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Is there an advantage to using the BSM versus the Heston model for pricing call warrants?
Please comment on draft Black-Scholes-Merton Excel spreadsheet & graphics application
BSM High Call Volume. Should I follow the trade.
Wall Street Week Ahead for the trading week beginning April 10th, 2023
Wall Street Week Ahead for the trading week beginning April 10th, 2023
Why B$M is a Strong Investment: A Look at its Financial and Historical Performance
Options are sophisticated gambling (for small players). Is it not?
Volatility and You: How Underhedging Creates Crashes
Spire Global ($SPIR) DD – Easy Peasy SPIR Squeezy
Why Retail Traders Should Avoid The Kelly Criterion Method
Math details - in what sense is Black Scholes optimal?
The hidden link between $GME and $PRPL
Mentions
Option prices are drived by two major blocks within BSM: intrinsic value and time value. Shorter dated options will always be cheaper (assuming no major changes in volatility) than longer dated ones. One-month dated options will still experience fast theta decay as this decay is not linear past the 30-45 day mark. It turns exponential. Buying options should be done after far more research done, not only having an eye on the expiration. Its all about the volatility environment you are in. I advice you to study and research a bit more about options pricing before entering large orders.
AI definitely gave me a more thorough and well rounded response with practical recommendations to my genetic profile. My endocrinologist who had a wait list in one of the most advanced parts of the country said he wasn’t even sure was VDR BSM was and rattled off something he googled in the moment. With AI you have to know what to ask. I can’t be bothered to explain more to you.
What about the higher floor but lower ceiling BSM?
BSM is for European but used for American all day long. Use an OTM fudge for the smile if you insist.
Use BSM as a wrong sized measuring tool, knowing that you are going to have a wrong measurement in absolute terms, but it will be somewhat correct in relative terms. The common heuristic is to use a multiple like 2X on the delta of each vanilla american option in the chain, and then create your one-touch surface based on that. If you have European vanilla options as the only choice, then your factor must be higher than 2. This post is giving me flashbacks of being a sophomore in college and writing a research paper on exotic options named something like "Survey of Exotic Options and Their Applications", and the professor who was an advisor in the Federal Reserve Bank shook his head in dismay as he read the title. I cringe to this day when I think about that moment, but it was an important life lesson he gave me with a single head shake and a bit of purposeful and exaggerated disgust.
Being able to leg into a covered call position is not the same thing as doing a buy-write. Regarding cash-secured puts - let me expand - when you write a put against cash - at RH - RH hold that cash as collateral. RH considers that cash as uninvested - so you cannot earn interest on it. When you write a put - you are impacted by Rho. That is the interest rate that impacts the premium in BSM pricing model. Option traders that write puts expect to have a way to generate at least the risk-free rate on that cash collateral to compensate for the interest rate drag. Most brokers do not lock up your cash when you write a put contract - afaik - RH is the only broker where you cannot generate the risk-free rate on cash when writing puts.
Correct, even though I don't blindly use the POP as calculated by broker software because they use the BSM variations which use gaussian distributions.
Huh? Nothing here makes sense. It peaked at 29.18 like the rest of the market in December 2024. It has grown pretty consistently over the last 5 years from a low of 10 in 2020. I threw some numbers in a random BSM calculator and came up with 1.951 for the stats on that particular option, it's not *incredibly far* off
I think the gist of your question seems to be: will the option decay at a different theta after close if the underlying moves. The short answer is: yes, afterhours moves in the underlying will affect the value (and therefore their greeks) of options even if they're not trading. The full answer is more complicated: All other inputs held constant, the Greeks are calculated based on the current underlying's price and the price of the option. If the underlying moves, the value of the option might not change according to the gamma and delta based on the BSM model (2nd order approximation). If the underlying increases, it may shift the volatility curve (sticky-delta). In other words, the IV may increase and theta according to the BSM model may increase too. Theta may also increase due to the higher value of the underlying (try plugging in the scenario you described into a BSM + Greeks calculator). For example, at K=100, S=100, V=0.5, T=1, rfr=0.05 theta is -8.10119006. With K=100, S=101, V=0.5, T=1, rfr=0.05 theta is -8.15781263. This is all to say that the change in theta is more dependent on the accuracy of the BSM model, the reason that the underlying moved afterhours, and whether the options chain has sticky-strike rather stick-delta-like behaviour.
Most stocks, indices, ETFs, etc. have some skew in their option-implied probability distribution. You'll see this reflected in the BSM delta when you trade straddles centered around the ATM or ATMf. You can fix this by either moving your put further OTM (in most cases) or by purchasing the underlying, as you mentioned. >but that moves the break evens in a skewed way. Yes, the breakevens will be skewed based on the option-implied forward probability distribution. In most cases, probability distribution of an underlying's future expected price is ***not*** symmetrical in log returns (lognormal) or simple returns (normal).
No because the "IV increase" is an artifact of how the greeks are calculated analytically in BSM, which doesn't take special events into account to correct for it you'd have to find the true expected variance of the stock price, usually by looking at the original differential equation with finite time The above is well known and priced in
Using BSM, i calculated the total trade value at $20M and profit of 520M at vix 50.
Me, but I am buying securities, not the market. For example I bought BSM for > 10% dividend yield this morning. Am I too early? Could it sell off and go to 15% yield? Of course it could. But I don't foresee it's dividend being reduced. So If it does sell off, I will live with my regrets and keep holding and collecting 10% divs for a looooong loooong time.
Your brokerage should provide a value of theta based on the Black-Scholes Model somewhere in the option chain. If not, there are most likely free websites that provide that information. Note that BSM is just a model; it is not 100% accurate. Realized theta decay will never match the output of any model perfectly (especially considering there's never really a scenario where all inputs are held constant while time to expiration ticks down).
The cost to borrow is a little bit trickier to calculate than that. The option you mentioned likely has more leverage than you think. You're paying $490 for probably around $3000 of exposure (I'm guessing without looking at the BSM delta) to the S&P 500. Looking at deeper ITM options will give you a better approximation. You can also calculate the option-implied forward price by finding the synthetic future (long call, short put) that has 0 upfront cost, and then derive the risk-free rate from the option-implied forward price.
Correct. BSM does not use option price as an input, however the price of an option based on demand within the market place is used in reverse to solve for Implied Volatility, which affects extrinsic value, which contributes to how wide or narrow the deltas will be with respect to price
"If i recall correctly Black-Scholes doesn't have option price as an input for calculating delta" You are way ahead of yourself asking about delta when you have no basic understanding about how BSM works.
BS/BSM is an option pricing model for European options and gives a good approximation for American-style options. It takes the risk-free rate, implied volatility, call price, share price, time to expiration, strike price and in case of BSM, the dividend of the stock. Like any mathematical equation, if you have five out of six, you can calculate the other. Not sure how your are connecting this with delta which is the change of the option price for $1 change in the underlying stock.
Theta is the same for calls and puts at the same strike by definition if using BSM model. Slight difference if options allow early exercise but the difference is usually insignificant.
"I'm told to use implied volatility" Who told you? It is silly advice, as you have already discovered. I think you may be confused because BSM in textbooks is structured to provide an option price based on known parameter values including volatility. But in the real world future volatility is unobservable so you cannot use BSM to solve for price. But the real world knows the option price because the market shows it. With price already known from the market, the volatility parameter value in BSM can be deduced, i.e. the value is "implied" by the option price (and the other known parameter values). Therefore, if you are trying to use BSM to calculate an option price you must \*select\* a value for the volatility parameter. Selecting (or guessing) that value properly is the key to options trading, the whole enchilada, the full 9 yards. You don't win the game by finding the best price, you win the game by finding "incorrect" implied volatility. In practice, doing that effectively requires skill and effort. Generally it involves a lot of sophisticated statistical modeling and even that is an imperfect science. If you can't do the complex modelling, you are forced to select a value some other way. Selecting historic vol might be a reasonable but imperfect choice.
I thought BSM is for pricing option relative to the HV. Delta of BSM and market price is IV, isnt it?
I had similar problems and Im not sure I overcame them. Isnt implied volatility just difference of BS model and fair market price? If that is not the case, I have a lot to re-learn. I made BS model calculator and its based on 365d volatility and I used risk-free short collar as risk-free rate. My reasoning is higher volatility = higher implied volatility = wider range on short collar (its always near 4%). Then BS model would give fair price for that stock based on HV. Then market price/BSM=IV Or I fucked up - idk
"fudge factor" I'm glad to see I'm not the only one referring to it in that phrase. IV is the one "unknown" in BSM -- you solve for it.
Because the BSM formula for probability of going ITM, N(d2), and the formula for delta, N(d1), differ by only σ√t. So as long as volatility is low or expiration is near, the two formulas are approximately the same.
Implied volatility, for someone telling someone who studied the BSM model for years to learn the basics maybe you should too
of course. understanding how options are priced is extremely important. BSM is theoretical and not reflective of markets, I would never use BSM as the decision point of determining if something is priced well. the reality is the ONLY unknown input into a pricing model is forecast volatility. most of us derive forecast vol from option pricing using BSM which is what IV is. there are literally armies of PhDs that toil over solving the problem of what forecast vol should be and they then compete on an open market. i do not believe any retail trader will be able to price forecast vol better than market consensus. that all being said, i still believe it's an ENTIRELY worthwhile exercise. I maintain volatility surfaces and attempt to forecast vol more out of interest and deepening my understanding of what other entities are doing to price.
Do you think any of the math around options actually works? If you find a massively mispriced option based on BSM, is that, in your experience, actually a good bet?
So, you are correct that I am anchoring off a theoretical gain based on BSM and IV reversion to HV. I don't actually know what the IV reversion will be, but I can say that the IV is happening. I recognize I won't get the maximum, but I should be getting more and definitely not breaking even. Watching the price moves after my closures today, it's clear that, at least for my positions today, I just needed to wait. This is probably a function of a herding effect on the price early in the trading day from lots of similar positions being closed. Also, I am buying back theta on the short leg and shouldn't do that. I should wait until after lunch, and make sure to add a delta hedge (based on prevailing market direction for the day) with shares just in case. I'll have to guess at this, but I don't need to get it perfect, just avoid taking a bath if the price moves against me. The long leg I think I should wait until the next trading day, since there's no hurry anyway.
wow thank you for such a detailed reply. This is awesome. And I like your approach. I am also tinkering with using models that don't assume normality or even log normality. I've read Benoit Mandelbrot's work The Misbehavior of Markets and trying to find models that allow for power laws and multifractality. If you'd like to pair up and exchange ideas and code let me know. I've done some work using bootstrapped past returns at the 1 day , 5 day, 20 days (trading days) intervals to find the tail boundaries, and using that to pick strikes. But want to start delving deeper into finding "theoretical" prices on models outside of BSM and comparing to active market prices.
Likewise. BSM does not accurately handle interactions between two options through time. My observation has been that the IV drop is not as well as isolated to the short option like you'd want, so you get symmetric value loss instead of asymmetric value loss. As a consequence the profitable region collapses and what you believed was a safe margin turns out not to be. On the other hand, I have seen some fascinating examples of mispricing on calendar spreads. For example, I had one on ZIM for weeks that sat several standard deviations away from the theoretical value for weeks, which just shouldn't happen, and then suddenly reverted and I made a good profit. I have no idea how to predict the reversion though.
Are you re-selling the new CSP at the higher strike, with the same expiration or a further out one? I suggest you learn/use a BSM calculator, that way you can play all sorts of "what ifs" and answer the question you raise in your penultimate paragraph. I have this as an open tab in my browser: [https://goodcalculators.com/black-scholes-calculator/](https://goodcalculators.com/black-scholes-calculator/) And this isn't really a tweak on the usual wheel...rolling either CSP or CC are normal/routine.
If delta is the same, so will be gamma. You can look at the following screenshot: [https://ibb.co/8gc1mwH](https://ibb.co/8gc1mwH) As you can see, the first two lines have a different spot, different strike, different IV and different time to expiry (TTE). Yet, delta is the same, as is Gamma. To be fair, it depends somewhat on the way Gamma is displayed / computed but there is always a unique and identical relationship between the two that only depends on the price of the underlying. **Detailed Explanation (copy pasted from a comment I made elsewhere)** If you have a certain Delta, the Percent Gamma will be the same. Sometimes the math may be daunting, but the closed form BSM solutions are simple to code and it's fundamentally the same for American options. The link [https://quant.stackexchange.com/a/78030/54838](https://quant.stackexchange.com/a/78030/54838) shows that the code matches Bloomberg to the decimal. The choice of numeraire is crucial here. The actual price of an option is in some currency, say USD because currency is a commonly accepted medium of exchange, and unit of account, whereas stocks are not. In fact, that is the reason stocks are quoted in currency in the first place (as opposed to say another stock or [noodles, tuna or cigarettes](https://www.theguardian.com/society/shortcuts/2016/aug/23/what-do-british-prisoners-use-as-currency)). Therefore, the natural choice with (stock) options is to express them in terms of currency as opposed to (fractions of) shares of stock. **All the major Greeks, except Delta, depend on the actual value of the underlying because they are expressed in CCY and not in (one) stock.** That is why adding simply BSM (Unit) Gamma to Delta does not get you close to the Delta you have after a 1% (or small) change in spot. The code below creates the table in the first screenshot. The formulas can be found on [Wikipedia](https://en.wikipedia.org/wiki/en:Greeks_(finance)?variant=zh-tw). `using Distributions, DataFrames, PrettyTables N(x) = cdf(Normal(0,1),x) n(x) = pdf(Normal(0,1),x)` `""" Calculate Black-Scholes european call option price` [`https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks`](https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks) `"""` `digits = 4 function BSM(S,K,t,rf,d,σ) d1 = ( log(S/K) + (rf - d + 1/2σ2)*t ) / (σsqrt(t)) d2 = d1 - σsqrt(t) c = exp(-dt)SN(d1) - exp(-rft)KN(d2) delta_c = exp(-dt)N(d1) gamma_c = exp(-dt)n(d1) / (Sσ sqrt(t)) return DataFrame("Spot" => S , "Strike" => K , "IV" => σ , "TTE in Days" => t365 , "Premium" => round(c, digits = digits) , "Price in Pct"=> round(c/S, digits = digits) , "Delta" => round(delta_c, digits = digits) , "Gamma Pct" => round(gamma_cs/100, digits = digits) , "Unit Gamma" => round(gamma_c, digits = digits)) end` `s,k,t,rf,d,σ, df = 138, 138, 90, 0, 0, 0.22, BSM.(s,k,t/365,rf,d,σ) df` The output for various different inputs looks like this: [https://ibb.co/cg2n8hr](https://ibb.co/cg2n8hr) The Gamma from BSM is frequently called unit gamma as it refers to a change in delta to a one unit change in the underlying. However, unit changes are a difficult thing (in finance) - something clearly visible when looking at the [standard deviation of prices](https://quant.stackexchange.com/a/65943/54838) (as opposed to returns). As you can see in the results, Gamma Pct can simply be added to Delta (as it is in the same Numeraire) and works for any 1 Pct change. Unit Delta on the other hand is smaller the larger the price of the underlying. The Blue line shows a 1 Pct change in an underlying which is worth 50k. In tems of Unit Gamma, that would still mean a change in the underlying ba $1, which is miniscule for 50K but gigantic for an underlying thats worth $1. That's why Gamma is usually Percent Gamma (change in Delta to a 1% change in price) by adjusting for the spot rate. [https://quant.stackexchange.com/a/65827/54838](https://quant.stackexchange.com/a/65827/54838) is showing that by default, Bloomberg displays PCT Gamma on their pricing engines unless you swap manually to display unit gamma (if you have BBG, click on Settings (in OVME) - User Settings - Pricing - Greeks section - Gamma). Long story short, Gamma will be identical for any given Delta for a 1 percent change in the underlying, irrespective of IV, TTE etc.
Here's an idea; don't know if it fits the parameters of what you're required to do. The only issue I personally see (and this may not matter for a year one bachelor's course) is that the four bullet points can be massive topics in and of themselves. So my idea is to more narrowly focus, and then, dig deep. For example, take "Credit Put Spreads". Subfocus: * The implications of the BSM model on the management of CPS. * Focus on the prime Greeks (Delta/Theta) while just giving a brief overview of the others. * Discuss under what market conditions to best use. * Bonus: compare and contrast to Debit Call Spreads using the same expiration and strikes. Just spit-balling.
Gather 'round regards. Technically this is wrong and not how options are actually priced. First you start with the market price. This is what the smart money is willing to pay to buy/sell an option. Then from the price, you use an options pricing model (say BSM), and see what it should be priced in theory. Then the IV is basically the difference between the mathematical model price and market price. The options market responds to supply and demand. This is why if you look in the options chain for known events that are months out, IV is higher there--because option sellers, working within their models, price the risk of the event accordingly. And then option buyers, working within their models, bid accordingly. So IV changes options prices all the time without changing the underlying stock price, because by definition it captures "everything", it is literally reflecting the market's opinion on the underlying on a point in time, not only "volatility". One way to see it is as a risk premium--how much risk is that position putting on given the probabilities in play during the event. So if you are an option seller, you need to increase your premiums to make bearing that risk worth it. This is why the underlying can hold steady and the IV term structure (IV across all expiry dates) can be downward/upward/neutral sloped depending on market expectations.
regarding buying skew or otm puts during crashes and the BSM model, I think otm puts are profitable given that you're nimble. vol is autocorrelated on the way up and on the way down. I think tasty trade recently admitted in one of their studies that it's a decent idea to go long vol early on in a vol spike. but your suggestion not to buy an itm call makes sense, maybe otm calls to get long delta exposure with little vega to avoid crush during a rally? Also another question, if you're looking to short vol on something, when selecting a tenor, does the vega of longer durations hold more weight or does the fact that the IV moves more on shorter durations matter more? especially when vol is spiked and my assumption is that it's coming down, you have backwardation and the shorter term IV swings more than longer term, but shorter terms have less vega...
Expiration and price paid? First, if you expect a $ for $ increase in your option, that won't happen unless your delta is 1. Not knowing the expiration date I can't see what it currently is. Regardless, with your strike being roughly ATM at the moment, I'm guessing your delta is about 0.55. Realize that there are a number of factors described by BSM. Movement of the underlying is primary (delta), with the erosion of time being second (theta). Say an option is burning $0.20/day, and has a delta of 0.60. Stock goes up $1. Your option will increase by $0.40 -- the $1 the stock went up times your delta of 0.60 gives $0.60, but then theta ate away $0.20.
It seems you also don’t understand the product that he’s trading. [my answer is the correct answer](https://www.reddit.com/r/options/s/dX2TkfJMZc) This isn’t a matter of BSM, it’s the nature of the vix options which neither of you seem to get. His february contract relates to the february /vx futures, not the number you see on the vix. It’s delta, Vega, theta, etc relate to the february contract. Don’t give advice if you don’t know what you’re talking about. It just confuses the issue and perpetuates misinformation.
For short: The underlying asset price follows a lognormal distribution in BSM. Logarithmic returns (of the prices) are normally distributed. More technical: A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables. Geometric Brownian motion (GBM), sometimes called exponential Brownian motion, is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (Wiener process) with drift. It's a stochastic process that satisfies a stochastic differential equation (SDE) and is used to model stock prices in the Black–Scholes model. The analytical solution to the SDE is a log-normally distributed random variable with finite expected value and variance. Furthermore, a variable x has a log-normal distribution if log(x) is normally distributed. See reason 5 on https://quant.stackexchange.com/a/64041/54838 for computer code demonstrating this.
There are no contract samples. Once a signal is established by your strategy (trendlines, indicators or support and resistance or chart patterns, or some fundamental trigger, or a coin flip....whatever) then you must select the contract that meets your desired level of probability and risk/reward, through delta and IV. If taking a short position you also want to look at expected move using either BSM or Binary model.
IV is an \*input\* to Black-Scholes, not an output. You can back it out of the formula given a price and all the other inputs, but that's not the way the formula is normally used. Also, neither one "drives" the other. MMs don't use the BSM; read "Why We Have Never Used the Black– Scholes–Merton Option Pricing Formula" by Haug and Taleb.
I find that high demand for winter coats in December = high IV to be a misguided analogy. Wrt the underlying in this case, it should be months of October or early March (say weather changes more drastically than usually expected, of which weather forecasts drive demand), depends on where one lives imho. For IV to have enough operative value, it should pertain more to the differential, both in external factors that affect demand for the underlying, and the differential in traders’ reaction to those external factors compared to how it would’ve been if only accounting for a more conventionally seasonal pattern. It’s true that IV does drive the price of an option (vanilla), but it’s more so in the downstream section of the market. Certain market makers who write barrier options and exotic options in essence write the price first, whether using BSM or not. For the more academically minded, I highly recommend Elie Ayache’s Blank Swan.
I find that high demand for winter coats in December = high IV to be a misguided analogy. Wrt the underlying in this case, it should be months of October or early March (say weather changes more drastically than usually expected, of which weather forecasts drive demand), depends on where one lives imho. For IV to have enough operative value, it should pertain more to the differential, both in external factors that affect demand for the underlying, and the differential in traders’ reaction to those external factors compared to how it would’ve been if only accounting for a more conventionally seasonal pattern. It’s true that IV does drive the price of an option (vanilla), but it’s more so in the downstream section of the market. Certain market makers who write barrier options and exotic options in essence write the price first, whether using BSM or not. For the more academically minded, I highly recommend Elie Ayache’s Blank Swan.
Just do what old school floor traders - plug a fake vol in the OTM right side so that it looks like one side Joker' smile and not a one sided normal person's smile. If you think it is power law, the use it - it will be better than BSM or any other "known" model that does not fit your distribution. Whether you are right that it should look like a power law curve is a different issue, out of scope here.
1. margin 2. for index valuations I look at [yardeni.com/charts/sp-500/](http://yardeni.com/charts/sp-500/) Figure 2 (P/E band), for single stocks well after reading Damodaran's textbook I have to say now I know how much I don't know. For IV there are two values: Relative i.e. Historical Volatility vs implied, IV rank & IV pencentile Absolute i.e. the number back-derived from option pricing using BSM formula I don't trade single stock options and only look at VIX(>20→iron condor) and VIX1Y(<20→calendar/diagonal). But IV Rank below 30% is a good starting point for debit buyers like you. And you should really learn to do diagonal call spreads if you don't manage losers actively. [tastylive.com/concepts-strategies/diagonal-spread](http://tastylive.com/concepts-strategies/diagonal-spread) This guide says buy far month atm sell front month otm but I usually buy far otm sell front more otm. Think of higher leverage as allowing you to make the same money with less risk if it works, not gambling with your retirement money believing it will always work.
I actually believe it's a great if not the greatest time to try to grow in the storm. Back in 1999 with similar valuations the VIX usually traded in the mid-20s despite the market grinding sideways, so no free lunch back then(aside from collars thanks to very high interest rates). But now you have a low IV, low term premium(!) and low smile(!!!). So this is a chance for "long to short", a spin on Spitznagel's idea that hedging enables more long equity exposure vs diversifying into low yielding bonds. I have mostly \~$0.7 notional with SPY 27Jan500P-495P spreads "hedged" by each $ in XDTE(if dotcom double, if GFC flat, Ken & Jane will take care of the VIX1D). Long BSM on the front month and long bimodal on the far month. Plus some SVOL for longer complacency. After this VIX "bull steepening" episode I added a bit of diagonal strangle of 2Yr475P-2Mo400P+Jul670C-Jun675C. COST 50% otm put spread cherry on top. What if this passive flow insanity continues? What if it unravels? What if it stalls? GREAT. What if the anti-proverbial escalator down happens? I'll be down 5%. Now my living cost is covered by positive theta, plus responsive delta and hundreds of Vega. Thank you to ZVOL dip buyers. Where I truly differ is I think the student loan wage garnishment/delinquency being reported to credit bureaus(anti-stimmy checks) a much higher marginal impact on velocity of money. Leftover pandemic stimuli in local govs' budgets forced them to mass hire people until they have to fire them. So I believe unemployment to eventually be the catalyst. But that's the beauty of convexity, you don't need to predict the reason for a prospective move.
So your option is worth less than 20 but theta is -45? You might need to work on your scaling. If you use standard BSM closed form, it's in 1 year, so you should really divide it by 365. Irrespective, this plot must be ATM or close to ATM and you can clearly see that theta accelerates (gets steeper and larger) the closer to expiry.
This BSM and adds certain other factors to the script. We’ll see Monday
Someone reinvents the BSM model and uses crayons to draw, not realizing BSM is wrong.
Yep, finally someone...nice too meet you, fellow non-believer in BSM and delta.
Delta is a VALUE. Delta is CALCULATED using the BSM, a publicly available formula. There is no such thing as "your delta" because that ISN'T DELTA Ffs please keep "trading options" on short squeeze bullshit
What’s the BSM for delta you’re talking about? Bondage sado-masochism? Bc I’m more of a whips n’ chains guy myself.
Too many words to agree with me that BSM is shit and so is delta.
"Let’s see your options pricing model that is more predictive and accurate then BSM there big brain" How much are you willing to pay?
1. I calculate my own delta using my own model 2. Unless your own model is do some mad shit with the original BSM pde or for some reason not using the gaussian cdf, delta has the same formula
Volatility is the only unknown variable in BSM Delta is pure objective math wtf are you on about? Let’s see your options pricing model that is more predictive and accurate then BSM there big brain
I largely disagree with the given answers so far. You can look at the following screenshot to see why: https://ibb.co/DD74NkC As you can see, the first two lines have a different spot, different strike, different IV and different time to expiry (TTE), Yet, delta is the same, as is Gamma. It depends somewhat on the way gamma is displayed / computed but there is always a unique and identical relationship between the two. **Detailed Explanation:** If you have a certain Delta, the Percent Gamma will be the same. The other answers provide ressources to Greeks and claim things like - Gamma changes based on things like time to expiry and implied vol - Delta varies by stock price, strike price, IV and TTE However, what the two statements clearly show is that - Not just Gamma, but also Delta both change based on the same things and therefore neither explanation suffices to show it's constant or not. Sometimes the math may be daunting, but the closed form BSM solutions are simple to code. The link https://quant.stackexchange.com/a/78030/54838 shows that the code matches Bloomberg to the decimal. The choice of numeraire is crucial here. The actual price of an option is in some currency, say USD because currency is a commonly accepted medium of exchange, and unit of account, whereas stocks are not. In fact, that is the reason stocks are quoted in currency in the first place (as opposed to say another stock or [noodles, tuna or cigarettes](https://www.theguardian.com/society/shortcuts/2016/aug/23/what-do-british-prisoners-use-as-currency)). Therefore, the natural choice with (stock) options is to express them in terms of currency as opposed to (fractions of) shares of stock. **All the major Greeks, except Delta, depend on the actual value of the underlying because they are expressed in CCY and not in (one) stock.** That is why adding simply BSM (Unit) Gamma to Delta does not get you close to the Delta you have after a 1% (or small) change in spot. The code below creates the table in the first screenshot. The formulas can be found on [Wikipedia](https://en.wikipedia.org/wiki/en:Greeks_(finance)?variant=zh-tw). using Distributions, DataFrames, PrettyTables N(x) = cdf(Normal(0,1),x) n(x) = pdf(Normal(0,1),x) """ Calculate Black-Scholes european call option price https://en.wikipedia.org/wiki/Greeks_(finance)#Formulas_for_European_option_Greeks """ digits = 4 function BSM(S,K,t,rf,d,σ) d1 = ( log(S/K) + (rf - d + 1/2*σ^2)*t ) / (σ*sqrt(t)) d2 = d1 - σ*sqrt(t) c = exp(-d*t)S*N(d1) - exp(-rf*t)*K*N(d2) delta_c = exp(-d*t)*N(d1) gamma_c = exp(-d*t)*n(d1) / (S*σ *sqrt(t)) return DataFrame("Spot" => S , "Strike" => K , "IV" => σ , "TTE in Days" => t*365 , "Premium" => round(c, digits = digits) , "Price in Pct"=> round(c/S, digits = digits) , "Delta" => round(delta_c, digits = digits) , "Gamma Pct" => round(gamma_c*s/100, digits = digits) , "Unit Gamma" => round(gamma_c, digits = digits)) end s,k,t,rf,d,σ, df = 138, 138, 90, 0, 0, 0.22, BSM.(s,k,t/365,rf,d,σ) df The output for various different inputs looks like this: https://ibb.co/cg2n8hr The Gamma from BSM is frequently called unit gamma as it refers to a change in delta to a one unit change in the underlying. However, unit changes are a difficult thing (in finance) - something clearly visible when looking at the [standard deviation of prices](https://quant.stackexchange.com/a/65943/54838 ) (as opposed to returns). As you can see in the results, Gamma Pct can simply be added to Delta (as it is in the same Numeraire) and works for any 1 Pct change. Unit Delta on the other hand is smaller the larger the price of the underlying. The Blue line shows a 1 Pct change in an underlying which is worth 50k. In tems of Unit Gamma, that would still mean a change in the underlying ba $1, which is miniscule for 50K but gigantic for an underlying thats worth $1. That's why Gamma is usually Percent Gamma (change in Delta to a 1% change in price) by adjusting for the spot rate. https://quant.stackexchange.com/a/65827/54838 is showing that by default, Bloomberg displays PCT Gamma on their pricing engines unless you swap manually to display unit gamma (if you have BBG, click on Settings (in OVME) - User Settings - Pricing - Greeks section - Gamma). Long story short, Gamma will be identical for any given Delta for a 1 percent change in the underlying, irrespective of IV, TTE etc.
Thanks for the suggestions and the tips, much appreciated. Dumb question maybe, but what do you mean by "moneyness". I guess that your comment replies the one proposing to study how big the fudge factor between real prices and BSM ones, am I wrong?
Yes - how big is the fudge factor in SPX option prices when compared to the classic BSM pricing, and how did it mover over the decades?
While you are building out that entire pricing model in Excel format, this one might interest you. I don’t use these, but this one accounts for dividends. That and the American options early exercise are separate adjustment calculations when the BSM is used. https://www.omnicalculator.com/finance/black-scholes
It's in the post, but I do not blame you for not wanting to read it all. The price data is from yahoo finance, and the VIX is from CBOE. For the strike and premium BSM was used (the basic pricing model of options)
TLDR: sucks at Excel, uses ChatGPT to crease BSM, makes up theoretical i.e. fake 1DTE options with BSM, "backtests", then talks nonsense.
Natural resources. Look at BSM
I guarantee I am, 100% (I am also using money I am willing to lose). I am also not using the BSM, but that is a basis for saying at least with no other info that the probability it goes above/below some price is X.
If you're using BSM in your trading and it's not delta hedged options, I would bet money you're doing something severely wrong
Yes, I will try. I borrowed the e-book on the library but it is expired now. I you are short volatility (And delta neutral) you can only profit from the variance premium (Difference between implied and realized volatility). When you trade strangles you have a much higher Pop/win rate (depending on the chosen strikes). The high win rate on a strangle can give you a bad feedback that you have an edge of predicting the variance premium. So the straddle gives a more clear feedback. He make some simulation based on BSM and geometric brownian motion processes for illustration. Here is a post from esinvest and link to a youtube where Eauan explains that [https://www.reddit.com/r/options/comments/zb25kj/finding\_edge\_with\_euan\_sinclair/?utm\_source=share&utm\_medium=web3x&utm\_name=web3xcss&utm\_term=1&utm\_content=share\_button](https://www.reddit.com/r/options/comments/zb25kj/finding_edge_with_euan_sinclair/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) youtube link: [https://youtu.be/YDA449Fkwj4?si=ROxebpL32JoayHr3&t=2292](https://youtu.be/YDA449Fkwj4?si=ROxebpL32JoayHr3&t=2292) I also find an article with simulation based on the Euan Sinclair book: [https://jonathankinlay.com/wp-content/uploads/Straddles-and-Strangles.pdf](https://jonathankinlay.com/wp-content/uploads/Straddles-and-Strangles.pdf) I will make the same simulations for better understanding the behaviour of the two strategies.
The point of an options pricing model is to provide a theoretical value to quote around for market makers. If there wasn’t a market, what would you quote any given option? Also, if the market price is giving a vol surface, and you’re a market maker trying to fit the market, a vanilla BSM model will likely not fit any given surface due to supply and demand causing kinks. This has large implications for derives Greeks and PnL. Imagine you’re using a vanilla BSM model and you’re modeling an option to have a 5 delta when the actual delta should be 6.2. If you’re long 1000 of those and hedged against 50 short futures, you’re neutral to your sheets but in reality you could be long 12 futures vs the “actual” market delta of the option. If futures move against you it could be very costly
Take a look at BSM. Right now its trading at a price lower than most insiders have bought for since 2017. It pays a dividend of 10% if you hold stock. Recent purchases by the CEO were larger than previous ones made throughout the year. Most importantly if you're following the insiders is that the stock is still priced around the insiders purchase price. With that in mind, the profits the business generates, and the natural gas prices dropping it's a good time to buy if you're of the belief that things natural gas prices will eventually go up again. I do and it seems maybe insiders seem to think so too. I've created a position but it's 1% of my portfolio. This is a way to do what you did but a little more safely.
IMO, no, you don't need to delve into how BSM was derived or how the various terms of the PDE function to define a fair price. However, it is useful to understand some of the assumptions *behind* the math, like that stock price is assumed to be random and that the distribution of prices over time are lognormal with skew. You can get a really good overview of those assumptions in this short video: https://optionalpha.com/lessons/understanding-the-math Then read these articles. All are designed to be easy to understand and require no more than high school math: https://www.reddit.com/r/options/comments/14kdmur/geometric_vs_arithmetic_mean_in_the_wild/ https://www.reddit.com/r/options/comments/14kdsrr/using_log_returns_and_volatility_to_normalize/ https://www.reddit.com/r/options/comments/14ll86k/the_volatility_drain/ https://www.reddit.com/r/options/comments/14hzdal/solving_a_compounding_riddle_with_blackscholes/ https://www.reddit.com/r/options/comments/14jo0er/lessons_from_the_50_delta_option/ https://www.reddit.com/r/options/comments/14llh32/convexity_is_misunderstood/ https://www.reddit.com/r/options/comments/14jo8ld/finding_vol_convexity/ https://www.reddit.com/r/options/comments/14joaei/understanding_vega_risk/ https://www.reddit.com/r/options/comments/14kdijb/what_you_can_expect/ https://moontowermeta.com/understanding-edge/ https://www.reddit.com/r/options/comments/14hzmxv/understanding_implied_forwards/ https://www.reddit.com/r/options/comments/14m6u0j/on_delta_hedging/ https://www.reddit.com/r/options/comments/15c3wew/distributional_edge_vs_carry/
You look like you know more than I thought but I'm impressed you know nothing about the Greeks and the BSM model. Until you don't look at options as an intellectual subject of interrest you're off-topic. I'll recommend you a few books on PM and other resources. You need a serious mindshift in relation to how you solve problems with options. Learn to solve them by yourself and stop coming to reddit. I never came here it's just out of curiosity. It's like seeking investment advice in a youtube comment... Seek real options education (that's how it's phrased, I'm not making it up). Forget about forums. That's not where you learned math or philosophy.
Except volatility is a core defining mechanism in the BSM as well as the equations that price American-style contracts. In fact, you can’t understand options without understand volatility which represents demand. OP is much more regarded than he sounds.
Works well enough with variance calculations for early exercise and dividends. The BSM output is, literally, the theoretical option value which then is combined with supply and demand aspects that determine price. Like real estate, automobiles, restaurant meals, a value in a free economy is what buyers are willing to pay.
Please correct me if I'm wrong. To use vol surfaces: 1 Given a certain pricing formula, an array of option prices are calculated for different strikes and DTEs. 2 Prices are converted into IVs. This array is called the vol surface. 3 To get option prices, an appropriate IV is taken from the surface and inputed into the BSM formula to get the price. Technically, the price is obtained with the BS formula but, the vol used isn't the expected vol. This could also be seen as black scholes is being used to "store" prices are calculated using other models.
⚠️⚠️⚠️ Vanilla options are NEVER priced with black scholes. BSM is, for the purposes of pricing, an outdated formula that contains many flaws. Black scholes has applications in the option market, but it isn't for pricing options. Newer, better models are used for option pricing.
its more about the 'supply/demand' rather than BSM
Black Scholes is the foundational equation upon which the modern world of options was built. It doesn't matter what a bunch of Reddit muppets think about it, you simply wouldn't be able to make a robust market without having a universally accepted way to calculate a fair price for the same set of pricing parameters. Without market makers there would be no markets. That being said, you don't need to understand the equation itself in great detail. The gist is that it weighs a P&L against a lognormal distribution to produce an expected value. The more you understand about the state of finance in the early 1970s, the more you should be able to appreciate the challenge behind expressing that in a single analytical equation. However, it was designed for European-style options, so it's not ideal for American-style options. Instead, there are methods that attempt to approximate what BSM would produce if it were able to account for early exercise. There are also legitimate arguments around distribution mechanics, especially around major events like earnings reports. But it's been 50 years and we're all still using it whether we admit it or not. Organizations may have their own tweaks on BSM for whatever reason, but the core innovation earned the Nobel for a reason.
Brokers use least square monte carlo simulation (LSMC) to approximate the vol surface, not FDM (finite difference methods)/BSM/CRR. Then they apply the best fit regression of vol (usually linear) to feed CRR estimations of the Greeks. Brokers also tend to use multiple models to study vol. Saying jargon doesn't make you automatically right.
Not sure what you mean for all assumptions but given volatility, there is no need for anything beyond the Black Scholes Merton (BSM) PDE for American and European (listed vanilla) options. Frequently, the Cox-Ross-Rubinstein (CRR) model is used to price American options but CRR is just a particular case of an explicit FDM scheme (a special case of the FDM for the BS PDE). It's in essence just a discrete time approximation of the continuous process underlying the BS model. Leisen and Reimer is just a tweak to CRR to speed up computation. In any case, a market maker has a vol surface, which will be plugged into a model based on the BSM PDE, either directly via PDE solvers or via discretized versions like CRR. All risk metrics (Greeks, PnL attribution, ...) will be based on this model. The vol surface itself is backed out from the market (see for example https://quant.stackexchange.com/a/73891/54838) and "enhanced" with in house data (inventory, positioning, custom vol calendar data etc).
BSM is already baked into the prices. No need for calculations unless you want to single out a specific variable
Doesn’t matter what the stock is, any option will drop in IV if it doesn’t meet the expected move. You can mathematically prove that using BSM. Vega just happens to be less for longer dated options.
Logic may be different, since there are different compulational models for option price, like BSM vs. CRR vs. Monte Carlo, but even if they use the same model and identical code, calculators can still differ in the assumptions they make about price quotes and the evolution of IV. For example, most calculators assume IV is constant over time, which is absurd and almost never happens in reality. But it's easier to write the calculator code if you assume IV never changes. Even if you wanted to allow IV to evolve, how would the user specify the evolution? What would the input look like? Would requiring each user to upload a spreadsheet of IV samples over time be an acceptable UI for most people?
Ok. So, IV is basically the parameter that describes how far the share price can deviate from todays value over the life of the option. If IV is 45% that implies a 3% per day move. If it’s 15% that’s more like 1%. So if you think of a just random walk process, obviously something moving 3% per day, after any number of day will have a wider dispersion, on average, than the 1% stock. What most of the calculators do it take the current implied value and use that to project the terminal price of the stock. That should end up with a value pretty close to the options value, as that’s basically what BSM is.
Haha, if we're going to be counterparties may your puts chip and shatter. As to BSM, I did look at the math. S, is a function of time. It can have any function dependence. In practice, it is never linear, but that doesn't mean the concept doesn't exist.
Not in BSM. Just look at the math. Anyway, arrogant transplants from other fields make great counterparties. Hopefully you’ll be mine!
That’s exactly what the fish were saying in the barrel when BSM first rolled out and a select few started using it. Those open call guys either got with the times or got got.
I would say that situation is impossible (or close enough to not spend time on) due to the imbalance in flows in option markets. Because of that MMs tend to be net short options, which means they need a pricing and risk management model, so I’d disagree and say a pricing model is essential. And yes, options traded before BSM, but spreads were much wider, liquidity lower and retail wasn’t really involved.
The whole delta hedging argument underpins BSM. It only works in a frictionless market where you can buy, sell without any trading costs. Ok, let’s assume that is true for market makers. It still wont work when there are jumps in stock prices. If you’re delta neutral and the stock jumps 20%, there is no way you can stay delta hedged for the 20% move. The good news for market makers is that their net delta is usually low (if they are any good at their jobs). This is because they will have sold nearly as many options as they will have bought, at the strike and expiry level, balancing most of it out. They will not hold a high net delta options position and hedge with a position in the underlying to become delta neutral, due to this jump problem. How do they do it? By moving their bid/ask prices around for specific strikes. Have you noticed that some options prices are wild (wide bid/ask) right at market open? Market makers do not want to sell you a bunch of risk and get screwed on hedges because of wild swings. You will see wide spreads on smaller names, not on NVDA. This isn’t because NVDA is not a volatile stock. It is volatile, but options volume is high in NVDA. That means market makers are filling both buys and sells so they are hedged without having to hedge using the stock. So they will gladly make a tighter market.
Truth be told, I’m more prone to skipping over all of this, believing all I might ever need to know is that a arbitrage can exist, and that’s only so I better understand the entire system. Arbitrage is a backwards facing, ever changing revelation. My trades are placed by my brokerage, I long ago gave up any illusion that I could personally make a difference in my profits by using directed options trading. By the time my comparatively minuscule directed order ticket was 1/2 filled out, that arbitrage is gone, exploited by larger orders placed at faster speeds. It’s all a bit of a cart/horse situation compounded by BSM not being the only pricing model used, or factors within it being tweeked to suit the decisions of whomever wrote the algorithm, or even the frequency the model used is updated. I didn’t read all the responses but liked the short one about options price first and are interpreted second. It makes sense that the more useful understanding is in acknowledgment of the calculating direction. And as you mentioned, it’s what the mm are doing.
Options are all calculated to and adjusted once to the risk-free rate as that Fed change occurs. There are many other factors that may cause arbitrage pricing between options exchanges. Surprisingly enough to me, I think the biggest one may be who ran the calc last. Below is a link that shows some of the math features of BSM without the symbols, much more normal math for all but the most narrow of math disciplines: https://m.economictimes.com/definition/black-scholes-model#:~:text=Definition%3A%20Black%2DScholes%20is%20a,%2C%20and%20risk%2Dfree%20rate.
Although it's frequently done (in Retail) you should be cautious when comparing IV to historical vol (HV) - also called realized volatility (RV) - because it is not necessarily useful for at least two reasons: - Empirically, IV tends to overestimate RV, commonly referred to as Volatility Risk Premium: https://quant.stackexchange.com/a/64076/54838 - IV is the only free parameter in the Black-Scholes-Merton (BSM) model. Higher IV can be a result of compensation for tail risk: https://quant.stackexchange.com/a/64384/54838 There is no direct impact but it's all connected. Just like an index is just a weighted combination of the underlyings, options on an index can be seen as a weighted combination of options of the underlying option. That's the basic idea behind so-called dispersion trades. Less important but still noteworthy: RV isn't observable and just an estimate. The choice of estimator will give you different results.
That was actually plausible enough that I briefly looked. Although I could find a reference to additional risk premium, not a rate increase, as that applies to to the capital asset pricing model for risky assets, I did not find a reference where that would apply to a derivative option where both ends of a contract would agree to the terms which then opens the contract. BSM got an honorable mention, so I looked in that direction also but could not find a reference to longer duration contracts being assigned higher than the risk-free rate, just that the BSM assumes the risk-free rate is constant and known. I did find the reference to a change in the risk-free rate between European & American options due to early exercise, but not within either kind otherwise. For the others. Calls increasing, puts decreasing in value is a one-time thing as the risk-free changes. This may account for some skew. But it’s entirely possible for the skew to go the other way due to factors associated with adjusting an underlying’s options as this happens every trading day. I’m going to stick with the biggest factor for what’s perceived as skew being the most obvious, the one hidden in plain sight, the trinity of delta moves, underlyings move, strikes don’t move, and that most of what is called skew is actually price movement calculated against the rigid strike. Once an underlying moves off ATM, equal distance strikes from ATM will have different deltas due to underlying movement. Movement of the underlying itself will cause a perception of skew unless ATM. This is not skew.
No, let’s not say anything. Throwing a lot of words and formulas around that few here understand won’t change the validity of delta that you’re simply trying to subvert. Your BSM request is ridiculous because you are requiring it do something it doesn’t do. And about this time in these discussions, I ask whomever is name dropping the BSM, which is just one of many options pricing models having nothing to do with the utilization of net sum delta hedge realities of retail trading, to demonstrate for me in math terms the point you are trying to make. Because as far as I can tell, you tend to carpet bomb the discussion with words and factors that are not relevant, and are currently being dismissive of math while also citing a math model that has no real connection to how delta is used in trading at the retail level.
lol. Still telling me what I would have been doing in a job I did? My grad work is in financial engineering, you hump. I’m quite capable of building those “programs” myself. BSM framework is used everyday, by every professional option trader. The Greeks that we risk manage off of come from that framework. The idea that we’re capturing the spread between IV and RV on residual positions comes from it. The value of an option is defined by its bid offer spread? So options with tight spreads are worth less than ones with wide spreads? Seriously think about what you’re saying here.
You would have monitored a level maintained by a program written by a mathematician. BSM is no more a direct part of your day then than now where the value of an option is still defined by the Bid Ask spread. The much touted BSM knowledge on this sub is nonsense for it really applies to no one here in any practical sense with the possible exception of newly released options or and equity who offers a whole chain for the first time. In addition, BSM is not the only metric used, just one.
Correct, I’m no longer a market maker. But I did the job for many years. Enough to get a pretty deep understanding of what market makers do and how. And the main thing we do is maintain a BSM vol surface for pricing. Even exotic options that you don’t prices with BSM, you’re starting point is BSM vol. Every automated market maker taking the other side of your retail trades, is setting their pricing off of a BSM vol surface. But continuing posting, anyone with any knowledge who reads your posts will be laughing. Like me!
It’s only meaningless to you because you don’t understand it. I also didn’t link to a video? So not sure what you’re gibbering about. The BSM framework is used by every market maker you interact with, so on one side of the trade, yeah it definitely plays a role.
Happens pretty much the same every time. BSM plays zero role in day to day options trading from the retail prospective, yet so many think they’re some authority. In your case, the cost to replicate an option? Nice meaningless statement, no collaborating source, and an added obtuse posture about an option’s spread being difficult to accept as the valuation means of an option. I’m going to pass on that video.