The Concept of Variance Time

Tick-tock, tick-tock. The rhythmic sound of a clock’s second hand moving steadily around its face is a sound that has become synonymous with our perception of time. We’ve been conditioned to think of time as a linear progression, measured in days, hours, minutes, and seconds.

While our day-to-day lives is measured linearly in time, the field of physics has prompted intriguing discussions challenging the notion of time as a strictly linear function. In the realm of General Relativity, massive objects like planets, stars, and black holes bend and curve the fabric of spacetime itself. This gravitational bending of spacetime leads to a fascinating concept known as ‘time dilation’. According to General Relativity, the closer you are to a massive object, where the gravitational field is stronger, the slower time appears to pass for you compared to someone in a weaker gravitational field.

How is this related to quantitative finance?

In this blog post, I want to introduce the concept of “variance time” which acknowledges that the volatility on any given day is not evenly distributed across the year. Business days are more volatile than weekends as most business go about their activities. And weekends are more volatile than holidays when everybody is away at the beach.

Yet, standard option pricing models such as the widely used Black-Scholes-Merton (BSM) model assume that a year comprises a fixed amount of days with each day being equally volatile. And since the stock market is only open on bank business days, many opt to measure a year in business days only.

For those who need a refresher, the BSM model takes in as input:

1. S: spot price of an underlying in $
2. K: strike price of the option contract in $
3. $\sigma$: implied volatility of the underlying in %
4. T: time before expiration of the option contract in years
5. r: risk free rate, or the interest earned via holding bonds instead of holding the stock measured in %
6. q: dividend rate, also in %.

And outputs the theoretical price of an options contract. By comparing that theoretical price to the price on the market, you can make a trade. Similarly, given the price of an option, one can back out the IV using the same model. In this sense, the IV and the price of the option are equivalent in describing the cheapness of an option. Ok that’s very cool but what does it have to do with time?

Consider the following situation: it’s 3:59pm ET on a Friday. The stock market is about to close for the weekend. You want to price an option which expires on Wednesday of the following week.

• A calendar day model will show that the time to expiration is 5 out of 365 days.
• A business day model will show that time as 3 out of 252 days.

Come Monday morning,

• The calendar day model now shows 3 days until expiry (3/365) – two days have elapsed
• The business day model still shows 3 days until expiry (3/252)– zero days have elapsed??

All things equal, on Monday morning, the market price of that option is lower because, afterall, time did move closer to expiration. And what practitioners observe is that the price almost always decays less than predicted by the calendar day model and decays more than predicted by the business day model.

Plugging market prices back into the BSM model, the calendar day model will mechanically return a higher IV every Monday, because the option trades at a higher price than predicted and the more volatile the underlying is, the more the option is worth.

Meanwhile, the IV backed out from a business day model will mechanically show a lower IV every Monday.

So if a delta hedging trader doesn’t distinguish busy days from the weekends and holidays, the strategy will favor underhedging because the model predicts too much decay.

And if a trader pretends like volatility only happens on business days, the strategy will favor overhedging because the model predicts no decay over the weekend.

But do it enough times and it will become clear that both hedging strategies will lose money.

Variance Time

So much like how time passes slower around massive objects in space, the passage of time slows down during periods of low economic activity. On the contrary, during periods where important economic numbers are released, time tends to move faster, reflecting the higher volatility embedded in the higher options prices.

In a following post, we will examine a sophisticated approach to characterize variance time.