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barefootpilgrim89
Topic Author
Posts: 16
Joined: April 3rd, 2020, 9:33 pm

### Delta contribution to an options premium in terms of BSM

Dear Quant Finance Community,

Delta is roughly described as a risk metric that measures the change in the premium of an option with a one point change in the underlying stock. A current delta value will only influence the premium with a subsequent one point move. In other words, a change needs to occur. The current delta value does not affect the current premium.

An option can be split into two components, intrinsic value (IV) and extrinsic value (EV), assuming an option is in-the-money (ITM). Does anyone have a crispy clear understanding of how Delta contributes the two premium components of an in-the-money option? All of the resources I have consulted do not clearly spell out how delta affects different components, or whether it only affects one of them. Resources only state that delta affects the "premium," which is vague.

If an option is out-the-money (OTM) or at-the-money (ATM), there is only EV. Any delta score would only contribute to the EV of an option with every point move in the underlying. How does this change when the option becomes ITM?

IV is a simple function of the difference between the current stock price and the strike. Does delta contribute to IV and EV, or only IV?

As an example, I used a rudimentary BSM calculator to spit out some call option figures and greeks to illustrate the question I am asking. I only change the stock price. I keep all other parameters constant when moving from initial to new assumptions. These figures are read verbatim off the options calculator coming from the interactive brokers website. I also used other BSM excel calculators to check the figures and they are pretty close.

Initial assumptions at time zero (T0):

current stock price: 100
strike price: 95
interest rate: 2
implied volatility: 25
days to expiration: 90
dividend: 0

The total call option premium = 8.026
The IV = 5 (100 - 95)
The EV = 3.026 (8.026 - 5)

Delta: 0.697

The delta is a "close" approximation to how the premium will change with a one point move in the underlying. For the next step, I am going to assume the stock price rises by one (1) point. This would mean that in accordance with the previous delta calculation, the premium should increase by 0.697, however it is not clear where this 0.697 will go (extrinsic or intrinsic value).

New assumptions at time one (T1, fractions of moment later):

current stock price: 101
strike price: 95
interest rate: 2
implied volatility: 25
days to expiration: 90
dividend: 0

The total call option premium = 8.768
The IV = 6 (101 - 95)
The EV = 2.768 (8.768 - 6)

I would like to unpack what just happened. I kept all parameters constant apart from increasing the underlying stock price by one point.

1. The total option premium increased by 0.742 (8.768 - 8.026)
2. The IV has increased by 1 (6-5)
3. The EV has decreased by 0.258 (3.026 - 2.768)

With these results it becomes confusing as to how the delta value (0.697) for the initial assumptions at T0 impacted the various components (IV and EV) of the premium AFTER the one point move took place at T1.

The total option premium increased which makes intuitive sense. That is not the question though.... The IV increased by a full point. The EV decreased. Is the delta fully contributing to IV? If so, there is 0.303 missing (1 - 0.697). Or is delta only contributing to EV? If so, why did EV decrease?

How did the 0.697 delta contribute to IV and EV? In which ratios? Did delta increase IV and decrease EV? Or did delta only decrease EV?

In a nutshell, where is the 0.697 delta going, and how is it influencing the two separate components?

Any help in understanding would be most appreciated!

Alan
Posts: 10497
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Delta contribution to an options premium in terms of BSM

If a function is the sum of two functions, so are its derivatives. (google: linear operator)

barefootpilgrim89
Topic Author
Posts: 16
Joined: April 3rd, 2020, 9:33 pm

### Re: Delta contribution to an options premium in terms of BSM

Hi Alan,

I trust you had a great Easter given the circumstances.

Thank you for your feedback, but as you no doubt have picked up, I am not a financial math specialist such as yourself. Although your profound mathematical response may seem common sense to you, it explains close to nothing to a layman such as myself. "Google it," is not all that constructive either. If it was that simple, textbooks and articles would simply state what delta affects in the fullest detail for all to understand. This isn't the case.

I am sure you would agree that textbooks and articles stating that delta affects option premium is rather vague. Which part of the premium (EV, IV or both)? Does moneyness play a role, etc.

Would you be open to rephrasing your response so that the layman can understand in the context of the questions I have asked?

Many thanks.

Alan
Posts: 10497
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Delta contribution to an options premium in terms of BSM

What is your math level? Did you ever take Calculus I? I ask because, if you don't know what a derivative is, you're going to have a hard time understanding Delta.

barefootpilgrim89
Topic Author
Posts: 16
Joined: April 3rd, 2020, 9:33 pm

### Re: Delta contribution to an options premium in terms of BSM

Hi Alan,

Firstly, thank you for your response. To answer your condescending question; I have an undergraduate level of calculus and statistics from two different degrees (biotechnology and commerce). I understand your point in that you personally understand derivative values from how differential equations spit out figures when assumptions change. That is great Alan. I hope its working out for you and that you are making loads of money...

I was hoping an educational thread like this would not resort to clever personal attacks, but it seems to me that you are more interested in defending your intellect and math then actually answering the questions in posts with integrity. What is the value in that? I didn't come here to wage an intellectual war with quants, I came here for the pearls of wisdom they can share, without them resorting to creating artificial barriers of intellectual entry (i.e. you don't know the math, so don't bother understanding derivatives...). Do you think the people that have been trading options for hundreds of years didn't understand the anatomy of the products they were buying and selling? Sure, the BSM equation and formula (solution from boundary conditions) was not formally available back then, but they bought and sold options anyways. Do you think for one moment that people stop trading when they don't have sophisticated financial math to back up their valuations? Again, do you think a market maker gives a hoot about your valuations? Nope, you accept his/her price whether you like it or not.

Of course I have exposure to what derivatives are, I have been studying them for the better part of the last 5 years of my life, and I have a library full of resources that point out how people who rely on financial math can make a real mess of things in the financial world. My question to you is whether you have personally risked your hard earned money in the markets using financial math as a talisman to make decisions. If so, how is it going? Are you predicting pricing outcomes and making a killing? Or, are you making hopeful forecasts that sometimes win, but for the most part don't? Or are you an academic that sits on the sidelines with sophisticated models and theories that serve to prop up your own ego, or earn a salary from your "boss,"  like the tautological "point" you tried to make in our previous threads? If you have skin in the game, your approach to this topic may be different and you may be humbled by how cut-throat this industry really is (read  Nassim Taleb's book, he is a quant after all, so perhaps you will take him seriously).

I try to understand derivatives from as many perspectives as I can, mathematical language being only one approach. Again, please read what I quoted about mathematics in our previous threads. Its a useful tool that can prove concepts based on our assumptions, provided our assumptions are in fact firmly rooted in reality (BSM is not...). Please don't compare Newton and Schrodinger equations to quant equations based on financial economics. Two different worlds completely. Einstein may have won a Nobel prize in physics for using Brownian motion and his calculations to prove the existence of particles, but he certainly didn't use Brownian motion as a model to predict or forecast future pricing in abstract financial instruments as if atoms and securities are related in any way (they are not if you were wondering...). Physicists use different assumptions based on what they can readily observe and measure with consistency (i.e. employing scientific method which economists cannot replicate).

To get back to the topic at hand in good faith of sharing ideas, I was not asking how delta works from a partial differential equation perspective with a solution in terms of boundary conditions. I was asking a simple question that I believe can be answered in plain intuitive language. Does delta affect both the IV and EV for an ITM option, or just EV or IV (not both). Yes or no, and why in plain English language. You do not have to be a quant with a masters or doctorate in financial math to understand how values (or their constituent parts) evolve in time in terms of change.

I changed one parameter in my example (the price of the underlying). No need to invoke mathematical gods and intellectual zeitgeist here. If you cant explain this in plain English, you don't understand it yourself, which may put you in the same boat as me. If so, welcome. Perhaps we can work together to get to the bottom of this in good faith.

You really don't need to respond to my questions, but if you want to respond, it would be most appreciated if you stop throwing around your mathematical intellect around with snide comments that add no value. Hiding behind the math and your intellect is not impressive at all. Showing your depth of understanding without having to rely on your math is a different story.

Perhaps this was not the correct forum for me to find answers to really interesting questions. If so, that was my mistake and I take responsibility for that. I turned to the quant community in the hopes that somebody (a specialist like yourself) could explain things in plain language. The textbooks don't do this with certain topics (like delta).

Thank you for your consideration and time. Peace.

Alan
Posts: 10497
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Delta contribution to an options premium in terms of BSM

Sorry, miscommunication. I meant "derivative" in the sense of calculus, not in the sense of "financial derivative". Anyway, it was an honest question, not an attack. But I am discouraged, so perhaps somebody else will help.

barefootpilgrim89
Topic Author
Posts: 16
Joined: April 3rd, 2020, 9:33 pm

### Re: Delta contribution to an options premium in terms of BSM

Hi Alan,

Yes, the topic for this thread is an option (nonlinear financial derivative). The option's anatomy is comprised of EV and IV (if ITM). The question was whether the delta figure affects these values in isolation (one or the other) or collectively. Said differently, it is not clear how SN(d1) affects EV, IV or both.

Your reluctance to share your informed view in the form of a "yes or no answer with an explanation" is perfectly understandable given the history of the content of all our previous posts. We are on different pages with clashing ideologies, and I should have been way more sensitive in my approach. You have taught me a valuable lesson in communication. I thank you for that. We live and we learn.

At the end of the day, I am the one who is looking for answers, so it's my problem.

No hard feelings man. All the best.

Athletico
Posts: 872
Joined: January 7th, 2002, 4:17 pm

### Re: Delta contribution to an options premium in terms of BSM

Hi Pilgrim -

"Resources only state that delta affects the premium, which is vague."

Where do you see this?  Maybe I can help interpret.  Raw model delta doesn't "affect" premium. (Similarly, your car's speed is not affected by its location.)

barefootpilgrim89
Topic Author
Posts: 16
Joined: April 3rd, 2020, 9:33 pm

### Re: Delta contribution to an options premium in terms of BSM

Hi Athletico,

Thank you for your response! I really appreciate the feedback. Would you be open to responding to the very first post in this thread in context? April 12th, 2020, 11:22 am

Please follow the steps and calculations I used in context. It would really help since we would be discussing the issue from the vantage point of the evolution of the option premium when one variable (stock price) was changed.

DavidJN
Posts: 1770
Joined: July 14th, 2002, 3:00 am

### Re: Delta contribution to an options premium in terms of BSM

The answer was provided in the first response to your post by Alan. I'll repeat it here:

"If a function is the sum of two functions, so are its derivatives. (google: linear operator)."

You are decomposing the total option premium into two parts - intrinsic value and time premium. No one uses the terminology "extrinsic value". Get that language out of your head. When in Rome speak as the Romans do.

The derivative of the total premium is the delta. The derivative of the intrinsic value is extremely simple. Subtract the latter derivative from the former and you have your answer.

Why are you having so much trouble with this?

barefootpilgrim89
Topic Author
Posts: 16
Joined: April 3rd, 2020, 9:33 pm

### Re: Delta contribution to an options premium in terms of BSM

Thank you for your response DavidJN.

I will rephrase my question in context:

Does the current delta at T0 affect the option premium at T1? Or does the delta at T0 serve as a book keeping device only at T0? If so, how can delta be useful to traders that need to make decisions to hedge their positions as and when market parameters change moving forward to T1?

fyvr
Posts: 26
Joined: November 15th, 2015, 3:10 pm

### Re: Delta contribution to an options premium in terms of BSM

Pilgrim, the reason people are getting frustrated is that your question is meaningless. The delta does not "affect" the MTM value, it is a derivative of the MTM value. If the MTM (option premium, in your words) changes between T0 and T1 - which it is likely to do, for a whole host of possible reasons - then yes, the delta will almost certainly also have moved by the time we get to T1. But it's not 'causal'; the change in the delta hasn't by itself 'affected' (your word) the option price, IV or TV.
At the risk of labouring this, maybe this is the answer you want - whatever has caused the delta to move, will also have caused the option price to move, and both the IV AND the TV components will be different.
If you actually understood how the IV of a European-style option was defined, that would be obvious to you.

SWilson
Posts: 76
Joined: February 13th, 2018, 5:27 pm

### Re: Delta contribution to an options premium in terms of BSM

Pilgrim, the reason people are getting frustrated is that your question is meaningless. The delta does not "affect" the MTM value, it is a derivative of the MTM value. If the MTM (option premium, in your words) changes between T0 and T1 - which it is likely to do, for a whole host of possible reasons - then yes, the delta will almost certainly also have moved by the time we get to T1. But it's not 'causal'; the change in the delta hasn't by itself 'affected' (your word) the option price, IV or TV.
At the risk of labouring this, maybe this is the answer you want - whatever has caused the delta to move, will also have caused the option price to move, and both the IV AND the TV components will be different.
If you actually understood how the IV of a European-style option was defined, that would be obvious to you.
Well said.

barefootpilgrim89
Topic Author
Posts: 16
Joined: April 3rd, 2020, 9:33 pm

### Re: Delta contribution to an options premium in terms of BSM

Thank you for your response SWilson!

I have been looking for a response that points out how incorrect the trading textbooks are in claiming that the Greeks will provide some insight into how the pricing of an option will evolve through time, to aid in constructing pre-emptive hedging or speculative strategies. Instead, the Greeks can only give you outdated parameters in hindsight of option pricing change. As you point out, there is no causal relationship. Very well articulated SWilson.

I thank everyone for being patient and indulging in this thread. I appreciate the feedback.

barefootpilgrim89
Topic Author
Posts: 16
Joined: April 3rd, 2020, 9:33 pm

### Re: Delta contribution to an options premium in terms of BSM

My apologies, my response was meant for Fyvr! Thanks Fyvr! And thanks for the confirmation SWilson.

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