Optimal order choice: evaluating uncertain discounted trading alternatives

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

• Current Assignee: Pendleton Trading Systems, Inc.
• Original Assignee: Pendleton Trading Systems, Inc.
• Inventors: Wald, John K., Horrigan, Holly T.
• Primary Examiner(s): Millin, Vincent
• Assistant Examiner(s): Kyle, Charles R.
• Application Number:
  US09/396,647
• Time in Patent Office: 1,182 Days
• Field of Search: 705/35, 705/36, 705/37
• US Class Current: 705/36.00R
• CPC Class Codes: G06Q 40/00 Finance; Insurance; Tax strategies; Processing of corporate or income taxes
• CPC Class Codes: G06Q 40/04 Trading; Exchange, e.g. stocks, commodities, derivatives or currency exchange
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis

Abstract

The present invention provides a method for determining whether to execute an order (or list of orders) immediately, or delay execution in exchange for a possible price savings. The method'"'"'s generality enables the investor to optimize order decisions given individual beliefs about expected security returns and variance, risk aversion, and portfolio investment goals. Starting from an expected utility framework, the method maximizes the expected gains associated with trading. The method encompasses the case in which the investor plans to trade the security within a specified trading window as well as the case in which trading occurs only at attractive prices. Additionally, under the assumption of constant absolute risk aversion, the method resembles a traditional mean-variance analysis commonly used in equity portfolio management. The method also generalizes to handle the case of multiple orders and enables an investor to consider an order strategy taking overall portfolio risk into account. The method also can be used in conjunction with dynamic cost control techniques.

The method of the invention is the first such method to consider the maximization of gains in an order context as a function of both returns and the probability of the order being executed. This method is also unique in that it simultaneously accounts for the opportunity costs and the adverse selection costs of using discounted, uncertain orders such as equity limit orders, POSIT® trades, equity principal order trading, etc.

Claims

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;

$\begin{array}{c}\underset{\mathrm{\Gamma <br><br>}}{\max }<br><br>\mathrm{EU}<br><br>\left(W\right)=\text{\hspace{1em}<br><br>}<br><br>\mathrm{EU}<br><br>\{{\left({\tilde{R}}_{\mathrm{Filled}}<br><br>P\right)}^T<br><br>\tilde{X}<br><br>\left(\mathrm{\Gamma <br><br>}\right)+{\left({\tilde{R}}^{\mathrm{†<br><br>}}<br><br>P\right)}^T<br><br>\left(i-\tilde{X}<br><br>\left(\mathrm{\Gamma <br><br>}\right)\right)+\\ \text{\hspace{1em}<br><br>}<br><br>{\left(i+{\tilde{R}}_w\right)}^T<br><br>W_{\mathrm{non}-\mathrm{trade}}+i^T<br><br>W_{\mathrm{trade}}\}\end{array}$

wherein

EU(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 (p_a,t)_j·

s_j,t when the order is a purchase and (p_b,t)_j when the order is a sale, wherein (p_a,t)_j and (p_b,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 s_j,t are the number of shares of security j and the s_j,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 ${\left(r_w\right)}_j=\frac{{\left(p_{m,t+1}\right)}_j-{\left(p_{a,t}\right)}_j}{{\left(p_{a,t}\right)}_j},$

W_non-trade is a column vector of length N whose elements, (w_non-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 (w_non-trade)_j independently are a positive number, zero, or a negative number; and

W_trade is a column vector of length N whose elements, (w_trade)_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 (w_trade)_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.

2. The method according to claim 1 wherein N=1 wherein:

$\begin{array}{c}\underset{\mathrm{\gamma <br><br>}}{\max }<br><br>\mathrm{EU}<br><br>\left(W\right)=\text{\hspace{1em}<br><br>}<br><br>\mathrm{EU}<br><br>\{{\tilde{r}}_{\mathrm{Filled}}<br><br>p_{a,t}<br><br>s<br><br>\tilde{x}<br><br>\left(\mathrm{\gamma <br><br>}\right)+{\tilde{r}}^{\mathrm{†<br><br>}}<br><br>p_{a,t}<br><br>s<br><br>\left(1-\tilde{x}<br><br>\left(\mathrm{\gamma <br><br>}\right)\right)+\\ \text{\hspace{1em}<br><br>}<br><br>\left(1+{\tilde{r}}_w\right)<br><br>w_{\mathrm{non}-\mathrm{trade}}+w_{\mathrm{trade}}\}\end{array}$

wherein

p^†

is p_a,t when the order is a purchase and p_b,t when the order is a sale;

w is a scalar equal to the total dollar value of assets held in the portfolio at time t;

{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 at time t+1;

{tilde over (r)}_Filled, {tilde over (r)}_Forced, and {tilde over (r)}_Optional are ({tilde over (r)}_Filled)_l,l, ({tilde over (r)}_Forced)_l,l, and ({tilde over (r)}_Optional)_l,l, respectively, and are the expected returns of the 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;

{tilde over (r)}_w is the portfolio return for the period t to t+1 of the assets held in the portfolio at time t;

p_a,t is the principal price of the security, adjusted for splits and dividends, for the purchase of the security;

p_b,t is the principal price of the security, adjusted for splits and dividends, for the sale of the security;

s is the number of shares of the security being traded; and

{tilde over (x)}(γ

) is the fraction of the order that is filled at discount γ

.

3. The method according to claim 1 wherein the elements ({tilde over (r)}_w)_j of {tilde over (R)}_w are constant or zero.

4. The method according to claim 1 wherein the {tilde over (x)}_j are independently 0 or 1.

5. The method according to claim 1 wherein each of the s_j,t are 1.

6. The method of claim 1 or 2 wherein the order is a purchase order and the diagonal elements of the returns are given by ${\left({\tilde{r}}_{\mathrm{Filled}}\right)}_{j,j}={\left({\tilde{r}}_{\mathrm{Filled\_P}}\right)}_{j,j}={\left({\tilde{r}}_P\right)}_j+\frac{{\gamma }_{}}{}$

j(pa,t)j,

(r~Forced)j,j=(r~Forced_P)j,j=(p~m,t+1)j-(p~a,t+1)j(pa,t)j, 



and

 



(r~Optional_P)j=0,

and

 



wherein

 



(r~P)j=(p~m,t+1)j-(pa,t)j(pa,t)j,

and ({tilde over (p)}_a,t+1)_j is the principal price of security j at time t+1, (p_a,t)_j is the principal price of security j a time t, and ({tilde over (p)}_m,t+1)_j is the valuation price at time t+1, and wherein all prices are adjusted for splits and dividends when the securities are equities.

7. The method of claim 1 or 2 wherein the order is a short sale order and the diagonal elements of the returns are given by ${\left({\tilde{r}}_{\mathrm{Filled}}\right)}_{j,j}={\left({\tilde{r}}_{\mathrm{Filled\_S}}\right)}_{j,j}={\left({\tilde{r}}_s\right)}_j+\frac{{\gamma }_{}}{}$

j(pb,t)j,

(r~Forced)j,j=(r~Forced_S)j,j=(p~b,t+1)j-(p~m,t+1)j(pb,t)j, 



and

(r~Optional)j,j=(r~Optional_S)j,=0,

and

 



wherein

 



(r~S)j=(pb,t)j-(p~m,t+1)j(pb,t)j,

({tilde over (p)}_b,t+1)_j is the principal price of security j at time t+1, (p_b,t)_j is the principal price of security j a time t, and ({tilde over (p)}_m,t+1)_j is the valuation price at time t+1, and wherein all prices are adjusted for splits and dividends when the securities are equities.

8. The method of claim 1 or 2 wherein the order is a long sale order and the diagonal elements of the returns are given by ${\left({\tilde{r}}_{\mathrm{Filled}}\right)}_{j,j}={\left({\tilde{r}}_{\mathrm{Filled\_LS}}\right)}_{j,}={\left({\tilde{r}}_{\mathrm{LS}}\right)}_j+\frac{{\gamma }_{}}{}$

j(pb,t)j,

(r~Forced)j,j=(r~Forced_LS)j,j=(p~b,t+1)j-(p~m,t)j(pb,t)j, 



and

(r~Optional)j,j=(r~Optional_LS)j,j=(p~m,t+1)j-(pm,t)j(pb,t)j,

(r~LS)j=0,

and wherein ({tilde over (p)}_b,t+1)_j is the principal price of security j at time t+1, (p_b,t)_j is the principal price of security j a time t, and ({tilde over (p)}_m,t+1)_j is the valuation price at time t+1, and wherein all prices are adjusted for splits and dividends when the securities are equities.

31. A computer readable medium having stored thereon instructions for causing a central processing unit to execute the method of any one of claims 1, 2, 9 or 10.

9. 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;

$\underset{\mathrm{\Gamma <br><br>}}{\max }<br><br>\mathrm{CE}=E<br><br>\left(\tilde{G}\right)-\frac{1}{2}<br><br>\mathrm{\Psi <br><br>}<br><br>\left(w\right)<br><br>E<br><br>\left({\tilde{G}}^2\right),$

wherein

10. The method according to claim 9 wherein N=1, and:

11. The method according to claim 9 wherein the elements of {tilde over (R)}_w are constant or zero.

12. The method according to claim 9 wherein the risk aversion parameter is a constant, λ

.

13. The method according to claim 9 wherein the {tilde over (x)}_j are independently 0 or 1.

14. The method according to claim 9 wherein each of the s_j,t are 1.

15. The method of claim 9 or 10 wherein the order is a purchase order and the diagonal elements of the returns are given by ${\left({\tilde{r}}_{\mathrm{Filled}}\right)}_{j,j}={\left({\tilde{r}}_{\mathrm{Filled\_P}}\right)}_{j,j}={\left({\tilde{r}}_P\right)}_j+\frac{{\gamma }_{}}{}$

j(pa,t)j,

(r~Forced)j,j=(r~Forced_P)j,j=(p~m,t+1)j-(p~a,t+1)j(pa,t)j, 



and

 



(r~Optional_P)j=0,

and

 



wherein

 



(r~P)j=(p~m,t+1)j-(pa,t)j(pa,t)j,

and ({tilde over (p)}_a,t+1)_j is the principal price of security j at time t+1, (p_a,t)_j is the principal price of security j a time t, and ({tilde over (p)}_m,t+1)_j is the valuation price at time t+1, and wherein all prices are adjusted for splits and dividends when the securities are equities.

16. The method of claim 9 or 10 wherein the order is a short sale order and the diagonal elements of the returns are given by ${\left({\tilde{r}}_{\mathrm{Filled}}\right)}_{j,j}={\left({\tilde{r}}_{\mathrm{Filled\_S}}\right)}_{j,j}={\left({\tilde{r}}_s\right)}_j+\frac{{\gamma }_{}}{}$

j(pb,t)j,

(r~Forced)j,j=(r~Forced_S)j,j=(p~b,t+1)j-(p~m,t+1)j(pb,t)j, 



and

 



(r~Optional)j,j=(r~Optional_S)j,=0,

and

 



wherein

 



(r~S)j=(pb,t)j-(p~m,t+1)j(pb,t)j,

({tilde over (p)}_b,t+1)_j is the principal price of security j at time t+1, (p_b,t)_j is the principal price of security j a time t, and ({tilde over (p)}_m,t+1)_j is the valuation price at time t+1, and wherein all prices are adjusted for splits and dividends when the securities are equities.

17. The method of claim 9 or 10 wherein the order is a long sale order and the diagonal elements of the returns are given by ${\left({\tilde{r}}_{\mathrm{Filled}}\right)}_{j,j}={\left({\tilde{r}}_{\mathrm{Filled\_LS}}\right)}_{j,}={\left({\tilde{r}}_{\mathrm{LS}}\right)}_j+\frac{{\gamma }_{}}{}$

j(pb,t)j,

(r~Forced)j,j=(r~Forced_LS)j,j=(p~b,t+1)j-(pm,t)j(pb,t)j, 



and

 



(r~Optional)j,j=(r~Optional_LS)j,j=(p~m,t+1)j-(pm,t)j(pb,t)j,

(r~LS)j=0,

and wherein ({tilde over (p)}_b,t+1)_j is the principal price of security j at time t+1, (p_b,t)_j is the principal price of security j a time t, and ({tilde over (p)}_m,t+1)_j is the valuation price at time t+1, and wherein all prices are adjusted for splits and dividends when the securities are equities.

18. The method according to claim 9, wherein E({tilde over (G)}) is given by $E$

(G~)=∫

01

∫

-∞

∞



G

(r~,x~)

 



F

(r~,x~)



r~



x~

wherein F({tilde over (r)},{tilde over (x)}) is a joint distribution of the random variables {tilde over (r)} and {tilde over (x)}(γ

).

19. The method according to claim 18 wherein N is 1 and $\begin{array}{c}\underset{\gamma }{\max }\end{array}$



CE= 



∑

m=1k

(∫

-∞

∞



g~

(r~,β

m)

 



f

(r~)

P(x~∈

Bm

γ

,r~)



r~- 



12

λ



∫

-∞

∞



g~2

(r~,β

m)

 



f

(r~)

P(x~∈

Bm

γ

,r~)



r~)

F({tilde over (r)},{tilde over (x)})=f({tilde over (r)})P({tilde over (x)}ε

B_m|γ

,{tilde over (r)});

f({tilde over (r)}) is a univariate distribution of {tilde over (r)};

{tilde over (x)}(γ

) is continuous in the closed interval [0,1]

P({tilde over (x)}ε

B_m|γ

,{tilde over (r)}) is the probability that {tilde over (x)}(γ

)ε

=B_m for a given {tilde over (r)} and γ

;

k is an integer equal to or greater than 2;

B_m is the fill rate range, wherein m is an integer of from 1 to k and the group of B_m from m=1 to m=k includes the entire interval [0,1] and each fill rate is contained within only one range; and

β

_m is a fill rate representative of the range in B_m in which β

_m is contained.

20. The method according to claim 18 wherein N is 1 and $\begin{array}{c}\underset{\gamma }{\max }\end{array}$



CE= 



∫

-∞

∞



g~

(r~,0)

 



f

(r~)

P(x~=0

γ

,r~)



r~+ 



∫

-∞

∞



g~

(r~,1)

 



f

(r~)

P(x~=1

γ

,r~)



r~- 



12

Ψ



(w)[∫

-∞

∞



g~2

(r~,0)

 



f

(r~)

P(x~=0

γ

,r~)



r~+ 



∫

-∞

∞



g~2

(r~,1)

 



f

(r~)

P(x~=1

γ

,r~)



r~];

F({tilde over (r)},{tilde over (x)})=f({tilde over (r)})P({tilde over (x)}=a|γ

,{tilde over (r)});

f({tilde over (r)}) is a univariate distribution of {tilde over (r)};

{tilde over (x)}(γ

) takes on two discrete outcomes, 0 or 1;

P({tilde over (x)}=a|γ

,{tilde over (r)}) is the probability that {tilde over (x)}(γ

)=a for a given {tilde over (r)} and γ

; and

a is 0 or 1.

21. 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 γ

_j, wherein γ

_j is the order discount for the j^th 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.

22. The method according to claim 21 wherein the order is a purchase order.

23. The method according to claim 21 wherein the order is a short sale order.

24. The method according to claim 21 wherein the order is a long sale order.

26. The method according to claim 24, wherein the certainty equivalent, CE({tilde over (G)}) is approximated by

27. The method according to claim 26 wherein the risk aversion parameter is constant.

25. 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 γ

_j, wherein γ

_j is the order discount for the j^th 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.

28. The method according to claim 25 wherein the order is a purchase order.

29. The method according to claim 25 wherein the order is a long sale order.

30. The method according to claim 25 wherein the order is a short sale order.