# Let’s lay out some intuition

It’s the beginning of the first day of the year and the straddle that expires on the evening of December 31st is trading for \$10.
• Regardless of whether you use a calendar day model or business day model, the straddle has the same time to expiry.
• Calendar day model: 365 days out of 365 days remain. T = 1
• Business day model: 251 days out of 251 days remain T = 1 Both models acknowledge 1 year until expiration. They will both generate the same implied volatility from the \$10 straddle.
One day elapses.

• Your time to expiry now depends on the model. Let’s assume the straddle is unchanged at \$10
• Calendar day model: 364 days out of 365 days remain. T = .99726
• Business day model: 250 days out of 251 days remain T = .99602 The straddle is \$10 but the business day model has slightly less time to expiry. To compensate, the model must imply a slightly higher volatility than the calendar day model. We can see, as soon as time starts ticking, the implied vols of the models will diverge as the proportion of the model year remaining is accruing differently. Already you can see the issue — 2 users each with their own model think they are trading at different vols than one another even though they are trading at the same option price.
• The difference in implied vols between the models is more pronounced in shorter-dated options. Since most option volume is relatively near-dated, this is a critical point.

Let’s consider a near-dated example. It’s Friday morning and a straddle expiring the coming Wednesday evening is trading for \$2.

• Calendar day model: 6 days until expiry
• T = 6/365 = .0164 years remaining
• Business day model: 4 days until expiry
• T = 4/251 = .0159 years remaining
• The business day model has less time to expiry, so the \$2 straddle must imply a higher volatility than the calendar day model.

Let’s examine this straddle after Friday and the weekend elapse.
• Calendar day model: 3 days until expiry
• T = 3/365 = .0082 years remaining
• Business day model: 3 days until expiry
• T = 3/251 = .0120 years remaining
There is 45% more time remaining in the business day model than the calendar day model. So whatever price the straddle is trading for is going to yield a much lower implied volatility for the business day model!

Witness what just happened. The trader using a calendar day model observes the volatility increasing dramatically relative to the user of the business day model but they are both looking at the same changes in the straddle price!

Let’s interject a qualitative discussion of theta or simple time decay to get a closer view of what’s happening.

• At the close of business on Friday:
• The calendar day model shows 5 out of 365 days until expiry.
• The business day model shows 3 out of 251 days until expiry.
• At Monday’s open:
• The calendar day model shows 3 days until expiry. 2 days have elapsed.
• The business model still shows 3 days until expiry — zero days have elapsed!
Unless something crazy happened in the world, the straddle will be worth less (this doesn’t mean the theta is an edge — the stock may have moved a lot while the “clean” implied vol might not have done much. We are building to an idea of “clean” vol so that statement is a bit of a preview.)
• If the business day model thinks there’s the same time to expiry as there was on Friday, then its implied vol must dive lower to match the Monday morning straddle price.
• The calendar day model time to expiry has fallen but the straddle will not have fallen by the amount of theta you would have expected for 2 days. You are going to understand why soon but a simple thought experiment can help you see for yourself: Suppose it’s Monday morning and a straddle expires Friday afternoon. There are 5 calendar days until expiry. Monday and Tuesday elapse.
• Do you think the straddle will erode more or less over these 2 days than in the prior example where the 2 days that elapse out of 5 were weekend days? (Toggle to see the answer)
The straddle that spans the weekend will erode less. If for a given volatility, a straddle decays less than you expect as time passes, implied volatility must have mechanically increased! If you are using a calendar day model, you will almost always notice that a straddle does not decay as much as the model’s theta would predict over a weekend and you will find the implied volatility the model outputs is higher on Monday morning than it was on Friday evening.