.5σ* σ*t) ). Hence d1 is slightly greater than 0.5 (used for call delta) and naturally, (d1 - 1) which is used for the put delta calculation, is slightly less than 0.5 irrespective of the model used.

As an example, here are the greeks calculated for your inputs in Black76 (suitable for Indian markets):

Call:
array
(size=6)
'delta' => float 0.50614867659299
'gamma' => float 0.00074208045206883
'vega' => float 14.116047503481
'theta' => float -2.9877335075589
'rho' => float -0.20307554867875
'premium' => float 211.77878647927

Put:
array
(size=6)
'delta' => float -0.48773312994262
'gamma' => float 0.00074208045206883
'vega' => float 14.116047503481
'theta' => float -2.9877335075589
'rho' => float -0.20307554867875
'premium' => float 211.77878647927

Note that while using the Black76 model, it shows the premium as same for both the call and put because we have input the strike as 11500 and the forward price as 11500 as well. However, the deltas are still different. Hope this helps.

Last edited:

#### mohan.sic

##### Well-Known Member
Iam using Zerodha B&S calculator to calculate greeks of options below for Nifty now..Why is that for the same Spot and strike price the Call option delta is more than the put option delta ? Ideally should'nt it be 0.5 both sides as probability for the ATM to go in the money is same both sides mathematically .. Can someone pls help .. Thanks

View attachment 34001

Hi,

Very good and detailed answer from rusty_banks. Hope its clear.

----------------------------------------------------------------------

So why there is difference is because, though you gave 11500 as spot, the B&S calculator will add the interest at given rate and consider the FUTURE value of Nifty in calculating the theoretical option prices. In this case the future comes to appx 11570 ( 11500 @ 6.4% interest @ 35 days )
You may check in the calculator by changing the spot to 11570 & interest to 0% and you will get appx same CE& PE values which you got above.

Now considering Nifty at 11570 and strike is 11500 - It means CE is ITM by 70 points. and PE is OTM.
And we know ITM options will have higher Delta than OTM options. And that is why CE delta > PE Delta.
That is the main reason for that reasonable gap between CE & PE Delta s of 11500 strike.

And secondly what he says is, Interest rate part is not the only factor in this Delta difference. Even if interest rate is Zero ( that is Spot and Future value is same ), still we see the difference in Delta of CE & PE as the formula varies. But this difference will be very less comparatively.

#### hairamki

##### New Member
The reasons due to which the deltas of the ATM call and put are different, as shown in your screenshot is because:
• The calculator that Zerodha uses is based on the BSM model. It means that the interest rate that you input is used by the model to calculate a forward price for the stock. The model then uses that forward price (and not the stock price that you input) as the underlying price. Going by this, the higher the interest rate and more time to remaining, the higher would be the call price and lower the put price and different deltas. This should be simple.
On a side note, the theoretical premium (in the screenshot) for the 11500 call and put is different. Hence, once can easily tell that it is the Black-Scholes-Merton (BSM) model.
• In addition to this, even if the model did not calculate the forward price (such as the Black76 model) and took the forward price as the actual input, it would show the deltas of the ATM calls and put different.
Why?

Because if you look at what the delta calculation formula for a call, it is: Exp(-1 * riskfree_rate * time_to_expiration) * N(d1) and for a put it is: : Exp(-1 * riskfree_rate * time_to_expiration) * N(d1-1)

Here N(x) is the standard cumulative normal distribution function. It returns the probability of getting an occurrence less than x by calculating the area under the standard normal distribution curve between the values of –∞ and x

d1 = tells us, in standard deviations, how far the exercise price is from the mean. This means that if in case, d1 = 0.5, then the absolute delta would be same for both the ATM call and put. But is d1 equal to 0.5 for ATM strike?

Thought it may seem that for an ATM option, d1 should be exactly half, but the caveat here is that stock prices do not follow normal distribution, but rather the log-normal distribution. There is a slight upward drift in Geometric Brownian motion.

To elaborate more, the actual formula for d1 that we use in option modelling is (ln(S/K)+(r+0.5σ* σ)(t))/σ√t
This is already adjusted for log-normal distribution ( by adding (0.5σ* σ*t) ). Hence d1 is slightly greater than 0.5 (used for call delta) and naturally, (d1 - 1) which is used for the put delta calculation, is slightly less than 0.5 irrespective of the model used.

As an example, here are the greeks calculated for your inputs in Black76 (suitable for Indian markets):

Call:
array
(size=6)
'delta' => float 0.50614867659299
'gamma' => float 0.00074208045206883
'vega' => float 14.116047503481
'theta' => float -2.9877335075589
'rho' => float -0.20307554867875
'premium' => float 211.77878647927

Put:
array
(size=6)
'delta' => float -0.48773312994262
'gamma' => float 0.00074208045206883
'vega' => float 14.116047503481
'theta' => float -2.9877335075589
'rho' => float -0.20307554867875
'premium' => float 211.77878647927

Note that while using the Black76 model, it shows the premium as same for both the call and put because we have input the strike as 11500 and the forward price as 11500 as well. However, the deltas are still different. Hope this helps.

Thanks a lot Rusty..
I have a follow up question ..I wanted to setup a delta neutral trade so that I can profit either way the underlying moves( specifically Nifty index )..but so far I have not been successful, because of this delta skew between call and put of same strike ..
I tried different combintaions like buying a near OTM put and buying a far OTM call so that delta's match, but that does not give me any profits if Nifty ( India market index ) moves in either direction.. The reason I feel is because far OTM call responds slow to price variation than near OTM put ..
So I started doubting the greeks provided by Zerodha calculator itself .. do you think it is accurate for Indian market ? Asking as look to be a pro in options.. If zerodha BS calculator is not accurate then which is accurate ?
So for the next directionless strategy Iam planning to combine a future with near OTM put and see whether that gives directionless returns .. buy 2 futures and buy 5 near OTM puts so that deltas nearly match.. Hope that will work . Any comments from your side ?
Also do you have any good directionless returns strategy for index or stock ?

#### hairamki

##### New Member
Hi,

Very good and detailed answer from rusty_banks. Hope its clear.

----------------------------------------------------------------------

So why there is difference is because, though you gave 11500 as spot, the B&S calculator will add the interest at given rate and consider the FUTURE value of Nifty in calculating the theoretical option prices. In this case the future comes to appx 11570 ( 11500 @ 6.4% interest @ 35 days )
You may check in the calculator by changing the spot to 11570 & interest to 0% and you will get appx same CE& PE values which you got above.

Now considering Nifty at 11570 and strike is 11500 - It means CE is ITM by 70 points. and PE is OTM.
And we know ITM options will have higher Delta than OTM options. And that is why CE delta > PE Delta.
That is the main reason for that reasonable gap between CE & PE Delta s of 11500 strike.

And secondly what he says is, Interest rate part is not the only factor in this Delta difference. Even if interest rate is Zero ( that is Spot and Future value is same ), still we see the difference in Delta of CE & PE as the formula varies. But this difference will be very less comparatively.

Thanks Sir.. the same question to you ..

I have a follow up question ..I wanted to setup a delta neutral trade so that I can profit either way the underlying moves( specifically Nifty index )..but so far I have not been successful, because of this delta skew between call and put of same strike ..
I tried different combintaions like buying a near OTM put and buying a far OTM call so that delta's match, but that does not give me any profits if Nifty ( India market index ) moves in either direction.. The reason I feel is because far OTM call responds slow to price variation than near OTM put ..
So I started doubting the greeks provided by Zerodha calculator itself .. do you think it is accurate for Indian market ? Asking as look to be a pro in options.. If zerodha BS calculator is not accurate then which is accurate ?
So for the next directionless strategy Iam planning to combine a future with near OTM put and see whether that gives directionless returns .. buy 2 futures and buy 5 near OTM puts so that deltas nearly match.. Hope that will work . Any comments from your side ?
Also do you have any good directionless returns strategy for index or stock ?

One more question sir..
Since you say it is calculating based on future price.. If I choose strikes equidistant from say Apr future price( Apr becuase far expiry theta will be less ) will that be delta neutral and will it give good returns anyways the underlying moves ?

#### hairamki

##### New Member
I have a follow up question ..I wanted to setup a delta neutral trade so that I can profit either way the underlying moves( specifically Nifty index )..but so far I have not been successful, because of this delta skew between call and put of same strike ..
I tried different combintaions like buying a near OTM put and buying a far OTM call so that delta's match, but that does not give me any profits if Nifty ( India market index ) moves in either direction.. The reason I feel is because far OTM call responds slow to price variation than near OTM put ..
So I started doubting the greeks provided by Zerodha calculator itself .. do you think it is accurate for Indian market ? Asking as look to be a pro in options.. If zerodha BS calculator is not accurate then which is accurate ?
So for the next directionless strategy Iam planning to combine a future with near OTM put and see whether that gives directionless returns .. buy 2 futures and buy 5 near OTM puts so that deltas nearly match.. Hope that will work . Any comments from your side ?
Also do you have any good directionless returns strategy for index or stock ?

One more question sir..
Since you say it is calculating based on future price.. If I choose strikes equidistant from say Apr future price( Apr becuase far expiry theta will be less ) will that be delta neutral and will it give good returns anyways the underlying moves ?

#### hairamki

##### New Member
@Rusty_Banks @mohan.sic

Another followup question
Regards Delta neutral trade in Nifty..
Current Nifty spot is 11521 so forward price is 11591 appx.. .. The OTM strikes where deltas are nearly equal as per Zerodha BS calculator is https://zerodha.com/tools/black-scholes/
PE - 11500
CE - 11700
Interest rate - 6.4
Expiry - 25-apr-19
Volatility - 14%

Will this trade be delta neutral and give profits if Nifty moves either direction by 10 - 20 points ??
Thanks in advance..

#### mohan.sic

##### Well-Known Member
@Rusty_Banks @mohan.sic

Another followup question
Regards Delta neutral trade in Nifty..
Current Nifty spot is 11521 so forward price is 11591 appx.. .. The OTM strikes where deltas are nearly equal as per Zerodha BS calculator is https://zerodha.com/tools/black-scholes/
PE - 11500
CE - 11700
Interest rate - 6.4
Expiry - 25-apr-19
Volatility - 14%

Will this trade be delta neutral and give profits if Nifty moves either direction by 10 - 20 points ??
Thanks in advance..

My feeling is, that your got fancied with this concept " Delta neutral " and missing some basics.

Will that above trade give profits -
where is a trade in that ? Please Give a example ( Fut/CE/PE - Buy/Sell ?)

No. Generally Nifty move of 10-20 points can not yield returns through hedging delta ( Expiry days may be exception to this )

Last edited:

#### mohan.sic

##### Well-Known Member
I have a follow up question ..I wanted to setup a delta neutral trade so that I can profit either way the underlying moves( specifically Nifty index )..but so far I have not been successful, because of this delta skew between call and put of same strike ..
I tried different combintaions like buying a near OTM put and buying a far OTM call so that delta's match, but that does not give me any profits if Nifty ( India market index ) moves in either direction.. The reason I feel is because far OTM call responds slow to price variation than near OTM put ..
So I started doubting the greeks provided by Zerodha calculator itself .. do you think it is accurate for Indian market ? Asking as look to be a pro in options.. If zerodha BS calculator is not accurate then which is accurate ?
So for the next directionless strategy Iam planning to combine a future with near OTM put and see whether that gives directionless returns .. buy 2 futures and buy 5 near OTM puts so that deltas nearly match.. Hope that will work . Any comments from your side ?
Also do you have any good directionless returns strategy for index or stock ?

One more question sir..
Since you say it is calculating based on future price.. If I choose strikes equidistant from say Apr future price( Apr becuase far expiry theta will be less ) will that be delta neutral and will it give good returns anyways the underlying moves ?
1) Just by initiating a delta neutral position ( in any combination of Future/CE/PE either Long or short ) there is no guarantee of profit.

2) Option calculator in zerodha should be 100% perfect. No need to doubt any calculator from any website/broker. Problem is in our approach and understanding.

3) If you Buy 2 Futures and 5 OTM. ( Agreed it is a Delta Neutral position )
Positional -Lets assume Nifty is very flat for 3 days or it went up/down and came back to same level.
Now we will be at break-even in the Future position. But Long put options position will loose Time value. Hence Net Loss.
Intraday - There is no use unless there is a decent directional Up-move. Even if that happens, there
are chances that you could end up in Net Loss Eod if put iv decreases ). And If Nifty is
in Down- move coupled with a decrease in put iv's, there are absolute chances you will end up in Loss.

So why do you think initiating a Delta neutral position is No Loss method.

Delta neutral trading is not one time entry and watch method. As the market moves, our positions net
delta keeps changing and adjustments have to be done to bring back delta to neutral position.
This helps when you are selling options positionally ( we keep adjusting the positions delta to neutral and gain from time decay) But again this is not as easy and simple as discussed on paper.

But from your writing, I see that you are planning intraday trades by Buying Options. How do we make
profit from a delta neutral position in intraday ?
A simple example can be buying a ATM call & put at appx at same rates.
Lets assume Nifty Future is at 11000 and we buy Strike 11000 CE & PE at Rs.80 each. In theory the Delta
of CE will be +0.5 & for pe -0.5 . As nifty moves upside - call is moving ITM and Put is moving OTM -
This means Delta of Ce increases and Delta of Pe decreases- Means this ITM CE price will increase at a
faster rate than the fall In the put price. So here you make profit only when there is a good and reasonable
directional move in Nifty + No fall in PE imp vol and No time decay effect.

Last edited:

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