Comparison between the Mean Variance Optimal and the Mean Quadratic Variation Optimal Trading Strategies

Algorithmic trade execution has become a standard technique

for institutional market players in recent years,

particularly in the equity market where electronic

trading is most prevalent. A trade execution algorithm

typically seeks to execute a trade decision optimally

upon receiving inputs from a human trader.

A common form of optimality criterion seeks to

strike a balance between minimizing pricing impact and

minimizing timing risk. For example, in the case of

selling a large number of shares, a fast liquidation will

cause the share price to drop, whereas a slow liquidation

will expose the seller to timing risk due to the

stochastic nature of the share price.

We compare optimal liquidation policies in continuous time in

the presence of trading impact using numerical solutions of

Hamilton Jacobi Bellman (HJB)partial differential equations

(PDE). In particular, we compare the time-consistent

mean-quadratic-variation strategy (Almgren and Chriss) with the

time-inconsistent (pre-commitment) mean-variance strategy.

The Almgren and Chriss strategy should be viewed as the

industry standard.

We show that the two different risk measures lead to very different

strategies and liquidation profiles.

In terms of the mean variance efficient frontier, the

original Almgren/Chriss strategy is signficently sub-optimal

compared to the (pre-commitment) mean-variance strategy.

This is joint work with Stephen Tse, Heath Windcliff and

Shannon Kennedy.