Are (AMM) limit orders options, and does it matter?

6 min readDec 1, 2022

There recently was an excellent Twitter Space on AMMs, organized by Max Resnik, with Alex Nezlobin and Guillaume Lambert, and one of the topics that came up was “Limit Orders” which is, for obvious reasons, close to my heart. One of the questions that came up was “Are limit orders options?”, and this question is harder than it looks because it relies on rather subtle assumptions about market microstructure. TLDR — the answer is “it depends”. Read on for details…

Are limit orders options…

…in a Black-Scholes setting?

In a Black-Scholes setting we have deep and liquid markets, and the spot price process follows a Brownian motion, typically a lognormal one but this does not matter in this case. Any Brownian motion will do.

They key characteristics of Brownian motion with respect to option pricing and hedging is that dx²=dt, ie it is rugged enough to create option value, but not rugged enough to prevent perfect replication. The key here is that Brownian motion is extremely violent in the very short term, but becomes more and more benign the longer the observation period. We see this in the chart below where the red line is the standard deviation of a Brownian motion as function of the time interval dt, and the black secants are the ratio dSD/dt which we can see go to infinite slope for dt to zero.

So the good news about Brownian motion is that you can fully delta hedge your risk. The bad news is that this gives rise to a frenetic buy-high-sell-low or vice versa, depending on the sign of your Gamma, which gives rise to the — deterministic! — option value. So, TLDR: under Brownian motion, all your risk is hedged (good!) but it costs you (bad!).

So what happens to a limit order under Brownian motion? The answer is: not much. It gets executed. At exactly its limit price. There is no time value attached: the time value in option comes from trading back and forth, and doing an infinite number of infinitesimally small trades somehow (via Ito’s lemma, to be precise) generates your option. If you don’t trade back and forth, there is no option value.

TLDR — if the world was a Black-Scholes world, limit orders would not have option value.

…in a jump-diffusion setting?

Now let’s move on to jump diffusion setting where the spot process follows a Brownian motion most of the time, but from time to time it jumps. The timing of jumps is often considered following a Poisson process, and the size can be constant (percentage) or stochastic. This does not really matter for us other than that the jumps must be reasonably benign, which means they should not happen too often (any finite time interval should almost certainly only contain a finite number of jumps) and should not be too violent (ideally all moments exists, but at the very least expectation and standard deviation exist).

Now the first thing to note is that this market is not complete, in the sense that perfect hedging is impossible: we may be able to hedge the Brownian (diffusion) component, but whenever there is a jump we can only rebalance after a finite sized move, and this means that the higher order terms (dx² etc) do not fully disappear, and therefore there is a residual risk. Option pricing is somewhat tricky here because in principle we need to think about risk premia. In practice, people tend to assume that they are risk neutral and just compute expected values under the unique measure that prices all forwards correctly (the “risk neutral measure”).

Calculating the value of this option analytically can be a bit complicated depending on the details of the model used, but in principle it is clear: it is something like the probability of ending up close to the limit, times to the expected loss that a jump takes us through the limit, and we buy at say 100 when the post-jump price is 98, and we therefore lose 2. And of course we can always use Monte Carlo simulation.

…in the real world?

The real world is often best described as a jump-diffusion world with relatively benign jumps, at least in liquid markets that are deep enough to execute our delta: we monitor the markets, and whenever our delta is off by too much we buy or sell to rebalance. We also keep our Gamma, Vega etc under control to hedge against out-of-model risks.

In the real world, limit orders operate pretty much like in the jump diffusion world: by definition they will be executed at the limit price or better — in practice this usually mean at the limit price. Let’s consider the following scenario: markets are at 110, limit-buy at 100. If markets move down slowly, tick by tick (109.99, 109.98, etc) then execution will be at 100 when the market is essentially at 100 (99.99 most likely). If markets move more quickly, say 110, 109, … then there will be some loss: the limit order will be executed at 100 when markets are at 99. Finally, if markets gap to say 50 in one go, then the order will be executed at 100 when markets are at 50. In other words — the option value is real and its related to the size of the gap.

Does it matter?

We made one implicit assumption above, which was that there is a deep and liquid market somewhere so that someone can buy the asset at market and sell it at 100 into the limit order. Now deep and liquid markets do not usually gap much, or at least they do not gap much very often, so our “big move” scenario is somewhat unlikely. So yes — in deep in liquid market there is some option value in a limit order, but chances are that the option value is small because you get lifted pretty much as soon as the order is in the money. Now markets may move further and you lose, but this happens after you’ve bought —this is simply what can happen when holding a position. Or markets may move back, in which case you make money. A priori there is no bias between those two scenarios, so either is equally likely.

When markets are not particularly liquid, first and foremost it is not entirely clear what we mean with market price, and gapping markets. Someone lifted us at 100. We have the asset we wanted, at the price we wanted. Later someone trades at a different price, maybe lower. Maybe markets gapped, maybe markets moved after we’ve executed, there is no way to tell.

Ultimately markets will be somewhat between those two extremes, and ultimately it will come down to information, and to act on it in a timely manner. If the limit “maker” and the ultimate “taker” both monitor the market and can pass instructions with similar latency, the game is balanced (not entirely, as the taker does not risk anything whilst the maker can lose). However, if maker latency is high, then they may see that toxic flow of takers with an information advantage that come in and lift the orders before the maker can adjust it. That has option value for the taker, and against the maker, and as Alex Nezlobin has pointed out, there is no way to charge for this option because any fee is simply moving the limit price. The only way to deal with this is in a recurring game scenario, by widening bid/ask spreads whenever the risk of toxic flow is high.