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barefootpilgrim89
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Nobel prize contributions and the birth of implied volatility

April 6th, 2020, 9:12 pm

Dear Quant Finance Community,

I have been reading a considerable amount of material regarding the BSM model and I am fascinated by the evolution of the model and options trading industry. However, from what I have read, there are some controversial topics that seem to have no definite conclusions. I was hoping that persons in the Quant Finance Community may be willing to share their insight.
 
What I have learned is that the notion of a self-financing replicating strategy (e.g. dynamic hedging) was not an original idea for Black and Scholes. They recognise this in their 1972 publication, pointing out that Thorp and Kassouf already discovered this (and were applying it in the markets way before options became standardised on the CBOE in 1973).
 
It is pointed out in various texts that the main difference between Thorpe and Kassoufs work and Black and Scholes work, is that Black and Scholes observed that a combination of “delta” long shares over a very (instantaneous) short period of time should earn a return equal to a riskless investment.
 
But what does not make sense to me is that Black and Scholes would not have “observed” this if Merton didn’t point it out to them by introducing the notion of continuous-time dynamic hedging? Only under the continuous-time and continuous trading assumption (along with other unrealistic assumptions) does the no-arbitrage hypothesis kick in, where only the risk-free rate remains as the expected return of a perfectly hedge portfolio?
 
This makes me question why the work of Black and Scholes took precedence over Thorp and Kassoufs work? It would seem that Black and Scholes didn’t actually invent anything novel? But they had the vital input from Merton which validated the general ideas at the time? I am making a presumption of course, and would appreciate constructive negative or positive feedback.
 
If my presumption is correct, what then was Black and Scholes real contribution? I can understand why Merton got the economic equivalent of the Nobel prize for his Ito calculus contribution, but why did Scholes also get the prize (Fischer had passed away but would also have receive the prize if alive)?
 
What I also find highly suspicious is that the BSM model only works in a risk-neutral world (where investors have no risk/reward preferences) and other unrealistic assumptions hold. Of course risk preferences are moot if one can perfectly dynamically hedge, but one cannot actually do this in real markets. Furthermore,  intuitively anyone who studies the problem understands that volatility is never constant and it cannot be predicted (it can only be forecasted which is a poor methodology for traders). If traders (buy side and sell side) intuitively understand that geometric Brownian motion is not a reflection of reality, and that volatility is not constant and cannot be predicted; then surely expected return based on risk preferences are actually very important?
 
The problem is that expected return cannot be observed or measured, which isn’t conducive to deriving a close-form solution for option pricing. Perhaps Black, Scholes and Merton knew this and had to find a way to “get rid” of the notion of expected returns?
 
This also makes me think that the market reintroduced the inescapable reality of risk preferences (i.e. expected return) into what we today call implied volatility? Now market makers and traders manipulate IVol to reflect their risk preferences related to unknown future underlying pricing outcomes. Market pricing of options reflects this. No longer does sigma represent volatility in accordance with GBM assumptions. Rather sigma represents whatever traders and market-makers believe the future may or may not hold which is purely subjective or based on sophisticated mathematical and statistical guesswork. Traders, market makers and quants are now financial alchemists in their own right?
 
Is there something wrong with my thinking? I would really appreciate your feedback if anyone has the time and interest to respond.

With thanks and appreciation in advance!
 
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Alan
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Re: Nobel prize contributions and the birth of implied volatility

April 7th, 2020, 12:20 am

Just a couple comments.

First, lots of people were working on the problem of option valuation. But, Black and Scholes (with help from Merton) made a key discovery and, IMO, were the correct 3 to get most of the accolades for advancing the state of knowledge. Rather than argue that in detail, I think Mark Rubinstein's account is fair, and I attach it for you below, with a little highlighting applied by me. The excerpt is from Rubinstein's  book: "A History of the Theory of Investments: My annotated bibliography".  

Second, on the problem of observing the expected rate of return, I have done a little recent work on that myself, found here.
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barefootpilgrim89
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Re: Nobel prize contributions and the birth of implied volatility

April 7th, 2020, 5:55 am

Thank you for your constructive feedback Alan!

I will give the resources you have attached a serious study.
 
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Alan
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Re: Nobel prize contributions and the birth of implied volatility

April 7th, 2020, 6:23 pm

You're welcome. Read my paper and I think (hope) it will answer your other posted question about the equity risk premium that is embedded in and inferrable from option prices. 
 
barefootpilgrim89
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Re: Nobel prize contributions and the birth of implied volatility

April 8th, 2020, 9:13 am

Dear Alan,

I have had a concentrated read of your work. Thank you! However I have to be open in saying that I am first and foremost a trader and learn as much as I can about quantitative finance to understand the products and formulas invented by quants (which I find fascinating). I am trying to wrap my head around theoretical options valuation versus practical marking pricing from a conceptual standpoint.

I am baffled by some of the terminology I have come across in various sources. 

May I ask what the difference is between drift, expected return and equity risk premium? Are these terms in any way related? Are they the same thing or different? Are these variables or parameters (depending on their interpretation) represented by the same symbol (µ)? Can these principles be explained in plain language?

Why does BSM literature always state that we do not need to concern ourselves with the determinants of µ in any detail? The textbook reason being "that the value of an option (or other derivative) dependent on a stock is, in general, independent of µ..." Why is this so conceptually speaking?

From what I understood in your paper, you are concerned about determining a value for equity risk premium that has a term structure. BSM sidestepped trying to directly determine a value of equity risk premium (if I understand the literature correctly in context). They inferred as per the Noble Prize website that that "it is not necessary to use any risk premium when valuing an option (which I am assuming is related to the equity risk premium your paper is all about). This does not mean that the risk premium disappears, but that it is already incorporated in the stock price."

BSM completely removed µ from their equation. Why and how? If one creates a perfectly replicating dynamically hedged portfolio that includes both the stock, option and a bond (so that the position is also self-financing); you are going to be long µ and short µ, that cancel each other out if the hedge ratio is precise. Basically, the act of perfectly dynamically hedging eliminates the abstract concept of µ. If the equation deals with the problem like this, no need to worry about the risk preferences of traders related to unknown future stock pricing outcomes (the actual problem). The µ which I am assuming relates to the risk premium that would be expected by a trader (i.e risk that cannot be diversified away) is incorporated in the current stock price in terms of the tenets of the EMH (the literature dances around the issue but that seems to be the only plausible rationale). Since the replicating portfolio is comprised of the stock whose current price magically incorporates all risk preferences in the present moment, and which gets instantaneously factored in every time the portfolio is re-balanced, the option must also magically incorporate all risk preferences since it is a derivative of the stock. The deltas cancel each other out (long versus short). Ultimately, the options risk premium is then calculated by the total costs of dynamically hedging. Magically, the premium of an option (which represents what option writers demand not knowing what the future may hold in terms of realized volatility) indirectly equates to the total costs of dynamically hedging.

Here is the rub. If dynamic hedging is based on unrealistic utopian fairy land assumptions (including Samuelson and Famas EMH in terms of rational economics), then it cannot work in reality. If it cannot work in reality, how can we expect traders to accept the BSM equation and formula? Furthermore, sigma (σ) is assumed to remain constant (a parameter) in terms of the fairy land assumption of brownian motion applied to the abstract notion of a stock price (which has nothing to do with physical processes such as atoms). Traders may or may not be convinced by the notion of Brownian motion, but they cannot accept that volatility is constant. Traders also realize that dynamic hedging doesn't work, so µ doesn't actually cancel out. It will crop up at some point during the discrete (not continuous) reality of trading (i.e. discretely dynamically hedging). If µ isn't cancelled out and σ isn't constant, then how could an options risk premium reflect an unknown future (risk) to the satisfaction of an option writer? It cant... Implied volatility is born. Implied volatility is the markets solution to take into account risk preferences as well as what traders think future volatility will be. Traders change σ as they see fit to reflect their risk preferences when facing an unknown future (future volatility). Of course, implied volatility is indirectly calculated through the BSM formula, but it is included in the market price of an option. Traders set the market price of the option to incorporate their risk preferences and views on future volatility. There is nothing scientific about this process. A supposedly scientific equation gets tainted by implied volatility to find the market price of an option. One may argue that there are sophisticated models and equations that determine what implied volatility should be, but nobody seems to agree. People use different models, different numerical inputs, different data and different calibration methods. Sounds pretty subjective to me, albeit that mathematics gets used in the process. Although I may be wrong (I hope so), I am not a specialist. 

And of course trading and price determination cannot be scientific. Why? If everyone used some universal formula or model, with the same volatility input assumed to be constant, and where subjective risk preferences didn't feature, then there would be no discrepancies in pricing. There would be no disagreement in pricing. There would be no market.... Everyone would arrive at the same pricing figure. No potential for profit or loss.

Alan, your thoughts would be most appreciated. I welcome any and all critique in this thought process mentioned above. If I am wrong with my understanding of these variables, parameters and the assumptions behind them, please let me know! Ultimately, I am trying to wrap my head around BSM. Why it was globally accepted to begin with, and subsequently why traders introduced implied volatility that doesn't gel with what BSM had in mind with their Nobel prize winning model, equation and formula.

If anyone else in the Quant community is interested and can shed some light, this too would be appreciated.
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Alan
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Re: Nobel prize contributions and the birth of implied volatility

April 8th, 2020, 5:06 pm

I think you have a very good practical understanding. Everything you say from "Here's the rub" and below is widely accepted.

I think what you don't appreciate is that all the practical limitations of the BSM formula were also well-appreciated early on by just about everybody, including the inventors. The model formula quickly morphed into just a convenient way to discuss market prices, via implied volatility. Everybody gets that.

So, you are beating a dead horse by pointing out these limitations.

The BSM formula is derived from a simplified model of the world. It does offer general insights: indeed, enough to be worthy of a Nobel prize. It was Nobel prize-worthy, despite the limitations. Remember, we are talking about the economics prize here, which is often, IMO, more of an academic lifetime achievement award.  :D

To really understand it conceptually, if that is your goal, that means mathematically. To do that, you need to spend some time learning stochastic calculus and its connection with parabolic partial differential equations. Without that understanding, anything anybody says will sound like hand-waving.

Alternatively, if your goal is to critique it, don't bother. Trust me that the traders understanding and the academic finance understanding of the many limitations of the model are quite compatible and well-known.

Hope it helps.   
 
barefootpilgrim89
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Re: Nobel prize contributions and the birth of implied volatility

April 8th, 2020, 7:36 pm

Dear Alan,

Thank you for your response! I have learned much with the interaction we have had over the last couple of days! Thank you for taking time out of your schedule to shed light on these issues! I agree 98% with all your comments, and I certainly have huge respect for the time, blood, sweat and tears that Black, Scholes and Merton sacrificed over their lifetimes to achieve Nobel Prize accolades. I doubt that many people would question whether they deserved it (perhaps Thorp would but he elected making money over academic prestige). 

However, in my studies of the great thinkers that have changed the world, they first made observations of the world around them. They try to understand the observation and use their imagination to conceptualize an idea or solve a problem. They then create a hypothesis or model, and use mathematics as a language to codify their thoughts. They then collect unbiased data and test their hypothesis or model which involve mathematics as a tool. To summarize, first comes conceptual understanding of the nature of the problem, and then conversion of the problem or idea into rigorous math to be tested with data. I am assuming that financial economists, financial engineers and quants consider themselves first and foremost "scientists" that use mathematics as a tool, and not the other way around. However, one must exercise caution not to fall down rabbit holes. Why?

"While mathematics and the sciences enjoy a close and mutually beneficial relationship, at heart they are completely different endeavors. Mathematics is all about proving things, but the things that mathematics proves are not necessarily true facts about the actual world. They are implications of various assumptions, axioms or interpretations. A mathematical demonstration shows that given a particular set of assumptions; certain statements inevitably follow. The statements we can prove based on explicitly stated axioms are known as theorems. But “theorem” doesn’t imply “something that is true;” it only means “something that definitely follows from the stated axioms.” For the conclusion of the theorem to be “true,” we would also require that the axioms themselves be true. That is not always the case." - Sean  M Carroll, physicist.  

So yes, coming to terms with the mathematics is very important, but I don't agree that the mathematics being pursued first is a good methodology for learning. First one needs to understand conceptually, then use the math to back it up. That is my opinion and I am sure that many in the Quant community would disagree with this form of heuristic learning. Many would first study the math and become experts, and then question the thought process after they understand the math (hoping that the many years they have sacrificed don't cause cognitive dissonance should the math they studied and adopted be incorrect). The problem with the math first approach is that one can easily fall victim to economic and financial ideology ("laws") backed up by elegant math that only works due to unrealistic assumptions that work in fairyland. I am sure many would agree that countless trees have been cut down to serve as paper for the thousands of publications regarding these topics. Models piled on top of models, maths on top of maths becoming ever more complicated, but never really addressing the problem. Again, this is my opinion, I may be wrong. What I do know is that If you are a trader you cannot afford intellectual zeitgeist. It works well in the classroom or around a campfire where money is not at stake (perhaps academic reputation is though). 

Alan, at this point I would understand if our communication has come to and end due to perhaps our differences in learning. However, if you feel so inclined, would you please indulge me based on my method of heuristic learning? I believe that certain ideas can be described in plain English and conceptually understood. Financial economists and quants had specific thought processes that surely can be described in words? After all, if one cannot describe something simply in plain English, but have to use contradictory circular logic and/or complicated math to explain something, then perhaps one doesn't really understand it? 

For clarity, would you please bring insight regarding the following questions? 

What the difference is between drift, expected return and equity risk premium? Are these terms in any way related? Are they the same thing or different? Are these variables or parameters (depending on their interpretation) represented by the same symbol (µ)? This is all in context of the underlying (e.g. stock) and options pricing.

Again, any feedback is most appreciated!
 
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Re: Nobel prize contributions and the birth of implied volatility

April 8th, 2020, 8:21 pm

1. Black and Scholes looked at Thorp & Kassouf's 'Beat the Market', which had a fair amount of warrant data, the dominant type of tradeable option at the time. They also followed up with comparisons of their formula with market data. So, you perhaps mis-characterize their work as 'math only'. My point with you is that, as a trader, you probably have a good understanding of the data. But you need to understand the math if you want to truly appreciate the BSM contribution.

2. Drift, expected return, and equity risk premium (ERP) are indeed closely related but not identical notions. The ERP is an expected stock return less a riskless rate. An expected return is the mean or expectation of some (asset) return probability distribution. A drift generally refers to the coefficient of the 'instantaneous' expectation of a diffusion process. Yes, they often are all represented by a [$]\mu[$], but sometimes other symbols are used as well, and the meaning of any symbol always depends upon the context. 
 
barefootpilgrim89
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Re: Nobel prize contributions and the birth of implied volatility

April 8th, 2020, 9:28 pm

Dear Alan,

1. You make an insightful valid point "you need to understand the math if you want to truly appreciate the BSM contribution." I don't specifically have a problem with the model, equation and formulas (solved from boundary conditions) since I don't apply it myself when trading. I see it as an interesting thought experiment justified by math which I do appreciate in terms of their story, but the assumptions of the thought experiment are quite frankly laughable. I appreciate the BSM contributions for what they are; a model, and that's where it ends. I just wanted to make sure that my reasons for finding the model amusing are sound. Like you point out, there is no use or productive value being a critique standing on the sidelines. My quest is to understand why things are the way they are (in reality). I need to understand why things are right or wrong, and I mean no offense at any juncture. 

2. Thank you for shedding light on ERP, exepected return and drift! It makes me think that the extrinsic value of an option based on BSM ideology, boils down to calculating or divining the mean of a probability distribution (gaussian). The expected return boils down to the historically calculated (or subjectively plugged in, i.e. implied volatility) mean in terms of a probability distribution. Said differently, whatever the numerical figure for sigma is deemed to be (statistically calculated from past historical pricing data, or subjectively plugged in), would then churn out an extrinsic value for the option using the BSM formula which is the best effort guess of what the future may hold (i.e. unknown future realized volatility) which justifies the risk that an option writer would be willing to accept by writing the option. Option traders are buying and selling "risk" as represented by extrinsic value compensation, based on the tenets of probability theory. Am I on the right track here?

Perhaps of interest was the observation that an option has a theoretical value or market price in the present moment. In accordance with BSM, the option has an instantaneous value in the present moment. However, in terms of dynamic hedging, the options value (with the option being one component of a financial instrument fruit salad where all risk is neutralized) is indirectly equated to the totals costs of hedging the aggregate position as and when the pricing of the underlying changes (i.e. through retrospective observation of underlying pricing change or realized volatility). From the perspective of the present moment, how do costs of hedging arise for measurement if there has been no change in the pricing of the underlying in the present moment? There can be no change in the present moment, since the present moment simply "is" a state of being with no ties to the past or future. What is there to hedge with no change in the pricing of the underlying in present moment? Hedging costs only arise retrospectively, yet the option has a deterministic value in the present moment... Said differently, an option writer cannot technically sell the value determined by calculating what happened historically because that would mean waiting and thus selling the past to some option buyer in the present (when the present arrives). Yet magically, the option writer does sell the present value in the present moment which magically gets calculated from the past that has not yet been recorded (no time has elapsed to calculate the costs of hedging, which should equate to the present value of the option). Mind boggling. 
 
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Alan
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Re: Nobel prize contributions and the birth of implied volatility

April 8th, 2020, 10:25 pm

Re 'laughable', I think that might go too far.

Here's a thought experiment. Pretend you are trading and also tasked with providing ballpark theoretical option values to support trading. It's a two-person shop: you and your boss. So, you can't pass the buck.  :D

Your boss gives you the following assignment: the firm is soon to begin making markets in Asian options on ticker XYZ, a security which has never before had such options trading. The firm needs a theoretical valuation method to get started. What do you come up with?  

It's an honest question with a point to it (later). 
 
barefootpilgrim89
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Re: Nobel prize contributions and the birth of implied volatility

April 8th, 2020, 11:58 pm

Dear Alan,

Don't you find the assumptions of BSM model humorous in that they (as you have pointed out) are clearly documented to be undoubtedly false? Don't answer this, I didnt ask a constructive question... I had no intention to provoke a debate in this thread. I just wanted to understand the BSM from a different perspective based on the knowledge and experience from the Quant community. And, I have learned much in the process and thank you for your considered input. 

However, your thought experiment is a great hypothetical challenge, but for the fact that I don't have a boss. I trade my own capital in a prop shop environment. I also don't make my own pricing OTC. Which bring us to the crux of the matter I believe is behind your thought experiment. Quants provide a crucial function in finding theoretical values for exotic OTC products. I never disputed this or challenged this. Whether the theoretical values get converted into actual pricing figures accepted by a buyer and seller is debatable. If the trade goes down, great.

However, as a trader that participates over regulated exchanges, I don't haggle pricing with a market-maker or broker-dealer. They make the price, and I accept it or not. I am sure you don't believe that a theoretical valuation (however mathematically sophisticated) actually matters to a market-maker on the other side of your trade. You either accept the market-makers pricing or stay out of the market... You don't debate theoretical valuations with counterparties over an exchange. 

Now of course, if you asked me whether I had a hypothetical boss that is a market-maker firm that has employed me to create an algorithm to price the underlying or derivatives, and disseminate the bids/offers over an exchange, that would be a different thought experiment. And here, I believe quants play a major role. Again, I am not disputing it. However, I don't believe that these quants use BSM. They use their own tailor made pricing equations, and they input whatever data and assumptions they deem fit. If these market-makers have enough impact and market share, other market participants will accept these bids/offers. If not, other market-makers will win the day.

Anyways, Alan, I have thoroughly enjoyed our conversation and I believe this thread has run its course. We disagree on some things, but I believe we agree on most things. Ultimately these forum discussions are great for stimulating thoughts and the sharing of ideas, which I think we have done. We live and we learn.

Thanks Alan.
 
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fyvr
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Re: Nobel prize contributions and the birth of implied volatility

May 2nd, 2020, 2:14 pm

"Quants provide a crucial function in finding theoretical values for exotic OTC products. I never disputed this or challenged this."
Well you ought to, because (given that you are specifically referencing exchange-traded options) it's nonsense. If stock XYZ 3M ATM calls are trading at $2.50, that didn't come from a quant's calculations any more than the price of gold or USDJPY spot. It's the market price, defined as the equilibrium price at which there are equal numbers of buyers and sellers.
What BSM (or any other approach) does is try to put that $2.50 in the context of some theoretical model that captures something interesting about the physics. In this case, the interesting physics is (inter alia) the implied volatility of the trajectory of the underlying stock, looking forward.
Why is this valuable? Here's two reasons for starters.
1) relative-value trading: suppose the $2.50 price is consistent with an implied vol of 20%. But another 3M option, with a different strike, is trading at a price consistent with an implied vol of 25%. Is that interesting? Yes, because the vol numbers are, on the face of it, contradictory. Is it a trading opportunity? That's not so clear (you need to understand what might be causing the strike-dependency of the vol) - but at least you've drilled down to something more physical.
2) Hedging. Simply knowing that an option is trading at $2.50 does absolutely nothing for me in terms of telling me how to hedge. But having a model (albeit an idealised one), that has some explanatory and predictive power, does.
 
barefootpilgrim89
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Re: Nobel prize contributions and the birth of implied volatility

May 3rd, 2020, 5:23 pm

Thank you for your feedback Fyvr, very interesting. However, I have some questions if you are interested in answering them.

Dont you think a broker-dealer acting in a market-making capacity has an algorithm programmed by a quant to spit out option pricing (bids/offers) over exchanges? Do you think the price-discovery process is fair in this regard (simply supply and demand)? Or do you think there are powerful players behind these quotes? What do you think they use to calculate their option premiums? Do you think these equations are programmed into algorithms or does a market-maker manually input a pricing figure thumb-sucked from the air? 

Of course quants provide a crucial function on the sell-side of OTC products, but quants also work for broker-dealers on the exchange side.

You mentioned that "It's the market price, defined as the equilibrium price at which there are equal numbers of buyers and sellers." What a wonderful economic notion that explains nothing constructive about the price-discovery process...  Do you really believe that the "market" miraculously just spits out the "equilibrium" "fair" price as an intersection between supply and demand without any powerful movers behind the pricing? Neoclassical and financial economists would give you straight A's Fyvr... Furthermore, the real market players would love to trade against you. You are the perfect counterparty to trade against... You are a cash cow.

More importantly Fyvr, please dont confuse the discipline of physics (a natural science) with economics and finance... There is nothing about economics or finance that involves the scientific method. If you were wondering, mathematics is not the scientific method, mathematics is a useful tool to all disciplines because it "proves things" given certain assumptions (let us hope the assumptions are correct). There is no physics in trading or valuing abstract securities with abstract pricing figures that have never landed on the physical plane. 

Where do you think implied volatility comes from? Did nature provide a law for it? Is it a physical constant? Did people run experiments and apply the scientific method to derive equations (as in the laws of physics) to deal with implied volatility? Did they come across implied volatility in nature to begin with? We arent dealing with laws of motion and equations of classical physics here Fyvr. Pricing doesnt follow predictable trajectories... Do you think NASA would use financial models to send people to the moon? I doubt they would even use financial models in an attempt to make money to fund their operations to send people to the moon...

Thank you for pointing out that an option's current pricing does absolutely nothing for you in telling you how to hedge. I cannot agree more. Yet you think a model that uses current pricing inputs or the inputs from a chain of options in reverse to calculate implied volatility will give you explanatory and predictive power? Do you think stochastic volatility models, implied volatility binomial trees and the like will give you predictive power? If so, put your cash in a brokerage account and start trading brother... you will make a killing! Perhaps you would even be kind enough to allow others to invest in your fund.... After all, LTCM did great for the first few years, then they lost everything and had to be bailed out after Scholes and Merton received the equivalent of the Nobel prize.

How amazing would it be if intelligence and a background in applied financial mathematics was all that one needed to make a killing on Wall Street. Sign me up...

Seriousness aside Fyyr, take what I say with a pinch of salt. You are free to believe whatever you like. I can be completely wrong or crazy. Im playing devils advocate here, and perhaps you will find the kinds of questions I pose interesting food for thought. Ultimately, quantitative finance serves an important purpose and I mean no disrespect to those who have sacrificed years to become specialists in the field. Its a remarkable achievement to master applied financial mathematics. Few can do it, and I certainly do not count myself as someone who comes even close to meeting that watermark. That's why I like asking quants questions. Always something to learn. 
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