Digital Option Market Making on Prediction Markets
I was supposed to write on fast message parsing using simd intrinsics and orderbook implementation, but I am abit too swamped at the moment. I picked a topic that is abit shorter and seems to be on the cusp of many’s interests.
Betting markets are all the rage. Of a particular kind are event contracts that are just digital options. They are on Kalshi. Our prediction market arb lectures take a slightly different approach;
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Now, into the stuff.
Digital Options/Binary Options
They go by different names but are the same beast. The payoff structure is simple, and hence so is their pricing (theoretically).
We will cover a cash-or-nothing binary call that pays $1 if the underlying stock price S_T exceeds the strike (K) at expiration (T), and $0 otherwise (ignoring pin risk at exactly (K)). We’ll use a notation where time is (y) (current time), stock price is S_y and τ = = T - y is the time to expiration. The fair price is denoted F_y.
Unlike vanilla options, which have a linear payoff beyond the strike, digitals pay a fixed amount (e.g., $1) if a specific condition is met at expiration. They are popular in betting markets, because well, isn’t it fun? If you like coin flipping, you will love this. But you should stick to coin flipping, I think.
From a market-making perspective, delta hedging is challenging, especially given how near-expiry these tend to be and the jumps in high frequency data. The result is that the bid-ask gets incredibly wide for near-expiry contracts. It’s like 0DTE but worse. On the plus side, you have a capped inventory loss since the loss is -1 or 0 instead of -max(S_T - K,0).
But if you were so interested as to read this post, I assume you are interested in these markets. You should at least be equipped with knowledge to be stated…
Fair Price Formula
Fair price of a digital option under the Black-Scholes model is derived from the risk-neutral probability that the option finishes in-the-money.
where Phi is CDF standard normal and d_2 is
the standard risk-free in crypto could just be the default borrow (popular value 0.0013%/H) for perp funding. Or, for short tenors we can treat r = 0 and get rid of the second term. Accordingly, we also get rid of the discount factor. This is your theoretical mid. Everything is known except for σ.
For a quick mental reference, I picked tau = 1 * 1/365 * 1/24 (1 hour) - d_2 is looks linear ATM because ln(1 + x) ~ x, remember? It is the log-moneyness that is exactly linear.
Spread
Before we go into finding vol, a reasonable theoretical spread should be the cost-of-hedging. This is scaled relative to
These are actually terms that appear in the Greeks of the fair price. Rather than go into the math, it is simple to intuit - each term is the uncertainty w.r.t the price changes. Vol speaks for itself. Sqrt of expiry is the amount of time volatility has left to act. The density function takes the shape of the Gamma curve and encodes how “in-the-money” or “out-of-money” it is.
In practice, this is determined by more factors, your hedging cost, vol calibration errors, inventory and adverse fill costs. The calibration to vol is a matter of choice;