[HN Gopher] The Black-Scholes/Merton equation [video]
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The Black-Scholes/Merton equation [video]
Author : surprisetalk
Score : 77 points
Date : 2024-03-03 13:50 UTC (9 hours ago)
(HTM) web link (www.youtube.com)
(TXT) w3m dump (www.youtube.com)
| mhh__ wrote:
| Black-scholes is a hedging argument, the eqn isn't the essence of
| it
| JumpCrisscross wrote:
| Eh, put-call parity is the hedging argument [1]. Black-
| Scholes-(Merton) _was_ a breakthrough because it lets one
| understand why the hedge works, and thereby hedge and price
| more precisely.
|
| [1] https://en.m.wikipedia.org/wiki/Put-call_parity
| sixers2329 wrote:
| What I never fully understood is there's a free parameter in
| the equation (Implied Volatility)- which has no solid
| definition besides "the number that makes the rest of the
| equation work". At that point... how much value are you
| really getting from the rest of the equation?
| lordnacho wrote:
| Former option trader here. The free parameter is actually
| the "thing" that you're actually trading when you trade an
| option. All the other parameters are just environmental,
| you look them up.
|
| The short story is that the implied vol is a sort of
| balancing price between how much the option loses in value
| over time vs how much you can make performing the hedge.
| MuffinFlavored wrote:
| Why do any of the other variables matter if the market is
| collectively fighting between "overpriced and
| underpriced" on premium/implied volatility?
| kflansburg wrote:
| You can find an IV that makes sense for a single option
| with invalid other parameters, but things will break down
| when you go to price other expirations / strikes.
|
| When trading, you don't want to wait to see an "updated"
| IV, you would want to respond directly to changes in
| important and well understood parameters like underlying
| price.
| sheepscreek wrote:
| The Black-Scholes equation describes the unconscious
| biases that influence the price of options. It is able to
| beautifully separate the one quantity that every trader
| prices differently from the rest. That's volatility.
|
| All the other factors, time including, are the same for
| everyone.
| lordnacho wrote:
| They don't really matter. They are just things you look
| up in order to a get a number out for what the option
| costs in dollars.
| eggdaft wrote:
| Was BS actually of any practical use? It just seems to be
| a fantasy like most maths in economics.
| lordnacho wrote:
| Yes, it has some use. First of all, if you don't have a
| common model, it becomes impossible to talk about vol. So
| even if everyone uses their own model, they convert back
| into BS vol to talk about vol. Second, all models are
| wrong, but some models are useful. BS is a good starting
| point because it captures something that is relevant in
| option pricing, namely that uncertainty matters.
| scherlock wrote:
| Implied volatility is really the standard deviation of the
| price over time. You can calculate it by look at prices in
| the market. Then interpolate values. Where banks get funky
| is that the market for options go out about 3 years, but a
| banks will write options going out much much further. For
| those options, they are really just guessing, no matter how
| much fancy math they do, it's all to dress up a guess. And
| the traders don't care since they won't be around when the
| option expires
| kgwgk wrote:
| > Implied volatility is really the standard deviation of
| the price over time.
|
| Maybe you mean that "implied volatility is really the
| implied standard deviation of the price over time".
| kflansburg wrote:
| It's not really implied volatility, it's the actual
| volatility between now and expiration. Everyone can only
| estimate at what that will be.
|
| IV is essentially using prevailing prices to understand
| what everyone else has estimated that forward volatility to
| be.
|
| Beyond that, you will also find that IV differs across
| strikes [1]. Still, being able to fit a vol smile from
| incomplete market data (and some other adjustments if you
| are very sophisticated) and then price an arbitrary option
| is pretty useful.
|
| [1] https://en.wikipedia.org/wiki/Volatility_smile
| dist-epoch wrote:
| This equation is an idealized option, a spherical cow. If
| one would actually use it to price options one would lose
| money.
|
| There are many empirical option pricing features that this
| equation can't explain - the "smile", the "skew", ...
| drexlspivey wrote:
| The parameter (sigma) is the historical volatility (stdev
| of annualized returns). Implied volatility is what you get
| if you run the formula backwards and input the observed
| price to solve for volatility. In practice though many
| people use implied volatility as the input making the whole
| thing circular.
| RandomLensman wrote:
| It's no really circular, just think of IV as the
| price/what is traded.
| dxbydt wrote:
| I don't believe any of the comments below address the meat
| of your question - what value are you getting from the
| equation ?
|
| I would answer- not much.
|
| You can think of BS as a curried function. Since all the
| other params are fixed, you can curry and get a reduced
| equation that only depends on IV and underlying. If you do
| that, then its just - you give me iv and underlying, i give
| you spot. So, for a given strike(fixed), with the
| prevailing time left(fixed theta) under current interest
| rate(fixed), given the underlying, the historical vol gives
| you the wrong spot. You fudge it until you get the right
| spot. Call the fudged quantity the IV. Now plot that fudged
| quantity for a few other strikes and you get a smile. Then
| you can mess with that smile, plot the vol surface etc but
| end of the day, does the BS equation matter if the price of
| spot is going to be off and you have to fudge it with IV ?
| Its a good question. From an operational standpoint, the
| equation doesn't matter. You can use bopm and get a more
| intuitive price anyways. Traders can trade the iv without
| knowing what effect BS has on the system.
|
| When I was in 5th grade, they took us to the top of a tall
| building. We dropped a ball and measured the time it took
| to hit the ground. So if you square that time and multiply
| by 5, that's how tall that building is. At that age I
| thought wow this is such magic! Then I grew up and reached
| 8th grade and worked out equations of motion with some
| basic differential calc, and derived the canonical equation
| s equals ut plus half at square. So since u is zero and a
| on planet earth happens to be g which is 9.8, half of which
| is about 5, that's why 5t^2.
|
| ok but does this equation matter ? I could have gone my
| whole life measuring height of buildings without knowing
| what is gravity.
| Fade_Dance wrote:
| Wasn't the main breakthrough it's utility and accessibility,
| not precision?
|
| It's still quite generalized in that it assumes a flat
| volatility surface, which even traders in the pits
| intuitively knew was wrong (thus the emergent volatility
| smile after '87). What it did allow was for a single number
| (implied volatility) to function as the single knob to be
| dialed to move quotes up and down for convex instruments.
| Therefore, instead of calling a trade desk and working out
| direct price quotes, you could have an immediate frame of
| reference ("this is trading at 31 vol") and move the offer
| to, say, "30 vol", leaving the calculation to the computer
| because both parties shared a language.
|
| As for the vol smile and lack of volatility surface
| uniformity in real markets, it wasn't an issue, because pit
| traders were fine pricing different strikes at different
| vols, and players deeper in the volatility space had their
| own more accurate models geared for each market/instrument.
|
| Knowing an underlying's iVol gives a good general overview of
| the pricing landscape with just a single number, and then if
| you need more precision, you can pull up the list of strikes
| and ivols for each strike and see the shape of the vol
| surface. That just takes a few more seconds. It's very quick
| and very useful, from the pit trader crews with the proto-
| handheld computer to the sell side and buy side deals working
| the phones. Utility!
|
| To expand a bit, it is also a great feature that the second
| level of granularity (breaking away from the theoretical flat
| vol surface by applying different iVol values to different
| strikes) isn't crammed into another overarching generalized
| model. It breaks the model and lets traders go, say after the
| '87 crash, "tail risk is trading much higher what it has been
| historically, and this stuff is staying permanently bid,
| looks like a regime shift. We don't have a generalized model
| for this yet but in the meantime, traders in the pit are
| working with this new pricing, we can all see it and speak
| the same language, and we'll work out the new generalized
| models at a later date." That's why these simple options
| pricing models are still useful today, even though there are
| far more known kinks in volatility surfaces than there were
| decades ago.
| mhh__ wrote:
| Taleb and Derman have argued that put call parity implies BS
| but other disagree quite strongly.
| lupire wrote:
| Under reasonable assumptions, put call parity is true so
| it's not an important statement to say that it implies BS.
| mhh__ wrote:
| You still need additional assumptions about the delta
| hedged PnL, no?
| RandomLensman wrote:
| Id say the underlying hedging argument is continuous delta
| hedging.
| btdmaster wrote:
| How does this square with "past market returns are do not
| (entirely) determine future market returns"? Surely the same
| applies to the historical stddev?
| fancyfredbot wrote:
| In practice the standard deviation used is implied from option
| prices, making the whole thing completely circular. To match
| the market you need to use different standard deviations
| (volatilities) for different options!
| dr1ver wrote:
| Volatility is a bit more predictable than price. And there are
| more complex formulae that also model volatility rather than
| treat it as a constant.
| lupire wrote:
| If volatility is predictable, it would quickly be traded
| until it became unpredictable and unprofitable.
| throwawayFinX wrote:
| dr1ver is correct here and you are not.
|
| Actual volatility (not implied!) is much easier to predict
| than price.
|
| It's also much more difficult to trade than price changes.
| So your intuition about this is correct though.
|
| It is not super difficult to predict tomorrows volatility
| sign (up/down compared to today) with +60% success. Even
| textbook GARCH models do well here.
|
| If you could do that with the price, you'd quickly become
| filthy rich.
| richrichie wrote:
| It is a well known empirical fact that volatility is mean
| reverting.
| jwsteigerwalt wrote:
| It's about using the best information you have to quantify and
| systematize risk.
| mhh__ wrote:
| This is why options traders (prototypical ones at least, I
| suspect most trading is still directional) trade on the implied
| volatility, as a projection of future volatility.
|
| You take a view on volatility by buying or selling an option,
| if you are right then you will make money proportional to the
| options gamma (i.e. the convexity of the option is where the
| money comes from)
| RandomLensman wrote:
| Need to separate two different situations here:
|
| 1) where there are pretty complete markets for implied
| volatility, looking at the past matters less to little, because
| there is a market for the "future volatility" you can hedge and
| interact with
|
| 2) when there isn't a good volatility market and hedging future
| volatility exposure is difficult, looking towards the past for
| some guidance increases in importance
|
| Both things can get complicated at times and in both cases it
| isn't strictly speaking the stddev you care about, but the
| quadratic variation (which can be the same under some
| assumptions).
| vikramkr wrote:
| Yep. You can make money off of using options as a way of
| betting on what the volatility measure itself will be. If you
| think the historical standard deviation is lower than what it
| will be because of some new change to the company or the world
| environment, and your view is different from the market's view.
| It's why sometimes very out of the money call options will
| paradoxically go up in price after really bad news - you're so
| far away from the price of the share that the increase in
| volatility from the price drop increases the option's value
| even though it's moved even more out of the money
| exclipy wrote:
| Is that a bug in the equation that one could take advantage
| of?
| ironSkillet wrote:
| In a way, yes. A lot of money follows these standard
| formulas for pricing which do not necessarily reflect
| accurate probabilities of the underlier price movement.
| After an idiosyncratic price shock (disappointing earnings,
| geopolitical news etc), people blindly following a trailing
| 1 month volatility or something will misprice the option as
| volatility reverts back to the mean. This probably has been
| arbed away to a large extent by trading algorithms.
| cubefox wrote:
| This got me thinking: Can the backpropagation algorithm be
| expressed as an equation?
|
| I assume this should always be possible using functional
| programming.
| eternauta3k wrote:
| Are you talking about backpropagation in neural networks? Do
| you mean as a differential equation? It already is a normal
| algebraic equation.
| cubefox wrote:
| Yes in neural networks. Backpropagation is probably worth
| more than the equation in the video. But it's an algorithm,
| which aren't usually expressed as equations.
| mhh__ wrote:
| Something the intro of this video missed a little bit is that
| most derivatives are linear -- no optionality.
|
| Options are a relatively tiny market, but are nonetheles key to
| reasoning about markets.
| curiousgal wrote:
| Pure Black Scholes is not really that useful nowadays because of
| its key limitations, some can be easily fixed (dividends, no risk
| free rate, etc.) other cannot (constant volatility) which makes
| it only useful in areas like vol targeting where you want the
| volatility to be constant.
| rvcdbn wrote:
| I've been trading for 4 years on the side mostly on intuition. I
| like to think one can emotionally program the powerful computer
| that is the unconscious mind to serve the distributed monster of
| global finance for profit. Sometimes I even do tarot. Seems to
| kinda work LoL what could possibly go wrong????
| ripjaygn wrote:
| Generated summary for those who can't or don't want to watch a
| long video(though I recommend Veritasium's story telling):
|
| This video talks about the impact of a mathematical equation
| derived by Louis Bachelier on the financial world. The equation
| is used to price financial instruments called options, and it has
| led to the creation of a multi-trillion dollar derivatives
| market.
|
| The video starts by mentioning how Jim Simons, a mathematician
| who founded Renaissance Technologies, used a different approach
| to make significant profits in the stock market. Then it goes on
| to explain what options are and how they are used. Options are
| contracts that give the buyer the right, but not the obligation,
| to buy or sell an underlying asset at a certain price by a
| certain time.
|
| The video then discusses the work of Louis Bachelier, who in
| 1900, derived a formula to price options. This formula, known as
| the Black-Scholes-Merton formula, revolutionized the financial
| industry by providing a way to accurately price options. The
| creation of this formula led to the development of the exchange-
| traded options market, which has grown to be a multi-trillion
| dollar industry.
|
| The video also talks about the impact of options on the financial
| system. Options can be used to hedge other investments, which can
| help to reduce risk. However, they can also be used to make
| leveraged bets, which can lead to significant losses.
|
| The video concludes by discussing the efficient market
| hypothesis, which states that all available information is
| already reflected in stock prices. The video suggests that the
| success of Renaissance Technologies, and other similar firms,
| challenges the efficient market hypothesis.
| JackFr wrote:
| There is some valuable intuition in that the value of an option
| can be broken into three components, the intrinsic value - that
| is the difference between the asset price and strike price on the
| option, the time value which is dependent on the time to expiry
| and the risk free rate, and the "insurance" value which is
| dependent on the volatility and the time to expiry.
|
| In the swaptions market, for instance quotes are typically for an
| "at-the-money forward" rate, which makes the first two components
| zero and all the value is tied to the vol.
| imtringued wrote:
| Fisher black has written some interesting books. For anyone who
| is interested, check out his Wikipedia page.
| NovemberWhiskey wrote:
| Just FYI: Fischer ... not Fisher.
| SirLJ wrote:
| If someone is interested in all this, I would strongly recommend
| looking into Ed Thorp, he discovered pretty much the same thing
| earlier, but instead of publishing, he made money with the
| knowledge...
|
| Great book about all this, 2017 Autobiography: "A Man for All
| Markets: From Las Vegas to Wall Street, How I Beat the Dealer and
| the Market"
| AlexCoventry wrote:
| The video refers to him.
| RichardCA wrote:
| There was a PBS NOVA episode about this.
|
| https://vimeo.com/302855460
|
| The question that seems obvious, but no one ever seems to talk
| about, is how the failure of LTCM paved the way for the subprime
| crisis of 2008.
|
| I mean, did the elite financial world fail to learn the lesson,
| or did it simply learn the wrong one?
| RandomLensman wrote:
| How would you connect the two events?
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