The primary challenge of market making in spot precious metals is navigating the liquidity that is mainly provided by futures contracts. The Exchange for Physical (EFP) spread, which is the price difference between futures and spot, plays a pivotal role and exhibits multiple modes of relaxation corresponding to the diverse trading horizons of market participants. In this paper, we model the EFP spread using a nested Ornstein-Uhlenbeck process, in the spirit of the two-factor Hull-White model for interest rates. We demonstrate the suitability of the framework for maximizing the expected P&L of a market maker while minimizing inventory risk across both spot and futures. Using a computationally efficient technique to approximate the solution of the Hamilton-Jacobi-Bellman equation associated with the corresponding stochastic optimal control problem, our methodology facilitates strategy optimization on demand in near real-time, paving the way for advanced algorithmic market making that capitalizes on the co-integration properties intrinsic to the precious metals sector.

Recent decades have seen a profound transformation in financial markets, driven largely by the widespread adoption of electronification. This digital evolution has been paralleled by the development of new decision-making tools for market makers and the emergence of systematic market making for most asset classes. The case of stocks traded through central limit order books has been discussed extensively (see [1] for a good example using stochastic optimal control), but OTC markets have also been addressed using the modelling framework introduced by Avellaneda and Stoikov in [2] (see for instance [3] for the FX market). These market making models have helped dealers to manage inventory risk through strategic hedging and quote skewing. In single-asset models, optimal quotes typically vary monotonically with current inventories, influenced by factors such as risk aversion, clients’ pricing sensitivity, and the liquidity on external platforms. Dealers aim to maximize client flow internalization to mitigate the cost of external execution and reduce market impact, hedging only when inventories exceed franchise-dependent thresholds (on the internalization-externalization dilemma, see for instance [4] and [5]).

Multi-asset extensions of market making models have been built to help dealers manage their risk at the portfolio level (see for instance [3] and [6]). Market makers often deal indeed with large portfolios of assets of very different liquidity. Illiquid assets may be difficult to internalize and costly to execute in the market. However, the risk associated with illiquid assets may sometimes be partially offset by positions in other instruments that are more liquid, leaving time to unwind the illiquid asset with less cost. In FX markets, illiquid instruments can also be traded via more liquid legs, introducing both complexities and opportunities for the dealer (see [3] and [7]). Most existing models consider the case of asset price dynamics driven by correlated Brownian motions.

Diverging from conventional approaches that focus solely on correlated asset price dynamics, our model capitalizes on the unique benefits of using co-integrated assets for hedging, regardless of whether the market maker proposes quotes in those assets. Co-integration offers indeed a special opportunity for market makers to tap into enhanced liquidity pools and benefit from mean reversion. This is particularly relevant in the case of a spot dealer hedging their position through futures, as in the case of precious metals market. While the interbank spot market does exist, precious metals futures are considerably more liquid, with tighter spreads and higher volumes. Since the price difference between futures and spot – so-called Exchange for Physical (EFP) spread – is primarily driven by swap rates which are relatively stable over the market maker’s intraday risk horizon, the co-integration assumption between spot and futures prices is very natural. There are other asset classes were the same mathematical ideas could be used. In FX, one can think of market making involving Non-Deliverable Forwards and onshore spot liquidity for instance.

Our modelling framework builds on the classical foundation laid by [2] and [8] (see [9] for a detailed discussion). In this framework, a single underlying instrument (spot) price is modelled by an arithmetic Brownian motion. The arrival rates and size distribution of client trades are modelled with predetermined intensity kernels. Going beyond the initial academic literature and building on recent advances, our model considers a dealer quoting in spot who can hedge (with costs) in both spot and futures. The novelty comes from the modelling of EFP by a nested Ornstein-Uhlenbeck (hereafter nested OU) process, as inspired by the observation of multiple relaxation times in the market ranging from hours to days, possibly related to different trading horizons of different types of traders, as illustrated in Figure 1. This model is in the spirit of the two-factor Hull-White model for interest rates [10]. However, the observed EFP volatility significantly exceeds the underlying interest rate volatility so the mechanism is not purely driven by interest rates but also influenced by the cost of physical delivery, apart from speculation. This article expands, therefore, the now classical stochastic optimal control framework for market making by incorporating the existence of co-integrated and liquid assets. Not only does it address a gap in the existing academic literature on market making models but also provides a new class of price dynamics compatible with the approximation techniques developed in [6].

In the forthcoming sections, we begin by introducing our model alongside the key equations that define the optimal strategies for market makers in the spot precious metals market. We also discuss the merits and limitations of utilizing nested OU processes. Subsequently, we demonstrate the applicability of the method from [6] to our general framework. We conclude with an in-depth numerical analysis focused on the gold market, illustrating the practical implications of our model.

State variables and optimal control problem

Solution

Remarks on nested OU processes

Approximating numerically the solution \( \theta\) of equation (2) using grid methods poses challenges due to the high dimensionality of the state space and the complex geometry of the frequently visited inventory states at optimality. To avoid grids, we adopt a methodology similar to that described in [6], which involves approximating equation (2) with a closely related equation. The solution to that equation, denoted by \( \check{\theta}\) , will be a quadratic polynomial and serve as an approximation to the original value function \( \theta\) . Consequently, \( \check{\theta}\) will be utilized in lieu of \( \theta\) in (3) to derive strategies that are nearly optimal.

To obtain our new equation, we employ instead of \( L^S\) and \( L^F\) the quadratic approximations \( \check{L}^S(v) = \eta^S v^2\) and \( \check{L}^F(v) = \eta^F v^2\) . Subsequently, we substitute \( \mathcal{H}^S\) and \( \mathcal{H}^F\) in (2) by

Also, following [6], we approximate the function \( H\) by a quadratic function

We then obtain the following partial differential equation:

with terminal condition \( \check \theta(T,q^S,q^F,E,D) = - K^S (q^S)^2 - K^F (q^F)^2 \) , whose solution \( \check \theta\) will be our approximation of \( \theta\) .

As announced above, the interest of the above approximation is that the solution to (4) is a polynomial of degree \( 2\) in \( q^S\) , \( q^F\) , \( E\) , and \( D\) . Let us make indeed the following ansatz:

where \( A:[0,T] \mapsto \mathcal S_4(\mathbb R)\) , \( B:[0,T] \mapsto \mathbb R^4\) , and \( C:[0,T] \mapsto \mathbb R\) are differentiable functions such that

Plugging this polynomial ansatz into (4) yields a system of ODEs for \( A\) , \( B\) , and \( C\) : We only report here the equations for \( A\) and \( B\) , as \( C\) is irrelevant for the computation of the optimal strategy.

where

This system of ODEs can easily be solved numerically to finally get \( \check \theta\) . Then, replacing \( \theta\) by \( \check \theta\) in (3) leads to the following approximations of the optimal controls:

where \( e^S = \begin{pmatrix} 1\\ 0\\ 0\\ 0 \end{pmatrix}\) and \( e^F = \begin{pmatrix} 0\\ 1\\ 0\\ 0 \end{pmatrix}\) .

For illustration, we consider market making in spot gold (XAUUSD) with access to futures liquidity.

A typical pricing ladder size discretization is set as \( 100\) , \( 200\) , \( 500\) , \( 1000\) , \( 2000\) and \( 5000\) oz. We assume a standard client trade intensity represented by

where price sensitivity parameters are taken to be \( \alpha = -0.8\) and \( \beta = 5\,\text{bp}^{-1}\) . Gold spot and EFP volatility parameters are set to \( \sigma_\text{S} = 140\,\text{bp} \cdot \text{day}^{-1/2}\) and \( \sigma_E = 5\, \text{bp} \cdot \text{day}^{-1/2}\) , respectively. Spot-EFP correlation is typically small and assumed to be zero in this example. EFP price relaxation rate is \( k_E = 8\,\text{day}^{-1}\) . We assume first, as a benchmark, a simple OU framework with \( k_D = \sigma_D = 0\) and \( \bar{D}\) is fixed at zero. Standard functional dependence of instantaneous market impact on execution rate is assumed: \( L^S(v) = \psi^S |v| + \eta^S v^2\) and \( L^F(v) = \psi^F |v| + \eta^F v^2\) with \( \psi^S = 0.4\)  bp, \( \psi^F = 0.2\) bp, \( \eta^S = 7\cdot10^{-8}\,\text{bp} \cdot \text{day} \cdot \text{oz}^{-1}\) and \( \eta^F = 3\cdot10^{-8}\, \text{bp} \cdot \text{day} \cdot \text{oz}^{-1}\) . The parameters have been selected by analyzing a subset of HSBC market making franchise. However, they should not be considered as representative of HSBC but rather of a typical institutional spot gold dealer. Daily turnover in this setup is approximately $1B. Basis point convention is used for price changes, which is an approximation insignificant in practical terms given the time frame of the problem (see [4]). We consider a time horizon of \( T=1\) hour that ensures convergence towards stationary quotes and hedging rates at time \( t=0\) . Terminal penalty coefficients are set to zero unless specified otherwise.

Figure 2 demonstrates the variation of optimal market making controls with spot inventory. When there are no futures in the book, the equilibrium spot position is zero, and the dealer will skew quotes to attract client flow in the direction towards this equilibrium. As noticed in other papers dealing with the internalization-externalization dilemma (see for instance [4]), there exists a pure internalization area where skewing will be considered as the only risk reducing option. Larger positions will, however, also involve hedging in the market. We can see that the onset for hedging with futures is earlier than for spot and the corresponding rate of execution is also higher for futures than for spot. This is totally understandable given the liquidity (and thus the cost of trading) difference. A non-zero position in futures leads to a shifted quasi-equilibrium in spot, corresponding to an EFP position (where each futures position is paired with another in spot in the opposite direction).

Figure 3 illustrates the expected inventory relaxation following a relatively large 1000 oz OTC trade for \( \gamma = 3\cdot 10^{-4}\) . It was calculated via Monte Carlo simulation (\( 2\cdot 10^4\) trajectories) of the model given the initial condition and optimal controls with terminal penalty The effect of terminal penalty is seen only near \( T\) , so the choice of \( K^S\) and \( K_F\) is not very important. However, this may change for less liquid markets. \( K^S = K^F = 10^{-3}\) . We see that part of the risk is hedged with futures very quickly. The remaining excess spot risk is unwound via skew (attracting the offsetting client flow) and spot execution. Figure 2 shows that skew can be quite aggressive for this level of risk aversion, with offers better than mid price. Effectively, during this fast relaxation stage, the dealer creates an EFP position which is later unwound very slowly. Figure 4 shows an inventory probability distribution extracted from a sufficiently long Monte Carlo trajectory (\( 10^7\) seconds). Since the effect of terminal penalty is rather local, this is practically equivalent to carrying simulation over multiple days. As expected, the system spends most of the time along the low-risk EFP diagonal.

EFP mean reversion influences optimal controls. EFP deviation from the expected mean shifts the equilibrium inventory along with the corresponding execution onsets and quote skew, as shown in Figures 5 and 6. Understandably, when the deviation is negative the dealer will tend to accumulate long EFP position, and vice versa. Figure 5 demonstrates that EFP mean reversion will only lead to rare direct arbitrage opportunities under strong risk aversion. This is related to the cost of opportunistically entering into an EFP position (one would have to cross two spreads). The dealer will instead skew quotes in the required direction (see Figure 6) and wait for the opportunity to materialize while making spread. With a lower risk aversion, the appetite to capitalize on EFP mean reversion increases leading to opportunistic execution at extreme deviations (where the upper futures execution onset is below the lower spot execution onset and, similarly, where the lower futures execution onset is above the upper spot execution onset). The skew will be repurposed from EFP risk management to opportunistic skew on EFP deviation, as shown in Figure 6.

We introduced nested OU processes in this paper to mitigate the latter repurposing which is mainly due to overconfidence when using the OU model. Uncertainty on mean EFP deviation through the introduction of the stochastic process \( (D_t)_{t\ge 0}\) creates additional risk that is taken into account through risk aversion, resulting in the dealer being less inclined to keep EFP position. Figure 7 demonstrates the effect of \( \sigma_D\) on the propensity to capitalize on EFP mean reversion. Here \( k_D = 0.2\,\text{day}^{-1}\) , \( \gamma = 3\cdot 10^{-4}\) , and other parameters are as defined earlier. As expected, the larger is \( \sigma_D\) , the smaller is the skew and the higher is the arbitrage onset.

The choice of risk aversion is ultimately in the hands of the dealer. The rule of thumb is to decrease \( \gamma\) while the expected benefit in P&L exceeds the expected increase in risk. Figure 8 demonstrates the effect of risk aversion on volume share of hedging, P&L and risk. In this specific example, it is clear that decreasing \( \gamma\) below \( 10^{-4}\) is questionable as the expected risk increases much faster than P&L. Similarly, increasing \( \gamma\) above \( 10^{-3}\) is not efficient.

Figure 9 illustrates the performance of the optimal strategy over a single trading day on real market data but with simulated client flow (uniform intensity throughout the day). EFP position clearly echoes the mean-reverting nature of EFP – the dealer aims to keep EFP risk against the direction of EFP deviation. Volume share of futures execution significantly exceeds that of spot due to lower cost. Introducing futures hedging into risk management of OTC spot improves Sharpe ratio by at least 30% (not shown), despite the additional round-trip cost of futures trading.

We have extended a stochastic optimal control framework for OTC market making by incorporating co-integrated liquidity for hedging. Using a computationally efficient approximation technique, our methodology facilitates strategy optimization on demand in near real-time, potentially benefiting both electronic and voice traders. A specific example of spot gold market is analyzed in detail demonstrating efficient risk management benefiting from access to the significantly more liquid gold futures with reduced transaction costs while capitalizing on EFP mean reversion. In the spirit of the two-factor Hull-White model for interest rates, EFP spread is modeled by a nested Ornstein-Uhlenbeck process describing the observed multiple modes of relaxation corresponding to the diverse trading horizons of market participants. Interestingly, nested OU processes also appear as a way to robustify pure OU strategies with uncertainty in the mean reversion parameters, a commonly encountered case. They could find application in generalizing most of the optimization problems involving classical OU processes (see for instance the trading problem of [12]).

The results presented in this paper are part of the research works carried out within the HSBC Research Initiative. The views expressed are those of the authors and do not necessarily reflect the views or the practices at HSBC. The authors are grateful to Richard Anthony, James Donaldson and Manuel Abellan-Lopez (HSBC) for helpful discussions and support throughout the project.