# Optimal Execution

FIG. 2: from book encyclopedia: Algo trading!

Price risk =  risk that the execution will be less favorable due to random movement of prices.

Shortfall = difference between effective execution pruce and arrival price.

Arrival price = prevailing price at the start of the execution period.

If we use the variance of shortfalls as a proxy for risk, a trader's aversion to risk establishes a risk/cost trade-off.

1st scenario --> trader pays a higher cost to eliminate risk -->  (8) - (1).

2nd scenario --> trader pays a lower cost but takes on a greater dispersion of shortfalls.

Risk aversion increases a trader's sense of urgency and makes it attractive to pay some premium to reduce risk.

The premium that trader pays is in the form of higher temporary market impact.

All other factors held constant:

--> Higher expected temporary market impact encourages slower trading.

--> Higher expected risk or risk aversion encourages faster trading.

Risk aversion embodies the notion that people dislike risk.

Risk-averse agent: utility of fair game < utility of having expected value of the game with certainty--> u(G) < E(u(G)).

The degree of risk aversion may be captured by the risk aversion parameter

This parameter is used to translate risk into a certain dollar cost equivalent - the smallest certain dollar amount that would be accepted instead of the uncertain payoff from the fair game.

For an agent with quadratic utility, the certain dollar cost equivalent is given by E(G) - Var(G).

Hence, his degree of risk aversion is characterized by the family of risk/return pairs with the same constant trade-off between the expected return and risk.

An annualized target return and standard deviation imply a risk aversion, and may be translated to a risk aversion parameter of the type used in some optimal execution algorithms.

Another factor that influences the decision to trade more quickly or more slowly is the expectation of price change.

For the purpose of execution:

--> Positive alpha is an expectation of profits per share per unit time for unexecuted shares.

A faster execution captures more of the profits associated with this expectation of price change.

-->Negative alpha is the expectation of losses per share per unit time for unexecuted shares.

A slower execution incurs less of the losses associated with this expectation of price change.

The general form of the optimal execution problem:

--> Finding the best trade-off between the effects of risk, market impact, and alpha

--> by minimizing risk-adjusted costs relative to a prespecified benchmark.

Common benchmarks are VWAP and arrival price.

Assuming a quadratic utility function, a general formulation of this problem takes the form:

where C(xt) is the cost of deviating from the benchmark.

The solution is given by the trade schedule xt that represents the number of shares that remains to buy/sell at time t.

The trader's optimal trade schedule is a function of his level of risk aversion, ≥ 0.

His level of aversion determines his urgency to trade and dictates the preferred trade-off between execution cost and risk.

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