Joint Calibration of SPX and VIX Options Using the Willow Tree Method

The willow tree method is a powerful technique with many applications in derivative pricing. We have discussed how it can be used to determine the implied volatilities of American options. It can also be applied to price convertible bonds by simultaneously [glossary_exclude]accounting [/glossary_exclude]for equity and credit risks. In addition, it is useful for calculating the value of complex path‐dependent derivatives and associated risk measures, such as Asian options and American moving average barrier options. Reference [1] proposed using the willow tree method to build a model that describes the volatility dynamics of both SPX and VIX options concurrently.

Among the methods commonly used to jointly calibrate SPX and VIX options, the non-parametric approach typically reconstructs the risk‐neutral density (RND) using only SPX option prices. The authors employed the implied willow tree (IWT) method to extract the RND of both SPX and VIX options, thereby accommodating both sets of market-observed option prices effectively. They pointed out,

In this study, we propose a novel nonparametric discrete‐time model called the joint implied willow tree (JIWT) approach to tackle the joint calibration challenge. The JIWT method bypasses the need for model‐based simulation techniques by using discrete‐time nonparametric methods to derive the risk‐neutral probabilities from observable SPX and VIX option prices. Our method offers three primary contributions. First, we delve into the conditional probability distributions between two maturities using both SPX and VIX option prices. While solely SPX option prices provide insight into SPX unconditional RNDs, they offer limited information on conditional densities. Leveraging the VIX definition (1) grounded in the SPX, we can identify conditional densities that align with VIX and its options prices. It enables us to capture the volatility smile in both the SPX and VIX markets, especially for short‐term maturity. Second, our JIWT method is simpler, more straightforward to implement, and efficiently extends to multiple maturities of SPX and VIX options as compared with the nonparametric discrete‐time model proposed by Guyon (2023)… Third, our JIWT method operates without the need for any prespecified mathematical model for SPX. Instead, it extracts the entire RNP directly from market‐observable option prices, making it both model‐free and data‐driven.

In short, this paper proposed a nonparametric method for calibrating SPX and VIX option prices simultaneously. The method has proven successful in addressing the joint calibration challenge of the SPX and VIX markets. As a result, it will enable risk and portfolio managers to identify new opportunities and manage risks more effectively.

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References

[1] Bing Dong, Wei Xu, Zhenyu Cui, Joint Implied Willow Tree: An Approach for Joint S&P 500/VIX Calibration, Journal of Futures Markets, 2025; 1–22

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