Multi-asset market making under the quadratic rough Heston
Multi-asset market making under the quadratic rough Heston
This paper addresses the multi-asset market making problem under the quadratic rough Heston (QRH) model, which jointly models SPX and VIX derivatives. The market maker quotes bid and ask limit orders on SPX, VIX futures, and SPX/VIX options, aiming to maximize expected terminal wealth while penalizing inventory risk. The QRH model's rough volatility and Zumbach effect are captured via a multi-factor approximation, but the resulting high-dimensional HJB equation is intractable. To overcome this, the authors propose two key approximations: (1) assuming constant volatility and asset deltas over the short market making horizon, and (2) a quadratic approximation of the Hamiltonian in the HJB equation. These approximations reduce the problem to a system of ODEs with an asymptotic closed-form solution. Numerical experiments demonstrate that the approximations are accurate and the closed-form solution enables efficient online recalibration. The work provides a practical framework for automated market making in multi-asset derivatives markets, balancing spread capture and inventory risk.
Highlights
- 1Develops a multi-asset market making framework under the quadratic rough Heston (QRH) model, jointly modeling SPX and VIX derivatives.
- 2Introduces approximations to reduce dimensionality: constant volatility and deltas over the market making horizon, and quadratic approximation of the HJB equation.
- 3Derives an asymptotic closed-form solution for the optimal quoting policy, enabling efficient recalibration.
- 4Validates the approximations through numerical experiments, showing accuracy and relevance for practical use.
Methods
- MMulti-factor approximation of the QRH model to achieve Markovian dynamics.
- MHamilton-Jacobi-Bellman (HJB) framework for stochastic optimal control with inventory risk penalties.
- MQuadratic approximation of the Hamiltonian to obtain closed-form asymptotic solutions.
- MMonte Carlo simulation and neural network-based computation of price sensitivities (deltas).
Results
- RThe value function reduces from (2+n+d) to (1+d) dimensions after approximations, making the problem tractable.
- RClosed-form asymptotic solution for the HJB equation is obtained, allowing fast online recalibration.
- RNumerical experiments confirm that constant delta and volatility approximations are reasonable over short horizons (e.g., daily).
- RThe proposed strategy effectively manages inventory risk while capturing spreads across multiple assets.
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