Optimal order choice: evaluating uncertain discounted trading alternatives

A method of determining the discount, Γ, from the principal price of each of N securities at which to place an order for one or more securities, wherein the order is subject to uncertain execution, for each security to maximize the expected utility of wealth of an investor, the method comprising determining the value of Γ for which the certainty equivalent, CE, of the risk is used in maximization of the utility function:

• max Γ ⁢ CE = E ⁡ ( G ~ ) - 1 2 ⁢ Ψ ⁡ ( w ) ⁢ E ⁡ ( G ~ 2 ) ,
• wherein
• {tilde over (G)}=({tilde over (R)}FilledP)T{tilde over (X)}(Γ)+{tilde over (R)}†PT(i−{tilde over (X)}(Γ))+{tilde over (R)}wTWnon-trade;
• E({tilde over (G)}) is the expectation value {tilde over (G)};
• N is the number of unique securities in the union of securities owned by the investor at time t and the securities for which orders are to be placed;
• Γ is a column vector of length N whose elements are the order discount, γj, for each security;
• P is a column vector of length N whose elements are (pa,t)j·sj,t when the order is a purchase and (pb,t)j when the order is a sale, wherein (pa,t)j and (pb,t)j are the principal price of security j of the N securities at time t for purchase orders and for sale orders, respectively, adjusted for splits and dividends when the securities are equities, and sj,t are the number of shares of security j at time t and the sj,t are independently a positive number or, when there is no order for security j, zero,
• {tilde over (X)} is a column vector of length N whose elements, {tilde over (x)}j, are contained in the closed interval [0,1] and are the fraction of the order that is executed at discount γj;
• {tilde over (R)}† is {tilde over (R)}Forced if execution of the order is forced at the end of the time period or {tilde over (R)}Optional if execution of the order is optional at the end of the time period, t+1;
• {tilde over (R)}Filled, {tilde over (R)}Forced, and {tilde over (R)}Optional are N×N diagonal matrices whose elements are real, random variables, ({tilde over (r)}Filled)j,j, ({tilde over (r)}Forced)j,j, and ({tilde over (r)}Optional)j,j, respectively, and are the expected returns of each of the N securities when the order is executed during the time period, forced to be executed by the end of the time period, and optionally executable by the end of the time period, respectively, at the order discount, γj;
• Ψ ⁡ ( w ) = U ″ ⁡ ( w ) U ′ ⁡ ( w )
• is the risk aversion parameter, wherein U(w) is the utility function;
• w is a scalar equal to the total dollar value of assets held in the portfolio at time t;
• Wnon-trade is a column vector of length N whose elements, (wnon-trade)j, are the dollar values of each of the N securities already in the investor's possession, net of desired orders, and wherein the (wnon-trade)j independently are a positive number, zero, or a negative number;
• Wtrade is a column vector of length N whose elements, (wtrade)j, are the dollar values of each of the N securities already in the investor's possession which are to be traded and wherein the (wtrade)j independently are a positive number, zero, or a negative number;
• and
• {tilde over (R)}w is a column vector of length N whose elements, ({tilde over (r)}w)j, are the returns at time t+1 on each of the j securities as given by ( r w ) j = ( p m , t + 1 ) j - ( p a , t ) j ( p a , t ) j ,
• i is a column vector of length N whose elements are each 1, N is an integer value of at least 1 or more, j is an integer from 1 to N, and the superscript T indicates the transpose of a matrix.

A method of determining the discounts, Γ, from the principal price of each of N securities at which to place an order during a time period starting at time t and ending at time t+1, wherein the order is subject to uncertain execution for each security, so as to maximize the expected utility of wealth of an investor, the method comprising determining the value of Γ for which EU(W) is a maximum, wherein

• N is an integer value of 1 or more;
• Γ is a vector having elements γj, wherein γj is the order discount for the jth security of the N securities for which an order is placed,
• EU(W) is the expected value of the utility function U;
• W is the wealth of the investor at time t+1 given by the sum of:
• a) the dollar value on assets held in the portfolio, but not traded, at time t+1;
• b) the dollar value, at time t, on assets held in the portfolio at time t, which are to be traded;
• c) the dollar value realized when the order is filled at discount Γ times the probability that the order for each of the securities will fill;
• and
• d) if the order did not fill before time t+1, the dollar value realized when (i) the order is forced at time t+1, or (ii) the order is optional at time t+1.

A method of determining the discount, Γ, from the principal price of each of N securities at which to place an order for one or more securities, wherein the order is subject to uncertain execution, for each security to maximize the expected utility of wealth of an investor, the method comprising determining the value of Γ for which the certainty equivalent, CE({tilde over (G)}), is a maximum, wherein

• N is an integer value of 1 or more;
• Γ is a vector having elements γj, wherein γj is the order discount for the jth security of the N securities for which an order is placed,
• {tilde over (G)} is the gains of the investor given by the sum of:
• a) the change in the dollar value between time t and t+1 of all securities owned by the investor but not traded;
• b) the dollar value realized when the order is filled at discount Γ times the probability that the order for each of the securities will fill;
• and
• c) if the order did not fill before time t+1, the dollar value realized when (i) the order is forced at time t+1, or (ii) the order is optional at time t+1.