How Risk Preferences Shape Optimal Liquidation Timing
Summary
The document frames closing an existing share position as a choice of liquidation time before a fixed deadline. Under its baseline assumptions—zero-drift geometric Brownian motion, known volatility, immediate full execution, and no market impact—it compares several possible interpretations: maximizing the observed price over a time window, minimizing risk through immediate sale, using a decision rule based on future price probabilities, or treating the position like an American option without dividends.
Its main lesson is that the objective must be specified before an “optimal” stopping rule can be identified. A price-only rule, a risk-adjusted PnL objective, and an option exercise analogy answer different questions. The document offers no empirical results or definitive resolution; it raises the concern that learning a rule under the simple assumptions may either be uninformative or amount to estimating risk aversion. Market impact, partial execution, and other relaxed assumptions could change the problem, but their effects are left open.
Key ideas
- The liquidation rule depends on the trader’s objective and treatment of risk.
- Under zero drift and frictionless execution, the document questions whether historical data can teach a useful stopping rule.
- A rule based on the best observed price differs from mean-variance optimization and option exercise.
- Adding market impact or partial execution may turn the problem into a broader execution problem.
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Full text
# Is this an optimal stopping problem?
# Is this an optimal stopping problem?
I am trying to work out how to approach a machine learning problem of 'learning' an optimal liquidation time/threshold, under some conditions, from historic data. The idea is a trader armed with this model will choose the right time to close an open position, to maximise his PnL over a large number of orders.
Consider a trader holding some position of shares $X(t)$ at time $t$ where $X(0) = x_0$. Price evolves as $p(t)$ and we assume shares were bought at price $p(0) = p_0$.
For now, we assume:
- Price follows geometric Brownian motion with zero drift and known variance.
- No market impact and infinite liquidity: the position is closed instantaneously at time $t_c$ where $0<t_c<T$.
- The full position is liquidated at once
- There is some finite end-time $T$ at which the trader is forced to close his position.
(Side-note: I think the problem generalises to optimal execution, if we include market impact and allow slices of $X(t)$ to be traded.)
The problem to solve is, what is the optimal time to trade?
I see a number of conflicting interpretations / solutions for this problem:
- Trade at the best price after some fraction of $T$ has passed - following the secretary problem (i.e. optimal stopping)
- Trade immediately (mean-variance optimisation with risk aversion)
- Trade with zero/negative risk aversion; allow time to pass and close the position depending on some other metric (e.g. $\operatorname{P} (p(T) > p(0) \,| \,p(t) )$.
- Trade at $T$, if we treat this as an American option without dividends
Side-note; if the answer is (2) or (3), this suggests that risk-aversion determines the optimal execution time; this seems to simply substitute one problem for another.
In conclusion, to 'learn' the optimal time seems to depend on the interpretation. If it is option (1) I don't think anything can be learned. If option (2) or (3), the problem becomes instead one of training the parameters of some risk-aversion model (and seeing which risk-aversion results in the largest PnL).
Which of the above interpretations is correct?
If it is an optimal stopping problem, is it even possible to learn/train anything - given the constraints/assumptions above? In this case, how would relaxing these constraints impact the conclusion?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.