[HN Gopher] The Black-Scholes formula, explained (2019)
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The Black-Scholes formula, explained (2019)
Author : jorgenveisdal
Score : 67 points
Date : 2021-04-11 19:42 UTC (3 hours ago)
(HTM) web link (www.cantorsparadise.com)
(TXT) w3m dump (www.cantorsparadise.com)
| beervirus wrote:
| Paywalled, no thanks.
| mikkom wrote:
| 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.
| ZephyrBlu wrote:
| Here's a full version: https://outline.com/328SDq
| silentsea90 wrote:
| 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 wrote:
| Don't feel bad, at least I chuckled :)
| mam2 wrote:
| Explain the joke plz
| peterdemic wrote:
| "...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 wrote:
| Ok
| wutangson1 wrote:
| the joke is the basis for the SNL skit of 'celebrity
| jeopardy' portraying Sean Connery.
| dheera wrote:
| 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!
| lordnacho wrote:
| 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.
| JumpCrisscross wrote:
| (Thank you.)
| blake1 wrote:
| 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.
| worik wrote:
| "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
| cpp_frog wrote:
| 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'.
| spekcular wrote:
| 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.
| vardaro wrote:
| 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 wrote:
| 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.
| jcfrei wrote:
| 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.
| JumpCrisscross wrote:
| > _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.)
| jevgeni wrote:
| Having two sets of assumptions for buy and sell side is
| pretty indicative of the quality of the model.
| JumpCrisscross wrote:
| > _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 wrote:
| 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 wrote:
| > _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 wrote:
| We never recalibrate the g constant in our calculations
| nor we wait a person to announce what the g for this
| quarter will be.
| JumpCrisscross wrote:
| > _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_.
| cpp_frog wrote:
| >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 wrote:
| > _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
| PhantomGremlin wrote:
| _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.
| JumpCrisscross wrote:
| > _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...
| whatever1 wrote:
| 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 wrote:
| > _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.
| ZephyrBlu wrote:
| 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?
| JumpCrisscross wrote:
| > _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.
| ZephyrBlu wrote:
| 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?
| JumpCrisscross wrote:
| 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.)
| economusty wrote:
| How can an individual trader use the formula?
| sprash wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| "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 wrote:
| 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.
| lovedswain wrote:
| 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 wrote:
| 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.
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