With a background in financial derivatives we’re going nerd out on some financial theory from time to time. I apologize in advance, but this is one of those times.

**Option Pricing Theory**

Stock options are equity derivatives that are frequently used for employee compensation or speculation within the finance realm. All of you tech bros out there should know what I’m talking about. A typical plain vanilla call option provides the upside of capital appreciation without any downside risk.

The upside potential provided by options frequently holds considerable value. Stock option are frequently valued using the Black-Scholes option pricing method, using variables such as the price of the underlying asset, the exercise price of the option, time to expiration, volatility and a risk-free interest rate.

For our purposes, we’re going to simplify things a bit by using a simple binomial option pricing model which determines option value by assuming the price of an asset can either increase or decrease by some estimated amount with some estimated probability.

Quick example: let’s assume Tesla is trading at $1,000 per share and they report earnings tomorrow. We know that depending on how many crappy trucks they sold, the price will either be $1,100 or $900 tomorrow with 50/50 probability. If one of your dull friends said “Hey, I’ll sell you my share for $1,000. Just let me know tomorrow if you want it” what should your reply be?

My reply would be, “Sure I’ll let you know tomorrow, chump.”And then I would wait to see how earnings went. If Elon sold a lot of crappy trucks (and the price increased to $1,100) I would go ahead and buy my friend’s share for $1,000. If earnings shit the bed, I would pass on my friend’s offer and not by the share. Basically, you have no downside, only upside. Visually that option looks as follows:

To value this option that our friend gave us, we would simply multiply the outcomes by the probability of each one occurring:

Basically, our friend gave a free $50 worth of option value.[1] Nice friend, but you might want to refer him to Cleat Street if he ever wants to start betting sports.

**Optionality in Sports Betting**

Another advantage that sports bettors hold is deciding when to bet. We will cover in further detail in a later post, but it’s important torecognize that lines are dynamic and frequently vary across sportsbooks. Sometimes the lines differ considerably across books and sometimes they are very similar.In the former scenario, bettors can get tremendous value from shopping lines[11] .In the latter, bettors might hold significant option value.

Let me demonstrate with an example.

This season, the New York Jets hosted the New England Patriots in a divisional clash on MNF. Let’s assume that your model suggests that there is value to betting the Jets (*barf*) on the moneyline. You get paid on Friday and you want to fire off your bet at one of your two sportsbook accounts that evening. Book 1 offers the Jets at +345 and Book 2 offers the Jets at +344. You should go ahead and place your bet at Book 1, right?

Not necessarily.

With odds that are nearly identical, your option value is worth more than the one penny in price difference (on a huge dog). If the line at either book moves up, you can get a better number. If the line at either book moves down, you slam the book that didn’t move. This is option value. The value of that optionality depends on 1) if the books generally move in tandem and 2) the expected magnitude of the line movement and 3) the amount of time remaining until the game starts.

With the historical lines of each book, you can determine the average discrepancy between the lines to figure out what the likely magnitude of a future line move. In the table below, you can see that there was often a considerable difference between the lines offered at these two books.

[1]We’ll ignore some of the technicalities of discounting that you would typically do with financial assets as the term (one day) is negligible and U.S.treasuries aren’t yielding shit.

From line release until 6pm ET on Friday night, there was an average difference of 10 cents between the books. Let’s assume that a 10-cent move is a reasonable estimate for the expected magnitude of the next line movement.

At this point, we don’t know if the next line movement will be in our favor or against us. Let’s assume that there is a 50% likelihood of the next line movement will be -10 cents and a 50% likelihood of the next line movement being +10 cents (on Book 2).[2]

[2]Doesn’t matter which book we assume will move next. The math is the same.

By waiting on a line, we can increase our expected odds from +345 to +349.5.

**Value of Optionality**

Now, we assume you are betting on the Jets ML because you believe there is an edge and that your expected win % exceeds the breakeven win%. As an example, let’s assume your expected win % is 24.0%. We can now determine your expected profit by 1) betting the odds at +345 or 2) waiting and getting expected odds of +349.5.

As calculated below, the expected profit for a $100 bettor increases from $6.80 to $7.88 by preserving your optionality and waiting to bet. As a result, the indicated value of the optionality is $1.08.

Now – a common critique might be that “hey, we can’t predict the future and there is a chance that both lines move down simultaneously” or that “the lines were volatile early in the week but since reached efficiency”.Certainly, it’s possible that lines move in lockstep, but given the historical spread between the lines, I wouldn’t count on it.[3]

The argument that the lines have settled (and are thus less volatile) can be disproved by the line movement from 6pm ET on Friday until kickoff. If lines have settled, we would expect a negligible difference between the lines going forward. This, however, is not the case as the average difference between lines averaged 11 cents from 6pm ET Friday until kickoff, frequently exhibiting a 20-cent difference and peaking at a difference of 30 cents around midnight on Monday.

[3]Let’s also not forget that you’re contemplating placing a wager 72 hours before kickoff. If there are only 5 minutes to kickoff, that’s a different story(clearly not as much option value).

So - what can we learn from this?

Big picture: if you have multiple sportsbooks with the same line, you’re generally better off waiting for one of the lines to move rather than pull the trigger. This especially holds true when there is a considerable amount of time before kickoff/first pitch/etc.

If you made it through this piece and enjoyed it – you’re in luck. There will be plenty more technical pieces coming your way in the coming weeks.

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