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Optimal Trading in a Dynamic Market Robert Almgren

Summary: Optimal Trading in a Dynamic Market
Robert Almgren
June 30, 2009
We consider the problem of mean-variance optimal agency execution strate-
gies, when the market liquidity and volatility vary randomly in time. Under
specific assumptions for the stochastic processes satisfied by these param-
eters, we construct a Hamilton-Jacobi-Bellman equation for the optimal cost
and strategy. We solve this equation numerically and illustrate optimal strate-
gies for varying risk aversion. These strategies adapt optimally to the instan-
taneous variations of market quality.
A fundamental part of agency algorithmic trading in equities and other asset
classes is trade scheduling. Given a trade target, that is, a number of shares
that must be bought or sold before a fixed time horizon, trade scheduling means
planning how many shares will be bought or or sold by each time instant between
the beginning of trading and the horizon.
Grinold and Kahn (1995) and Almgren and Chriss (2000) suggested that the
optimal trajectory could be determined by balancing market impact cost, which
leads toward slow trading, against volatility risk which pushes toward rapid com-
pletion of the order. This framework leads to an "efficient frontier," in which the


Source: Almgren, Robert F. - Courant Institute of Mathematical Sciences, New York University


Collections: Mathematics