[HN Gopher] An Intuitive Explanation of Black-Scholes
___________________________________________________________________
An Intuitive Explanation of Black-Scholes
Author : alexmolas
Score : 204 points
Date : 2024-10-03 13:28 UTC (3 days ago)
(HTM) web link (gregorygundersen.com)
(TXT) w3m dump (gregorygundersen.com)
| ComplexSystems wrote:
| If you found a stock price that actually follows the geometric
| Brownian motion pattern this model is built on, wouldn't that
| basically just print you an infinite amount of money? The
| expected value of the price movement one time-unit later would be
| positive.
| pfdietz wrote:
| I think these models are self-defeating, since they stop
| working when enough people try to exploit them.
| dragontamer wrote:
| The opposite.
|
| We have huge numbers of people 'rocking the boat' trying to
| create say.... a Gamma Squeeze.
|
| The only reason everyone trusts a Gamma Squeeze can happen is
| because they trust the math in Black Scholes. The may not
| even understand the math, just trust that the YouTuber who
| told them about Gamma Squeezes had enough of an understanding
|
| ------------
|
| Today's problem IMO, is now a bunch of malicious players who
| are willing to waste their money are trying to make
| 'interesting' things happen in the market, almost out of
| shear boredom. Rather than necessarily trying to find the
| right prices of various things.
|
| Knowing that other groups follow say, Black Scholes, is taken
| as an opportunity to mess with market makers.
| rubyn00bie wrote:
| I used to think charting was bullshit for day and swing
| trading. Because on paper it sure seems to be, but in reality
| so many other players are also charting that it becomes
| useful and somewhat predictive. Largely because you're all
| using the same signals. Sure it's impossible difficult to
| time things perfectly, but perfect is the enemy of profit.
| You don't need to catch the absolute bottom and you don't
| need to catch the absolute top.
|
| Specific to Black-Sholes the best option plays, when going
| long, are the ones which have incorrect assumptions about the
| volatility of the underlying. You can have far outta the
| money options, absolutely print, with a sufficient spike in
| the underlying. Even if the strike price will never be met
| (though you'll also give that back if you ride them to
| expiration or let things settle down).
| pram wrote:
| The competitive advantage is lessened because everyone knows it
| already. It's "priced in" as they say
| yold__ wrote:
| No, this doesn't imply an "infinite amount of money", it's just
| a pricing model. You still need the parameters of the
| distribution (brownian motion / random walk), and these are
| unobservable. You can try to estimate them, but there is a lot
| of practical problems in doing so, primarily that volatility /
| variance isn't constant.
| bjornsing wrote:
| Yes. That's basically how the stock market works. If you buy
| and hold an S&P 500 index fund you can expect to make an
| infinite amount of money, in an infinite amount of time. But
| few have the patience for that.
| tehjoker wrote:
| We'll hit the limit in a few decades or at most a couple
| centuries due to ecological limits on growth though (unless a
| robust space economy develops).
| twoodfin wrote:
| Sorry, which limits? How do those apply to the increasing
| economic value of turning the same amount of sand into
| faster and faster GPUs, for example?
| sokoloff wrote:
| There are still finite people willing to buy whatever the
| intermediate or end product of that fancy sand is. And
| finite energy and space. And only 5 billion years until
| the sun goes red giant.
|
| The limits may be very large, but they aren't infinite.
| twoodfin wrote:
| Yeah but a couple centuries? What's the evidence for
| exhaustion of demand and supply for economic goods on
| that kind of time horizon.
| lazide wrote:
| What you're describing is just a variant of Malthusianism
| [https://www.intelligenteconomist.com/malthusian-theory/],
| which may not be wrong - but has not proven right either
| (in modern times) with advances in technology.
|
| Especially improvements in energy generation, fertilizer
| production, and efficient usage of both (often through
| information technology).
|
| Given any stable state of technology/energy/space, a
| society will generally reach a high point, then go through
| cycles of growth/retraction.
|
| But improvements in technology and energy generation means
| it won't be at a stable state, eh?
| immibis wrote:
| Or just poor people hitting the zero bound.
| melenaboija wrote:
| This is a pricing model, i.e. what is the value according to
| the assumptions the model does (which btw are known to be weak
| for BS) but as anything else the price is what you are going to
| pay in the market for whatever other reasons.
|
| Imagine you have a model that establishes the price of used
| cars, it can be really really good but if you go to the market
| to buy one you will pay whatever is been asked for not what
| your model says.
|
| EDIT: Although pricing models do not have direct affectation to
| market prices they do in an indirect manner. To manage risk are
| needed pricing models which somehow condition market
| participants and therefore prices indirectly. In the simile
| with cars, you can buy as many cars as you want at the price
| you want, but what you do when you have them and if you want to
| take wise decisions with them you have to know something about
| their value.
| listenallyall wrote:
| Yes, but also no. Because you don't have to buy a mispriced
| asset (mispriced against you) and also, in many cases, you
| can construct what you want from pieces of other assets.
|
| One car dealer trying to sell a 2023 Honda Accord with 60,000
| miles can't just decide, independently, to forget the high
| mileage and price the car based solely on it being 1 year
| old. Sure that's "whatever is being asked" but that car will
| never sell until he brings the price down in line with other
| 60k mile cars - and that is because the pricing models are
| essentially agreed upon by all market participants.
| melenaboija wrote:
| Yes, but also no. The value of things is only what the
| market wants to pay for it, and it does not matter if it is
| a 2023 Honda Accord or a financial product, currency... In
| one you might trust the engine reliability and on the other
| on the government behind the currency, whoever is writing
| the option, issuing the bond, ... But still, it is a matter
| of faith and bid/ask.
| stackghost wrote:
| Indeed, hence the meme "stocks only go up". There's a grain of
| truth to the meme, though. The safest bet I can think of to
| make is that, on average, the S&P 500 will be higher in the
| future than today. Obviously there are temporary down trends
| but on a time horizon of years to decades I can't think of a
| safer bet.
| pigeons wrote:
| However the company stocks included in the S&P 500 aren't the
| same.
| smabie wrote:
| Safer bet would be to hold short term treasures.
| stackghost wrote:
| I'd argue that's not a bet.
| sokoloff wrote:
| It's a bet on the continued existence, and
| willingness/ability to honor its obligations, of the US
| federal government.
|
| If that bets goes bad, the typical investor in Treasuries
| has perhaps bigger problems to worry about, but it's
| still a bet IMO (and one which will inevitably eventually
| go bad).
| gyudin wrote:
| Considering they only choose top performers and inflation
| sounds like a safe guess :D
| nicolapede wrote:
| No. Just look at equations 6 and 7 in the link. The expected
| value of the move can be either positive or negative depending
| on the model parameters.
| tel wrote:
| Generally these parameters are unknown and the drift parameter
| is often quite a bit smaller than the volatility. As a
| consequence, you cannot be sure your investment is secure and
| its value is likely to wobble significantly in the short term
| even if it ultimately produces value in the long term.
|
| If you actually knew that the drift on a certain investment was
| positive, you still have to be prepared to survive the losses
| you might accumulate on the way to profit. The greater the
| volatility the more painful this process can be. If you can
| just sock away your investment and not look at it for a long
| time it will become more valuable. On a day-to-day time scale,
| as an actual human watching this risky bet you've made wobble
| back and forth, it can require a lot of fortitude to remain
| invested even as the value dips significantly.
| eclark wrote:
| How does this hold on assets that trend today wards the whole
| market if we assume that governments will not let markets
| crash too long before printing money?
|
| What I mean is that if we can assume that the wiggle for VTI
| or SPY on the long term is positive because of outside
| factors, does that make options on those larger market assets
| become a game of who has a large enough reserve
| smabie wrote:
| Why? the mean can be negative?
| bibouthegreat wrote:
| 2 parts:
|
| 1. Interest rates can be negative 2. Volatility reduces the
| average. Take an example of +10% then -10% (1+0.1)*(1-0.1) =
| 1 - 0.12 = 0.99 < 1. It's due to the "log normal returns"
| crystal_revenge wrote:
| No. In fact, the fundamental principle of all quantitative
| finance is that your results in the ideal scenario are
| _arbitrage-free_ meaning that nobody stands to make any money
| off any transaction. That 's how you determine the ideal price
| given the ideal asset.
|
| edit: To address your specific observation, that the price of
| the stock is expected to go up, it's assumed that if the stock
| goes up, so do all other assets. In mathematical finance you
| never keep you money as cash, so if you sell the stock you put
| that money in an account that expected to grow at the "risk-
| free" rate. The major difference between the "risk-free"
| account and the stock is the variance of these asset prices.
|
| However, in your scenario, you wouldn't need Black-scholes for
| the price of the stock itself since that should be
| theoretically equal to it's _expected_ (in the mathematical
| sense of "expectation") future value assuming the risk-free
| rate.
|
| Black-Scholes is used to price the _variance_ of the underlying
| asset over time for the use of pricing derivatives. But again,
| if the stock moved exactly as modeled then the model would give
| you the perfect price such that neither the buyer nor the
| seller of the derivative was at a disadvantage.
|
| The way you would make use of such a perfectly priced stock
| would be to search for cases where either buyers or sellers had
| _mispriced_ the derivative and then take the opposite end of
| the mispriced position.
|
| However you don't need a perfect ideal stock to make use of
| Black-Scholes (this is a common misconception). Black-Scholes
| can also be used to price the _implied volatility_ of a given
| derivative. Again, derivatives fundamentally derive their
| values from the _volatility_ /variance of an asset, not it's
| expectation. By using Black-Scholes you can assess what the
| market beliefs are regarding the future volatility. Based on
| this, and presumably your own models, you can determine whether
| you believe the market has mispriced the future volatility and
| purchase accordingly.
|
| One final misconception of Black-Scholes is that it's always
| incorrect because stock price volatility is "fat-tailed" and
| has more variance than assumed under Black-Scholes. This _was_
| the case in the mid-80s and people did exploit this to make
| money, but today this is well understood. The "fat-tailed"
| nature of assets prices is modeled in the "Volatility smile"
| where the implied volatility is different at different prices
| points (which would _not_ be expected under pure geometric
| Brownian motion), but this volatility can still be determined
| using Black-Scholes for any given derivative.
|
| tl;dr Buying stocks is about your estimate of the expected
| future value of a stock, but Black-Scholes is used to price
| _derivatives_ of a stock where you actually care about the
| expected future variance of a stock. Even in an unideal world
| you can still use Black-Scholes to quantify what the market
| believes about future behavior and buy /sell where you think
| you have an advantage.
| FabHK wrote:
| Well, yes. If you buy a stock with positive drift and hold it,
| the model predicts "infinite growth" (in the sense that for any
| number N you give me I can give you a time t at which the
| E[S(t)] > N).
|
| But it might take quite some time, and it's still random, it
| might be much smaller or much bigger.
|
| You could be tempted to employ leverage. However, that
| introduces the chance of being wiped out.
|
| ETA: Real rates are normally positive. So you can achieve the
| same result by investing in long term bonds with less risk.
| Just have to wait even longer.
| zyklu5 wrote:
| This guy's other notes are also well thought through and written.
| Thanks for the link.
| yieldcrv wrote:
| the creators of Black-Scholes destroyed their options selling
| fund based on their flawed belief that everyone else had
| mispriced options, or the black swan possibility should have been
| part of the formula
|
| also Black-Scholes doesnt factor in the liquidity of the
| underlying asset, in modern times I think this is relevant in
| determining the utility of an options contract
|
| there are other options pricing formulas
| smabie wrote:
| LTCM wasn't really an options selling fund though selling
| equity options did become a big trade for them
|
| Also they were more of advisors in the fund then anything else
| yieldcrv wrote:
| You're judged by the company you keep
| mhh__ wrote:
| If you mean LTCM then the story is far more dull (i.e. too much
| leverage, fund goes boom)
|
| Ed Thorpe _did_ originally want to setup an options fund (he
| was the first to trade the model) that he later estimated would
| 've blown up due to various market conditions at the time IIRC
| javitury wrote:
| Great article and very intuitive explanation.
|
| I also wanted to point out a (minor) typo. On equation 3, dZt is
| multiplied by sigma squared, but it should be multiplied just by
| sigma instead.
| gwgundersen wrote:
| Thanks! I'll fix this.
| ncclporterror wrote:
| In modern finance the Black-Scholes formula is not used to
| "price" options in any meaningful sense. The price of options is
| given by supply and demand. Black-Scholes is used in the opposite
| way: traders deduce the implied volatility from the observed
| option prices. This volatility is a representation of the risk-
| neutral probability distribution that the markets puts on the
| underlying returns. From that distribution we can price other
| financial products for which prices are not directly observable.
| mikeyouse wrote:
| It's still used as an input into illiquid 409a valuations.
| nknealk wrote:
| It's also frequently used to price stock options given to
| employees at publicly traded companies.
| dumah wrote:
| Black-Scholes assumes constant volatility and cannot
| compute option prices without a volatility input.
|
| This volatility is backed out of nearby options prices,
| often using the formula for European options.
|
| There isn't any purely theoretical option price because an
| assumption depends on observed prices.
| klysm wrote:
| Kinda, but it's not great because of the volatility smile
| wavemode wrote:
| Sure, but isn't most of supply and demand in the market driven
| by large investors who use such formulas to derive the fair
| price of the option?
|
| That is, if the real price ever differred significantly from
| what Black-Scholes predicts, wouldn't algorithmic trading very
| quickly correct this deviation?
| onerandompotato wrote:
| If there was a way to directly formulate every parameter of
| the black Scholes formula you would be correct. The problem
| that you run into is how to calculate volatility itself?
| Without the volatility value, your algorithm cannot trade on
| it.
|
| Using history of volatility is insufficient, because
| volatility is a forward looking measure. Just because the
| stock was volatile in the past does not mean it will be in
| the future, and vice versa. There are even more nuances with
| this, as volatility is a smile (or a surface), not a singular
| number https://en.wikipedia.org/wiki/Volatility_smile.
|
| TLDR Trading in volatility is a very complicated topic.
| However, volatility is a useful parameter, and black Scholes
| is typically used to deduce the forward looking volatility
| from option price, in addition to volatility -> option price.
| thaumasiotes wrote:
| > There are even more nuances with this, as volatility is a
| smile (or a surface), not a singular number
| https://en.wikipedia.org/wiki/Volatility_smile.
|
| That article says that _implied volatility_ is
| inconsistent, with options at strike prices that are very
| far from the current market price having costs that imply a
| different level of volatility than options at strike prices
| that are close to the current price. The cute question here
| is "should an option be priced according to the actual
| level of volatility in the price of the underlying asset,
| or should it be priced according to the level of volatility
| that exercising the option would require?"
|
| Volatility is just a quantity.
| FabHK wrote:
| In a sense, BS and the option market enable trading in
| volatility itself.
|
| Specifically, you trade in the estimate of the stock's
| volatility over the time from now to expiry of the option.
|
| If you don't want to trade options directly to do that (it is
| cumbersome, as it involves "continuous" delta hedging), you
| can trade in VIX futures for the same purpose. Or variance
| swaps.
| ndesaulniers wrote:
| Also, isn't it only used for European style options, not
| American?
| FabHK wrote:
| European and American calls cost the same on non-dividend
| paying stocks (on dividend paying stocks, it might make sense
| to exercise an American just before the ex-date).
|
| Either way, as was pointed out, in reality BS is used as a
| deterministic one-to-one mapping between option prices and BS
| vols. Then, from market quotes (either as prices or as BS
| vols) a vol-surface is fitted (as a function of strike and
| expiry time), from which a stochastic process is fitted that
| correctly re-prices all these points (using a model such as
| "local vol" or "stochastic vol" or a combination of those
| two, or others), and then everything is priced of that.
| dahfizz wrote:
| > European and American calls cost the same on non-dividend
| paying stocks
|
| All else being equal, I would prefer to buy an option
| contract I can exercise at any time vs one I can only
| exercise on a certain date. It doesn't make intuitive sense
| they would be priced the same, can you please elaborate?
| Renevith wrote:
| The parent is assuming that you can always sell your
| option to someone else for its fair value. If that's the
| case, there would never be a time where it's optimal to
| exercise a call option, because the optionality will
| always make the option value higher that the value of
| owning the stock.
|
| This is shown in the article: the curved lines
| representing the option value are always above the
| straight lines of the final option payoff (the value if
| exercised).
|
| This is not necessarily true for put options or for call
| options if the stock pays dividends. In those cases the
| option value can be below the payoff line and early
| exercise would be better than selling the option.
| sokoloff wrote:
| American style options are inherently more valuable.
| Imagine you had options on a stock that experienced a sharp
| but possibly temporary move. As a holder of an American
| style option, you could benefit from that temporary move,
| making it more valuable.
| im3w1l wrote:
| The way the market is typically modeled, temporary moves
| are not a thing.
| sokoloff wrote:
| The way the market _actually exists_ , temporary moves
| are definitely a thing.
| mhh__ wrote:
| The real purpose of models is risk anyway e.g. implied vol is
| handy, delta is essential.
| IAmGraydon wrote:
| >In modern finance the Black-Scholes formula is not used to
| "price" options in any meaningful sense. The price of options
| is given by supply and demand.
|
| I'm not sure what your point is. Yes, actual market prices are
| determined by...the market. The Black-Scholes formula is widely
| used in modern finance to MODEL the price of an option given
| different sets of inputs in theoretical situations.
| loveparade wrote:
| And it's a cycle. Supply and demand are partially driven by
| pricing models used by hedge funds, and variants of Black
| Scholes is one of those.
| ncclporterror wrote:
| The way the article is written, it appears that the formula
| is used as: 1. Observe market parameters (volatility of the
| underlying and risk free rate) 2. Plug into formula 3. Deduce
| a price for the option.
|
| My point is that it is used in the opposite way: observe
| prices to deduce market parameters. You claim my point is
| obvious, but I'm not sure it would be obvious to a reader
| unfamiliar with modern finance reading this article, which is
| the target audience.
| blitzar wrote:
| > 1. Observe market parameters (volatility of the
| underlying and risk free rate) 2. Plug into formula 3.
| Deduce a price for the option.
|
| In the FX market (interbank), the quoted and "traded"
| number is Implied Vol - the price of the option then
| follows from there (via the Black-Scholes model).
| e-master wrote:
| I've seen it used for OTC option pricing - there's no liquid
| market, so you are more of a market maker than a market taker.
| RayVR wrote:
| Black Scholes is not used for any otc option pricing, except
| perhaps to provide an instantaneous estimate to get in the
| ballpark, but no one would use it for the final price.
| RandomLensman wrote:
| For valuing financial products with no directly observable
| price, BS or its descendants matter quite a lot. For actually
| pricing a transaction on those, it becomes more complex but
| model value is typically an important input.
| mcdeltat wrote:
| I have seen the insides of an options market maker, and can say
| this is not really true (at least for some regions of the
| market). Black-Scholes is used to derive theoretical prices for
| options. Good option traders will have an opinion on volatility
| and won't just take whatever the market says.
|
| _However_ , one of the interesting aspects of serious option
| trading is that Black-Scholes is merely your bread and butter.
| There is a lot of information that goes into option pricing,
| including supply/demand signals. The mix of signals also
| depends on the time scale on which you are trading.
|
| What rings true to me with this comment is the correlation
| between products. Option traders are often concerned with many
| relationships between product pricing: between underlying and
| option, across expiries, across strikes, between products in
| indices, between products in sectors, etc .
| MuffinFlavored wrote:
| > traders deduce the implied volatility from the observed
| option prices.
|
| I've only ever seen one thing:
|
| Black-Scholes models say IV should be less but your
| broker/brokerage/the market are overpaying for it.
|
| I always figured it was closer to a Vegas juice/vig.
|
| I never understood the benefit really.
|
| Complicated math to tell you lots of people want to play
| roulette on NVDA earnings and whatever you are going to pay for
| it is going to be "overpriced/overvalued" in at least one way.
|
| I've never seen the opposite where it helps you find an edge
| and something was undervalued.
| keithalewis wrote:
| Here is a replacement for the Black-Scholes/Merton model:
| https://keithalewis.github.io/math/um1.html#black-scholesmer...
| erehweb wrote:
| You can also use nonstandard analysis to derive Black-Scholes,
| replacing stochastic calculus by a random walk with infinitesimal
| steps. https://ieeexplore.ieee.org/document/261595 (don't see an
| ungated version)
| jesuslop wrote:
| The "Loeb Measures in Practice" book also by Cutland has a
| survey chapter.
| jesuslop wrote:
| I jotted a time ago a Sage snippet for options pricing in
| elementary calculus terms, pasted here
| https://pastebin.com/tTMp6fPk.
|
| The idea is that the clean picture is done in terms of log-prices
| (not prices). Probability of log-prices follows a diffusion with
| an initial Dirac delta at-the-money. At expiration the profit
| function is deterministic (0 out of the money, a ramp if in the
| money) and the probability is certain gaussian. The expectancy of
| the value of a function applied to a random var of given density
| is like a weighted sum of the values, weighted by the
| frequency/density, as in a dot product (an integral here). Add to
| that the "time value of money" (see Investopedia) that works as
| linear drift, and you are done.
| charlie0 wrote:
| Brownian motion is what happens when people lose their life
| savings on meme stocks.
| bryan0 wrote:
| Another good explanation from Terence Tao's blog:
| https://terrytao.wordpress.com/2008/07/01/the-black-scholes-...
| bee_rider wrote:
| Of course, Black-Scholes is a very famous and important
| mathematical model. However, it is Saturday night, so let's be a
| little silly.
|
| I've always thought that one reason it became so well known is
| that it sounds kind of badass. A shoal is, of course, a shallow
| bit of water, general associated with running aground and that
| sort of thing. Black-Shoals sounds like an area where Blackbeard
| the pirate will hang out steal all your stuff if you get stuck.
| I've always thought quants secretly want to be pirates, but of
| course the era of going around pirating is over, so they learned
| how to do it on the market instead.
|
| In the time of piracy, they could probably have been navigators,
| that job was pretty mathy. The would have presumably gone around
| the Black-Shoals.
| putcher_willow wrote:
| You're not alone in finding the name poetic: see
| https://www.blackshoals.net/
|
| "Black Shoals Stock Market Planetarium is an art project
| created by Joshua Portway and Lise Autogena.
|
| The project takes the form of a darkened room with a domed
| ceiling upon which a computer display is projected, like a
| planetarium. Audiences are immersed in a world of real-time
| stock market activity, represented as the night sky, full of
| stars that glow as trading takes place on particular stocks.
|
| In Black Shoals each traded company is represented by a star,
| flickering and glowing as shares are traded. The stars slowly
| drift in response to the complex currents of the market, while
| outlining shapes of different industries and the huge
| multinational conglomerates like the signs of the zodiac. The
| movement of the stocks is based on calculated correlations
| between the histories of each stock and those of its near
| neighbours. The stronger the correlation between the histories
| of the stock prices of any two companies, the more powerful the
| gravitational attraction between them. Although they start out
| randomly distributed in the planetarium, over time the stars
| clot together and drift into slowly changing constellations,
| nebulae and clusters. Through this technique different
| industries naturally start to emerge as galaxies. Any general
| disturbance in a section of the market will have a visible
| effect on the sky - the collapse of Enron, for instance, would
| have caused a sort of black hole - all the companies affected
| would glow very brightly due to the level of trading and would
| be pulled in to a single point in a very powerful vortex."
|
| It goes on...
| marxisttemp wrote:
| Like all economics, this uses massive oversimplifications that
| never apply in the real world to imply some incontrovertible
| nature to free markets that simply does not exist. Spherical cows
| indeed.
|
| There was an article posted here recently about "mathy" equations
| that this reminds me of.
|
| Anyways read Das Kapital if you want to actually understand
| economies.
| LudwigNagasena wrote:
| > this uses massive oversimplifications that never apply in the
| real world
|
| If you've read Das Capital, you have noticed it also uses
| massive oversimplifications in its models.
|
| > imply some incontrovertible nature to free markets that
| simply does not exist. Spherical cows indeed.
|
| Das Kapital (as one can guess from its name) also studies the
| spherical cow of the free market. The implication of
| incontrovertible nature, that's something in people's heads
| though, not in the models.
|
| > There was an article posted here recently about "mathy"
| equations that this reminds me of.
|
| Any math model (including models described in Das Kapital) is
| either going to be oversimplified or "mathy". The only other
| choice is non-math models, which doesn't seem very useful if
| you want to talk about money, prices, profits and other
| numerical stuff.
| 1htfp wrote:
| One of the most fascinating things about working on a trading
| floor is that models such as BSM transcend their normative aspect
| and become mental models. Pricing an option? Basically only two
| things matter: where the underlying asset forward price is at
| maturity (this is related to the concept of drift) and what the
| volatility is. At any time, your job is choose "bumps" (which you
| add to market prices) in order to maximize your odds of making
| money on a trade subject to beating your competition on price.
| There are some people who make a living making these markets who
| likely have never heard of "Ito's lemma" or diffusion equations.
| MuffinFlavored wrote:
| > where the underlying asset forward price is at maturity
|
| What models do people use for SPY/SPX forward price?...
| blitzar wrote:
| Futures. Decomposes down to price + dividend + time
| value/cost of money
| MuffinFlavored wrote:
| How often is front-month /ES not right around 20-50 points
| ahead of whatever SPX is trading at?
| FabHK wrote:
| A few points:
|
| 1) Very nice exposition.
|
| 2) Near eq. (4) it is claimed that one cannot compute the delta
| \frac{\del C}{\del S} without stochastic calculus, since S is
| stochastic. That doesn't strike me as correct: C is just a
| deterministic continuous function of S, C, K, T, t, r, sigma; and
| computing partial derivatives does not require stochastic
| calculus.
|
| 3) It captures the notion that when you hedge, you use risk-
| neutral probabilities.
|
| 4) Generally, in practice, BS is written as follows:
| C = df ( F N(d1) - K N(d2) ), where d1 = (ln(F/K) + 1/2 s^2)/s,
| d2 = d1 - s, s = sqrt(sigma^2 (T-t)), df is the discount factor,
| and F is the forward price of S.
|
| This abstracts away the whole discounting business.
|
| Note that sigma never occurs except in the expression sigma^2
| (T-t), which is dimension less, thus sigma has physical dimension
| 1/sqrt(year), usually ("annualised vol"). C has the same
| dimension as F and K.
| gwgundersen wrote:
| 2) Thanks for pointing this out. I've fixed.
| lowkey wrote:
| I've always found it strange that BSM is used for calculating
| implied volatility of American style options when it was
| specifically designed only for European style options.
|
| Can anyone comment if there are more suitable models for American
| style options?
| FabHK wrote:
| Generally, you back out local vols (as a function of S, t) of
| the BS vols (as a function of K, T) by a process described
| first by Dupire, and then you price American options (and other
| products that are not sensitive to vol of vol) with that using
| a numerical PDE solver.
|
| https://en.wikipedia.org/wiki/Local_volatility
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