L. Ingber

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Research Papers in Economics

Trading in Risk Dimensions (TRD)
Lester Ingber
Lester Ingber Research (LIR)
ingber@ingber.com
http://www.ingber.com/ • http://alumnus.caltech.edu/˜ingber
As horter updated paper with this title appears in the Handbook of Trading: Strategies for
Navigating and Profiting from Currenc y,B ond, and Stock Markets, published by Wiley( 2010).
Only the Table of Contents and the Abstract appear in http://www.ingber.com/markets05_trd.pdf, as
the publisher has requested all previous papers be removedf rom the internet. The full report,
markets05_report.pdf, is nowap rivate paper; contact the author for more information.
$Id: markets05_trd.ms,v 1.219 2011/12/15 22:58:48 ingber Exp ingber $Copyright © 2003-2009 Lester Ingber.A ll Rights Reserved. TRD-2
Contents
Abstract ............................. 3
1. Background ........................... 4
1.1. Adaptive Simulated Annealing (ASA) ................. 4
1.2. Generic Approach ....................... 4
2. Data .............................. 4
3. Exponential Marginal Distribution Models .................. 4
3.1. Fitting Data ......................... 5
4. Copula Transformation ........................ 5
4.1. Aside on Use of Gaussian Copula .................. 5
4.2. Transformation to Gaussian Marginal Distributions ............ 6
4.3. Including Correlations ...................... 6
4.3.1. Stable Covariance Matrices ................. 7
4.4. Fitting Data ......................... 8
4.5. Copula of Multivariate Correlated Distribution .............. 8
5. Portfolio Distribution ........................ 8
5.1. Recursive Risk-Management in Trading Systems ............. 9
6. Risk Management ......................... 9
7. Sampling Multivariate Normal Distribution for Events .............. 1 0
7.1. Transformation to Independent Variables ................ 1 0
8. Numerical Development of Portfolio Returns ................. 1 0
8.1. C From Sampled Events Into Bins .................. 1 0
8.1.1. Events Without Bins .................... 1 1
8.2. C Directly From Moments of Events ................. 1 1
8.2.1. Moments With No Trading Functions .............. 1 1
8.2.1.1. Variate Corrections ................. 1 2
8.2.2. Moments With Trading Functions ............... 1 2
9. NC Models ............................ 1 3
9.1. Individual-Market VaRi Prefilter .................. 1 3
9.2. Candidate Postfilters ...................... 1 4
9.2.1. Diversification of dNC Including Correlations ........... 1 4
9.2.2. Diversification of dNC Including Covariances ........... 1 4
9.2.3. Maximization of P/L ................... 1 4
9.2.4. Maximization of Sharpe Ratio ................ 1 4
10. Multiple Trading Systems ....................... 1 4
10.1. Canonical Momenta Indicator (CMI) Trading System ........... 1 5
10.1.1. Consistent Use of Distributions ................ 1 5
10.1.1.1. Use of Projected Distributions ............. 1 5
10.2. Trend & Volatility,P rice and CMI, Value-Weighting, LevelI I, etc. ....... 1 5
10.3. Sets of Multiple Optimal Trading Parameters .............. 1 6
10.3.1. Recursive ASA Fits ................... 1 6
10.4. Adaptive Logic ....................... 1 6
11. TRD Highlights .......................... 1 6
11.1. Relative Time Scales of Fits ................... 1 7
11.2. Inter-Day Trading ....................... 1 7
11.3. Post-Processing of Multiple Optima/“Mutations” ............. 1 7
11.4. STREAM For All Time Resolutions ................. 1 8
12. Conclusion ........................... 1 8
13. Acknowledgments ......................... 1 8
14. REFERENCES .......................... 1 9Copyright © 2003-2009 Lester Ingber.A ll Rights Reserved. TRD-3
Abstract
Previous work, mostly published, developed two-shell recursive trading systems. An inner-shell of
Canonical Momenta Indicators (CMI) is adaptively fit to incoming market data. Ap arameterized trading-
rule outer-shell uses the global optimization code Adaptive Simulated Annealing (ASA) to fit the trading
system to historical data. As imple fitting algorithm, usually not requiring ASA, is used for the inner-
shell fit. An additional risk-management middle-shell has been added to create a three-shell recursive
optimization/sampling/fitting algorithm. Portfolio-leveld istributions of copula-transformed multivariate
distributions (with constituent markets possessing different marginal distributions in returns space) are
generated by Monte Carlo samplings. ASA is used to importance-sample weightings of these markets.
The core code, Trading in Risk Dimensions (TRD), processes Training and Testing trading systems on
historical data, and consistently interacts with RealTime trading platforms at minute resolutions, but this
scale can be modified. This approach transforms constituent probability distributions into a common
space where it makes sense to develop correlations to further develop probability distributions and
risk/uncertainty analyses of the full portfolio. ASA is used for importance-sampling these distributions
and for optimizing system parameters.

Trading in Risk Dimensions (TRD)

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