Moving the Scales of Time
Vol #179: May 16th, 2024
Like the centimeter or pound, there is a precise definition for how long a second, minute, and hour last. Showing up fifteen minutes late to a rendez-vous is objectively one quarter hour past the mark.
But when someone says, ‘I’m spending a week in Istanbul’, you’re unlikely to hold them to all 10,080 minutes. Colloquially you might even get a full day of fudge factor - Sunday afternoon to Saturday morning.
Even precise measurements conflict. A day is the sum of twenty four, sixty minute segments; but also the amount of time that the earth takes to travel around the sun. My day is not your day.
The difference between a complete rotation of the earth and the movement on your watch hands can be as different as thirty seconds. Because of the tilt of the earth and elliptical nature of its orbit, the hours for the sun to reach the exact same point on this rock varies over the year, and is exactly twenty four on only four occasions.
This represents a little under three basis points of error for the day. Far more efficient than VRP pricing. But like interest, it compounds over seasons, and combined with latitude and longitude, the difference between when the sun is perfectly in the south (or north) and what your clock says as noon can be upwards of fifteen minutes.
The equation of time measures this, and mostly matters if you buy $20,000 watches or frequently use sundials. The Zoom still starts at high noon.
Time is complicated at an universal level, but we can also intuitively understand that not all seconds are created equal. The retour of a round trip always feels quicker. A three hour lunch is not a three hour meeting. Friday afternoon at the CBOE is very different from Tuesday morning.
The clock is crucial for options traders because it’s the flip side of volatility. Even Gamestop’s vol doesn’t matter with zero seconds to go. How we decline the space between now and settlement has profound implications for options pricing.
Implied volatility is a single number that summarizes all the expectations of movement for a given strike and expiry. If you keep the parameters constant, and subtract one day from the calculation, the option will be worth less. That’s theta. Less time, less potential - but the vol remains the same.
There are many things that might change your estimate of volatility. It could be demand driven, or perhaps price action in related and underlying markets. But a riskless guarantee is that time will move forward regardless, and that unyielding force changes the value of your positions.
Friday afternoons on the floor are a grind because of long weekends and big lunch prints. While the gut reaction might be to move volatility lower in the absence of activity, what really needs to happen is fast forward the clock into the weekend.
Saturday and Sunday are still closed for trading, so there’s no mechanism for prices to move. (Other than meme stonk coins.) Those are still days in the calendar, but they don’t deserve the same weight as a Tuesday or Wednesday. Because something could still happen over the weekend.
If you lower implied volatility in reaction to the fact that the weekend melt is already happening, Monday will be a cold bath. The current clock time with a lower vol will underprice options that are now back on their regularly scheduled variance process. Expected movement didn’t change, time just slouched into the days off.
Even if you’re fortunate enough not to trade at a naive level, adjustments to volatility have an impact on risk measures like theta. It also impacts deltas and hedging.
But it’s more than the Greek geeks that need good pricing. An equity risk manager will look at historical volatility to compute Value at Risk. Accountants use it to estimate the cost of stock options grants, which affects the quantity given to employees and the financial statements.
Black Scholes and most other models require a clock time, so we have to shoehorn in the adjustments. Weeknights are be treated differently than weekends, and events decay on their own schedule. Moontower is an awesome resource on the specifics of this.
Calendar and business day time is a straight forward adjustment because it’s right in line. The bare minimum for any options valuation these days. But days to expiration matters because of what can happen in that space. “What” is price action, and price action is dictated by volume. Prices don’t move ex-nihilo, they go up when there are more buyers and down with more sellers.
For equity trading, volume weighted average price (VWAP) is a common benchmark. Rather than just taking every minute’s close price and dividing by 390, weighting trade prices by their sizes delivers far more information about a day’s activity. Where are the levels at which real size changes hands?
Stock algorithms will use VWAP as an execution target. Whether you’re a market maker hedging or an institutional account moving around positions, getting the “fair average price on the day” is a politically and theoretically defensible execution. Because if you can consistently tick above or below that, you can probably also tell me about motor yachts.
In the short term, there is a lot of noise in the markets. Flows dominate the tick by tick values, and it’s only over the slightly longer term that imbalances normalize. A few minutes to hours in the most liquid names, days and weeks in the illiquid. The artifacts of imbalance decorate market data everywhere.
Volumes tend to be largest at the beginning and end of day - the saddle curve. Market on close orders have become increasingly popular because they offer a two fold guarantee. Not only are you locked on a fill for your size, but you also rebalanced the book at the last official price. Specialists move the closing print based on supply and demand, which is determined by the set of participants who show up.
Measuring volatility simply on a close to close basis means there’s likely to be some additional noise. There are interesting arguments that the stock’s VWAP price is a better descriptor of historical volatility, and that close to close estimates show persistent overestimation, contributing to the variance risk premium.
Alas, options still settle on the closing price. Over longer terms the movement of the stock might be best summarized in an abstract sense by its volume weighted average prices, but that’s not necessarily perfectly aligned with the contracts that settle on expiration day. Before you get too clever, understand the use case.
Volume weighting is interesting because it tackles part of the liquidity question. Alongside spread width, trading activity is the most important calculation in the LIQ index. Extreme prints happen in low liquidity, but they’re mostly inaccessible to participants and thus represent less meaningful information.
If VWAP is the weighted average price where stocks trade, we can also consider a volatility metric that weights price variance by the deviation from average and the respective volume trading at that level. The idea here is that variance matters proportional to the volume that creates it.
In this tomorrow’s edition of Portfolio Design, we’re going to look at how indicators for covered calls and cash puts behave in higher and lower volume environments.
A major advantage of volume weighted volatility is that it is more statistically normal than other measures. It demonstrates less kurtosis, alongside very close mean and median values. Interesting as a theoretical model and useful for certain applications. But fair warning, everyone that’s seen end of day convexity or FOMO stampedes knows not to ignore kurtosis for practical purposes.
The benefits here depend on your use case. As a multinational issuing restricted stock units, less noisy, high level predictors are probably better for risk and cost estimations. For investors buffering equity with a GULL, trading best practices like slicing up the order and rolling over time mean VWAP is also a more relevant indicator.
When your options strategy’s edge is risk management and outcome structuring, abstract volatility is the framework.
For actively managed books, the respect for time on the clock must be more nuanced, and reflective of anything can and does happen with the settlement of listed contracts. Seventeen minutes difference to solar time matters immensely. Close to close overestimations are far from an odd lot, rough edge to scrub - they’re the nickel you accidentally sold.