Optimize Trade Schedule Trading Strategy for Basket

This example shows how to optimize the strategy for a basket by minimizing trading costs using transaction cost analysis from the Kissell Research Group. Using this optimization, you determine the optimal order slicing strategy for the basket based on the trade-off between trading cost, risk, and the specified level of risk aversion. The optimization minimizes trading costs associated with the trade schedule trading strategy and a specified risk aversion parameter Lambda. The trading cost minimization is expressed as

$\min [(M I + P A) + L a m b d a \cdot T R],$

where trading costs are market impact MI, price appreciation PA, and timing risk TR.

To access the example code, enter edit KRGTradeOptimizationExample.m at the command line. In this example, you can run this code using a trade schedule trading strategy or a percentage of volume trading strategy. This example shows the trade schedule trading strategy. An exponential function determines the optimal trade schedule.

After executing the code in this example, you can submit an order for execution using Bloomberg^®, for example.

This example requires an Optimization Toolbox™ license. For background information, see Optimization Theory Overview (Optimization Toolbox).

Retrieve Market-Impact Parameters and Load Data

Retrieve the market-impact data from the Kissell Research Group FTP site. Connect to the FTP site using the ftp function with a user name and password. Navigate to the MI_Parameters folder and retrieve the market-impact data in the MI_Encrypted_Parameters.csv file. miData contains the encrypted market-impact date, code, and parameters.

f = ftp('ftp.kissellresearch.com','username','pwd');
mget(f,'MI_Encrypted_Parameters.csv');
close(f)

miData = readtable('MI_Encrypted_Parameters.csv','delimiter', ...
    ',','ReadRowNames',false,'ReadVariableNames',true);

Create a Kissell Research Group transaction cost analysis object k. Specify initial settings for the date, market-impact code, and number of trading days.

k = krg(miData,datetime('today'),1,250);

Load the example data TradeDataTradeOpt and the covariance data CovarianceTradeOpt from the file KRGExampleData.mat, which is included with the Datafeed Toolbox™.

load KRGExampleData TradeDataTradeOpt CovarianceTradeOpt

For a description of the example data, see Kissell Research Group Data Sets.

Define Optimization Parameters

Specify initial values for risk, trading periods, portfolio value, and covariance matrix. Convert to a buy-only problem. Set the initial trade schedule.

% Convert table to array
CovarianceTradeOpt = table2array(CovarianceTradeOpt);

% Use total trading time of 1 day with 13 trading periods 
totalDays = 1;
periodsPerDay = 13;

% Set risk aversion level
Lambda = 0.5;

% Set minimum and maximum percentage of volume
minPOV = 0.00;
maxPOV = 0.60;

% total number of trading periods
totalNumberPeriods = totalDays * periodsPerDay;

% Portfolio Value
PortfolioValue = TradeDataTradeOpt.Price'*TradeDataTradeOpt.Shares;

% Number of stocks
numberStocks = height(TradeDataTradeOpt);

% Covariance matrix is annualized covariance matrix in decimals.
% Convert to ($/Shares)^2 units for the trade period; this matrix is for a
% two-sided portfolio, buys and sells or long and short.
CC = diag(TradeDataTradeOpt.Price) * CovarianceTradeOpt * ...
    diag(TradeDataTradeOpt.Price);        

% Scale to one trading period
CC = CC / periodsPerDay / k.TradeDaysInYear;       

% Convert to buy-only problem (e.g., one-sided problem)
CC = TradeDataTradeOpt.SideIndicator * TradeDataTradeOpt.SideIndicator' .* CC;      

% Convert Alpha_bp from basis points per day to cents/share per period
TradeDataTradeOpt.Alpha_bp = TradeDataTradeOpt.Alpha_bp / 1000 .* ...
    TradeDataTradeOpt.Price / totalNumberPeriods;

% Set the initial trade schedule or POV values
theta0 = rand(numberStocks,1);

Define optimization options using the optimset function. For details about these options, see Optimization Options Reference (Optimization Toolbox).

optionsold = optimset;
options = optimset(optionsold,'LargeScale','on','GradObj','off', ...
    'DerivativeCheck','off','FinDiffType','central','FinDiffRelStep',1E-12, ...
    'TolFun',10E-5,'TolX',10E-12,'TolCon',10E-12,'TolPCG',10E-12, ...
    'MaxFunEvals',20000,'MaxIter',20000,'DiffMinChange',10E-04);

Define lower and upper bounds of shares traded per interval for optimization.

LB = zeros(numberStocks,1);
UB = 100 * ones(numberStocks,1);

Minimize Trading Costs for Trade Strategy

Minimize the trading costs for the trade schedule strategy. fmincon finds the optimal value for the trade schedule trade strategy based on the lower and upper bound values. It does this by finding a local minimum for the trading cost. Use the objective function optimizeTradingSchedule. To access the code for this function, enter edit KRGTradeOptimizationExample.m.

[theta,fval,exitflag,output] = fmincon(@optimizeTradingSchedule,theta0,[], ...
    [],[],[],LB,UB,[],options,totalNumberPeriods,numberStocks,periodsPerDay, ...
    TradeDataTradeOpt,CC,Lambda,k);

To check whether fmincon found a local minimum, display the reason why the function stopped.

exitflag

exitflag =

     1.00

fmincon returns 1 when it finds a local minimum. For details, see exitflag (Optimization Toolbox).

Calculate shares to trade, residual shares, price appreciation, and timing risk. Then, calculate the average percentage of volume rate and trade time.

numPeriods = 1:totalNumberPeriods;
K_Matrix = repmat(numPeriods,numberStocks,1);
Theta_Matrix = repmat(theta,1,totalNumberPeriods);
Volume_Matrix = repmat(TradeDataTradeOpt.ADV/periodsPerDay,1, ...
    totalNumberPeriods);
TradeDataTradeOpt.VolumeProfile = Volume_Matrix;
Shares_Matrix = repmat(TradeDataTradeOpt.Shares,1,totalNumberPeriods);

% X = Shares to trade in period i
Xpct = (exp(-K_Matrix .* Theta_Matrix) .* (exp(Theta_Matrix)-1)) ./ ...
    (1 - exp(-totalNumberPeriods * Theta_Matrix));
X = repmat(TradeDataTradeOpt.Shares,1,totalNumberPeriods) .* Xpct;
TradeDataTradeOpt.TradeSchedule = X;
  
% R = Residual Shares at beginning of period i
Rpct = (exp(-(K_Matrix-1).*Theta_Matrix) - exp(-totalNumberPeriods.*Theta_Matrix)) ./ ...
    (1-exp(-totalNumberPeriods.*Theta_Matrix));
R = repmat(TradeDataTradeOpt.Shares,1,totalNumberPeriods) .* Rpct;

% Price Appreciations in Dollars
PA = sum(R,2) .* TradeDataTradeOpt.Alpha_bp;
  
% Market Impact in Dollars
MI = marketImpact(k,TradeDataTradeOpt) .* TradeDataTradeOpt.Shares .* ...
    TradeDataTradeOpt.Price ./10000;
  
% Timing Risk in Dollars
TR = sqrt(sum(R.^2,2) .* diag(CC));
TR_bp = TR ./ (TradeDataTradeOpt.Shares .* TradeDataTradeOpt.Price) * 10000;
    
% Avg POV Rate
kTR = ((TR_bp/10000*1./TradeDataTradeOpt.Volatility).^2).*(k.TradeDaysInYear*3 ./ ...
    (TradeDataTradeOpt.Shares./TradeDataTradeOpt.ADV));
POV = 1./(1+kTR);
POV = max(POV,TradeDataTradeOpt.Shares./(TradeDataTradeOpt.Shares+totalDays .* ...
    TradeDataTradeOpt.ADV));
    
% TradeTime
TradeDataTradeOpt.TradeTime = TradeDataTradeOpt.Shares./TradeDataTradeOpt.ADV .* ...
    (1-POV)./POV;

Estimate total trading costs using the optimized trade strategy.

TotMI = sum(MI) / (TradeDataTradeOpt.Shares' * TradeDataTradeOpt.Price) ...
    .* 10000;       % bp
TotPA = sum(PA) / (TradeDataTradeOpt.Shares' * TradeDataTradeOpt.Price) ...
    .* 10000;       % bp
TotTR = sqrt(trace(R'*CC*R)) ./ (TradeDataTradeOpt.Shares' * ...
    TradeDataTradeOpt.Price) * 10000;

Display total market-impact cost, price appreciation, and timing risk.

totalcosts = [TotMI TotPA TotTR]

totalcosts =

   38.2902         0   26.5900

For details about the preceding calculations, contact the Kissell Research Group.

References

[1] Kissell, Robert. The Science of Algorithmic Trading and Portfolio Management. Cambridge, MA: Elsevier/Academic Press, 2013.

[2] Malamut, Roberto. “Multi-Period Optimization Techniques for Trade Scheduling.” Presentation at the QWAFAFEW New York Conference, April 2002.

[3] Kissell, Robert, and Morton Glantz. Optimal Trading Strategies. New York, NY: AMACOM, Inc., 2003.