10 ms·

The Black-Scholes formula, explained (2019)

93 points by jorgenveisdal 5y ago

worik 5y ago
"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
- JumpCrisscross 5y ago
  > 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.)
  - cpp_frog 5y ago
    >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.
    - JumpCrisscross 5y ago
      > 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
      - whatever1 5y ago
        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.
        JumpCrisscross 5y ago
        > 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.
        5y ago
  - jevgeni 5y ago
    Having two sets of assumptions for buy and sell side is pretty indicative of the quality of the model.
    - JumpCrisscross 5y ago
      > 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.
      - jevgeni 5y ago
        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.
        JumpCrisscross 5y ago
        > 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.
        whatever1 5y ago
        We never recalibrate the g constant in our calculations nor we wait a person to announce what the g for this quarter will be.
- economusty 5y ago
  How can an individual trader use the formula?
- sprash 5y ago
  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.
  - jevgeni 5y ago
    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.
    - sprash 5y ago
      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.
      - jevgeni 5y ago
        “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.
    - kolbe 5y ago
      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.
- spekcular 5y ago
  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 https://en.wikipedia.org/wiki/Volatility_smile.
  - vardaro 5y ago
    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
    - ivalm 5y ago
      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.
  - FabHK 5y ago
    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.
- jcfrei 5y ago
  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.
  - xtracto 5y ago
    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.
    - FabHK 5y ago
      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.
- cpp_frog 5y ago
  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'.
  - worik 5y ago
    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.
beervirus 5y ago
Paywalled, no thanks.
mikkom 5y ago
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.
- silentsea90 5y ago
  Paywalling gender change information seems weird indeed. Edit: docked for bad sense of humor (mine or of downvoters - of that I am not sure)
  - fegu 5y ago
    Don't feel bad, at least I chuckled :)
  - mam2 5y ago
    Explain the joke plz
    - peterdemic 5y ago
      "...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.
      - mam2 5y ago
        Ok
    - wutangson1 5y ago
      the joke is the basis for the SNL skit of 'celebrity jeopardy' portraying Sean Connery.
- ZephyrBlu 5y ago
  Here's a full version: https://outline.com/328SDq https://outline.com/328SDq
lordnacho 5y ago
Former options trader here. This all checks out correctly, but there's maybe some intuition that enlightens it. BTW option traders are often called volatility traders, because when you look at the formula there's this one free variable (all the rest are somehow given by the market). So when you're trading options, you're trading vol and the actual price is just a sort of formality. 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.
lovedswain 5y ago
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.
- blake1 5y ago
  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.
freebee56 5y ago
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)
vmception 5y ago
The Black-Scholes model does not account for the liquidity of the underlying assets (usually shares) and therefore the probable slippage

93 points by jorgenveisdal 5y ago

"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

JumpCrisscross 5y ago
> 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.)
- cpp_frog 5y ago
  >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.
  - JumpCrisscross 5y ago
    > 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
    - whatever1 5y ago
      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.
      - JumpCrisscross 5y ago
        > 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.
        5y ago
- jevgeni 5y ago
  Having two sets of assumptions for buy and sell side is pretty indicative of the quality of the model.
  - JumpCrisscross 5y ago
    > 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.
    - jevgeni 5y ago
      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.
      - JumpCrisscross 5y ago
        > 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.
        whatever1 5y ago
        We never recalibrate the g constant in our calculations nor we wait a person to announce what the g for this quarter will be.
economusty 5y ago
How can an individual trader use the formula?
sprash 5y ago
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.
- jevgeni 5y ago
  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.
  - sprash 5y ago
    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.
    - jevgeni 5y ago
      “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.
  - kolbe 5y ago
    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.
spekcular 5y ago
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 https://en.wikipedia.org/wiki/Volatility_smile.
- vardaro 5y ago
  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
  - ivalm 5y ago
    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.
- FabHK 5y ago
  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.
jcfrei 5y ago
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.
- xtracto 5y ago
  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.
  - FabHK 5y ago
    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.
cpp_frog 5y ago
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'.
- worik 5y ago
  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.

beervirus 5y ago

Paywalled, no thanks.

mikkom 5y ago

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.

silentsea90 5y ago
Paywalling gender change information seems weird indeed. Edit: docked for bad sense of humor (mine or of downvoters - of that I am not sure)
- fegu 5y ago
  Don't feel bad, at least I chuckled :)
- mam2 5y ago
  Explain the joke plz
  - peterdemic 5y ago
    "...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.
    - mam2 5y ago
      Ok
  - wutangson1 5y ago
    the joke is the basis for the SNL skit of 'celebrity jeopardy' portraying Sean Connery.
ZephyrBlu 5y ago
Here's a full version: https://outline.com/328SDq https://outline.com/328SDq

lordnacho 5y ago

Former options trader here. This all checks out correctly, but there's maybe some intuition that enlightens it. BTW option traders are often called volatility traders, because when you look at the formula there's this one free variable (all the rest are somehow given by the market). So when you're trading options, you're trading vol and the actual price is just a sort of formality. 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.

lovedswain 5y ago

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.

blake1 5y ago
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.

freebee56 5y ago

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)

vmception 5y ago

The Black-Scholes model does not account for the liquidity of the underlying assets (usually shares) and therefore the probable slippage