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
Just for fun, plug in the following to a Black-Scholes calculator: * Underlying price: $100 * Strike Price: $10 * Time to expiration: 1 year * Dividend rate: 5% * Risk-free rate: 3% * Volatility: 20% You may notice that the BSM price is less than the intrinsic value. No one would ever sell an American option for less than intrinsic because the buyer could immediately execute it. That means the ask must be at least the intrinsic value and the bid would approximate the fair value.
I should note that taking the limit as something approaches infinity isn't exactly useful in this context but... Under BSM: All of the puts would have a value equal to their strike price discounted by the rfr. And all of the calls would have a value equal to the current underlying price. This is fairly easy to prove by taking the limit of a call or put's price as volatility goes to 100. d1 is equal to s \* sqrt(t) / 2 (which is infinity). Where s is the volatility over the option's period and t is the time to expiration. d2 is equal to -s \* sqrt(t) / 2 (which is negative infinity). So: C = N(inf)S - N(-inf)Ke\^(-rt) = S and P = -N(-inf)S + N(inf)Ke\^(-rt) = Ke\^(-rt) Note: The math above is imprecise (but accurate) to keep my response short.
It might be a harder question than it appears, but it is a good exercise on thinking about calendars. The only definitive payoff line is at expiration; for early time points you need a model. If you choose to use BS, you'll need to either handle non-local volatility or ignore volatility smile. You can alternatively use jump diffusion (Merton) or stochastic volatility (Bates, Heston) instead of non-local IV. I know IBKR uses non-local volatility by fitting the IV curve to a surface, then use that non-local IV on their models for their pricing lines. So there's isn't a simple answer to whether there's a function to do that. Yes, there's a function and libraries (or just calculate BSM yourself), it's just that those results might not be meaningful. That said, payoff lines for calendar spreads aren't always that helpful because they are so vega dependent. If IV doesn't change, simple models' lines might be ok, but the whole point of a calendar is to take advantage of IV changes over time independent of changes to the underlying. You probably need a payoff surface instead of a line (payoff as a function of underlying and IV).
> I’m trading Iron Ore (IODEX) european options on the SGX I'm not familiar with that market, so there might be differences between that market and what I am familiar with, which is US standard exchange-traded options. What follows may not apply to your market. > when trying to reconcile my theoretical prices (using black) with broker quotes. Uh ... that is never a useful thing to do. Instead, you should use BSM to back out IV from the "broker quotes". There is no expectation that BSM, with some random assumption for volatility, should match the market price. You'd have to get insanely lucky to guess a vol that just happens to match a market price. > I’ve noticed that to make my prices line up, I need to manually tweak the Days to Expiry (DTE) — basically adjusting how I count time for theta This is further evidence that your assumption for vol does not correspond to the market price. It's your vol assumption you are supposed to tweak, not DTE. > So my questions Seem to have an underlying assumption that the "broker quotes" are driven entirely by BSM. That would not be the case in the US. In the US, some modeling would be done by *market makers*, and then in most cases, some adjustment to modeled price ranges would be done to build in a profit margin to quotes. Unless the market is very active, like ATM front month SPY calls, the stub orders that establish the bid/ask spread quotes for a low-volume contract would be **discounted** from the modeled price range. This is why it is often possible to fill for a price that is between the bid and ask.
I'm not sure if my post was clear. I never said Monte Carlo was in itself a pricing model, just that it can be used for pricing just like your said. I also said there are closed form solutions to BSM for American options and gave a link to the same citations you gave (a different Wikipedia page that cites the same sources) Everything you said is true, I agree with it, and seems compatible with what I wrote, just with much more (and better) exposition.
Understand how options are actually priced. Learn the BSM model, know how Delta and Vega move with price and volatility, and how all the Greeks interact. Once you understand what truly drives premium, not just direction, you’ll stop guessing and start managing risk. Options are all about risk.
For most liquid underlyings and near-ATM options, Black-Scholes will usually price within a few cents of more complex models. That said, those small price differences aren’t meaningless. They come from how each model treats the volatility surface and distribution tails, and those differences compound when you’re managing multiple strikes, maturities, or a hedged book. Even a few basis-points of mis-pricing per leg can add up in delta-hedged or vol-arb portfolios. Where it really shows, though, is in the shape of the risk landscape. Black-Scholes assumes symmetry: Vega and Gamma peak exactly at ATM, Theta decays evenly. Stochastic-vol or local-vol models reshape that geometry; volatility skew, smile curvature, and term-structure all emerge naturally. So the goal isn’t just to “beat” BSM on one theoretical price, but to see how different dynamics bend both prices *and* Greeks across moneyness and time. That insight is what matters when sizing or hedging real risk.
BSM has always been good enough for me. All the modeling I've done was so close to real prices that it's not worth it to use anything else. Only times where it doesn't work that well is low supply/demand on bad equity, but you'll never get fair price for those.
> skew in the near months(nov-jan) appears to be priced off of a log-normal distribution model(BS). Do you have this backwards? It looks like the higher-DTE options have a flatter IV curve which would imply their forward distribution is closer to log-normal. >Out to 27-28 there is substantial discounting in tmv calls. assuming caused by volatility drag(Heston model). What led you to the idea that tmv calls are discounted? Also, volatility drag can exist outside of the Heston model. In the case of LETFs, I'd argue that the volatility drag reduces the drift rate of the underlying. >Gamma on levered inverse etf options is higher than their levered(non inverse)counterparts when you translate the option’s delta back into the original units. I'd have to think about this more to give a better answer, but my intuition is that this is a model problem. It shouldn't be possible to get cheaper gamma. My guess is that the gamma isn't cheaper, but it appears that way. In order to increase the "price" of gamma exposure, the price of the options goes up, which causes the BSM gamma to go up too. However, this is only meaningful if the option's prices behave how the BSM model 'expects' them to, which you already know isn't true based on your first question about there being skew on some chains. I'm familiar with LETFs, and familiar with options, but tbh, I'm not too familiar with how the two interact. The fact that LETFs have volatility drift that depends on the LETF's target index's return distribution can cause very strange behavior that's difficult to model in the context of options. I'll maybe look into this topic more over the weekend.
Greeks are not linear, so multiplying delta by the change in the underlying's price is only an estimate of how much the option should change. Same thing with theta: Theta itself increases as time to expiration increases. Lastly, Black-Scholes (the equation that is used to calculate Delta, Theta, etc.) is a model of option prices that makes assumptions that aren't true. Options aren't bound to behave consistently with the BSM Greeks.
Yea, 5 delta and lower are hard to price. They don't work well in the BSM model due to these black swan events. Selling them is dangerous for this reason. Buying them can be a cheap hedge, but generally won't pay off unless something crazy happens, like 2008 GCF or covid or other massive events. They almost never pay,but when they do, boy howdy do they print.
IV is a free parameter in the BSM, in other words it's just a number derived from market price, one way you can interpret this number is as the predicted volatility, but it also includes so many other factors i. E skew, vanna, risk reversal , etc that it's really not just as simple as "predicted volatility". IV does not cause volatility, IV is a derived number from supply/demand agreed prices.
"looking at the daily change in options": which daily change are you looking at? If you are looking at "last trade," it could have been a matched ask compared to bid. Options can have a large bid-ask spread. Also they are traded infrequently. One could have happened at opening and the other at closing. The general answer to your question about why an option between two can appear to move less than the others is "don't look at last trade price as an indication of anything because trades happen infrequently and there can be a very large bid-ask spread." Current bid-ask midpoint is probably better. Or roll your own pricing model (e.g., fit the volatility smile then invert it). As for "how are options priced in practice?" Market makers run statistical models (more advanced that BSM) given a set of parameter probability distributions. That gives them probability distributions of outcomes and sensitivity to changes (i.e. the "greeks"). They then pick bids and asks that balance their overall risk while also hedging with the underlying. As prices of underlying and market demand for specific options change, their parameters change and the models are updated, giving new prices. While that's happening, other traders are placing bids and asks inside of the spread. So it's not a simple answer, and confounded by random trades within the spread.
Thanks for pointing out the Deep OTM label.. the optionality of exercise in american options is not being accounted for in this mini study, but why would the optionality of exercise do anything more than guarantee liquidity? I’ve always seen that as a liquidity clause the forces the price to be at minimum intrinsic. My hypothesis for the discrepancy is from the lognormal tails implied by BSM, but your interpretation is interesting. Could you explain more!
Of Course BSM is wrong! It assumes a Gaussian distribuiton of price moves, whereas in the real world, the distribution is much more kurtotic and fat-tailed. Have you tried a Cauchy distribution? It's a much better fit. Or just construct the actual distribution, and interpolate in it.
i chose GME because its known to not be “well-behaved” which is what i seeked to improve with my implied modeling. SPY is obv more standard. Brier loss is the probability equivalent of mean square error on discrete outcomes 1 and 0. I did this to compare the implied probabilities made by delta vs Priced-In (my model). Log loss is a more standard metric, but its pretty sensitive to being far off, and since being ITM is binary… it gives pretty large losses in this case. Its useful here to demonstrate how BSM can be confidently wrong about implied probability/volatility especially at Deep OTM/ITM.
Delta (sensitivity to underlying) actually becomes implied probability if the model is lognormal or binary (degenerate). The way black scholes merton maps to “real world probabilities” is if you believed the current options market pricings were the “best” possible pricing under uncertainty. Except, BSM additionally assumes the underlying to be lognormal. Let me rephrase… lets say I had a “better” way to compute option greeks (the implied sensitivity of the options premium in relation to certain variables), how would you prove that this new model was superior to BSM? What burden of proof would you require? Sorry if my jargon is confusing.
Your question has more to do with discrete time vs continuous time calculations. Assuming your definitions of IV is the 30d ATM IV (which is the default on every platform), it "appears" to increase because the discrete pricing event (i.e. earnings) becomes closer. In reality, the quants pricing the options have always and will always price in discrete events because they're modeling reality outside of the outdated BSM math.
You seem to regurgitate the most common objections to BSM without getting into the crux of the model - it is a yardstick to measure a certain dimension of options, let's call it market demand, and we all know that the yardstick is wrong, but that it can at least offer a value on the dimension on a relative basis. When trading spreads, you care about relative values, and not much about absolutely correct absolute values. "delta is not the implied probability of being ITM" It is commonly used by practitioners as a proxy. Do you even trade or are you a student of accounting? "In other words, you are using the wrong metric, with an imperfect model which doesn't use real-world probabilities." In other other words, I said what I said above - delta as a commonly used proxy for the probability of the option being ITM by expiration can not be used to properly size bets, let along a series of bets with a constantly changing cross correlation.
Page 389, Chapter 18 of Natenburg is a God Send. It will clarify all the doubts. Roughly: ***{N(d1)\* Forward Price of the Stock continuously compounded}*** gives us the average value of all stock prices above the exercise price. Think of this as Conditional VaR where we compute the avg losses once VaR is breached. Similarly ***{N(d1)\* Forward Price of the Stock continuously compounded}*** gives us the average value of all stock prices at expiry date. We then calculate ***{N(d2)\*X}*** which is the average amount we will pay at expiry if we exercise the option \[N(d2) is the probability of option being in the money and X is the exercise price\] Expected value of a call option at expiry would be = ***{N(d1)\* Forward Price of the Stock continuously compounded}*** \- ***{N(d2)\*X} = {N(d1)\* Se\^rt}*** \- ***{N(d2)\*X}*** Now, we need the value of the option today, hence we discount it from expiry day payoff to today Expected value of a call option **TODAY**= *e\^-rt {N(d1)\* Se\^rt* \- *N(d2)\*X}* ***= S\*N(d1) - X\*e\^-rt\*N(d2)*** This is the BSM formula!
Page 389, Chapter 18 of Natenburg is a God Send. It will clarify all the doubts. Roughly: ***{N(d1)\* Forward Price of the Stock continuously compounded}*** gives us the average value of all stock prices above the exercise price. Think of this as Conditional VaR where we compute the avg losses once VaR is breached. Similarly ***{N(d1)\* Forward Price of the Stock continuously compounded}*** gives us the average value of all stock prices at expiry date. We then calculate ***{N(d2)\*X}*** which is the average amount we will pay at expiry if we exercise the option \[N(d2) is the probability of option being in the money and X is the exercise price\] Expected value of a call option at expiry would be = ***{N(d1)\* Forward Price of the Stock continuously compounded}*** \- ***{N(d2)\*X} = {N(d1)\* Se\^rt}*** \- ***{N(d2)\*X}*** Now, we need the value of the option today, hence we discount it from expiry day payoff to today Expected value of a call option **TODAY**= *e\^-rt {N(d1)\* Se\^rt* \- *N(d2)\*X}* ***= S\*N(d1) - X\*e\^-rt\*N(d2)*** This is the BSM formula!
but as the stock goes back and forth you can trade you delta back and forth. Sell high, buy low. This is called gamma 'scalping'. This is what theta decay is accounting for. (plus dividends, borrow and voting rights, yes, yes... I know... cost of capital... alllll parts of BSM model or binomial)
what do you mean by rates? at 10% youre obviously not talking about the risk free rate baked into BSM - you‘re talking about a target return on capital rate?
I'll try to summarise: 1. Download historic EoD OHLC price data for indices, futures, & volatility indices 2. Built BSM to calculate options prices 3. Overlay the logic for the strategy I'm testing e.g. if vol moves above average buy the spread 4. Run the backtest over the historic period e.g. 10 years 5. Summarise results to calculate equity curve & Sharpe Ratio I also use MonteCarlo Simulation to run scenarios for varying price/vol but that's another discussion.
Implied volatility is derived through options market pricing model such as the BSM. It reflects the market’s view on the underlying’s future price volatility.
Part 1 of 2. > What exactly determines the price of an option? The market. As for all exchange trade assets (stocks, funds, bonds, futures, options). If I don't mention a bullet point, assume that I agree that the point is false. > Then I realized that the BS model is mostly used for simplification and explanation. Market makers use more sophisticated models when a new option is first “released” to the market. That's close, but more wrong than right. Many things in the real world are too complex to understand well enough to make useful predictions, including weather, employment numbers, traffic congestion, and option prices. So instead, for each of those complex real world systems, smart people come up with statistical models that have a statistically significant correlation to the real world phenomena, making them *partially* accurate for making predictions. "Statistically significant" does not mean 100% accurate for all time, however. Market makers use BSM for some cases and other models for other cases. They basically pick the most accurate model for the specific use case, with the best trade-offs. For example, one MM may want computational efficiency more than accuracy, because they make markets for thousands of contracts that need updates every tick, while another MM may want more accuracy than computational efficiency, because they specialize in only 2 or 3 series of contracts. It doesn't have anything to do with first release or a contract that is over a year old. > That’s when I realized that a model works backwards, trying to figure out how relevant each of those factors is. That's over-exaggerated. Only one greek, IV, is back-solved out of the model. There's no need to "figure out" whether a greek is relevant or not. It's relevant to the extent that the rate applies to something the user wants to understand. Like if you are driving a car, is the gas gauge more or less relevant than the speedometer? It depends on what you care about in that moment, right? If you got a full tank of gas, who cares what the gas gauge says? But if the warning light of low fuel is blinking, the gas gauge is the most relevant dial on your dashboard. > The thing I thought was the most difficult actually started to make more sense: IV. IV is affected by supply and demand and is somewhat unpredictable. But time, strike, etc, are already known so I assumed the option price must have some backend algorithm that adjusts its value based on maturity, moneyness, and interest rates. BUT how exactly is that determined? IV is affected by **market price** and market price is affected by supply/demand, as well as other things, like sentiment. IV is the *least* predictable greek, because it is affected by market price, which is the least predictable observable in the whole system. > But time, strike, etc, are already known so I assumed the option price must have some backend algorithm that adjusts its value based on maturity, moneyness, and interest rates. BUT how exactly is that determined? 100% false. The market is the only determinant of price. > Trying to answer those questions proved even more difficult. I tried chatgpt, but it told me that the price depends on the Greeks (back to square one). This only proves how persistent this misconception is. Market price comes **first**, everything else, like greeks, come after. This also proves you shouldn't trust LLMs to explain complicated things where there is a lot of misinformation in the training data. > But it’s the market that defines that price the same goes for all the other factors. **FINALLY!** It took a while, but we finally arrived at the one and only truth. (to be continued)
Use a BSM calculator and vary the IV or other parameters like DTE
No one is answering because people are clueless. They are either giving you wrong answers or are making bad jokes to hide their ignorance. The answer is simple: pick your option and see what your broker says the IV is. Then pick the same strike but the first expiration before the event happens - check that options IV. The delta is going to be the expected IV crush. IF you want to translate that into dollars, remove some time value from the post-event option, a week's worth of premium. Or, use the pre-event IV and plug into BSM to price the post-event option and you will get the price you need. Each option will be different because each option contract is its own market place, and some options might be higher or lower in demand, i.e. price, i.e. IV.
No, I’m saying there’s almost no reason for retail to trade because they likely won’t beat a passive index which means they’ve wasted their time, and when they do manage to, it generally is with far worse risk adjusted returns than a leveraged version of the passive investment. That time spent learning to “trade” would bear far more fruit if applied to their primary career. “If you're primarily trading volatility, sure.” Build a stock price simulator, keep it simple, rates and divs at zero, and run 10,000 paths at 20, 40 and 60% vol and compute the value of an ATM put option at the end of those 3 vol regimes. And compare that to the BSM price and Vega of an ATM option priced at 40%. You’ll quickly see that the return is completely driven by the realized volatility over the life of the option vs the implied. You’ll also see that some of the paths under the 20% vol regime will be profitable even if you paid 40%. That just means you got lucky on a particular path. If you buy or sell options you are making a bet on volatility. You should have some basis to believe your estimate of volatility is better than the market’s before you do so. “Implied Volatility is not my estimate vs yours, it's the volatility implied by the current price of the options. You are probably thinking of future realized volatility, which is the truly valuable metric to be able to estimate. Again, even someone with a couple years of professional options experience would understand this well” So here’s what you don’t understand, the market trades in IV terms not in dollar price terms. IV is a more stable parameter where as dollar price changes with every tick of the stock. Professionals build out their vol surfaces and then compute dollar prices.
prolly you want to learn about selling theta at Tastytrade's YT and options in general. For in general, people always recommend big books by McMillan and Natenberg. There's other books that assume basic college math (probability, mostly, but BSM model is dif eq's sidebar this sub and wiki at /r/thetagang will point you at big books but also smaller books by Euan Sinclair, Passarelli and J. Spina.
Beating the market is not some part time thing to do in your spare time. There’s a whole industry trying to get alpha out there. If you do it half heartedly you should expect it to take decades. Nobody will tell you what makes them money because it erodes their advantage. Anyone that tells you how to make money is lying to you, because if it worked they do that instead of selling you something. If you still want to have a crack read the thick books people avoid (options as a strategic investment, BSM model, etc) take all the tasty trade courses and start with backtesting too. Once you have an idea of how to get an edge dip your toes in. Check my posts for other small help
I wouldn't rely on that idea. 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. 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. 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)
Boundary conditions. At T=0 (expiration), we explicitly know the payoff function of a portfolio consisting of an option and its underlying as a function of the underlying price. That allows solving the BSM differential equation to get price when T !=0.
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.