Hawkes-Driven OTC Market Making: Volterra-Riccati Approximation

We formulate an over-the-counter (OTC) market-making problem in which request-for-quote (RFQ) arrivals are modelled by general Hawkes kernels and fills are controlled thinnings of the exogenous request flow. The modelling choice is motivated by spot-FX RFQ data: after filtering and transforming to seasonality-adjusted RFQ activity time, two-way activity in major currency pairs exhibits large fitted branching ratios and multi-scale persistence. For general Hawkes kernels the control problem is path-dependent: the relevant state contains the order-flow history, or equivalently the forward curve of conditional future RFQ intensities. Exact Markovian lifting is available for exponential kernels, but it becomes high-dimensional for mixtures of exponentials and impractical for long-memory kernels. We therefore develop a hierarchy of Volterra-Riccati approximations. The first level replaces random future request flow by its conditional Volterra forecast; the second adds a covariance correction for intensity uncertainty; the third updates the quote rule with the realized Hawkes memory, or equivalently with the post-request conditional forecast curve. The approximation hierarchy is validated in an exponential Hawkes benchmark, where the exact lifted HJB can be solved numerically. The state-feedback Volterra-Riccati policy closely tracks the exact benchmark, especially in directional regimes, while a memory-free Poisson policy suffers substantial regret. We then apply the same state-feedback rule to a power-law-like RFQ memory model. A directional RFQ burst changes the conditional forecast of future flow and is converted by the continuation-value shadow price into a persistent quote skew. The resulting endogenous OTC quote impact inherits the long-memory decay of the RFQ forecast response and improves inventory and P&L risk control relative to a no-conditioning Poisson benchmark.