Folie 1

1
Empirical Analysis of an Online Trading Algorithm

• Günter Schmidt12) and Esther Mohr1)
• Saarland University, Saarbrücken, Germany
• University of Liechtenstein, Vaduz, Liechtenstein
• gs,em_at_itm.uni-sb.de

18th Annual Conference, South African Finance
Association, January 14-16, 2009
2
Agenda

• Motivation
• Problem Formulation
• Simulation Design
• Experimental and Worst Case Results
• Conclusions

3
Motivation

• We focus electronic markets where trading is
  carried out automatically (online)
• Trading policies which have the potential to
  operate without human interaction
• are often based on technical analysis She02,
  RL99, RS03
• are implemented from the perspective of
  artificial intelligence, software agents or
  neural networks CE08, FRY04, SR05
• We are interested in the analysis of policies
  based on online algorithms from a worst case and
  an average case point of view

4
Trading Algorithms

• Three approaches for performance analysis of
  trading algorithms in the literature
• Stochastic Analysis a given probability
  distribution for asset prices is a basic
  assumption
• Competitive Analysis assuming uncertainty about
  asset prices and analyzing worst case outcomes
• Heuristic Approach algorithms are implemented
  and the analysis is done on historic data by
  simulation runs
• For our performance analysis we combine
  approaches (2) and (3) when solving a multiple
  trade problem

5
Agenda

• Motivation
• Problem Formulation
• Simulation Design
• Experimental and Worst Case Results
• Conclusions

6
Problem Formulation

• Search Problem
• Buy or sell a single asset at minimum or maximum
  price
• (1) a trader obtains price quotations p(t), m lt
  p(t) lt M at points of time t1,2,,T
• (2) every time t, the trader must decide whether
  or not to accept p(t)
• (2.1) p(t) is accepted trading is closed
• (2.2) p(t) is not accepted the trader obtains
  the next price quotation if no price is accepted
  until T-1 then p(T) has to be accepted which
  might be m or M.
• Single Trade Problem (k1)
• Buy and sell a single asset Search for its
  minimum price m and its maximum price M in a time
  series of T prices p(t)
• -gt try to buy the asset at m and sell it at M
• Multiple Trade Problem (kgt1)
• Trade k assets sequentially Try to buy asset u
  at m(u), sell u at M(u) buy asset v at m(v),
  sell v at M(v), etc

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Competitive Analysis of Online Algorithms

• Algorithm ON computes online if for each
  j1,,n-1, it computes an output for j before the
  input for j1 is given
• Algorithm OPT computes offline if it computes a
  feasible output given the entire input sequence
  j1,,n-1
• ON is c-competitive for a MAX problem if for any
  input I
• ON(I) gt 1/c OPT(I)
• -gt any c-competitive Algorithm ON is guaranteed
  a return of at least a fraction 1/c of the
  optimal offline return OPT
• -gt c is a worst-case performance ratio OPT(I) /
  ON(I) lt c
• the smaller c the better is Algorithm ON

8
Search Problem

• The selling rule for the MAX search problem
  Yan98
• accept the first price greater or equal to
• the reservation price
• has a competitive ratio
• i.e. OPT / ON lt
• -gt This result is generalized for the case of
  single and multiple trade problems in Sch08

9
Single Trade Problem (k1)

• In the single trade problem search is carried out
  twice
• (1) Buying decision
• (2) Selling decision
• Policy for trading assets
• Buy the asset at the first price smaller or
  equal to
• reservation price p
• AND
• Sell the asset at the first price greater or
  equal to
• reservation price p
• In the worst case we get a competitive return
  ratio of
• ct M / m

10
Multiple Trade Problem (kgt1)

• The ratio ct(k) can be interpreted as the rate of
  return OPT achieves in comparison to ON buying
  and selling assets k-times
• ct(k) IIi1,, k M(i) / m(i)
• i.e. ct(k) (M / m)k
• iff m and M are constant for all trades Sch08
• Multiple Trade reservation price policy is
  evaluated empirically and represented by a
  Market Timing (MT) Trader

11
Agenda

• Motivation
• Problem Formulation
• Simulation Design
• Experimental and Worst Case Results
• Conclusions

12
Traders

• Simulation runs with three types of ON policies
• Market Timer (MT) applies reservation prices p
• Buy Holder (BH)
• Index BH holds an index
• MT BH holds the asset which MT bought
• Random (Rand) points of time for buying and
  selling are chosen randomly
• and with OPT as a benchmark
• Market (MA) buys at minimum possible price pmin
  gt m and sells at maximum possible price pmax lt M
  within each period

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Questions to be Answered

• (1) Does MT show a superior behavior to BH
  policies?
• (2) What is the influence of m and M on the
  performance of MT?
• (3) What is the average case performance of MT
• (4) and how does it compare to the worst case
  competitive ratio ct(k)?

14
Test Design

• Simulation based on historical data from XETRA
  DAX for the interval 01.01.1998 06.30.2008
  (01.01.2007 12.31.2007 )
• Time horizon is divided into trading periods of
  different length each trading period consists of
  a constant number of days
• Six types of periods with length 1, 2 weeks and
  1, 3, 6, 12 months
• Two different test sets
• Historic estimates for m and M (Historic m, M)
• Precise estimates for m and M (Clairvoyant m, M)
• -gt 12 different simulation runs

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Assumptions for all policies

• There is the same initial portfolio value greater
  than zero for all traders
• In each period i at least one buying and one
  selling transaction must be executed
• At each point of time t there is at most one
  asset in the portfolio
• In period i 1 all policies buy the same asset j
  on the same day t at the same price pj(t)
  (calculated by MT policy) in periods i gt 1
  trades are carried out according to the different
  trading policies in the last period there is
  only a selling transaction.
• Policies are always invested cash has a return
  rate of 3

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Assumptions for all policies
Policies are allowed to trade according to the
following pattern
T1
1
. . .
Buy u
Sell u
Buy w
lt
lt
lt
MT
Buy v
Sell v
Buy w
Buy u
Sell u
MA
lt
lt
lt
Buy v
Sell v
lt
lt

MT-BH
Buy u
Sell u
Buy w
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Agenda

• Motivation
• Problem Formulation
• Simulation Design
• Experimental and Worst Case Results
• Conclusions

18
Experimental vs. Worst Case Results

• Performance of policies evaluated empiricaly with
  annualized return rates as the performance
  measure
• The following results for the ten year interval
• 01.01.1998, 06.30.2008 are achieved
  (transaction costs included)
• (1) Annualized returns for historic estimates
  of m and M including transaction costs
• (2) Annualized returns for precise estimates of
  m and M
• (3) Comparison of worst case competitive return
  rates and empirical return rates

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(1) Does MT show a superior behavior to BH
policies?
Tab. 1 Historic 1998-2008 annualized returns
MT generates the best annual return rate (4
weeks) outperforms Index BH in three of six
cases (4 weeks and lower) outperforms MT BH in
all cases MT BH generates the worst annual
return rate in five cases (except 4 weeks)
Average performance 6.53 for MT, 1.99 for
Index BH, -2.64 for MT BH -gt MT shows on
average a superior behavior to buy-and-hold
policies. For shorter periods MT shows a
superior behavior to all buy-and-hold policies.
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(2) What is the influence of m and M on the
performance of MT?
Tab. 2 Clairvoyant 1998-2008 annualized returns

• Clairvoyant precise estimates for m and M
• MT outperforms MT BH and Index BH in all cases
• -gt the better the estimates for m and M the
  better the performance of MT

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(3) What is the average performance of MA and MT?
Tab 3 Example Trades in 2007
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(3) What is the average performance of MA and MT?
Tab 4 Competitive ratios for 1998-2008
-gt MT reaches at least 65.47, at most 98.73
and on average 82.02 of the return of
MA (MT/MA) -gt the shorter the periods, the more
achieves MT in comparison to MA (MT/MA) -gt the
average case bound reaches at least 13.43, at
most 42.42 and on average 22.95 of the
worst case bound
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(1) Historic Data 2007 vs. 1998 2008
Tab 5 Historic 2007 and 1998-2008 annualized
returns
-gt The shorter the periods the better is the
return for MT relative to buy-and-hold policies
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(2) Clairvoyant Data 2007 vs. 1998 2008
Tab. 6 Clairvoyant 2007 and 1998-2008 annualized
returns
-gt MT clearly outperforms all buy-and-hold
policies for all period lengths
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(2) Clairvoyant Data 2007 vs. 1998 2008
Tab. 6 Clairvoyant 2007 and 1998-2008 annualized
returns
MT/MA decrease 2007 versus (increase) 1998-2008 ?
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(3) Competitive Ratios 2007 vs. 1998 2008
Tab. 7 Competitive Ratio 2007 and 1998-2008 for
clairvoyant estimates without transaction costs
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Agenda

• Motivation
• Problem Formulation
• Simulation Design
• Experimental and Worst Case Results
• Conclusions

28
Conclusions

• Clairvoyant m and M
• - MT outperforms BH in all cases (including
  transaction costs)
• (2) Historical estimates of m and M
• - MT outperforms BH in half of the cases and on
  average
• - if the period length is short MT
  outperforms BH
• (3) Obvious
• - the better the estimates of m and M the
  better the
• performance of MT
• - the longer the periods the worse the
  historical estimates of m
• and M
• -gt the performance of MT (and of MA) gets worse
  the longer the periods because of bad estimates
• (4) Fortunately It is very difficult to reach
  the (analytical) worst cases under (simulated)
  real-life conditions.

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Appendix

• Detailed 2007 Results

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(1) Does MT show a superior behavior to BH
policies?
Tab. 8 Annualized 2007 Returns
MT generates the best annual return rate in two
cases (1 and 4 weeks) outperforms Index BH in
four of six cases outperforms MT BH in three of
six cases (4 weeks and lower) MT BH generates
the worst annual return rate in three cases (4
weeks and lower) improves its performance
proportional to the length of the periods.
outperforms MT in cases period length 3 months
and higher Average performance 27.86 for
MT, 20.78 for Index BH, 20.63 for MT BH
-gt MT shows on average a superior behavior
to buy-and-hold policies. MT shows for
shorter periods a superior behavior to MT BH.
31
(1) Historic Data 2007 vs. 1998 2008
Tab. 11 Historic 2007 and 1998-2008 Annualized
Returns
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(2) What is the influence of m and M on the
performance of MT?
Tab. 9 Clairvoyant Annualized 2007 Returns
Clairvoyant precise estimates for m and
M MT outperforms MT BH and Index BH in all
cases -gt the better the estimates for m and M
the better the performance of MT
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(2) Clairvoyant Data 2007 vs. 1998 2008
Tab. 12 Clairvoyant 2007 and 1998-2008
Annualized Returns
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(3) What is the average performance of MA and MT?
Tab.10 Competitive Ratios for 2007
-gt MT reaches at least 27.33, at most 77.22 and
on average 45.67 of the return of MA (MT /
MA) -gt the shorter the periods, the less
achieves MT in comparison to MA (MT/MA) -gt the
average case bound reaches at least 49.06, at
most 64.56 and on average 57.07 of the
worst case bound