Reddit Posts
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
Look at the oil royalty companies like CRT, VOC, etc. The gold royalty companies are hot now with gold going parabolic. WPM, FNV, SAND etc. I also like GNT and BSM now.
Thanks for the idea from 9 days ago… also, your “model” LOL, copy paste BSM?
BSM takes in the current risk free rate. If you think that the risk free rate will go up further, the model output will only change when the actual risk free rate changes. Hence to actually monetize this view, you will need to 1) know what is the rho on the option, 2) know when the risk free rate changes and by how much, 3) ensure that 1&2 are sufficiently accurate such that your gain via rho outweighs your decay + spot move. That seems like a pretty tall order.
Gold trends for a while then it reverses. Gold/silver royalty companies like FNV WPM SAND MTA etc. give you income. At some point income becomes more important than capital appreciation. I also like GNT. BSM for other minerals.
What part of BSM are you recommending?
I think what you’re really asking is how do I determine the future value of an option I BTO/STO’d at a certain strike on a certain day. The answer is to use a BSM calculator or an app that uses an accurate option pricing model and just plug in the numbers
IMHO companies with real assets, will not go to zero and will rebound. Look at BSM, GDX, XLE, LAND, etc.
It's a bit misleading to label these as "7 main." Main according to whom? And why seven? What was the selection criteria? What models were rejected? Each one of those has multiple variations in the literature, so why are these seven so special compared to the dozens of variations? I would not be surprised if each institutional trading house and boutique quant fund has their own homebrewed model that is some combination of variations of these. I think it's pretty common practice to use some variation of BSM for near the money European contracts and a different variation of Binomial for OTM American contracts. And there are more variants of Binomial, not just CRR, like Jarrow-Rudd and Leisen-Reimer. Then even within any specific implementation, identical implementations that are configured for different numbers of steps will give very different results. The usual reason for publishing a new and improved version of model X in an academic paper is to achieve some optimization goal or to address a weakness in the model. Who is to say which of those is the "main" and which is the "variant?"
Diversify into areas like minerals, oil, gold, and land. Hard assets. Look at BSM, XLE, GDX, LAND.
I’ve lost track of how many times I’ve discarded your opinion. And you’ve clearly not read all the posts, some which do indicate exactly what I’ve warned against, as well as the original op eluding to that. You now agree but wish to move into a more controversial area of delta not equating to probability. Please explain your opinion, which also conveniently ignores my assertion of probability IN THAT MOMENT, to the creators of BSM and every Google source that dumbs that down for mere mortals. Your use of the ad hominem attack clearly indicates a goodly amount of personal insecurity.
The volatility smile plain simple shows that BSM is not the way to understand how pricing works for options. BSM also assumes some hedging that is impossible to execute on practice. Good first approximation but that’s about it. I have been downvoted many times for saying that you can be a successful trader without understanding any Greeks. You don’t need to be a mechanic to be a professional driver (though it probably helps)
BSM is like the most vanilla model, option market makers will have their own models used to price contracts since they want to make money when they make markets. The thing about IV is that it's when you plug in the underlying and option price into the model and solve for volatility. So the market maker may make a market using one assumed volatility over til expiry, and then as price drifts, the implied volatility will drift as a result.
Hey! Yes, I use PyMC and either a big CPU/GPU or some patience. This is a very effective method for gaming out scenarios and interrogating the median and low percentile outcomes. I’d look into stochastic volatility models. You can do much better than BSM if you’re running Monte Carlo simulations.
What do you mean by saying BSM’s accuracy is debatable, and compared to what? The greeks are derived from Black-Scholes.
Over the course of 1998, LTCM lost $4.8 billion, which breaks down as follows (Yalincak, Li, and Tong 2005): $1.6 billion in swaps $1.3 billion in equity and volatility $430 million in Russia and other emerging markets $371 million in directional trades in developed countries $286 million in equity pairs $215 million in yield curve arbitrage $203 million in S&P 500 stocks $100 million in junk bonds So the answer to their failure is more complex than ‘BSM could not be used for the types of trades that broke their neck’. It’s more like they hanged themselves by believing their models were correct when in fact they underestimated risk or the fat tail or black swan. But ~40% of their loss was attributable to equity exposure while 60% was off their interest rate and currency swaps. They might have survived the equity meltdown losing only 70% of their capital but I would hardly call that an endorsement for BSM. Quants might take this as an oversimplification but the fact that real markets don’t have a limited set of probabilities. For their interest rate and currency bets they would not have used BSM, but a set of differential equations that smelled similar in order to calculate their VaR. From Ron Rimkus CFA Institute: (By the way they have a number of good papers analyzing various market failures) “LTCM used a combination of Value at Risk (VaR), stress testing, and scenario analysis to manage risk……VaR is a statistical technique to estimate the amount of losses a portfolio might incur over a given day or week (or longer) based on historical price movements. There are many nuanced ways to implement VaR, but they all essentially involve computations of historical volatility, estimates of future volatility, a time horizon, a chosen probability distribution (e.g., normal distribution, Student’s t, chi-square), correlations, and a confidence interval (95%, 99%, etc.). The particular approach to VaR that LTCM used was based on daily standard deviation of security prices and correlations among all securities owned across the entire company over the preceding 500 trading days (estimated) assessed at the 95% confidence interval” “VaR, by virtue of using a probability distribution, treats the markets as a closed system, like a lottery drawing. A typical lottery drawing that chooses 1 of 60 or so balls in a bin is an example of a random, closed system. There are always a certain number of balls to choose from, and the chosen ball is determined through a randomized process. Therefore, the probability of selecting a single ball is exactly the same each time: namely 1 in 60.”
All of above, as I said before. look @ my previous comment on this thread. *To eliminate all confusions* : BSM came to give some reference to Option price. -European style. You plug in all five "free" variables and solve for option theoretical price. By using existing Option current price as an input -- you solve for IV. see?
Implied vol is the plug in the BSM. Options prices are set by supply and demand, and implied vol is backed out from the prevailing market price. Given that the BSM only has IV as the plug, you can use it to compare options with different specs to each other as IV is a standardized metric.
The risk free rate is not a free parameter on BSM. RFR swaps are market quoted and companies like the [CME](https://www.cmegroup.com/trading/interest-rates/files/discounting-transition-proposal-aug-2020.pdf) and LCH moved to SOFR PAI and discounting on [Oct. 16 2020](https://www.newyorkfed.org/medialibrary/microsites/arrc/files/libor-timeline.pdf). For EUR cleared, major CCPs did this since July 27 2020. The RFR is just like spot, dividends, time, and strike something that is fixed by the contract term and market data. Also, I don't think you understand what risk neutral pricing means. It does not mean you don't expose yourself to risk.
The risk free rate is not a free parameter on BSM. RFR swaps are market quoted and companies like the [CME](https://www.cmegroup.com/trading/interest-rates/files/discounting-transition-proposal-aug-2020.pdf) and LCH moved to SOFR PAI and discounting on [Oct. 16 2020](https://www.newyorkfed.org/medialibrary/microsites/arrc/files/libor-timeline.pdf). For EUR cleared, major CCPs did this since July 27 2020. The RFR is just like spot, dividends, time, and strike something that is fixed by the contract term and market data. Also, I don't think you understand what risk neutral pricing means. It does not mean you don't expose yourself to risk.
The risk free rate is not a free parameter on BSM. RFR swaps are market quoted and companies like the [CME](https://www.cmegroup.com/trading/interest-rates/files/discounting-transition-proposal-aug-2020.pdf) and LCH moved to SOFR PAI and discounting on [Oct. 16 2020](https://www.newyorkfed.org/medialibrary/microsites/arrc/files/libor-timeline.pdf). For EUR cleared, major CCPs did this since July 27 2020. The RFR is just like spot, dividends, time, and strike something that is fixed by the contract term and market data. Also, I don't think you understand what risk neutral pricing means. It does not mean you don't expose yourself to risk.
Look at BSM and GDX for mineral exposure.
You’re talking about options, OP is talking about equities. Equities work kinda like how op explained, they provide liquidity if there isn’t any. They can also make money from exchanges by a maker-taker or taker-maker model of fees. For options, the options desk has pricing models that try to price is better than typical BSM models
Having zero clue or understand of BSM pricing models doesn’t stop them from trading options lol soooo
You're welcome. I'm happy to help however I can. May I try to to clarify some concepts? A lot of people seem to be confused about a view of the market, a strategy, an edge, a type of play, a trade structure, and a play. **View of the Market**: An assessment of whether the market, overall, or the subset of it that's relevant to your potential play, is likely to be green, red, or neutral during the course of a potential play. This is complicated, but a core aspect of it involves knowing how to analyze, interpret, and act on the evolving behavior of futures, e.g. /ES, /NQ, and /VX, including volume. /VX is incredibly important. **Strategy**: An algorithm for trading that has an edge. I like the dictionary definition: "a general plan to achieve one or more long-term or overall goals under conditions of uncertainty." Implementing a strategy involves selecting a trade structure and creating a play, but a strategy is not a trade structure. An example of a strategy is this: wait for SPX to close in the red (nearly) four days in a row. (This is rare.) Then, within five minutes of the close of the fourth day, short a put spread with an upper striking price 1.00% below the spot price. Due to some degree of mean reversion, this is likely to be a viable strategy with an edge, but the opportunity to execute it would be very infrequent. Many strategies these days are derived empirically using data analytics, but you have to be careful because when market conditions change, they might no longer have an edge. **Edge**: A sustainable probability of profit higher than what can be had by buying and holding SPY or QQQ that results from the product of one's win rate and the average profit from winners compared to the average loss from losers as the sample size grows large (at least 100 trades, but ideally many more). Although most people wouldn't bring SPY or QQQ into the idea of an edge at all, I think that it's imperative, because it doesn't make sense to trade unless you can do much better than buying SPY or QQQ. **Type of Play**: A category of play, such as earnings crashes, pre-earnings run-ups, binary earnings bets, directional bets, volatility bets, opportunistic news catalyst bets (such as the DOJ criminal investigation into BA for the ALK midair door blowout), opportunistic macroeconomic data bets, rangers, undervaluation or overvaluation bets, purely statistical bets, sympathetic bets, and many others. (Think of templates in C++.) **Trade Structure**: A single legged option or a combination of legs into a more complicated structure to express your view of the market and serve as the template for a bullish, bearish, or neutral play. (Think of this as you would a class in Java or Python.) You can see examples of the typical trade structures, each of which have common names, here: [https://www.optionsplaybook.com/option-strategies/](https://www.optionsplaybook.com/option-strategies/). I read a statistic that some astronomical percentage of options traders only trade long calls and long puts; it was well over 80%, and possibly over 90%. Only a very, very small percentage knows how to trade, and does trade, multi-legged trade structures. Also, it's worth pointing out that professional traders generally trade volatility, not direction. **Play**: A fully specified instance of a trade structure with concrete striking price(s) and expiration date(s). (A play instantiates a trade structure.) At least, this is how I try to keep things straight in my own mind. :) It's also very important to understand how the Black-Scholes-Merton (BSM) differential equations work, conceptually, and what the (literally dozens of) parameters, the most well-known of which are delta, gamma, theta, vega, and rho, which describe option price sensitivity to various factors, mean, and how they can be used to give you a preliminary idea of risk, among other benefits. As a software engineer, you have a great advantage. If you can get the data that you need, you can develop Python code (or use R, if you prefer it; Python is easier and more flexible, especially if you ask Google Bard for help!) to explore potential strategies. It's amazing how many incredibly useful Open Source Python financial libraries there are. The learning curve is steep and requires a background in statistics, not to mention a very, very high threshold for frustration. The main barrier to entry is that it's (very) expensive to buy granular data from a data provider, and you would need not only underlying, e.g. SPX, data, but comprehensive options chain data. Generally, the sampling frequency is every minute, which isn't great, but you can also try to create your own fine-grained miniature data lake by intercepting data from a retail brokerage and streaming it to a flat file in near-real-time. I work with a very small team that does this type of work. Finally, there is another, almost mystical, side to this: discretionary trading. This is predicated on an individual brain's ability to recognize meta-patterns from years of intensive observation that would be difficult to turn into an imperative algorithm. Discretionary traders might call this intuition. If you want to learn to trade options profitably, it's going to take a long time. There are many different ways of trading, and each can have an edge in the right market conditions. Part of what you should do will depend on your particular personality and skills. Follow your strengths. The initial step is to learn how all of this works conceptually, bit by bit, and start developing concrete skills through repetitive drills. These are some of the kinds of things that I'm going to teach in the LearnOptions subreddit. Feel free to join, if you like. The journey of a thousand miles begins with a single step. After that last mile, I hope that you'll find financial freedom. Artyom
> I completely understand all the terms used and math behind the concept All? You are 99th percentile, then. I couldn't backsolve IV out of the BSM PDEs to save my life, but I know enough to know that's a thing. I'm not being sarcastic, I'm being legit impressed. > what is the best way to choose assets to take options positions on? It really depends on what edge you are trying to exploit. If it's a straightforward "stonks only go up" play, I like to roll ATM calls on XSP (little brother index option to SPX), because I'm poor (so the size of the contract matters), I like cash-settlement, and the market is comparatively liquid, though not the super start that SPX is. Come to think of it, a good way to start is screen out all underlyings with bad option liquidity. That's like 80% of the universe of underlyings eliminated, making your job of selection much easier, since the list of stocks will be much shorter. I like to see the bid/ask spread of the ATM monthly calls being no more than 10% of the bid, as a general rule of thumb for good liquidity. If you are playing for reversion to the mean of IV over middling terms, like 30 to 60 days, you'll want stocks with historically high IV. You can filter for that with IV Rank or IV Percentile. If you are trading 0 DTE gamma to minimize theta, that's a whole other set of criteria (although here, the answer is usually SPX). Etc.
Just a clarification, there were standard option pricing models before BSM. The concept of random walk/normally distributed returns was implemented by Louis Bachelier in the 1800s, and before that traders would consider the fair price to be where the risk/reward was equal for the buyer and seller. BSM is obviously better, but even today there are still improvements to option pricing being made since BSM does make some assumptions that don't reflect reality.
> How exactly are prices calculated and then if you need the volatility to calculate the prices what volatility is one using? What prices do you mean? The prices quoted by your broker were NOT calculated. They were set by the market. The price calculated by a model is the *modelled price*, or *theoretical price*. That is not the same thing as the *market price*, which is what you see quoted by your broker. BSM takes volatility as an input and outputs the theoretical price. To calculate IV, you **input** the market price and back-solve the PDE for volatility. That's IV. To get a theoretical price without using IV, you can use the standard deviation of the closing price of the stock over the previous year (calendar or trading year, up to you to pick).
following the Hilpisch book Derivatives Analytics with Python, my IV computation is the same with BSM, Trees, or Montecarlo for European options, but not for heston / stochastic jump
Although it's done frequently, one should be very cautious when looking at IV as a predictor like this. IV is just a result of solving an option pricing model for the volatility that gives the current market price. The way this expected move is computed in your logic is relying on a symmetric (normal) distribution and essentially ignores drift. Returns are neither normally distributed nor is there no drift. Ignoring this fact, looking at IV like that is not necessarily useful either for at least two reasons: 1 ) Empirically, IV tends to overestimate RV, commonly referred to as [Volatility Risk Premium](https://towardsdatascience.com/what-is-the-volatility-risk-premium-634dba2160ad) 2 ) IV is the only free parameter in the Black-Scholes-Merton (BSM) model. Higher IV can be a result of compensation for tail risk. That's also why every strike has a different IV, although all options refer to the same underlying. A lot of retail option brokers provide numbers like this because it sounds good and can generate nice stories for trade ideas. However, professional systems like Bloomberg usually don't even compute these numbers. The only calculation Bloomberg provides is the risk neutral probability density function implied by options (on OVME for equity if you tick the box in the scenario tab). A lot more details about IV can be found [here](https://quant.stackexchange.com/a/76367/54838).
Yep, ultimately the premium is what the market is willing to buy/sell the contract at the current moment. You could use fancy equations akin to BSM etc to derive a close approximation to the premium for a particular contract, but that's out the window when buyers are jumping asks or selling are slamming bids. I'd recommend anyone entering the option market to use the break even as the gauge for where profitability lies (i.e for long calls : premium+ strike= break even). This is a simplified concept since IV can spike the contracts to become profitable, but it at least gives a rough gauge for where profitability is in the trade. However i'd argue the premium consists of greeks and market sentiment for the underlying. Sky high premiums just indicates certain greeks are elevated (Vega mostly here), but also buyers are more willing to over pay for contracts. It is not necessarily a bad thing if price action meets and exceeds the implied move. I agree SMCI options are extremely expensive, but the volatility on the underlying does warrant extreme premiums. Literally 200-300 dollar intra-day moves are insane, but there's a lot of money that can be made (or lost) in entering those positions. So, like any trade and any good trader here, risk management is key. Options should be used as hedging, and even though I made 5000%+ on naked long calls, I strong discourage that type of trade.
1) There are many strategies that traders use to make gains. The choices are based on the capital available, risk tolerance, knowledge and experience, stocks/ETFs selected, and a number of other factors. There is not one strategy that all use. 2) The brokers provide pricing and indicators with at least some of these based on the BSM. 3) The Greeks are like the gauges of your car in that they show what the trade is doing. They do not impact the share price. 4) Looking at the IV formula there is nothing that accounts for volume, but I’m not an IV expert so someone else may be able to answer in more detail. You may want to post over at r/algotrading where there are many working to do the same thing you are trying to do.
Read the OP and imagine you need to help an undergraduate student, ffs. Would you not just accept a single model like BSM and a single method of calculating historic volatility to illustrate a point, or would you get into the whole "but many people use different models and all of them are wrong" sort of spiel? This sort of spiteful argumentation tactic is clouding your reasoning and is causing issues in your trading....and career. And yes, I am more qualified in math, finance and option pricing than someone looking for soft PM roles in "trading", i.e. a failed trader who is transitioning into a "management" role.
How did you just prove my point and call me an idiot? One theoretical value implies that there are no degrees of freedom in terms of assumptions. As soon as you say you’re assuming BSM and a specific volatility and, it follows, a specific interest rate then of course there is one specific theoretical value at that point. BSM is a deterministic function so if you make all the assumptions about variables then it’s going to provide only one theoretical value. But you saying there is **only one** theoretical value for a given option is insane. You can make an argument for plenty of different volatilities and interest rates to plug into the BSM model, not to mention an argument for plenty of other models being more accurate than BSM.
For the sake of the argument on THIS topic and thought process, lets assume that we would be using BSM. "Is it from a 252 day realized volatility or a 126 or a 60 or a 20?" I said "Calculate the standard deviation of the daily returns for the past year for both stocks" "(Hopefully this shows you how dumb the idea of there being one correct theoretical value is)" I will refrain from calling you an idiot, even though I am itching inside.
How do you propose there being one and only one correct theoretical price for an option when there are tens if not hundreds of options models? Is the one theoretical value from a BSM or binomial or trinomial or sabr model? Is it from a 252 day realized volatility or a 126 or a 60 or a 20? Is it based on the prevailing zero coupon bond or the interest rate on marginal capital in your specific account? Curious to hear how this **one** correct price is calculated.
Take two companies, with the same stock price, say $100. Calculate the standard deviation of the daily returns for the past year for both stocks, and somehow, they end up being the same. Their options should be priced the same. However, for stock A, the at-the-money call expiring in 30 days is priced at $10 and for stock B, it is priced at $20. Why? People who are clueless will say "Well it's because the IV is higher!!" In reality, IV is just a plug in the BSM model to come up with the market price of the option, which is different than the theoretical price. The theoretical prices for options on stock A and B are the same. So, the bottom line is that supply and demand for the option itself are driving the option prices, and for stock B, there is some reason that the demand is higher and people perceive the likelihood of that option ending up ITM, versus stock A which is perceived to have less of a chance to go up, in the eyes and pockets of the option traders.
This represents the profit/loss (P/L) that can be attributed to the difference between the implied volatility (σ_implied) and the realized volatility (σ_realized) of the underlying asset, scaled by the option's Vega. Vega measures the sensitivity of the option's value to changes in the volatility of the underlying asset. This is derived from BSM. I found this very well explained in the 1st chapter of Euan Sinclair's "Positional Option Trading" and it stuck with me ever since.
Yeah, seems to be a mistake. In this case just a general errata. That said, Natenberg isn't exactly great in general when it comes to (mathematical) rigor. For example: - He claims that "negative gamma always goes hand in hand with a positive theta". However it's perfectly reasonable to have positive Theta, with a non-negative (0) Gamma as shown [here](https://quant.stackexchange.com/a/73629/54838) with an interactive gif and charts. - He also claims that delta is only an approximation of the probability because interest considerations and dividends may distort this interpretation. In reality, it's not delta but N(d2) and it's also only a risk neutral implied probability which isn't really related to a real world probability in any case. - He explains that it's possible for an option to have a positive theta such that if nothing changes, the option will be worth more tomorrow than it is today because of the depressing effect of interest rate and shows a call option chart. While positive theta is possible for a long call, it's because of dividends, not rates. On the other hand, the book was likely written in a way that builds the intuitions needed to be a successful trader, not a quant. The book offers a great hands on approach with a commonsense point of view. It is sufficient to know that theta can be positive, that delta is approximately equal to the probability and that usually gamma and theta have opposite signs. Anything more complicated than the concepts mentioned in the books will likely not help you much on a trading floor. In the words of [Euan Sinclair on P.XV of the preface in his book called Option Trading](https://www.amazon.com/Option-Trading-Volatility-Strategies-Techniques/dp/0470497106): "Many successful traders have had only a basic understanding of the mathematics behind BSM and certainly have no idea of the differences between calculus and stochastic calculus."
I'm trying to apply Euan Sinclair's formula to real world options, while I'm reading his book. The formula and excerpt are linked in the post. I'm not trying to model BSM. I'm just not knowledgable enough with the actual math to do this confidently. Perhaps I should have posted this in a math-based sub.
Are you trying to use the model to hypothesize different inputs? In my opinion, BSM is rather formulaic so just pulling it from somewhere else is fine (even if tracking it over time) unless you are trying to model different assumptions for their effect. Never forget the guys who invented it got the Nobel Prize, opened a hedge fund called Long Term Capital (kind of hilarious in retrospect) based on their model and promptly went bankrupt almost taking down several of the major investment banks.
FWIW, since this section is just recovering Black Scholes from first principles, you could just find a Black Scholes model spreadsheet and use it. Or use an online BSM calculator, like either of these: https://www.wolframalpha.com/input?i=black+scholes https://www.optionseducation.org/toolsoptionquotes/options-calculator But if you really want to re-invent the wheel, the conventional way to use σ in BSM is on the same time-base as the time variables, which are a fraction of a calendar year, for example, 0.5 would mean 6 months. So that means take the standard deviation of daily price (closing price or opening price) for the trailing 52 weeks.
This is a great question. The option prices move differently. And obviously, with a long call or long put, you're paying up-front rather than being paid, when you short a put or call spread. When you buy a long call or long put contract, you're making a directional bet, and time is against you. In a sideways market, this can ruin your play. It's hard to be right about direction, especially in these market conditions where we're at an all-time high and there are all sorts of macroeconomic catalysts that can cause unexpected reversals midday. A long call or put is also an unforgiving play compared to a long butterfly, for example. When you short a put spread, you're also making a directional bet, but with a margin for error built in. Let me step back a little bit. the ATM Straddle price, at any given time, is a proxy measure for how much the market maker expects the price of the underlying to move 80% of the time. (When the market maker is wrong, it tends to be really wrong.) So, let's say that you're trading XSP as the underlying, and you see that the ATM Straddle price is 2.00. If there's a mini-crash at the open that drops 3.0 standard deviations down from the 240-minute (assuming you're 0 DTE trading) moving average, if you're able to short a put spread with an upper striking price that's 4.00 lower than the underlying's spot price when you believe it's at its lowest point (this is why seeing the standard deviation curves is helpful), you have a good chance of winning the play. Just make sure to add a stop-loss that's 3x the credit received. This type of play might not come up very often, but when it does, it has a high probability of succeeding. You should, however, monitor and truly understand the macroeconomic catalysts (see: [https://tradingeconomics.com/calendar](https://tradingeconomics.com/calendar)), not front-running them, but paying attention to how the market responds, to guide your decision-making, such as whether and when to execute a shorted put or shorted call spread. What you really want with this type of play is for the price to move violently (quickly and a relatively large move) in one direction, around 3 stdev away from the 240-period moving average. The strategy relies on the case that the market panics from time to time, but can't sustain its panic and will reverse at least to some degree intraday. One more thing: In a non-trending market, a long call or long put has horrible performance characteristics compared to other trade structures. You can see this for yourself by purchasing CBOE data on SPX and then using Python (or even Excel) to calculate how long call or put prices move relative to, say, a long butterfly. Nothing will give you greater insight than playing with the actual data, so that you're not trying to guess. Learning to trade options effectively really requires becoming intimately familiar with how the prices of different trade structures that express the same view of the market move. Let me step back a bit again. You should never start with a trade structure, but a view of the market. What's that? It's your belief about where the underlying will go, and by when, based on macroeconomic catalysts on the economic calendar, the overnight move in futures (/ES) trading, high-level patterns that form, how different sectors are performing, the advancing/declining ratio, the underlying's options chain skew, large block orders at various striking prices to try to discern likely support and resistance levels, and various descriptive statistics, and the nuances of technical analysis, which you can learn from Adam Grimes's book, The Art and Science of Technical Analysis, which you can find at [http://libgen.li](http://libgen.li) by searching for it. Multiple trade structures can express your view of the market. The challenge is to pick the right one for the market conditions. In a strong bull market, a long call makes sense. But that doesn't happen very often. In more challenging market conditions, such as we have now, use a more forgiving trade structure, such as the long butterfly or a ratio spread. Learning to trade options involves a lot of knowledge. 95% of retail traders have only heard of "call and put" options, but they've never heard of option pricing models such as Black-Scholes-Merton (BSM) or the binomial model. This is a serious liability, because you're trading against algos and professionals. It also puts you at a significant disadvantage to not know a dozen different trade structures and which is the best to use for your view of the market, when there are multiple possibilities. The only way to know, again, is to compare how the prices for those different trade structures move with the underlying, and the best way to do that that I personally know of is by acquiring data and writing a Python script to parse it and create a visualization. You can use Google Bard (better) or ChatGPT (not quite as good) to help with developing the code. Finally, I've personally found it helpful to learn from someone who knows a lot more and has been trading for decades. You not only need to be able to follow an expert's trades, but be able to ask him detailed questions. One free way of doing this is to look at the videos that ShadowTrader puts out on YouTube. Pay attention to those times when he shows his trading spreadsheet in a video. Stop the video and zoom in. Write the trades down. Look at the chart for those time periods. Try to understand why he did what he did, and how it played out. Pay attention to adjusted trades. In my opinion, it's easier to start with longer-dated trades, and then shorted the time frame as you gain experience. (The shorter the time frame, more capital-efficient you become.) At some point later this year, I might start tweeting free trades in real time to help others out. If I do, it won't be a get rich quick scheme. There's a big difference between professional trading and gambling. I go after high-probability wins, not high-gain moonshots to impress people. The latter is a proven path to blowing up an account over time. I hope that this helps a bit!
For volatility: TPL HPK For dividends: BSM DMLP
And to your point, not how options are priced for that exact reason. We know markets are not lognormally (Gaussian) distributed, but lognormal with kurtosis and skew. BSM was a revolutionary concept that led the charge but even after Merton adjusted for variable rates, we’ve moved far past their pricing models into jump diffusion E/GJR/T GARCH, etc.
Hello Thanks for the pointer on py_vollib. Given the BSM assumption of constant volatility across strikes, is there another Python pricing model that can take similar inputs but generate prices in accordance with the IV smile? Thanks
Simple put, volatility is where the edge is. 1. Yes 2. Most people can't. Collective pricing is the market buying and selling based on demand, models, etc. Most brokers will show you the IV value that will produce the current market price for a contract when plugged into the Black Scholes Model, but there are many other models out there. BSM is just the most common and most accepted. 3. Trading on IV alone is the equivalent to trading with just the quoted price if you are using BSM. (not exactly, but I don't think that's your point in this question) 4. Yes, options are zero sum. One side must win and one side must lose, but some market actors expect to lose (hedging), some use complex positions where they don't care about price movement, so if you are trading only on direction their actions may not be adversarial to yours or even create opportunities for you. However, yes, at the end of the day its all 0.
Note that VIX can be unpredictable since it's measuring the IV of SPY which is kinda the endogenous variable in the BSM model, the IV of VIX is even more unpredictable. So selling naked calls or puts is very risky because if you engage in a short call and IV spikes +30% in a day also its IV will increase and you can rapidly go -100% without triggering your order to sell asap
Each basically tries to address a short-coming of BSM, like computational efficiency, handling early expiration, better assumptions for volatility, etc. I don't know of an objective comparative study and I'm not sure one would be possible. Since each successive improvement is an optimization of some deficiency in BSM, you can't really compare them blindly. You'd have to first group them by whatever they are trying to optimize and then evaluate them against each other on how well they achieve that optimization. But that means you can't necessarily compare a model that handles skew and kurtosis better vs. one that handles early exercise better. In any case, here are a few models to look into. These will at least give you some starting points that you can base further research on and maybe find head-to-head comparisons. https://www.wallstreetmojo.com/option-pricing-2/ https://www.macroption.com/cox-ross-rubinstein-formulas/ https://www.macroption.com/leisen-reimer-formulas/ https://medium.com/@polanitzer/option-skew-part-8-modified-corrado-su-2004-and-haugs-skewness-kurtosis-put-call-b9c048c150ee https://fastercapital.com/content/Option-pricing--Valuing-Options-using-Monte-Carlo-Simulation.html
Don't feel bad. I've seen other finpros post on here that they also use BSM in certain cases. Are there better models? Sure, but sometimes good enough is good enough.
Ouch! Our place uses BSM with additionally defined Greeks. Can't say we are the epitome of sophistication though. For market price, you define a model over market data and use it. Most naive model could be mid point of top level prices. Since your use case is not for hft and presumably works at a 1s or 1m data then just smoothen the data via moving average of mid points [between two sample timestamps] and that should be good enough for the start. It all depends on the resolution or precision you want in the data. Any further changes to the model can be considered as "alphas".
I've been working on my own volatility and portfolio risk modeling package for a bit. It's not intended for HFT market making but I would like to identify vol mispricing fairly quickly. One of the things that I see CBOE LiveVol provides is "theoretical IV". Given that IV requires a market price as input, what market price would be used to calculate this IV? Or is this where the ultra-secret proprietary pricing models come into play? On that note - I'm guessing that no successful options MM is using just vanilla BSM, but how far off are these internal pricing models? Are they slight modifications to BSM, significantly modified BSM or not related to BSM at all? Or is it a mix of various models including (or not) BSM?
Why would you even want a word or metric for volume, when the word "volume" describes it pretty accurately? Also, an options strategy can be created around theta/vega/delta/gamma, but you cannot create a strategy based on volume. The price of an option is not in any way based on volume (I'm talking in terms of BSM models and not about wide bid-ask spreads).
No. Greeks are not a "tool" either. You got it right when you said the Greeks are not a strategy, but then the rest of your post was essentially describing flawed *strategies* based on the Greeks ("Seeing high theta?...sell a put", "low delta?... consider a neutral strategy". Both examples are absolutely *not* what traders should do, btw). Greeks are just properties of an option at a particular point in time. That's all. Just as a share has a last price, current bid & ask, volume, plus various other calculated values/indicators; an option has it's Greeks. Most of the greeks are calculations using BSM or Binomial Tree based on current premium, etc, so they are, *at best,* extrapolations of observed data that you can then use to estimate future premiums under various scenarios or use them to observe trends (such as IV vs real vol over time) to aid in trading decisions. Your intent is good - you are making a point that new traders with training wheels on shouldn't be smashing words together that include the greeks and sharing the resulting word casserole with the world as though it is a divine message. But you are also clearly not entirely knowledgeable wrt options, and yet here you are making your own word salad right here in this very post. Be careful that you are not the subject of your own warnings.
I know, like I said, I understand what you meant. It's just that your original comment reads like BSM is the *only model with a vol flaw*, and that would be misleading. One might conclude from your comment that no modern institution or market maker would use BSM for any purpose, but that is patently false. Are there models that do vol better than BSM (not to mention dividends and early exercise)? Yes, absolutely. But "better" is not the same as "flawless."
BSM assumes constant vol across strikes, the main flaw of the model. This is well known and I don't see how it could be interpreted as misleading, certainly not false. The point is that you shouldn't take a model and run with it expecting everything to work out, if anything, not mentioning the downsides of models would be misleading.
I understand what you mean, but the way you stated it is at best misleading, and at worst patently false. BSM is not "flawed" because of vol. It accounts for vol in a perfectly logical way. What you mean is, no model can accurately predict vol because vol is intrinsically unpredictable. So it's not like BSM is flawed but some other model is perfect. They all have that same flaw to a greater or lesser extent.
The calculation you describe is essentially an approximation of the same calculation done via integration inside the BSM model. The only real difference is you are approximating an integral with a 1000 individual points. If you did an infinite number of points, your EV would converge to the BSM price. Also doing a binomial tree with infinite steps will also give you the same EV. Selling vs buying the same option will have exactly opposite EV's by definition. Otherwise you could simultaneously buy and sell the same option (in separate accounts?) and make money. That ain't going to happen. As others have commented, BSM and its IV are knowingly flawed representations of the real world of options. You will always get an incorrect price from an incorrect model If you can determine the true future price probability distribution then your idea would work. You would also win a Nobel prize.
Yes, this is something that you should consider when trading. Keep in mind, though, that the EV is based around probability which is expressed through volatility. BSM is flawed when it comes to vol, so any numbers you get out of it will not exactly be accurate. It takes a lot of research and exp to build a decent vol model, and it still won't be very good most likely. This is what the big shops do differently and better than retail.
BSM= algebra? It involves some elementary calculus. Also BSM does not account for term structure of interest rates and volatility + a few other matters. I don't know anyone who uses BSM except to understand the basics. It's a starting point.
When I did my first analysis with just calls, the option data I was using ended in May I think. If you looks at the first day in my data set to last day, the market was down big. So the trend would likely have meant more puts ITM vs calls. Could definitely look at a lot more variables, but I think I leave it for now. Going to try and learn more about the BSM and other indicators. Specially going to look at put/call ratios grouped by ITM/OTM/Total and DTE windows. For instance if I use DTE from 1-14 days if that has any better correlation for future prices 7, 14, 30… days out vs perhaps a DTE window of say 30-60.
Not sure why you take the time to do this study, if you unpack the solution to BSM (delta is one term of the PDE) you can get the probability of ITM given BSM's assumptions. (hint: Delta is not probability of ITM, and the probability of ITM as computed by BSM is also wrong, because it assumes a gaussian distribution of returns) Start here: [https://en.wikipedia.org/wiki/Greeks\_(finance)#Formulae\_for\_European\_option\_Greeks](https://en.wikipedia.org/wiki/Greeks_(finance)#Formulae_for_European_option_Greeks) From the second term in the solution, you can clearly see it's the expected present value of the strike, and if you know what expected value is you would know that the probability that S > K is N(d2), where N is the standard normal cumulative distribution function. N(d2) = (ln(S/K) + (r - q + 0.5 \* σ\^2) \* T) / (σ \* √T) That is the BSM probability that your option finishes ITM. If you really want to have fun, now you can say if the option is trading at fair value, what does that imply about the inputs to BSM? Turns out there is only one free parameter - volatility. Do you agree with the implied vol? Why or why not? That might get you to a place where you understand if options should be sold/bought
BSM is special because they were able to derive an exact formula for a European call option, whereas before there was always an approximation/tree involved (90% sure this is right, learned about the history a while ago at uni). Not disagreeing at all, just wanted to add. > BSM is not made for optimizing directional trades. Definitely, volatility pricing models are used to trade, surprise, volatility. BSM actually makes the assumption that stock returns are perfectly log normal, which isn't the case, as well as some other inaccuracies like how it handles vol at different strikes. If you are using options to leverage delta, you want to look at actual stock models. You can look for underpriced options and buy those unhedged, or sell overpriced ones, but at that point you're just trading volatility poorly.
BSM is in the end just a model to help you make sense of the factors that affect option prices and it's especially useful (in its more modern variants) for market makers to hedge their option portfolio and remove big arbitrage opportunities from the market. Options were traded on exchanges for hundreds of years prior to the formulation of BSM, albeit at much smaller scales. You can get a good intuition of options without BSM, but BSM surely helps you get that intuition if you are fluent with relatively simple algebra. I prefer to look at options with probability trees. I find it more useful for constructing my option portfolio because I can more easily integrate conditional correlations between my various underlyings. I'm not really looking at Greeks, I'm looking at branches in the probability tree of multiple asset prices that look undervalued to me given my pricing and understanding of the market. I will consult Greeks just to make sure I'm not taking on too much delta and too much gamma risk, but that's it. All that to say, there's not a unique way to trade and what's best is to get your hands dirty with very little risk for at least half a year or to paper trade until you're comfortable enough to trade for real and find your own edge. Don't try to copy other traders too much because you probably won't be able to copy what makes their edge work, they're unlikely to share it or to even be able to explain it even if they tried.
The Black-Scholes-Merton (BSM) vs Heston debate really hinges on your setup. 1. Volatility: • BSM assumes constant volatility, not ideal for long-term warrants. • Heston adapts to changing volatility, better for longer expiries. 2. Efficiency: • BSM is computationally easier, good for a quick pricing. • Heston is more accurate but demands more computational power. 3. Market Data: • Scant volatility data for private warrants? BSM might struggle, Heston could cope better. 4. Ease of Implementation: • BSM is simpler, Heston is more complex but could offer a deeper insight. 5. Monte Carlo: • Another option, but maybe not ideal given the data constraints.
Sorry, this is for a equity warrant portfolio I assist managing. It currently uses BSM and I real the model validation findings and it mentioned that other models could be utilized, which I then wanted to know which model would be more efficient for our valuation purposes. It is not an actively traded portfolio, but simply call warrants that we may or may not exercise.
Why wouldn't MCS be relevant here? BSM is just foundational theory at this point. Everyone use newer models or custom models that make adjustments - especially when the options are European. All models have assumptions and drawbacks.
BSM isn't that useful in modern options. It can't handle dividends (modeling a constant dividend rate is incorrect), changing interest rates, American expiration (matters in the wings), or non-constant volatility. Binomial is much more useful.
If you want to make it even easier, I'm sure there are a ton of BSM excel sheets pre-made out there. It's a fairly common thing to make for derivatives courses.
BSM has been my favorite for a few months now, long term growth with strong dividend.
The Python library, you mean? Yes, as it uses the BSM model. But I only trade SPX options, so I'm good.
Just apply BSM to the underlying price of your stop loss adjusting regularly for change in DTE and using a volatility estimate based on the resulting move.
Well, there's Black Scholes and then there's Black Scholes. As I understand it, it's standard practice to use a PDE solver with Black-Scholes PDEs for near the money strikes, since those are unlikely to be exercised early, which would violate a pre-condition about BSM. However, you mentioned that your stock has a big dividend. That also violates a pre-condition for BSM, unless there is no ex-div date between now and expiration.
yes, there are ways of achieving this. forget about the so called "strategies" those are just option positions (iron condor, credit spread). you need to find an statistical edge. and that statistical edge comes from the option pricing. the BSM assumes that you can hedge a call option with the underlying to make the PnL=0 then solves the equation. the only missing component is volatility. you need to enter volatility in the formula to get the "fair" price. but you can get prices from the market and solve for volatility. now the question is, is that volatility "fair"? well there is plenty of evidence that shows that volatility sellers have to be paid to take unlimited risk, that is called "variance risk premium". Basically states that nobody will sell you volatility if you don't paid them a premium. that premium is set by the implied volatility at the time of the transaction. and on expiration or trade close the difference between the historical volatility and the implied volatility defined at open time will define if that premium was positive or negative. in the papers bellow the talk about a negative premium, implying that option buyers pay a premium to options sellers. [https://deepblue.lib.umich.edu/bitstream/handle/2027.42/74142/0022-1082.00352.pdf;jsessionid=F4F5B31628FA6FB519993D62E03E7056?sequence=1](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/74142/0022-1082.00352.pdf;jsessionid=F4F5B31628FA6FB519993D62E03E7056?sequence=1) [https://ssrn.com/abstract=577222](https://ssrn.com/abstract=577222) ​ this is profitable in "average". which is the same that saying that a roller coaster has an average speed of 15km/h. manage your risk, be consistent and you should be profitable. strangles are friendlier than straddles to retail investors.
Oil royalty companies like CRT, look at BSM too.
this is so wrong it's not even worth a reply, but I'll do it anyways to stop the misinformation from spreading. The deltas from the brokers are binomial for American and European options, sometimes BSM is used for European. For market makers, they use jump diffusion models or other variants on it since they price in the gap risks.
> The binomial model is not used much these days for (American) option pricing. If you say so. I heard this from a couple of current or former MMs in AMA threads within the last year. IIRC, one of the MMs said they use CRR or BSM (though they might have meant PDE solvers for BSM), depending on moneyness and American vs. European. One of the commenters on an AMA called CRR the "gold standard" used in all of the various jobs they've had in the industry. That was in fact the first time I'd even heard of CRR, just like PDE solvers today. I hadn't heard of PDE solvers/grids before, but after doing a quick search, it does look like the greeks fall out for free, which has to be way more efficient that CRR.
>IV is an input into the option pricing model that provides the market price of the option you look at. This is backwards. The market price is given and BSM is calculated with an iteration of candidates for the volatility parameter until the output price is "close enough" to the given market price. Newton-Raphson is one common method for calculating IV, another uses brute force bisection.
As mentioned in a comment, a straddle does not represent a 1SD move as suggested elsewhere because the price of straddle is equal to the mean absolute deviation (MAD) of the stock price. You can see an explanation [here](https://money.stackexchange.com/a/155064/109107). Also, the whole idea that IV can be turned into a symmetric range is flawed for several reasons. ou can see find a decent, non technical explanations of IV [here](https://quant.stackexchange.com/a/76367/54838). In a nutshell, since every moneyness level (strike) has a different IV, although you have only one underlying, you cannot back out the range, as it would be different for every option you look at. There are two simply explanationsL 1 ) Empirically, IV tends to overestimate RV, commonly referred to as [Volatility Risk Premium](https://towardsdatascience.com/what-is-the-volatility-risk-premium-634dba2160ad) 2 ) IV is the only free parameter in the Black-Scholes-Merton (BSM) model. Higher IV can be a result of compensation for tail risk. The second point is best explained by quoting from [Just What You Need To Know About Variance Swaps - JP Morgan Equity Derivatives](https://www.sk3w.co/documents/volatility_trading.pdf) > For each strike and maturity there is a different implied volatility > which can be interpreted as the market’s expectation of future > volatility between today and the maturity date in the scenario implied > by the strike. For instance, out-of-the money puts are natural hedges > against a market dislocation (such as caused by the 9/11 attacks on > the World Trade Center) which entail a spike in volatility; the > implied volatility of out-of-the money puts is thus higher than > in-the-money puts. Breeden Litzenberger (B&L) showed how you can get the conditional risk-neutral density of the underlying using the second derivative of the option price with respect to the strike. Essentially, traditional Arrow-Debreu prices pay 1 dollar in one particular state of nature, so B&L rescale that butterfly strategy. The link I gave above where IV is explained shows the resulting probabibilty density function alongside a vol surface in a GIF. Last but not least, even this is just a risk neutral expectation that is more likely than not different from a real world expectation. However, that is all you can get from option prices.
Under BSM assumptions the straddle is less than the implied one sigma move, it's ~0.8
If you work at a cash register, should you know how to add numbers. Not necessarily, because a computer does it for you. Knowing that 1.7 + 6 isn't 23 will help you figure out if something looks off. Everything is based on the ideas of Black Scholes and Merton. If you don't care to not understand why prices change the way they do, you don't need to know it. On the other hand, I'd claim there is a reason institutions ask you a lot of details about BSM if you apply for a job as an options trader.
I recommend you read the volatility smile by Derman and Miller and the volatility surface by Gatheral. The first book as an introductory book to volatility models, BSM, Heston model, jump diffusion, local volatility model.
Yes, sorta. Options pricing models attempt to find the "fair" price of an option, IE the price where both buyer and seller are expected to gain as much as they lose. That would be the midpoint of the spread. Long term, if you have no edge, you should expect to gain/lose the same amount, netting to $0, according to BSM. Picking a different underlying, different DTEs, or different distances OTM/ITM doesn't affect this. Market makers are taking a cut of the money on the table, so it's actually a negative sum game for both the buyer/seller of options.
Yes. It's works with the strike price inputed into the BSM parameters. The implied volatility solution just requires the listed call and put prices be for the same strike price. The strike needn't be at the money. Any strike will work as long as the BSM strike input corresponds to your listed options. Just email me for a copy. ProJ-Money
Black Scholes Merton Theory is an interpretation of market prices. A curve does not violate the BSM theory.
How does this read? I'm new to reddit in general and in particular r/options. I've lurked for years but finally decided to join and post. I’ve written a handy Black Scholes Merton pricing model in Excel. Yes, I know there are several available on here and the web already. I’ve focused on the tables and graphics in my spreadsheet more the others. Google sheets won’t work because certain graphical formats and surface graphs don’t convert. The inputs are the usual Black-Scholes-Merton requirements; security price, strike price, volatility, risk-free rate, security yield, days to expiration. The spreadsheet calculates the BSM option price, parity price, delta, gamma, vega, rho, theta, ∆ Hedge, and forward security price for both the call and put. These values are reflected in the accompanying 3D bar graph at the top of the spreadsheet. There is also an implied volatility computation for corresponding observed call or put option prices (Newton-Raphson). You can solve for either of these implied volatilities after inputting the observed call and put prices. You can then paste either implied volatility value into the input cell for volatility. I also provided an implied security price computation based on an observed call and put option prices using the put-call parity theorem. It will need both the corresponding observed call and put price inputs. Table 1 calculates option prices, the Greeks, and forward security price for a range of security prices for the given strike price. You do need to input the initial security price and security price increment. Table 2 calculates the same option values as table 1 but for a range of time decreases for the given strike price. You do need to input the initial days to expiration and the daily increment. Table 3 calculates the same option values as the previous tables but for a range of volatility decreases for the given strike price. You do need to input the initial volatility to expiration and the volatility increment. Table 4 calculates call options prices on a range of security prices and strike prices. You do need to input the initial strike price and it uses the range of stock prices supplied from Table 1. Table 5 calculates call option deltas on a range of security prices and strike prices. No input is needed. Table 6 calculates put options prices on a range of security prices and strike prices. No input is needed. Table 7 calculates put options deltas on a range of security prices and strike prices. No input is needed. There are also a series of graphs in the sheets depicting selected data from the various tables; the option Greeks, prices, deltas, time, and volatility. What I need are volunteers to check my spreadsheet computational accuracy, embedded graphics appeal, and the overall look and feel. The final version post evaluation will be made available for free on request. Not sure how I’ll distribute it other than as an email attachment. If you’d like to give me a hand, just let me know. Thanks, Prof J-Money
> Isn’t any difference between the theoretical and actual option price just a consequence of different values of IV? Assuming all else equal, like interest rates (which is not a realistic assumption these days), yes. The actual input to BSM is σ, which is based on the annualized historical standard deviation of the underlying. So basically, the theoretical model uses an estimate based on historical volatility. But you don't have to use that. If you come up with a secret sauce proprietary algorithm that makes a better forecast of volatility, you may find an edge, because your theoretical price will be more accurate.
If current 1m IV is for the next 30 days, it takes you 30 days to figure out what historical / realized vol in that period was. If you now shift IV back, you will have a two month mismatch. So you either shift RV back, or IV forward in your time series. You can see a chart demonstrating this [here](https://money.stackexchange.com/a/154961/109107). One warning though. Some people interpret IV as a forward looking measure of standard deviation, just like the commonly used definition of historical / realized vol which is computed as the sample standard deviation of log return as shown [here](https://quant.stackexchange.com/a/71790/54838). However, one 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: 1 ) Empirically, IV tends to overestimate RV, commonly referred to as [Volatility Risk Premium](https://towardsdatascience.com/what-is-the-volatility-risk-premium-634dba2160ad) A simple explanation is that market participants tend to overestimate the likelihood of a significant market crash (or are risk averse / seeking insurance against large decline in their long positions) which results in an increased demand for put options. 2 ) IV is the only free parameter in the Black-Scholes-Merton (BSM) model. Higher IV can be a result of compensation for tail risk. As a result, there is no general IV for an option. Quoting from [Just What You Need To Know About Variance Swaps - JP Morgan Equity Derivatives](https://www.sk3w.co/documents/volatility_trading.pdf) For each strike and maturity there is a different implied volatility which can be interpreted as the market’s expectation of future volatility between today and the maturity date in the scenario implied by the strike. For instance, out-of-the money puts are natural hedges against a market dislocation (such as caused by the 9/11 attacks on the World Trade Center) which entail a spike in volatility; the implied volatility of out-of-the money puts is thus higher than in-the-money puts.
I think, though I can't prove, that every one of your assumptions about options are wrong. Options is a zero sum game, I will accept that assertion for now. I believe you have also heard some hogwash like "80% of options contracts expire OTM" or something similar. You then translate this into your mind as "80% of option sellers are making money, while 20% of option buyers are". That is too far of a logical leap. Even if 80% of options expire OTM, and, therefore, 20% of them expire ITM, this is in no way related to returns due to options activity. As a hypothetical example, what if 80% of options sold were 20 strikes OTM while the other 20% of options sold were 19 strikes OTM. All of these options would sell for pennies due to distance OTM. If the lower 20%, 19 strikes DOTM, all expired ITM, the account that bought the options would see extremely high CAGR. Trading $0.19 for $19.00 over and over will do that to you. So what if the 80% that expired OTM led to you trading $1 for $19 over and over? For the same reason, the hypothetical options seller from above, though they have a win rate of 80%, is losing money hand over fist. Enter the Black Scholes options pricing model, or the BSM. This is the most widely used options pricing model, and the outputs of similar pricing models tend not to deviate widely from those of the BSM. What the BSM attempts to do is to find the "correct" price of an option, such that neither the buyer nor the seller has positive EVs on the contract. That means that all options both bought and sold have an "expected gain" of $0 over the long term. Delta doesn't matter. DTEs don't matter. All of that is priced into the model. That means that "80% of options expiring worthless" is the "pathway" to zero EVs. Even if it was 50% or 90%, that would still be the "pathway" to zero EVs. The money paid out and/or taken in at contract origination is "fair". It's exactly the amount that the seller is likely to lose over many contract iterations and exactly the amount the buyer is likely to gain over many contract iterations. Infinite iterations into the future, it's 50% likely you are ahead of the expectation and 50% likely that you are below it. That means that you are just as likely to have >$0 net returns as <$0 net returns after 1000 contracts if you sell at 50 delta, 40 delta, 30 delta, 20 delta, 10 delta, 5 delta, 1 delta, etc. The same is true if you buy at any of those deltas. Selling at 50 delta, you are going to take more credits in at origination than a 5 delta contract seller will take in. You also have a higher likelihood of the contract being ITM and you will probably have to pay more back out as well, due to how many more ITM contracts there will be at expiration. These additional credits in pay back out in proportion. Similarly, when you sell 5 DTE options, you will be taking in pennies and paying out dollars when you do have to pay out. That's the balance. Even though you win more often, you still pay back out everything you took in after thousands of contracts change hands. I will hope that the above will stand for the time being. If you don't believe me, look up information about random walk and the BSM. The goal is literally net zero EVs for both sides. You also have incorrect assumptions about option sellers in general. They aren't all hedging at the levels you assume they are. WSB mentality is everywhere. Thetagang is just WSB on the other side of the trade. Both sides are exerting serious effort with the goal of losing all their money. Bet big and hope you get lucky, that's what "investing" is coming to mean to the average person in 2023. "Gambling" and "Investing" might as well have the same definition to the average person. If you don't think so, look at the digital coin fad. The only thing preventing every digital coin from being worth $0 is the "Bigger Idiot Theory", essentially that there is always somebody more stupid than you. Options don't have "good premiums" or "bad premiums". As above, the target is to have premiums that result in a net zero EV regardless of market conditions, strikes, DTEs, or what have you. If anything, all options have "bad premiums", because options are never priced such that either side ever "gains". The vast majority of "retail" options activity does not follow from a good understanding of the above information. The vast majority of options traders, if they could come to accept the above as true, would stop trading options, both buying and selling them.
The chasing can be good if you actively manage the chase. $BSM & $AGNC >14%
OK, thanks. May have been worth to mention this detail. When the question is about theta, there is no constant exponent. Likewise, in standard BSM there is also no constant. ​ For anyone interested, here is a Julia code replicating this answer: \- initially the packages are loaded, and the cdf and ndf are defined. \- Afterwards Black Scholes Merton with theta is coded. \- Tenor is an array with 1w, 4w and 9w as in the example (the trading days till expiration - TDTE). \- The constant is the computed value that makes the premium = constant \* TDTE\^ .5 \- lastly, I added actual theta to the dataframe `using Distributions,DataFrames` `N(x) = cdf(Normal(0,1),x)` `n(x) = pdf(Normal(0,1),x)` `#define Black Scholes with theta (cp = 1 is call, -1 is put)` `function BSM(S,K,t,r,d,σ, cp)` `d1 = ( log(S/K) + (r - d + 1/2*σ^2)*t ) / (σ*sqrt(t))` `d2 = d1 - σ*sqrt(t)` `opt = cp*exp(-d*t)S*N(cp*d1) - cp*exp(-r*t)*K*N(cp*d2)` `theta = (-(S * exp(-d*t)*n(d1)* σ )/ (2 * sqrt(t)) - r * cp * K * exp(-r*t) * N(cp *d2) +cp*d * S * exp(-d*t)*N(cp *d1))/365` `return opt, theta` `end` `w = 7` `m = 4*w` `w9 = 9*w` `tenor = [w9,m,w]` `constant = round(BSM(s,k,w/365,r,d,σ, 1)[1]/w^0.5, digits = 1)` `DataFrame("Days to Expiry" => tenor, "Option Value" => [BSM(s,k,t/365,r,d,σ, 1)[1] for t in tenor], "Rule of thumb" =>[constant*t^0.5 for t in tenor], "Theta" => [BSM(s,k,t/365,r,d,σ, 1)[2] for t in tenor])` ​ \[Result\]([https://i.stack.imgur.com/Zl8RL.png](https://i.stack.imgur.com/Zl8RL.png)).
I never claimed you need stochastic calculus in my post. In fact, the books I recommen at the end all use essentially no stochastic calculus. Euan Sinclair in his excellent book "Option Trading: Pricing and Volatility Strategies and Techniques" states that \*many successful traders have had only a basic understanding of the mathematics behind BSM and certainly have no idea of the differences between calculus and stochastic calculus.\* However, the OP asked for a deep dive into the details, which will require stochastic calculus in my opinion. Also, it is not true that big institutional traders do not need to know stochastic calculus. One of my former workmates wrote a summary of all the stochastic calculus he learnt while working at the Interest Rate Derivatives trading desk at Nomura, in London and made it \[available online\]([https://derivativesmaths.wordpress.com/](https://derivativesmaths.wordpress.com/)). Most job posting for derivatives traders have it as a requirement and some companies even list it for interest rate swap traders, like for example \[Barclays\]([https://www.linkedin.com/jobs/view/interest-rates-swaps-trader-at-barclays-corporate-investment-bank-3643622735?refId=GxVLz0ayp65UzfSwgH%2Fnxg%3D%3D&trackingId=fumkKVb0yVSbJ0B%2BK8Vhsw%3D%3D&position=4&pageNum=0&trk=public\_jobs\_jserp-result\_search-card](https://www.linkedin.com/jobs/view/interest-rates-swaps-trader-at-barclays-corporate-investment-bank-3643622735?refId=GxVLz0ayp65UzfSwgH%2Fnxg%3D%3D&trackingId=fumkKVb0yVSbJ0B%2BK8Vhsw%3D%3D&position=4&pageNum=0&trk=public_jobs_jserp-result_search-card)). The link will break in the future but it is a job advertisement for an Interest Rates Swaps Trader at Barclays Corporate & Investment Bank New York, NY. The requirements state that you need to be able \*to use stochastic optimization, differential equations, calculus and other numerical methods and quantitative models to discern pricing options.\*
Yes - if you are writing puts, it is common to use collateral which are at least generating the risk-free rate. Some traders will invest in equities or bonds for higher risk/reward. If you think about how options pricing works and how premium on puts work - think about BSM model - Rho is the risk free rate. If rates are high, you theoratically are getting less premium to account for Rho when you write your put contract. So that's one of the reasons why you always would want to invest your collateral in at least a risk free rate instrument. T-bills are fine - so are money market funds. Or even box spreas if you have portfolio margin. It should be in an instrument that is ideally margin efficient and liquid. Choices will also depend on the type of margin you have. And your broker's margin house rules.
Some people interpret IV as a forward looking measure of standard deviation, just like the commonly used definition of historical / realized vol which is computed as the sample standard deviation of log return as shown [here](https://quant.stackexchange.com/a/71790/54838). However, one 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: 1 ) Empirically, IV tends to overestimate RV, commonly referred to as [Volatility Risk Premium](https://towardsdatascience.com/what-is-the-volatility-risk-premium-634dba2160ad) 2 ) IV is the only free parameter in the Black-Scholes-Merton (BSM) model. Higher IV can be a result of compensation for tail risk. A simple explanation is that market participants tend to overestimate the likelihood of a significant market crash, which results in an increased demand for options as protection against an equity portfolio. So no, it's generally not a good idea. You can read plenty more details about this topic, including several Bloomberg screenshots, and computer code go replicate vol surface construction and so forth in this [quant SE](https://money.stackexchange.com/a/154961/109107) answer.
I don't know thinkorswim. What you could definitely do is to compute it yourself. I will hard code the dividend and risk free rate to zero to avoid daycount complications and changing it to appropriate continuous analogue in the example code. For details, you can look at this [OVML](https://quant.stackexchange.com/questions/70259/quantlib-day-by-day-evaluation-of-option-value/70296#70296) example. Spot, and IV are also hard coded to avoid any decimal precision error that may occur in the GUI (which is rounded unless you manually set to higher precision). I'll just copy the formula for charm from [wikipedia](https://en.wikipedia.org/wiki/Greeks_(finance)#Formulae_for_European_option_Greeks). For example, using [Julia](https://julialang.org/), the complete code looks like this. using Distributions,DataFrames N(x) = cdf(Normal(0,1),x) n(x) = pdf(Normal(0,1),x) function BSM(S,K,t,rf,d,σ, cp_flag) d1 = ( log(S/K) + (rf - d + 1/2*σ^2)*t ) / (σ*sqrt(t)) d2 = d1 - σ*sqrt(t) value = cp_flag*exp(-d*t)S*N(cp_flag*d1) - exp(-rf*t)*cp_flag*K*N(cp_flag*d2) charm = d*exp(-d*t)*N(d1) - exp(-d*t)*n(d1)*((2*rf-d)*t -d2*σ*sqrt(t)) / (2*t*σ*sqrt(t)) return value, charm/365*100 end s,k , t, σ, r, d = 270, 270, 90/365, 0.5, 0.0, 0.0 res = BSM(s,k,t,r,d,σ, 1) DataFrame(Price = res[1], Charm = res[2]) [Julia code with output](https://i.stack.imgur.com/4HUkK.png) Which matches Bloomberg to the decimal as you can see [here](https://i.stack.imgur.com/pCPqm.png). With regards to the scaling, theta (and as such charm) is expressed in value per year and usually divided by the number of days in a year. Moreover, delta is frequently expressed in percent (40 instead of 0.4). The same applies to the way Bloomberg displays charm, although technically what is displayed depends on your setting. You can read more detail in this [answer](https://money.stackexchange.com/a/154958/109107). With American options there is risk of early exercise and as long as your underlying pays dividends you cannot use the closed form solution. Provided you have a solver, you can do it with bump if you have delta, reduce the day by one and compute the finite difference to obtain charm. See [here](https://i.stack.imgur.com/ADqdn.png) for an example code where I added delta, (exp(-d*t)*N(d1)), to the return statement of the BSM function.
Well in my opinion this is relevant because understanding the fact that a dealer is the counterparty in these trades, and the dealer warehousing risk in the aggregate is what creates these changes to the vol surface is key to answering OPs question. If you know this, then it's easy to make sense of OPs confusion by understanding that some counterparty somewhere still inventories the risk that is inverse to the profitable position for the trader. The market isn't just "dealing" with shortcomings of BSM, lol. Each dealer is going to quote a price based on some model - yes. Supply and demand is also going to affect the quotes beyond that. The fact that the vol curve is affected by inventory means these dealers are holding open positions (risk) that they need to be mindful of if filling new orders. See PapaCharlie9's comment because he seems to be understanding the relevant mechanics of this. Simply put, imho, dealer inventory + supply and demand influence vol skew more than anything else... and I get that OP didn't specifically ask about skew, but not having a grasp of this framework is the same reason why OP could not make sense of "where the money is coming from". This is the same reason I brought it up in a previous comment and why I think you're still missing the point here
The original question was about some individual option contracts being sold and bought before expiry. That has nothing to do with vol skew apart from the fact that strike and expiry will determine the vol being implied by pricing. The decision of a single option buyers buying or selling has nothing to do with the counterparty being a market maker or not. I never claimed skew has nothing to do with supply and demand. I claimed the action in the single option contract purchase has nothing to do with this. The reason for the shape of the vol surface itself is simply a way the market deals with all the shortcomings of BSM. Quoting from [Just What You Need To Know About Variance Swaps - JP Morgan Equity Derivatives](https://www.sk3w.co/documents/volatility_trading.pdf) "For each strike and maturity there is a different implied volatility which can be interpreted as the market’s expectation of future volatility between today and the maturity date in the scenario implied by the strike. For instance, out-of-the money puts are natural hedges against a market dislocation (such as caused by the 9/11 attacks on the World Trade Center) which entail a spike in volatility; the implied volatility of out-of-the money puts is thus higher than in-the-money puts." Since more people seek protection but the number of buyers must meet sellers you will have an "insurance" premium but that in fact is just another way of saying market participants tend to overestimate the likelihood of a significant market crash, which results in an increased demand for options as protection. Either way, in my opinion, this is completely unrelated to the question asked.
There is no general IV for an option. Quoting from [Just What You Need To Know About Variance Swaps - JP Morgan Equity Derivatives](https://www.sk3w.co/documents/volatility_trading.pdf) " For each strike and maturity there is a different implied volatility which can be interpreted as the market’s expectation of future volatility between today and the maturity date in the scenario implied by the strike. For instance, out-of-the money puts are natural hedges against a market dislocation (such as caused by the 9/11 attacks on Yhe World Trade Center) which entail a spike in volatility; the Implied volatility of out-of-the money puts is thus higher than in-the-money puts". So based only on the fact that there exists a smile you cannot expect to simply trade based on a difference between IV and HV. Some people, also in this thread already, interpret IV as a forward looking measure of standard deviation, just like the commonly used definition of historical / realized vol which is computed as the sample standard deviation of log return as shown [here](https://quant.stackexchange.com/a/71790/54838). However, one 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: 1 ) Empirically, IV tends to overestimate RV, commonly referred to as [Volatility Risk Premium](https://towardsdatascience.com/what-is-the-volatility-risk-premium-634dba2160ad) 2 ) IV is the only free parameter in the Black-Scholes-Merton (BSM) model. Higher IV can be a result of compensation for tail risk. You can look at IV percentile, which is usually computed for certain moneyness levels and selected expiry dates compared to some set number of past days. For example, a 19% ATM 3M IV percentile means that ~20% of days had lower, ~80% of days had higher ATM IV for that expiry in the observed time span, with a possible range between 0 and 100. You can read some details [here](https://money.stackexchange.com/a/154961/109107), where the smile is explained, SVI fitting is shown with computer code and some empirical facts are discussed in more detail. Overall, I would be extremely cautious of making trades based on HV vs IV (if not ignore it).
It's really hard to compare these price model numbers unless you use identical models. Brokers often make trade-offs in accuracy in order to save on computational effort (time to next update). Is this call you are looking at no early exercise? Or is it American style with early exercise? If early exercise is possible, a discrepancy is to be expected between a BSM calculation and whatever the broker is using, probably CRR with less than 10 steps. Even if both calculations assumed no early exercise, discrepancies can still crop up just by using slightly different calculation steps.
The option pricing models are correct. There is nothing inherently wrong with them. The only thing that is unknown is volatility, which is true. Volatility is determined by market counter-party risks which are typically market makers or the like. The pricing of the options dictates the associated implied volatility and it feeds into itself and volatility is driven by counter-party risk management by which vol rises as option prices do following the options pricing models. Predict vol and you can win. This is why vol mean reversion is so heavily utilized in trading. It is statistical arbitrage-like in nature although managing positional size is very important as well as proper hedging. Understanding where the risks lie for a market maker helps inform about the market more than the BOPM, BSM, or any other pricing model IMHO.