[Paper Review] Consistent Pricing of VIX and Equity Derivatives with the 4/2 Stochastic Volatility Plus Jumps Model

Wei Lin, Shenghong Li+2 more|arXiv (Cornell University)|Oct 5, 2015

Stochastic processes and financial applications23 references3 citations

TL;DR

This paper introduces the 4/2 stochastic volatility model with jumps, combining features of the Heston and 3/2 models via a volatility process of the form $ a\sqrt{V_t} + b/\sqrt{V_t} $, enabling consistent pricing of both equity and VIX derivatives. Using Lie symmetries, the authors derive a closed-form solution for the joint Fourier-Laplace transform, achieving superior out-of-sample performance in pricing VIX futures and options compared to Heston, 3/2, and Black-Scholes models.

ABSTRACT

In this paper, we develop a 4/2 stochastic volatility plus jumps model, namely, a new stochastic volatility model including the Heston model and 3/2 model as special cases. Our model is highly tractable by applying the Lie symmetries theory for PDEs, which means that the pricing procedure can be performed efficiently. In fact, we obtain a closed-form solution for the joint Fourier-Laplace transform so that equity and realized-variance derivatives can be priced. We also employ our model to consistently price equity and VIX derivatives. In this process, the quasi-closed-form solutions for future and option prices are derived. Furthermore, through adopting data on daily VIX future and option prices, we investigate our model along with the Heston model and 3/2 model and compare their different performance in practice. Our result illustrates that the 4/2 model with an instantaneous volatility of the form $(a\sqrt{V_t}+b/\sqrt{V_t})$ for some constants $a, b$ presents considerable advantages in pricing VIX derivatives.

Motivation & Objective

• To develop a unified stochastic volatility model that extends both the Heston and 3/2 models while preserving analytical tractability.
• To enable consistent pricing of equity and VIX derivatives by modeling the instantaneous variance as a superposition of 1/2 and 3/2 terms.
• To apply Lie symmetries theory to derive a closed-form solution for the joint Fourier-Laplace transform of the model.
• To empirically evaluate the 4/2 model’s performance in pricing VIX futures and options against Heston, 3/2, and Black-Scholes models.
• To assess the model’s out-of-sample predictive accuracy and robustness in capturing market dynamics.

Proposed method

• Propose a 4/2 stochastic volatility model with jumps, where the instantaneous variance follows $ dV_t = \kappa(\theta - V_t)dt + (a\sqrt{V_t} + b/\sqrt{V_t})dW_t $, combining Heston (1/2) and 3/2 dynamics.
• Apply Lie symmetries theory to the Fokker-Planck PDE associated with the model to derive a closed-form solution for the joint characteristic function.
• Use the derived transform to price equity options, VIX futures, and VIX options via inverse Fourier-Laplace transforms.
• Calibrate the model to daily VIX futures and options data from March 2014, using parameter estimation from in-sample data.
• Perform out-of-sample testing by applying estimated parameters to predict option prices on March 14, 2014, and compute ARPE for evaluation.
• Compare model performance using ARPE (Average Relative Pricing Error) across Black-Scholes, Heston, 3/2, and 4/2 models on in-sample and out-of-sample data.

Experimental results

Research questions

• RQ1 Can a unified stochastic volatility model combining Heston and 3/2 dynamics consistently price both equity and VIX derivatives?
• RQ2 Does the 4/2 model with a volatility process of the form $ a\sqrt{V_t} + b/\sqrt{V_t} $ provide better pricing accuracy than Heston or 3/2 models for VIX derivatives?
• RQ3 To what extent does the 4/2 model’s closed-form solution, derived via Lie symmetries, enhance computational efficiency and accuracy in derivative pricing?
• RQ4 How does the 4/2 model perform in out-of-sample pricing of VIX options compared to benchmark models?
• RQ5 Does the inclusion of jumps in the underlying index improve the model’s fit to observed VIX option market data?

Key findings

• The 4/2 model achieves the lowest out-of-sample ARPE of 14.55% across all tested models, significantly outperforming Heston (15.22%), 3/2 (16.36%), and Black-Scholes (30.19%).
• For at-the-money (ATM) options, the 4/2 model records an ARPE of 14.23%, which is lower than Heston’s 8.76% but higher than 3/2’s 10.13%, indicating strong overall robustness.
• The 4/2 model delivers the best performance in pricing in-the-money (ITM) options, with significantly lower errors than Heston and 3/2 models.
• The model’s closed-form solution via Lie symmetries enables efficient and accurate pricing of equity and realized-variance derivatives.
• The 4/2 model shows a notable improvement in fitting VIX option implied volatility skews, particularly in capturing upward-sloping skews consistent with market data.
• The 4/2 model maintains strong predictive power in out-of-sample testing, with only a slight increase in ARPE compared to in-sample, suggesting low overfitting and high robustness.