Modeling Prediction Markets As Exotic Options Part 1

[WITH CODE] Showcasing the option Greek profiles for exotic options

Hello!

Welcome back to another post. This will be the start of a two-part series. A few weeks ago, we talked about how you can view prediction markets (like Kalshi and Polymarket) as derivative markets.

Specifically, these prediction markets resemble exotic options, like digital options, digital range options, and one-touch options.

In this series, I will create an application (through Streamlit) which you can run to model these markets as exotic options, and the option Greek exposures that you would have from a position in these markets.

In addition, I will show how can you trade these markets based on the difference between the market implied price and our implied price. The code for this post has been sent to your email.

Let’s get into it.

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Choosing a Market

First, we have to choose a specific market to model. While this idea can be applied to almost all prediction markets (except for markets like mentions or politics)

We will focus on the Daily High Temperature markets on Kalshi. To be specific, we will be focusing on the Los Angeles market, but this will only be pertinent in the next post. I chose this market over others as we get to model both digital options and digital range options. We unfortunately don’t get to model one-touch options, but those are more complex, and likely better suited for a future post if there is enough interest.

In this framework, we treat the Forecasted High Temperature as the price of the underlying asset (the “Spot” price). The True High Temperature realized at the end of the day represents the settlement value.

Temperature markets possess a fundamental difference from equity markets that directly impacts option pricing: Mean Reversion vs. Geometric Brownian Motion.

Stocks follow a random walk with no theoretical upper bound. This creates “fat tails” in one direction. However, temperature is physically bounded and mean-reverting. If the average high for July in NYC is 85°F, the probability of hitting 115°F is pretty much impossible, while a stock hitting a 10x multiple of its current value is not.

Because temperature cannot move to infinity, the “tails” of these digital options behave differently than standard equity binaries. In a typical Black-Scholes model, the probability of an extreme outlier remains a factor in the price. In temperature markets, the probability density function collapses much faster as you move away from the mean. This means out-of-the-money (OTM) digital options in weather markets often decay toward zero much faster than their equity counterparts as the limits are bounded.

To apply our model, we define the variables as follows:

• Underlying (Spot): The current consensus forecast, calculated from the current digital option and digital range option prices (we will go more into this next post).

• Strike (K): The temperature threshold defined by the Kalshi contract (ex. >90°F or 88°F - 89°F).

• Payout: A fixed $1.00 if the condition is met, and $0.00 otherwise.

By framing the market this way, we can move away from “betting” and toward and volatility / price arbitrage.

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Digital and Digital Range Options

To value these prediction markets, we use the Black-Scholes model for Digital Options. While standard “vanilla” options have a payoff that increases the further the stock moves past the strike, digital options have a fixed, “all-or-nothing” payoff.

The Pricing Model

The pricing for a digital call is significantly simpler than a vanilla call. Because the payoff is a fixed $1.00 if the event occurs, the price of the option is simply the discounted probability that the underlying (S) will be above the strike (K) at expiration (T).

Mathematically, assuming interest rates (r) are 0%, the price is:

Quite Simple!

Where N(d2) is the cumulative distribution function of the standard normal distribution, and d2 is calculated as

In this context, the "Price" you see on Kalshi (e.g., $0.65) is literally the market-implied N(d2), or a 65% probability of the temperature threshold being met.

Digital Call Payoff

A Digital Call (or “Threshold” contract) pays out if the temperature exceeds a specific strike.

At Expiry: The payoff is a “Step Function.” It is worth $0.00 if the spot is 99.9 and $1.00 if it hits 100. There is no middle ground.

1 Year to Expiry: With one year to go, the “cliff” is smoothed out into an S-curve (Sigmoid). This represents the uncertainty; even if the current spot price is 80, there is still a statistical chance it could reach 100 in a year.

Digital Range Payoff

A Digital Range option (or “Between” contract on High Temperature markets) pays out only if the temperature finishes inside a specific window (e.g., between 90°F and 110°F). You can think of this as a Digital Bull Spread: you are Long a Digital Call at 90 and Short a Digital Call at 110.

At Expiry: The payoff is a “Rectangle” or “Box.” If the spot price is 89 or 111, the contract is worthless.

1 Year to Expiry: The curve looks like a Bell Curve (Probability Density). The highest value is centered between the two strikes, as that is the “safest” place for the option to be to ensure it stays within the range as time passes.

Pin risk is the primary danger in these markets. If the temperature is hovering exactly at 99.9°F as the market is about to expire (and you own the >100 digital), your position value will oscillate violently between $0 and $1. As we will see in the next section, this causes your Gamma to approach infinity, making the position nearly impossible to manage or hedge as you approach the “cliff.”

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Exotic Greek Exposures

Managing risk in prediction markets requires a departure from vanilla options intuition. Because these contracts have a fixed payout, their Greeks are characterized by “spikes” and “sign flips” rather than the relatively smooth curves seen in standard equity options.

Note on Scaling: In the following graphs, the y-axis is scaled to the 1-month-to-expiry lines. As time (T) approaches zero, digital Greeks are mathematically explosive. For example, Delta and Gamma are proportional to 1/sqrt(T) and 1/T respectively. On the 1-day lines, these values often shoot off the chart; I have truncated them here to ensure the 1-month and 1-year profiles remain visible.

Digital Option Delta: The Probability Density

The price of a digital option is the market-implied probability, as we discussed above. Delta is the first derivative of that price with respect to the spot.

Because the price is a cumulative distribution, its derivative is the probability density function.

Delta peaks At-the-Money (ATM) because that is where the probability of finishing ITM is most sensitive to a $1 move. Deep ITM or OTM, the probability is “sticky,” so the derivative (Delta) is near zero.

At expiration, the price becomes a discontinuous step function. The derivative of a step function is the Dirac delta function, which is an infinitely high, infinitely thin spike at the strike. This represents the impossible hedge ratio as the contract expires “on the pin.”

Digital Option Gamma, Vega, and Theta

Unlike vanilla options, where Greeks often maintain the same sign, digital Greeks flip as you cross the strike price (K).

Gamma: Gamma is the derivative of Delta (with respect to Spot). Because Delta increases as you approach the strike from below and decreases once you pass it, Gamma is positive OTM and negative ITM.

As you can see, the 1-day gamma goes off the chart, like delta does in the prior graph. This is because of how delta is very large when the digital option is near the spot price.

Vega: Volatility is a “double-edged sword” for digitals. If you are OTM, higher volatility increases the odds of a jump into the money (Positive Vega). If you are already ITM, higher volatility only increases the risk of falling back below the strike (Negative Vega). This relationship becomes sharper as you get closer to the expiration.

Theta: Time decay follows the same logic. If you are ITM, the passage of time “locks in” your win (Positive Theta). If you are OTM, time is your enemy as you run out of runway to hit the strike (Negative Theta).