Theta in Plain Sight

When learning the Black-Scholes option pricing formula (and refinements and variations thereof) in grad school, the (T-t) term was always presented as "trading days to expiry". It was never explicitly stated, but I never considered it anything other than an integer. There was never an exercise of pricing an option on the expiry date where this would have been questioned.

What am I getting at? Eurex recently introduced end-of-day index options (OEXP) on the Euro STOXX50 index. The weekly series covers expiries for the next 5 trading days. For example, on Monday 20230911, there were contracts trading which expire end-of-day each weekday of the same week. (There are also contracts expiring at the end of Sept, Oct, and November.) When looking at options whose expiry is weeks or months apart, dividends (incl. tax rates) and discount rates muddy the water. Having option expiries on consecutive days makes it easy to isolate θ.

If (T-t) was an integer, i.e., constant over a trading day, then the price difference between a contract expiring on day D and a contract expiring on day D+1 would be (almost) constant over a trading day.

But as Fig. 1 below shows, this is not the case. The dark blue line, for example, shows the price difference on 20230911 between the at-the-money call (strike price 4260) contracts expiring at the EOD and the next EOD. We see how the longer expiry increases in value relative to the expiring contract throughout the day.

The close spacing highlights two effects:

• The difference changes increasingly less intraday between Tuesday & Wednesday, Wednesday & Thursday, and Thursday & Friday expiries. The difference between 4.99 and 4 matters less than that between 0.99 and 0.
• The absolute difference similarly becomes smaller the further away the two expiries are. In the limit of large N, the value of an option expiring in N days and one expiring in N+1 days should asymptotically approach zero.

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