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Hello,

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welcome back to Papers with Backtest podcast.

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Today we dive into another algo trading research paper.

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And today we're tackling something really central to market making.

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Absolutely.

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How to handle inventory risk.

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You know,

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the risk you face just by holding assets while prices are bouncing around.

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That's the core problem,

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isn't it?

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You want to capture that bid ask spread.

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That's your profit engine.

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Right.

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But every time you fill an order,

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you take on inventory.

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And if the price moves against you,

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Well,

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that spread profit can disappear fast.

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It's a real balancing act.

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And the paper we're digging into today really formalizes this.

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It's Dealing with the Inventory Risk from 2012 by Guillain,

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Lahaye,

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and Fernandez Tapia.

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A classic in the field.

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Definitely.

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So for you listening,

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our mission here is to really get under the hood,

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understand the main trading rules they derive,

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and crucially,

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how those rules held up when they were backtested.

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Yeah,

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the backtest is key.

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It's where the theory meets,

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well,

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reality,

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sort of.

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So how do they even frame this problem?

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It sounds complex.

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They approach it as a stochastic control problem,

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which sounds a bit academic,

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but the intuition is quite neat.

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Okay.

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Think about steering a ship in choppy waters.

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You're constantly adjusting the rudder based on the waves and wind the randomness to stay on course.

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So the market maker is the ship captain and the choppy waters are the price fluctuations.

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Exactly.

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The paper models that underlying reference price,

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the sort of...

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true price as a brownie in motion,

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that captures the,

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you know,

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the unpredictable wiggles.

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And that randomness is precisely what creates the inventory risk we're talking about.

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Precisely.

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And then there's the order flow.

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How often do buyers and sellers show up?

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The model assumes this isn't constant.

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Makes sense.

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If your price is way off market,

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fewer people will trade with you.

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Right.

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They use an exponential function for that.

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The further your quote,

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your bid or ask,

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is from that reference price,

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the lower the arrival rate of orders.

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An alha,

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they call them.

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And there are parameters A and K in there.

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Yeah,

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A is like the overall market activity level,

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and A measures how sensitive traders are to your price difference.

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High A means people really react to you being even slightly off.

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Okay,

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got it.

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So the market maker wants to make money,

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obviously,

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but they're also managing this risk.

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How is that modeled?

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Their goal isn't just raw profit maximization.

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It's maximizing the expected utility of their P&L.

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Utility,

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so incorporating risk preference.

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Exactly.

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They use a tech ARA utility function,

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constant absolute risk aversion.

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Basically,

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it means the trader dislikes risk by the same amount,

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whether they're currently up a million or down a million.

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Interesting.

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And the inventory itself,

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the Q.

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That's central.

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How much stock are you holding right now?

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And there are limits.

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Q and minus Q,

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a maximum you're willing to hold and a maximum short position.

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And these limits directly affect quoting.

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Absolutely.

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If you hit inventory Q,

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the model says you stop quoting a bid price altogether.

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You don't want any more.

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Right.

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And vice versa,

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if you hit minus Q,

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you stop offering to sell.

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Correct.

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It forces you to manage that inventory level actively through your quotes.

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So that's the setup.

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How do they get to actual usable trading rules?

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What should the bid and ask be?

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This is where the math gets pretty involved.

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They use Hamilton-Jacobi-Bellman equations,

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HEB equations.

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Standard toolkit for these dynamic optimization problems under uncertainty.

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Right.

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But solving them directly is tough.

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They use a clever variable change that,

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well...

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simplifies things considerably.

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It turns the HJB equation into a system of linear ordinary differential equations,

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much more tractable.

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Okay,

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so the math machinery churns away and out pop the optimal quotes.

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Essentially,

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yes.

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They derive formulas for the optimal bid and ask,

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or rather the optimal distance of the bid and ask from the reference price.

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And what do these optimal distances depend on?

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They depend on the time left t,

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Your current inventory Q and these functions VQT that come from solving those differential equations.

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Ah,

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the VQT functions.

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What's the intuition there?

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You can think of VQT as sort of the value or expected future utility if you have inventory Q with time P remaining.

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The optimal quotes balance earning the spread now versus adjusting inventory for a better future state.

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And the formulas also involve the risk aversion and that price sensitivity K.

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Yes,

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those parameters directly influence how wide or tight your optimal quote should be.

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So what about the resulting bid-ask spread,

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the difference between say and SB?

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They derive that too.

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It also depends on time and inventory,

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again,

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through those VQ functions.

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It's dynamically adjusting based on your risk profile and market conditions.

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Does it stay dynamic or does it settle down if you're,

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say,

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far from the end of the trading day?

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That's a good point.

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They look at the asymptotic behavior.

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what happens when time t is far away.

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The optimal quotes become almost independent of time.

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They find these limits,

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day and depu.

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Simple rules for the long run,

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basically.

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Sort of.

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The derivation involves eigenvectors and eigenvalues of a specific matrix,

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so still mathematically grounded,

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but it points towards a more stable quoting strategy when you're not up against a closing deadline.

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Can these asymptotic quotes be simplified even more,

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like a practical formula?

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Yes,

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they manage to get closed-form approximations for these asymptotic quotes.

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That's really useful.

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Why useful?

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Because these approximations explicitly show how the optimal distances depend on things like volatility,

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risk aversion,

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the arrival rate A,

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sensitivity K,

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and of course your inventory Q.

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So you can see the direct impact of changing market conditions or your own risk setting.

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Exactly.

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clearer intuition.

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For example,

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what happens if volatility squared goes up?

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My intuition says you'd widen the spread.

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More risk,

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right?

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Generally,

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yes.

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The approximation confirms that.

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Higher psi usually means a wider optimal spread.

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The exact adjustment on bid and ask separately might depend a bit on your current inventory queue,

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though.

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What about other parameters,

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like

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A, the order arrival rate?

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I or A,

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meaning more active trading,

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tends to have the opposite effect of volatility.

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It generally allows for a narrower spread.

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More flow means you can manage inventory more easily,

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perhaps.

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And risk aversion.

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That one's a bit more ambiguous in the model,

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actually.

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There are competing effects.

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Okay.

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And the Che,

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the sensitivity to price.

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Also complex.

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But the general expectation is that as K increases,

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meaning traders are more sensitive to your price,

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you'd likely need to narrow your spread to stay competitive.

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Makes sense.

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Okay,

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so we have the theory,

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the asymptotics,

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the approximations.

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How did this translate into a backtest?

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Did it actually work?

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Right,

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the crucial part.

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They took this continuous time model and adapted it for discrete,

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real-world data.

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Wow.

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They used tick data for France Telecom from one day in March 2012.

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They had to discretize prices to the nearest tick size and time into small intervals,

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representing how long an order stays active.

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And the model parameters,

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a,

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k,

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h.

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a and k were calibrated from the limit order book data itself on that day.

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They tried to estimate them from market activity.

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And gay,

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the risk aversion.

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That was chosen more subjectively,

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aiming to keep the simulated inventory within a reasonable range,

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like Medica 10 plus 10 average trade sizes,

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ATS.

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Okay,

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so what was the actual strategy being tested?

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The model would calculate its optimal

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SB and SI.

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It would then place orders of a fixed size,

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the average trade size,

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ATS,

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at those prices.

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And how long did the orders last?

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They stayed active for that short duration night.

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If filled,

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great.

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If not,

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they expired.

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The model would then recalculate and potentially place new orders.

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Quotes were updated after any trade or after it passed.

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Got it.

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And the results?

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The paper shows some figures,

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right?

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Yes.

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Figures 10 and 11 are key.

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Figure 10 shows the inventory path of the strategy throughout the day.

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You can see it fluctuating as it trades.

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Trying to stay in year zero,

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presumably,

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but getting pushed around by the flow.

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Exactly.

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And figure 11 shows the cumulative P&L of this

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model-based strategy.

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Looks like it generally trended upwards.

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It did on that day,

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yes.

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But the really interesting part is figure 12.

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What's that one show?

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It compares the model's P&L to a naive trader.

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And the naive trader?

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Just always post orders at the current best bid and best ask in the market.

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A very simple,

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passive approach.

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And how did they compare?

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The model-based strategy significantly outperformed the naive one in their back test.

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Figure 11 shows a much better P&L than figure 12.

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So,

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actively managing the quotes based on inventory and risk using their derived rules actually paid off compared to just sitting on the BBO.

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That's what the backtest suggests,

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yes.

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Figure 13 even gives a close-up view over one hour showing how the model's quotes,

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SB and so,

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moved relative to the market and its own inventory.

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That's pretty compelling evidence for the model,

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at least on that specific dataset.

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It is encouraging.

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Of course,

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the authors rightly point out this is illustrative.

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What are the caveats?

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Well,

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real-world implementation is harder.

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For one,

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parameters like A and K probably aren't constant throughout the day.

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They might change with market regime.

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True.

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Market dynamics shift.

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And there are other factors,

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like passive market impact,

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how your resting orders subtly affect others,

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which weren't explicitly in this version of the model.

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So a powerful framework,

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demonstrated potential in a backtest,

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but still room for refinement and further research.

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For real world trading.

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Precisely.

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They mentioned generalizing the intensity functions and adding passive impact as important next steps.

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Great stuff.

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A really insightful look into managing that fundamental market making challenge.

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Definitely a foundational piece of work in this area.

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Thank you for tuning in to Papers with Backtest podcast.

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We hope today's episode gave you useful insights.

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Join us next time as we break down more research.

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And for more papers and backtests,

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find us at agenttps.paperswithbacktest.com.

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Happy Trading.