A birdie asked how to model a 1-day option

Thoughts about pricing 0DTEs that respect how volatility unfolds

It’s going to shock you to hear it, but I get emailed a lot of option questions. I’ve gotten some that are pages long with commentary, prices, blood type.

This is one of the reasons I put the shingle up for calls. I’m definitely not reading all that free, but also I can talk through options stuff way faster than I can read and write about it. You’ll say 500 words about what your doing and I’ll collapse it into floor trader grunts in the time it takes to pick up a handset and say “sold”.

I spent my whole adult life having to make option decisions in a few seconds. This is not anything special — talk with any market-maker and their fluency with calls and puts seems like a parlor trick. Options are just another language and being fluent in it means you think in that language natively. The old floor folk can even sign in options as fast as you can talk. (I can read the signing but I was the tail end of futures being traded in the pit and not on Globex. I never developed a large hand signal vocabulary).

That said, if a sub sends me a question that I can peck on my phone in a few minutes I’ll just answer it. (Also I read every email I get and try to respond to all of them even if it’s to acknowledge that both I and the sender are humans worthy of not being ignored.)

If responding to an email takes more than a few minutes I’ll pass it through the “would other people care about the answer to this question?” filter. If so, I can kill 2 birds with one stone. Oh my god, F me that’s a horrid idiom. The first person who ever said that should have been locked up on the spot cause that’s I-knew-he-was-a-serial-killer-when-he-was-nine level of clue. I never thought about the literal meaning of that expression until I typed it.

Re-phrasing — today is one of those moontowers where I can pick up the 7-10 split by sharing a response to a reader question that you might find useful.

Reader question:

Do you know of any way to model an option’s price intraday?

My response:

2 things I'd think about:

1. Intraday I would think in terms of straddle prices and price changes and compare that to tick vols (but the tick vols themselves can also be in price space not vol space)

2. Modelling the fraction of a day's variance that typically accumulates every hour (for example the open represent more of the days volatility then any random 15 minute interval)

Reader reply:

When you say you would think in terms of straddle prices and price changes, how would that be used to model the price of a specific option? For example, if someone wanted to model the price of the 190 strike call 5/24 by the hour tomorrow relative to AAPL's price, what should they do to get a rough idea? 

I’d start with the question of:

“What do you think the straddle is worth every hour?”

A straddle represents the mean absolute deviation (MAD) which is 80% of the volatility or standard deviation of return.

If you think AAPL moves 1% per day then the straddle is worth 1% at the start of the day. If strike is ATM then the call is worth 50 bps.

The value of the straddle changes by root time (assuming vol is unchanged). 

[See Visual Derivation Of The Straddle Approximation]

So if half the day is gone, the straddle is worth: 1% * sqrt(.5) = .71%

The question is at what time do you think only 1/2 the day's volatility remains?

This question applies to every hour of the day. 

The entire concept of "intraday decay curves" is area of active inquiry for any market-maker so I don't have an answer key but the problem is familiar.

In practice, I’ve tackled this with a blend of lazy guessing and leaning on some quant research.

The lazy way

The 80/20 solution or guess would be to assume volatility transpires at the same rate volume unfolds over the course of the day.

I prompted perplexity.ai with “vwap volume distribution over the day”. To my delight it didn’t send me down the circus internet, but actually said something smart:

The volume-weighted average price (VWAP) is calculated by dividing the total dollar amount traded for a security over a specific time period by the total volume traded during that same period. This means that prices at which larger volumes were traded have a greater impact on the VWAP calculation than prices with smaller volumes.

To calculate VWAP accurately, it is important to consider the volume distribution over the trading day. Historically, volume is not evenly distributed throughout the day - there are typically periods of higher and lower trading activity.

Many trading algorithms account for this by using historical volume profiles to predict the expected volume distribution over the upcoming trading day. The algorithm then slices the total order into smaller "child orders" that are released at predetermined times based on the forecasted volume distribution.

For example, if 17% of the day's total volume historically trades in the first hour, the algorithm would aim to execute 17% of the total order during that first hour period.

Get your hands on historical volume profiles and you have a solid start. VWAP algos are commoditized and rest on that research so it shouldn’t be hard to track down.

The quant way

You can use tick data to compute realized variance for each hour and divide by the sum of all the variances for the day to see the proportion by interval. You can use many days data as well as many names to get a cross-sectional perspective.

You will need to treat the period from the prior close to the open in some coherent manner as well. Like you could take the point-to-point variance from the previous close to open divided by the close to close variance over many samples and names and then you can make a statement like “25% of the variance happens overnight.”

That means the remaining hourly variances are then divided by a variance of only .75 of the expected daily variance. Over many days of doing this you will likely get a strong sense of when on average half a day's variance has transpired. 

Extra thoughts

  1. Computing tick vols is a quant rabbit hole of its own. When you come across the words "bid-ask bounce" and "volatility signature plot" you are reading the right stuff.
  2. I'm not a quant researcher. I'm a hacker. I throw numbers in spreadsheets or if I'm really ambitious Python, and turn the crank until I see the shape of the problem. So my methodology above is a zoomed out answer but once you make contact with data the specific details will not go smoothly. Nature of the beast. But the approach is directionally correct you just have to savor the data-wrangling gruel. For example, how many data points are enough? I don’t know — keep adding more until the variations in proportions seem to stabilize at some quantity.

    An instinct one develops with enough practice is to know whether your cobbled-together “tape and twine” analysis has a rigor that is proportional to required precision of your use-case. If whatever I’m doing is going to break because I don’t truly understand what “degrees of freedom” means then I just need enough taste to know that I should get a quant’s help.

    Discerning how rigorous you need to be is part of being an efficient resource allocator. How much time do you spend on pricing vs execution vs figuring out how to hedge less vs exploring names like not AAPL or other high volume names where Citadel & SIG’s market-makers are trading from the cockpit of F-22s?