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Optimal order choice: evaluating uncertain discounted trading alternatives

DC
  • US 6,493,682 B1
  • Filed: 09/15/1999
  • Issued: 12/10/2002
  • Est. Priority Date: 09/15/1998
  • Status: Expired due to Fees
First Claim
Patent Images

1. 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 from the equation;

    maxΓ



    EU

    (W)
    =



    EU

    {(R~Filled

    P
    )
    T


    X~

    (Γ

    )
    +(R~



    P
    )
    T


    (i-X~

    (Γ

    )
    )
    +




    (i+R~w)T

    Wnon-trade
    +iT

    Wtrade
    }
    embedded imagewhereinEU(W) is the expected value of the utility function U;

    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 whose elements are the order discount, γ

    j, 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 prices 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 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 γ

    j;

    {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 non-diagonal elements are zero and whose diagonal 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 filled 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;

    {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 (rw)j=(pm,t+1)j-(pa,t)j(pa,t)j,embedded imageWnon-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; and

    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

    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.

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