BofA quants propose new model for when to hold, when to sell

Closed-form formula helps market-makers optimise exit strategies

The goal of every market-maker is to buy low and sell high. The hard part is knowing exactly when to sell.

Ideally, a market-maker will exit a position when a client meets the ask price or the price hits an upper barrier – the take profit level. Less ideally, a position will be liquidated when it reaches the end of its holding period or hits the lower price level, the stop loss.

Over the past two years, David Shelton, who leads a team of quants focused on electronic trading of foreign exchange and fixed income products at Bank of America in London, has been working on a method to generate clear signals on when market-makers should liquidate positions. The resulting solution is presented in a paper he co-wrote with Carlos Veiga, a senior quantitative financial analyst at BofA, which was published in Risk.net earlier this month.

Their model uses a closed-form formula to assign probabilities to each of the four exit scenarios and then computes the expected return of each outcome. “Once you know these probabilities, you know everything you need to know,” says Shelton.

What seems psychologically or intuitively reasonable to do isn’t [always] actually the best thing to do from an expected return point of view

David Shelton, BofA

The inputs for the model are the drift and standard deviation of the profit and loss (P&L) derived from bid/ask spreads and the arrival rate – or frequency – of client orders. The market-maker only has to define the maximum holding period and the take profit and stop loss levels. As soon as the P&L hits one of those barriers, the position is liquidated.

The author’s note their model is very similar to Louis Bachelier’s options pricing model, published in 1900. “If we were to remove the event where the client takes the risk away, this would fall back to the Bachelier results,” says Veiga.

The information about frequency of client orders makes all the difference, however. “The ideal scenario for a market-maker is that the client takes the risk away from you,” Shelton explains. “The starting point was to try and have a simple model for how we can explore those trade-offs and use it to inform the decision making of algorithms.”

Veiga argues that a market-maker with a good handle on the relationship between the spreads they charge and the arrival rate of client orders – and indeed the adverse selection they are exposed to – can make optimal trading decisions based on the outputs of the model.

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“It helped us with the thought process around our risk management decisions,” says Shelton. “What seems psychologically or intuitively reasonable to do isn’t [always] actually the best thing to do from an expected return point of view,” says Shelton.  

One of the advantages of the model is that it doesn’t rely on dynamic control methods, which can be more complex to handle. And because it is a closed-form solution, it doesn’t require numerical methods or simulations, which are computationally more expensive.

Shelton and Veiga only began collaborating relatively recently. In 2022, Shelton gave an internal presentation on methods for calculating hard thresholds for FX trades, and asked if anyone in the audience would be interested in contributing to the research. “We were genuinely trying to understand how we were making decisions in the electronic FX business,” says Shelton. Veiga volunteered.

The pair intend to continue working together. Their next project looks at mitigating risk by selecting the best bid/ask prices and order types when market-making. “If I take FX as an example, I can place passive orders, or I can place aggressive orders, or skew to one set of clients or another,” says Shelton. “I’ve got many different ways in which I can reduce risk, but we’ve still got the problem of how to score those different decisions and make sure I pick the best decision I can.”