Unwinding or liquidating a position is a trade off. Liquidate too quickly and you may suffer price slippage as the market order walks the book. Liquidate too slowly with more conservative limit orders, and you are exposed to the risk of adverse price moves. The concept of splitting a large order into a number of smaller orders to be executed over a certain time period is well-known to traders. Exchanges and many other market participants are therefore motivated to develop liquidation algorithms which behave optimally. In this post we’ll discuss the Almgren-Chriss model. For more details consult The Financial Mathematics of Market Liquidity by Gueant.
We assume a trader wants to unwind \(q_0\) trades in a time interval \([0,T]\). Writing \(q_t\) for the trader’s inventory at time \(t\), we write
\[dq_t = v_t dt, \]
where \(v_t < 0\) is the rate of liquidation. If the trades were exercised in a finite number of discrete blocks, then \(v_t\) would be a sum of delta functions, for example. The mid price of the stock is modelled as
\[ dS_t = \sigma dW_t + kv_t dt\]
for \(k>0\). The first term is simply Brownian motion, although note that the decision is made for simplicity to assume a normally distributed price instead of the usual lognormally distributed price. The second term means that the price drops by an amount proportional to the number of stocks our trader executes. This is the permanent market impact.
But the most significant equation here is the equation representing how the rate of liquidation \(v_t\) affects the price obtained for the shares. This is the instantaneous part of the market impact, which in the model has no permanent impact on the market price. We assume that the price obtained for the shares executed at time \(t\) is
\[S_t + g\left(\frac{v_t}{V_t}\right),\]
where \(V_t\) represents the total market volume and \(g<0\) when \(v_t < 0\). The choice of increasing function \(g\) is actually the key to the model. It quantifies how much worse the average price obtained for the shares trades at time \(t\) is when the rate of liquidation \(v_t\) is higher (i.e. more negative). The original model of Almgren and Chriss chose the function $g$ to be linear. This means that if the trader liquidates twice as many shares at time \(t\), the average price obtained for those shares will be twice as far from the mid price. The cash earnt by the trader is then simply the number of shares liquidated multiplied by the average price obtained, i.e.
\[dX_t = – v_t\left( S_t + g\left(\frac{v_t}{V_t}\right) \right) dt.\]
If the midprice were assumed to be close to constant over time, the optimal strategy would be to liquidate as slowly as possible. This would mean that the shares would all be sold at close to the mid price. However, liquidators are not only unwilling to wait forever, but also typically wish to liquidate the portfolio at close to the current market price. Liquidating over a longer time interval means that the price may fluctuate away from the current price. Some kind of “risk appetite” consideration must therefore be included in the model.
This requirement is not actually encoded in the differential equation for \(X_t\) above. Rather, it is encoded in the quantity we wish to optimize. The way this is done is to not simply optimize the final cash holding \(X_T\), but also to penalise its variance. This can be done by choosing the function to be optimized as something like \(\mathbb{E}(X_T) – \frac{\gamma}{2} \mathbb{V}(X_T)\) or \(\mathbb{E}(-e^{- \gamma X_T})\), for some constant \(\gamma > 0\). How much one penalises variance by choosing \(\gamma\) is essentially an arbitrary decision in the model. Of course, longer trading horizons give rise to more variance in \(X_T\) because \(S_t\) becomes less predictable when allowed more time to drift. Thus this parameter will determine the rate of liquidation based on risk appetite.
Finding the optimal trading strategy \(q(t)\) is a variational problem which requires minimising the function
\[J(q) = \int_0^T{\left(V_tL\left(\frac{q'(t)}{V_t}\right) + \frac{1}{2} \gamma \sigma^2 q(t)^2\right)dt},\]
where \(L(\rho) = \rho g(\rho) \).
Gueant also discusses several extensions of the model, including:
- Incorporating a drift term into the equation for the evolution of the stock price to allow the trader an opinion on the future trajectory of the stock
- Placing a lower and/or upper bound on the liquidation rate
- Considering the liquidation of portfolios of multiple stocks