Suppose that you are a DMM (designated market maker) for the value of a d20. You are required to always show markets with a width of 1 to keep your DMM status, but that is the only requirement. Suppose that you come to the realization that, for whatever reason, retail just wants to buy (maybe they haven’t figured out how to short in their Robinhood accounts yet or [some other silly reason to make this question work]). You realize that retail buys with probability 0.9, and sells with probability 0.1. What is the best market to show?
Suppose for the purpose of this question that you are managing a book of 100 dollars (lol) and you have log utility of wealth.
Okay, so first things first, the EV of a d20 is trivially 10.5, so if you were making markets around fair value, then you’d do 10@11. However, we know that there’s going to be substantially more buying flow so we should definitely be fading up here.
Napkin math approach
Recall that for a binary payout, Kelly tell us
Let’s assume that the d20 always settles to it’s EV of 10.5 (obviously this is impossible, but just go along with it for the approximation) and that we show a pick’em market (bid==ask), which is a reasonable approximation for a 1 wide market here.
Per the rules, we do not need to be within the bounds of the actual d20— if you think that’s cheesy or whatever, well keep in mind that this is a hypothetical situation with unrealistic price insensitivity and made up DMM rules. We can therefore set p=0.9. Let’s say that our pick’em market is X, we have B = (X - 10.5) / (X - 10.5) = 1. Putting this altogether we get f* = 0.8, which means that from this we would show a market of about 90@91, but since we have removed variance of what the d20 settles to, we know that this will in fact be a slight over bet to what Kelly would actually say so we can ‘round down’ a bit and say that the Kelly optimal market to show is roughly 89@90 per our napkin math. (We risk ~80% of our bankroll with X=90.5 because this settles to 10.5.) I think that maybe it’s hand wavy to say that we just lower our market by 1 due to the variance, but I think it’s important to realize that almost all of our variance comes from whether or not we buy or sell rather than what the d20 settles to, so the additional impact of the total variance there is quite minimal. Yes, I know that this is probably higher than you were expecting, and is admittedly higher than my first gut intuition answer as well.
Formal approach
Let’s define our bid to be X (and therefore our ask is X+1) we seek to maximize g(X)
There’s actually a pretty cool trick here that let’s us replace sums of logarithms. We can use the log gamma function, which becomes the digamma function when differentiated once, and the trigamma function when differentiated twice. I think that it would distract from the post to explain how it works (plenty of material online about it that can explain it better than I can anyway), and it’s also worth noting that it’s not needed here—it only makes our objective function ‘cleaner’.
Now that we have the first and second derivative of our objective function, we can use Newton-Raphson to find the local maximum:
and g’’(X*) < 0, so this is in fact a maximum.
The true optimal market to show is 88.94147@89.94147.
Wow very cool, good job buddy!