# Part 1: Overview of Risk Neutral Probabilities

• Win \$100 when the Lakers win the ’22 championship. We think there is a 30% chance this will happen
• Lose \$20 in all other cases
• The Expected Value at Expiry = (\$100 x 0.30) + (-\$20 x (1–0.3)) = \$16.00
• Using a 0% interest rate for simplicity, this security should be valued right now at \$16.00

# Key Takeaways

1. Calculate the risk neutral density using the formula as indicated in Appendix B. This is done by feeding in various option input parameters (ie: strike, underlying price, volatility, time to maturity, and interest rate).
2. At the start of each case study choose an option maturity to analyze across time. For example, in the case of the March 2020 Black Thursday analysis, we start the analysis on February 25, 2020 and end on March 27, 2020. As a result, we’re referring to a ~30 day maturity at inception which will gradually expire by March 27.
3. For each daily observation, we calculate the risk neutral probabilities across every strike in the selected maturity. Given the raw Deribit data only has discrete strikes, we’ll have to linearly interpolate the strikes and implied volatilities to allow for a smoother curve when calculating the implied probability distribution.
4. Repeating this same process for each day allows us to generate a 3D plot of the probability distributions across time.
• We can refer to the risk neutral density as “strike gamma” because we are taking the second derivative of the option price w.r.t strike. With regular spot gamma we are taking the second derivative of the option price w.r.t spot.
• Similar to spot gamma, the risk neutral density will be the same for both calls and puts (provided the strikes are the same)

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## Zeta ζ

Zeta (ζ) is the premier under-collateralized DeFi derivatives exchange, providing liquid derivatives trading to individuals and institutions alike.