The Black-Scholes formula, explained (2019)(cantorsparadise.com)
cantorsparadise.com
The Black-Scholes formula, explained (2019)
https://www.cantorsparadise.com/the-black-scholes-formula-explained-9e05b7865d8a
65 comments
> higher vol [...] makes the option worth more.
> Similarly having more time to expiry makes the option worth more [...]
These two observations become unified when one observes that the two variables sigma (vol) and (T-t) (time to expiry) only ever enter the Black-Scholes formula together, as sigma^2 (T-t) (which is the variance of the log return to expiry).
(That leads to the observation that the unit of vol is 1/sqrt(time), normally a^(-1/2) [a=annum=year]).
Next, you alluded to it, but let's make explicit the way to trade implied vol: Suppose (WLOG) that you think implied vol is too low. Then you buy it (buy cheap, sell expensive...) in the market (buy buying calls and/or puts). Now you're long gamma, but short theta, and long vol in two senses: a) if the stock doesn't move, but implied vol goes up, you can turn around and sell the option back for a profit. b) you can delta hedge the option. Every day, you lose money on theta, but gain money on the movements of the stock. If it doesn't move at all, you lose the theta. If the stock moves "as expected" (=as implied by the implied vol), you'll just break even. But if the stock moves more than expected (that is, realised vol is higher than the implied you bought it at), then you make money. If the market comes around to your estimate of the future vol, you can then sell the option at a profit, or you can hold it to the end and keep delta hedging, and that will be profitable if realised vol turns out to be higher than implied.
These two observations become unified when one observes that the two variables sigma (vol) and (T-t) (time to expiry) only ever enter the Black-Scholes formula together, as sigma^2 (T-t) (which is the variance of the log return to expiry).
(That leads to the observation that the unit of vol is 1/sqrt(time), normally a^(-1/2) [a=annum=year]).
Next, you alluded to it, but let's make explicit the way to trade implied vol: Suppose (WLOG) that you think implied vol is too low. Then you buy it (buy cheap, sell expensive...) in the market (buy buying calls and/or puts). Now you're long gamma, but short theta, and long vol in two senses: a) if the stock doesn't move, but implied vol goes up, you can turn around and sell the option back for a profit. b) you can delta hedge the option. Every day, you lose money on theta, but gain money on the movements of the stock. If it doesn't move at all, you lose the theta. If the stock moves "as expected" (=as implied by the implied vol), you'll just break even. But if the stock moves more than expected (that is, realised vol is higher than the implied you bought it at), then you make money. If the market comes around to your estimate of the future vol, you can then sell the option at a profit, or you can hold it to the end and keep delta hedging, and that will be profitable if realised vol turns out to be higher than implied.
All good points. All I would add is that it costs money to flatten the delta every day, or delta hedging your option. Each time the stock changes direction, you pay the bid-ask spread. The number of times this happens is proportional in some way to the volatility—that free variable you mentioned—so you can speculate that the market's number is too high or low.
If the market is too high on volatility: sell the option and you'll end up paying less in delta-hedging costs.
If that market is too low on volatility: do the opposite.
If the market is too high on volatility: sell the option and you'll end up paying less in delta-hedging costs.
If that market is too low on volatility: do the opposite.
(Thank you.)
"Since its introduction in 1973 and refinement in the 1970s and 80s, the model has become the de-facto standard for estimating the price of stock options"
...and has caused a lot of catastrophic losses. The formula depends on a normal distribution and financial returns are random but not independent. They are not normal.
The formula works, mostly, but when it does not it is worse than useless. Financial gains and losses are in the tails, and the tails are no where near normal
...and has caused a lot of catastrophic losses. The formula depends on a normal distribution and financial returns are random but not independent. They are not normal.
The formula works, mostly, but when it does not it is worse than useless. Financial gains and losses are in the tails, and the tails are no where near normal
> formula depends on a normal distribution and financial returns are random but not independent...worse than useless.
This is sort of like throwing out physics because Newtonian mechanics don’t account for fluid dynamics.
Yes, the original theory assumed that away. And yes, the original theory is taught in undergrad. But the work has been developed far past its original, adjusting for or incorporating away those initial assumptions, and—to a large degree—having been shown, empirically, to work.
(This is not your fault. Popular writing on the topic is terrible, elevating drama over accuracy. Against the Gods is one of the better ones, and doesn’t require much math.)
This is sort of like throwing out physics because Newtonian mechanics don’t account for fluid dynamics.
Yes, the original theory assumed that away. And yes, the original theory is taught in undergrad. But the work has been developed far past its original, adjusting for or incorporating away those initial assumptions, and—to a large degree—having been shown, empirically, to work.
(This is not your fault. Popular writing on the topic is terrible, elevating drama over accuracy. Against the Gods is one of the better ones, and doesn’t require much math.)
>This is sort of like throwing out physics because Newtonian mechanics don’t account for fluid dynamics.
>having been shown, empirically, to work.
What exactly do you mean by this? That they are correct most of the time? Or that the person that uses them won't go bust?
This parallel between physical theories and assumptions about how the market works is bogus. In trading you can have strategies that are correct most of the time yet when they fail the impact of the loss can take you out.
>having been shown, empirically, to work.
What exactly do you mean by this? That they are correct most of the time? Or that the person that uses them won't go bust?
This parallel between physical theories and assumptions about how the market works is bogus. In trading you can have strategies that are correct most of the time yet when they fail the impact of the loss can take you out.
> That they are correct most of the time?
Yup. Options market makers, the critical mass of dynamic hedgers, don’t blow up any more [1]. I left the business ten years ago, and was probably among the last well-paid people to do it. There isn’t much risk anymore which means there isn’t room for ingenuity—it’s execution, mechanical.
Between market circuit breakers limiting instantaneous price moves; the tremendous amount of liquidity in option-covered symbols; tail-risk estimating options models; and fully electronic options, equities and money markets, there simply isn’t empirical evidence for hidden risks in the model. Cash equities execution was once super complicated. It’s now commoditised. Same for options.
It sells books to claim otherwise. But you largely need to re-tell stories from the 90s, where LTCM bet on short-term Russian debt, or recount crisis-era structured products on illiquid mortgages to fill the pages.
[1] American OCC-cleared options market makers who aren’t making directional bets but manufacturing options and hedging their books
Yup. Options market makers, the critical mass of dynamic hedgers, don’t blow up any more [1]. I left the business ten years ago, and was probably among the last well-paid people to do it. There isn’t much risk anymore which means there isn’t room for ingenuity—it’s execution, mechanical.
Between market circuit breakers limiting instantaneous price moves; the tremendous amount of liquidity in option-covered symbols; tail-risk estimating options models; and fully electronic options, equities and money markets, there simply isn’t empirical evidence for hidden risks in the model. Cash equities execution was once super complicated. It’s now commoditised. Same for options.
It sells books to claim otherwise. But you largely need to re-tell stories from the 90s, where LTCM bet on short-term Russian debt, or recount crisis-era structured products on illiquid mortgages to fill the pages.
[1] American OCC-cleared options market makers who aren’t making directional bets but manufacturing options and hedging their books
Options market makers, the critical mass of dynamic hedgers, don’t blow up any more
I wonder if that's true in insanely volatile stocks like GME? People were buying way OTM calls on that stock. Then the stock would move 50% in one day. A market maker would have to be very good at dynamic hedging to keep up with that.
Of course options market makers have one incredible thing going for them. While they have market risk for every individual option they write, their net exposure can potentially be very small. That only works for a market maker, not for someone making a directional bet because YOLO.
Edit: The few times I tried to study what was going on with GME options, I saw a lot of "no bid" on many OTM strikes. So it looks like the market makers were simply stepping away. Which totally changes the market dynamics. If there's no liquidity in an option, a punter's only choice is to hold to expiration? That's financially very risky and also counter to everything we've come to believe about an "efficient" market.
I wonder if that's true in insanely volatile stocks like GME? People were buying way OTM calls on that stock. Then the stock would move 50% in one day. A market maker would have to be very good at dynamic hedging to keep up with that.
Of course options market makers have one incredible thing going for them. While they have market risk for every individual option they write, their net exposure can potentially be very small. That only works for a market maker, not for someone making a directional bet because YOLO.
Edit: The few times I tried to study what was going on with GME options, I saw a lot of "no bid" on many OTM strikes. So it looks like the market makers were simply stepping away. Which totally changes the market dynamics. If there's no liquidity in an option, a punter's only choice is to hold to expiration? That's financially very risky and also counter to everything we've come to believe about an "efficient" market.
> lot of "no bid" on many OTM strikes. So it looks like the market makers were simply stepping away
Bingo. Self help [1] and circuit breakers [2] negate the unsolvable edge case: large, instantaneous price movements.
[1] https://www.reuters.com/article/usa-options-cboe-idUSL2N1H40...
[2] https://www.npr.org/2020/03/09/813682567/how-stock-market-ci...
Bingo. Self help [1] and circuit breakers [2] negate the unsolvable edge case: large, instantaneous price movements.
[1] https://www.reuters.com/article/usa-options-cboe-idUSL2N1H40...
[2] https://www.npr.org/2020/03/09/813682567/how-stock-market-ci...
All that promise that innovation like HFT among other things was fine because of the value liquidity they can provide and now they can just choose to not play if they don't want to.
Near the money GME options are super liquid with tight spreads, exactly as promised. If you're holding GME, but you're worried about an earning surprise, you can put on a collar for like zero cost, which is pretty crazy considering the insane volatility. Exactly what the people who said they wanted the markets to support meaningful transactions instead of speculation asked for. Nobody ever promised you'd get easy access to 1000:1 yolo lottery tickets.
This comment is really interesting.
> There isn’t much risk anymore which means there isn’t room for ingenuity—it’s execution, mechanical
What do you mean by ingenuity here? Like coming up with your own model that was better than other people's models, or new strategies, etc?
Also, what the heck is an "aerospace investment banker"? Someone in IB who only works on aerospace stuff?
> There isn’t much risk anymore which means there isn’t room for ingenuity—it’s execution, mechanical
What do you mean by ingenuity here? Like coming up with your own model that was better than other people's models, or new strategies, etc?
Also, what the heck is an "aerospace investment banker"? Someone in IB who only works on aerospace stuff?
> What do you mean by ingenuity here?
Creativity. What you said. No more 10x improvement opportunities. Just marginal adjustments. Maintenance. Running the same model a bit more efficiently, carving off minuscule edge cases here and there.
> what the heck is an "aerospace investment banker"
A made-up moniker. I raised money—and did deals, e.g. IP licensing, M&A, PPP, et cetera—for rocket, satellite, drone and adjacent start-ups before that was a thing. I had enough technical knowledge to know we were and are on a precipice. Computer-aided design, singularly, as well as new fluid dynamics numerical methods being unsung and recent game changers; falling launch costs our transcontinental railroad. But not enough to do the work myself. So I sold and structured, things I am good at, while reading Banks and Nivens and K.S. Robinson on the weekends.
Really rewarding work. Didn’t pay that much, unfortunately, though the resulting equity changed my life.
Creativity. What you said. No more 10x improvement opportunities. Just marginal adjustments. Maintenance. Running the same model a bit more efficiently, carving off minuscule edge cases here and there.
> what the heck is an "aerospace investment banker"
A made-up moniker. I raised money—and did deals, e.g. IP licensing, M&A, PPP, et cetera—for rocket, satellite, drone and adjacent start-ups before that was a thing. I had enough technical knowledge to know we were and are on a precipice. Computer-aided design, singularly, as well as new fluid dynamics numerical methods being unsung and recent game changers; falling launch costs our transcontinental railroad. But not enough to do the work myself. So I sold and structured, things I am good at, while reading Banks and Nivens and K.S. Robinson on the weekends.
Really rewarding work. Didn’t pay that much, unfortunately, though the resulting equity changed my life.
Interesting job!
No creativity makes it sound like the market has been figured out. I know that isn't true, so does that mean that the risk/reward of strategies has flattened off (I.e. same risk for less reward) because there are less opportunities to exploit?
No creativity makes it sound like the market has been figured out. I know that isn't true, so does that mean that the risk/reward of strategies has flattened off (I.e. same risk for less reward) because there are less opportunities to exploit?
Thank you! It was.
> No creativity makes it sound like the market has been figured out
That was my bet. It has, so far, been a good one.
> does that mean that the risk/reward of strategies has flattened off (I.e. same risk for less reward) because there are less opportunities to exploit?
Your instinct was on point. You don’t need someone with a feel for volatility to make money (or not lose it) in options market making anymore. The work consists of, and will for some time, re-implementing existing ideas. There are still mis-pricings. But they aren’t inherent to the pricing model. The commodity component can be isolated and run with an eye towards costs and economies of scale.
(This has been a fun conversation, by the way. Thank you.)
> No creativity makes it sound like the market has been figured out
That was my bet. It has, so far, been a good one.
> does that mean that the risk/reward of strategies has flattened off (I.e. same risk for less reward) because there are less opportunities to exploit?
Your instinct was on point. You don’t need someone with a feel for volatility to make money (or not lose it) in options market making anymore. The work consists of, and will for some time, re-implementing existing ideas. There are still mis-pricings. But they aren’t inherent to the pricing model. The commodity component can be isolated and run with an eye towards costs and economies of scale.
(This has been a fun conversation, by the way. Thank you.)
Thanks for your perspective and no worries! It's always interesting to hear the opinions of people with domain expertise.
All of the inputs of the BS model are forecasts. All of them can be wrong, and they have been wrong countless times. It’s like saying that your linear extrapolation for the stock market mostly works, except for the times it doesn’t.
> the inputs of the BS model are forecasts
In the same way a rocket flight model is forecasting the arrangement of air molecules it’s about to run into. They’re instantaneous forecasts that are dynamically updated. No long-term forecasting involved.
At the end of the day, options market makers haven’t blown up since the early noughties. (LTCM got sunk by non-options bets.) They are low-margin, low-risk businesses. It’s fun to talk about them like they’re black boxes. And traders trying to defend their compensation will keep pitching them that way to senior management. But options pricing is a boring, largely solved—if still interesting—problem.
In the same way a rocket flight model is forecasting the arrangement of air molecules it’s about to run into. They’re instantaneous forecasts that are dynamically updated. No long-term forecasting involved.
At the end of the day, options market makers haven’t blown up since the early noughties. (LTCM got sunk by non-options bets.) They are low-margin, low-risk businesses. It’s fun to talk about them like they’re black boxes. And traders trying to defend their compensation will keep pitching them that way to senior management. But options pricing is a boring, largely solved—if still interesting—problem.
If options pricing is largely solved, do you think exchanges could provide an API where instead of specifying the price and quantity for hundreds or thousands of options on a stock, market makers could send a much smaller message containing their latest risk and model parameters, and have the exchange run a standard model internally to generate the quotes?
Since the model to convert parameters to prices and quantities would be known in advance to all traders, the next step could be providing a feed of current parameters which traders could use to build their own view of the public book. They'd still need plenty of quotes published piecemeal as now (as not all quotes would be model-generated) but the messaging could be much more efficient, and that could lead to more options trading.
Since the model to convert parameters to prices and quantities would be known in advance to all traders, the next step could be providing a feed of current parameters which traders could use to build their own view of the public book. They'd still need plenty of quotes published piecemeal as now (as not all quotes would be model-generated) but the messaging could be much more efficient, and that could lead to more options trading.
> market makers could send a much smaller message containing their latest risk and model parameters, and have the exchange run a standard model internally to generate the quotes?
No market works like this. The smaller-still message of a price is sent and disseminated. When you buy a flight, you want the price of the ticket—not the airline’s fuel and tariff costs.
No market works like this. The smaller-still message of a price is sent and disseminated. When you buy a flight, you want the price of the ticket—not the airline’s fuel and tariff costs.
None of the inputs to the BS model are forecasts, except the volatility. What the BS model allows, then, is trading this volatility. (Just as other financial products require other inputs, and thus make them, in a sense, tradable: Cross currency swaps make the cross currency basis tradable, credit default swaps make credit risk tradable, trading index vol versus single stock vol makes correlation tradable, etc.)
The only thing it offers, is a common denominator for trading. The value predicted by the model itself is completely arbitrary. You have no clue today about tomorrows volatility or the the interest of a “risk free” instrument aka the central banker can show up tomorrow announcing -10% interest rates.
Btw when interest rates went negative for the first time, many trading shops were caught pants down, because of course their models did not have a provision for negative interest rates.
Btw when interest rates went negative for the first time, many trading shops were caught pants down, because of course their models did not have a provision for negative interest rates.
Having two sets of assumptions for buy and sell side is pretty indicative of the quality of the model.
> two set of assumptions for buy and sell side is pretty indicative of the quality of the model
What does this refer to? And, no. Disagreement on inputs doesn’t convey much about a model—it’s a negotiation. Any model will have procurer and vendor using different inputs when negotiating purchase and sale. That Boeing and steel mill don’t agree on tensile strength assumptions doesn’t mean aircraft designers are winging it.
What does this refer to? And, no. Disagreement on inputs doesn’t convey much about a model—it’s a negotiation. Any model will have procurer and vendor using different inputs when negotiating purchase and sale. That Boeing and steel mill don’t agree on tensile strength assumptions doesn’t mean aircraft designers are winging it.
Buy-side firms don’t use BS with the same assumptions as market makers. If you’re a fund manager, you are interested in objective estimates of the value. If you’re a market maker you want to show that you can perfectly replicate and thusly hedge your derivatives. Never mind that you have to recalibrate your model every 20 minutes.
For a model that is supposed to measure the objective value of an asset, that’s clear failure. BS is at this point just a vague market consensus which stinks more and more, the farther you stray away from vanilla European options.
For a model that is supposed to measure the objective value of an asset, that’s clear failure. BS is at this point just a vague market consensus which stinks more and more, the farther you stray away from vanilla European options.
> Never mind that you have to recalibrate your model every 20 minutes
Let me know when we make a plane or rocket that doesn’t need to recalibrate it’s flight model every millisecond.
As I said, market neutral market makers will use different assumptions from directional buy side shops. To say nothing of their vastly different funding costs and structures. Using input heterogeneity or calibration frequency as an estimator of model quality is...odd.
Let me know when we make a plane or rocket that doesn’t need to recalibrate it’s flight model every millisecond.
As I said, market neutral market makers will use different assumptions from directional buy side shops. To say nothing of their vastly different funding costs and structures. Using input heterogeneity or calibration frequency as an estimator of model quality is...odd.
We never recalibrate the g constant in our calculations nor we wait a person to announce what the g for this quarter will be.
> We never recalibrate the g constant in our calculations nor we wait a person to announce what the g for this quarter will be
Sure. But we do update all manner of atmospheric, gravitometric and similar factors in our flight and orbital models. Once again, calibration frequency is a poor predictor of model quality. There are useless models in every domain involving immutable constants. And there are very good numerical methods that have no constants per se.
Sure. But we do update all manner of atmospheric, gravitometric and similar factors in our flight and orbital models. Once again, calibration frequency is a poor predictor of model quality. There are useless models in every domain involving immutable constants. And there are very good numerical methods that have no constants per se.
Planes don't crash all at once every few years and resume flying only when the airports renegotiated the basic laws of physics. What you are referring to as "re-calibration" in planes and rockets is actually stochastic control and is a completely different topic to calibrating a stochastic model. But hey, I guess you like your metaphors like your risk models: incorrect.
Wilmott covered model robustness and calibration frequency. Any time you have to adjust an unobserved term in your SDE (like market price of risk, for example) you are changing the model. So if your model has to be changed every x minutes else it gives wrong results, you're not "adjusting your inputs", you are just using T/x shitty models.
Wilmott covered model robustness and calibration frequency. Any time you have to adjust an unobserved term in your SDE (like market price of risk, for example) you are changing the model. So if your model has to be changed every x minutes else it gives wrong results, you're not "adjusting your inputs", you are just using T/x shitty models.
Some sense.
Comparing financial markets to physics is pretty much what made it all explode last two times.
It is getting ready to explode again.
The asset market has been sedately rising for a decade. Which makes models like really good.
\sarcasm{ON} This time is different \sarcasm{OFF}
No it is not.
Comparing financial markets to physics is pretty much what made it all explode last two times.
It is getting ready to explode again.
The asset market has been sedately rising for a decade. Which makes models like really good.
\sarcasm{ON} This time is different \sarcasm{OFF}
No it is not.
One should clarify that basically nobody tries to "predict" anything with Black Scholes, just as basically nobody "predicts" share prices by doing a discounted cash flow analysis. Yet, they each provide the conceptual framework for trading (of options and stocks, respectively) in such a way that those that have better predictive power make money, on average, at the expense of those traders that don't. With that, they make the market efficient for traders that trade for other reasons (hedging, investing, gambling, etc.).
It's a market convention, with as much predictive power as knowing options used to be quoted in vol points.
I think the phrase "the de-facto standard for estimating the price of stock options" is just imprecise. It's the standard for generating the statistics like implied volatility, etc. But it's definitely not used to estimate the fair price of a new option, there are much newer models and methods to do that.
This. Back in 2006 when i was doing my PhD in CompSci + Options markets, the Binomial model was the state of the art. IIRC Black-Scholes was usef for historical references, and to understand the underlying variables given the simple assumptions "'closed world" it has. For example, the fact that it serves only for European options.
One should maybe distinguish the model (eg Black Scholes market (fixed vol), Dupire local vol, Heston stochastic vol, Merton jump diffusion, etc.) from the technique one uses to compute prices within the model (analytic closed form, PDE, tree (binomial or trinomial), Monte Carlo, other numeric methods).
All of the models (and techniques) I mentioned were well known and in use by 2000.
All of the models (and techniques) I mentioned were well known and in use by 2000.
I'm not in finance, but my impression from reading literature from those who are is that no one uses vanilla B–S for pricing options. One reason is the volatility smile: https://en.wikipedia.org/wiki/Volatility_smile.
Well, everybody uses standard B-S to quote option prices, just like everyone uses interest rates to quote bond prices.
But nobody prices options while assuming all the good old innocent assumptions underlying the original derivation of the formula. That, indeed, can be seen from the fact that different vols will be quoted for different strikes at the same expiry.
But nobody prices options while assuming all the good old innocent assumptions underlying the original derivation of the formula. That, indeed, can be seen from the fact that different vols will be quoted for different strikes at the same expiry.
The vol smile is mostly a byproduct of greater demand for far OTM options to hedge tail risk, alongside more sellers for ATM options which depresses the middle portion of the curve. I am not sure why this is a problem
It's an example of how real life pricing deviates from Black-Scholes. At the same time the pricing is correct in a sense that tail risks are greater than would be expected from a random walk.
The fact that the implied returns distribution is not normal is more or less "priced in". This is why you get volatility "smiles" and "skews". From the volatility surface (Volatility in respect to strike and time until settlement) you can easily calculate the propability density function for what the market assumes to be the future price. This is rarely if ever Gaussian, true, but it is not fundamentally wrong.
The financial returns are not normal.
It is not "priced in"
It goes up up up... crash.
Every generation has to learn this. I just hope we put the bankers in jail where they belong next time rather than bail them out.
It is not "priced in"
It goes up up up... crash.
Every generation has to learn this. I just hope we put the bankers in jail where they belong next time rather than bail them out.
That makes BS essentially a very expensive interpolation method, where you get to pretend to the auditors that you can hedge away your delta perfectly.
I don’t know why you’re getting downvoted. While what you said isn’t exactly correct, it’s pretty close.
One reason for black scholes today is that it is a decent interpolation function. It is significantly easier to create an implied volatility function to interpolate with than it is to create a price function to interpolate with directly. Another is that regardless of the smile, the real delta of an option is pretty damn close to black-scholes delta. So, you can maintain prices in real time as a function of the underlying price pretty accurately. A third (and this is important) is that trading systems have it built in as a way to interface with them. People know black-scholes and it isn’t proprietary. So you can do all sorts of research on the dynamics of a volatility smile, and it can be orthoganol to someone doing research on expected dividends or what the actual value of the underlying is. And you can bring all those pieces back together via the black scholes equation. A fourth reason tied into machine learning: implied volatilities behave just much better than raw interpolated prices when running them through predictive algorithms.
One reason for black scholes today is that it is a decent interpolation function. It is significantly easier to create an implied volatility function to interpolate with than it is to create a price function to interpolate with directly. Another is that regardless of the smile, the real delta of an option is pretty damn close to black-scholes delta. So, you can maintain prices in real time as a function of the underlying price pretty accurately. A third (and this is important) is that trading systems have it built in as a way to interface with them. People know black-scholes and it isn’t proprietary. So you can do all sorts of research on the dynamics of a volatility smile, and it can be orthoganol to someone doing research on expected dividends or what the actual value of the underlying is. And you can bring all those pieces back together via the black scholes equation. A fourth reason tied into machine learning: implied volatilities behave just much better than raw interpolated prices when running them through predictive algorithms.
I am being flippant about a topic where HN readership thinks that their cursory knowledge of it makes them experts.
I mean, I agree with you. But to me, the whole complexity is just moved to vol modeling. BS with its economic assumptions is just an empty shell now, so to speak.
I mean, I agree with you. But to me, the whole complexity is just moved to vol modeling. BS with its economic assumptions is just an empty shell now, so to speak.
You are flippant.
I am deeply cynical
I am deeply cynical
This only means that the real probability density function is parameterized by a sum of many Gaussian functions. Considering that the real implied returns are a skewed "gaussian-like thing" this is not the worst thing to do. Truly, using BS in this context is more or less historically motivated but I doubt there are far less "expensive" ways out there to find a suitable parameterization, what ever "expensive" means.
“Expensive” (at least for market makers) means you have to maintain a staff of quants whose job is to essentially create a curtain of rigor and hide the fact that traders usually rely on a bunch of simpler models to judge the broad dynamics and their gut for actual business decisions.
I remember that in a graduate class the professor told that among the important contributions of the theory was the BS formula. He never told us precisely what you wrote: P/L is in the tails. I wonder if he knew that LTCM went bust, while taking pride in being advised 'by two Nobel Prize recipients'.
Yes. The tails.
LTCM went bust because they thought they new better than the market and they were very very greedy. Very.
There is no formula for the market. The EMH in its weak form is correct. Has not been proved, but it is like P!=NP. True.
I am dismayed but unsurprised that financail models get so much support here.
You get money by working. Investments are savings. Just because there is some fool driving a Ferrari does not make that untrue, you cannot see the rest of the finance geeks flipping burgers.
Greed. Hubris. Bankruptcy.
LTCM went bust because they thought they new better than the market and they were very very greedy. Very.
There is no formula for the market. The EMH in its weak form is correct. Has not been proved, but it is like P!=NP. True.
I am dismayed but unsurprised that financail models get so much support here.
You get money by working. Investments are savings. Just because there is some fool driving a Ferrari does not make that untrue, you cannot see the rest of the finance geeks flipping burgers.
Greed. Hubris. Bankruptcy.
The article is dated and somewhat misleading,
> Since its introduction in 1973 and refinement in the 1970s and 80s, the model has become the de-facto standard for estimating the price of stock options
The only contemporary use for BS by professionals is as a convention for quoting volatility. As a pricing model it does not account for key effects such as the permanent "volatility smile" appearing in the aftermath of the 1987 crash (significantly increased price of downside options), and well understood behaviours like jumps and volatility clustering.
> Since its introduction in 1973 and refinement in the 1970s and 80s, the model has become the de-facto standard for estimating the price of stock options
The only contemporary use for BS by professionals is as a convention for quoting volatility. As a pricing model it does not account for key effects such as the permanent "volatility smile" appearing in the aftermath of the 1987 crash (significantly increased price of downside options), and well understood behaviours like jumps and volatility clustering.
It's still useful in options with very long maturities. Then the law of large numbers becomes important, and the vol smile flattens out over decades. These aren't listed, but are occasionally traded over-the-counter or embedded in some financial contracts, like executive stock options, insurance contracts, or convertible bonds.
I used to work on options MM desk. Even though BS is not correct in magnitude (e.g. our quants used some black magic to get deltas at the tails, which even then we're quite off from what CME was giving) it is directionally correct. Just having an intellectual grasp of what caused a shift in the price can be very useful. The problem IMHO is that way too much energy has been spent improving it by old school quants vs exploring other approaches ( e.g. a limes regression works quite well for daily fx option movements)
The Black-Scholes model does not account for the liquidity of the underlying assets (usually shares) and therefore the probable slippage
Oh how I hate these Medium posts that are not readable without doing something (registering/installing app/paying.. whatever)
I feel like Medium is the new expertsexchange. I remember how much I hated the site always when I ended there and I seem to have very similar feelings towards Medium.
I feel like Medium is the new expertsexchange. I remember how much I hated the site always when I ended there and I seem to have very similar feelings towards Medium.
Here's a full version: https://outline.com/328SDq
Block all cookies from Medium and you can read all you want. Or clear on browser exit.
Their fault for violating GDPR by trying to put cookies without my permission! Ha!
Their fault for violating GDPR by trying to put cookies without my permission! Ha!
Paywalling gender change information seems weird indeed.
Edit: docked for bad sense of humor (mine or of downvoters - of that I am not sure)
Edit: docked for bad sense of humor (mine or of downvoters - of that I am not sure)
Don't feel bad, at least I chuckled :)
Explain the joke plz
"...Medium is the new expertsexchange".
Experts Exchange used to be a popular site for q&a (the stack overflow of the olden days). Without the a proper hyphenation the site url expertsexchange could be construed to read something quite different which I believe the OP is referring to.
Ok
the joke is the basis for the SNL skit of 'celebrity jeopardy' portraying Sean Connery.
Thoughts:
- Since you have a right but not an obligation to buy/sell, that creates asymmetry. Since it's asymmetric, a wider range of outcomes, ie higher vol (imagine your gaussian curve on top of the hockey stick), makes the option worth more.
- Similarly having more time to expiry makes the option worth more, the range of outcomes is more spread out.
- There's a whole bunch of Greeks that the books will go through, but the intuition is the same for all of them. You can work out what's good or bad for you from thinking about how the distribution of outcomes is affected by a change in whatever.
- To trade the vol and not a mix of the vol and the direction, you flatten your delta by trading the underlying. If you do this at some point on the option price vs underlying price curve, you can get the graph to be flat, ie neutral to small price moves. But you can't make it flat everywhere with a hedge, because of course the graph is bendy.
- Near the strike where the bend is in the hockey stick is where it curves the most. On one extreme the option is worthless, on the other it's the same as having the underlying.
- As time passes it's got to get more curvey at the strike, less curvey on the sides.
- Curveyness on the price graph is called gamma. This is the gamma that ended up biting with the GME squeeze, by the sound of it. The problem is if you are short options, the graph looks like an upside down parabola, so if the underlying moves up a lot you will be short and getting shorter. If it moves down a lot, you'll be getting longer and longer. This is bad.
- It doesn't actually matter whether you are buying the right to buy or the right to sell. If you're buying, you have a positive gamma. But how? Well since owning a put and shorting a call of the same strike (or vice versa) should give you no curvature (looks like a straight line) they must have the same curvature. In the business people just call options with higher strikes than the current underlying price "calls" and options with lower strikes "puts" regardless of what they actually are. In-the-moneys just have some more premium attached to them, but act the same (in terms of everything other than delta) as their partner out-of-the-money option at that strike.
- Why do people get short gamma, knowing that movement is bad for them? Of course the option costs something to the guy who buys them. As long as the movement isn't too much it might be worthwhile to be short. In fact, most of the time the movement isn't enough to justify the price.