1 optimisation of trading strategies using arbs

8/10/2019 1 Optimisation of Trading Strategies Using arbs

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Optimisation of Trading Strategies using

Parameterised Decision Rules

N. Towers and A. N. Burgess

Department of Decision Sciences

London Business School

Sussex Place, Regents Park,

London, NW1 4SA, UK.

E-mail: {ntowers, nburgess}@lbs.ac.uk

Abstract.In the context of a dynamic trading strategy, the ultimate purpose of

any forecasting model is to choose actions which result in the optimisation of

the trading objective. In this paper we develop a methodology for optimising an

objective function, using a parameterised decision rule, for a given forecastingmodel. We simulate the expected trading performance for different decision

parameters and levels of prediction accuracy. We then apply the technique to a

forecasting model of mispricing within a group of equity indices. We show that

optimisation of the proposed decision rule can increase the annualised Sharpe

Ratio by a factor of 1.7 over a naive decision rule.

1 Introduction

In recent years considerable research effort has been devoted to the development of

financial forecasting models which attempt to exploit the dynamics of financial

markets. Research in financial forecasting has applied and often extended techniques

developed in the fields of machine learning, non-parametric statistics and time series

modelling.

In the context of a dynamic trading strategy the ultimate purpose of any financial

forecasting model is to choose actions (i.e. trading positions) which result in

optimisation of a specific trading objective. A standard approach to decision making

under uncertainty is to decompose the process into two independent steps: first the

building of a predictive model and then secondly, the selection of an optimal decision

rule which converts the prediction into an action.

The financial forecasting community has mostly neglected optimisation of this

second step with trading performance measured using an arbitrary decision rule.

Research that does optimise trading systems use methodologies that maximise the

trading objective by building models that combine the forecasting and decision

making steps. Examples of these include using reinforcement learning by Moody inPerspectives on Financial Engineering and Data Mining, Proc. IDEAL 98. Edited

by L. Xu et al, Springer-Verlag


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[1] and neural networks by Choey & Weigend in [2]. In this paper we consider the

forecasting and decision modules as independent tasks and explore issues involved in

the selection and optimisation of trading rules for the decision making step. We show

how this choice affects the performance of the whole decision making process.

In section 2 we present an example which motivates the importance of the choice of

decision rule by showing that the same forecasting model can significantly outperform

or even underperform depending on the decision rule. We construct a forecasting

model based upon relative value between the UK and Swiss equity indices and show

that the mispricing exhibits predictability. We then apply three trading rules that

convert the relative value signal into a trading position. We show how the different

trading rules significantly effect the investment performance.

In section 3 we investigate the optimisation and selection of trading rules for

predictive models. We devise a simplified framework (i.e. no transaction costs,

borrowing costs, etc) and use it to optimise a parameterised decision rule given a

forecasting model. To build the simplified framework a methodology is developed for

generating synthetic forecasts from a predictive model with variable but known

prediction accuracy. A parameterised set of trading rules are devised which represents

the trading position of a single risky asset given the predicted percentage return of theasset. In section 4 we present simulation results for selected values of the trading rule

parameters and levels of prediction accuracy. A typical trading objective (annualised

Sharpe Ratio) is maximised by optimising the trading rule parameter given the

prediction accuracy of the model.

In section 5 we apply the technique to a real world forecasting model that estimates

the relative mispricing between a group of equity indices. We show that optimising the

decision step using a parameterised trading rule increases the annualised Sharpe Ratio

by a factor of 1.7 over a naive decision rule.

2 Example

To motivate the importance of the choice of decision rule for a relative value price

series was generated between the Swiss and UK equity indices. Data was collected at

10-minute intervals over a 20-day period and the percentage relative price estimated as

shown in figure 1.Relative Price: FTSE v SSMI

-4.00%

-3.00%

-2.00%

-1.00%

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 6 11 16 21 26 31 36 41 46

VR1

VR2

VR5

VR10

Fig. 1.The left hand graph shows the relative price series between FTSE and SSMI indices.

The right hand graph shows the variance ratio profile for the series.


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The variance ratio profile in figure 1 shows the values of the variance ratio up to 50

time periods. For a random walk process we expect the variance of the periodincrements to approximately equal times the variance of the single period increments

so the variance ratio will be close to one. In figure 1 the variance ratio function is

significantly below one which indicates negative autocorrelation and so mean-

reverting or cyclical behaviour.

Three simple but plausible trading rules where devised which converts the

mispricing into a trading position as follows:

Dtm

t -- m

t(1)

where D is the trading position and m the mispricing at time t. For trading rule 1, and are zero. For rules 2 and 3, and are 1 and 100, and 1 and 10 respectively.

Figure 2 shows the equity curves, annualised Sharpe Ratio and profit (ignoring

transaction costs) for the 3 different trading rules. Comparison of the three rules shows

that they generate significantly different equity curves with the cumulative profit

varying from -4.63% to 10.21%. Note that, rule 1 is the most profitable, rule 3 most

consistent in terms of Sharpe ratio, and rule 2 illustrates that negative returns can be

achieved for forecasting models with positive prediction accuracy. Overall thisexample shows that the choice of decision rule can dramatically effect the

performance of a trading strategy.

Annualised Sharpe Ratio

(1) (2) (3)

3.6 -2.2 3.8

Profit

(1) (2) (3)

10.54% -4.63% 3.16%

Fig. 2. shows the equity curves, Sharpe ratio and profit of 3 different trading rules (ignoring

transaction costs).

3 Decision Framework

In this section we design a dynamic decision framework to represent a synthetic

trading system for one risky asset. We show how to generate predictions of the risky

asset, which simulate the output from a well-specified forecasting model of known

prediction accuracy. A parameterised set of decision rules are proposed and theirconsequences are given in terms of expected return. The method assumes the

Equity Curves

-15%

-10%

-5%

0%

5%

10%

15%

PL(Stat)

PL(Cyc100)

PL(Cyc10)


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generated predictions have a level of prediction accuracy, measured in terms of the

correlation coefficient between predicted and actual asset returns.

Let an asset price series, y, an explanatory variable, x, and a noise variable,, be

defined by

yt= x

t+

t

where xt

~ NID(0,x

2)

and t~ NID(0,

2)

(2)

where represents a constant information coefficient between the explanatoryvariable and the asset price. The explanatory variable reflects the deterministic part of

the price signal and the noise term refers to the s tochastic part.

Suppose the variance between the explanatory variable is equal to the variance of

the actual returns. Standardising the normal distributions for the explanatory variable

and the asset return allows the variance of the noise term to be defined as

2=1 -

2(3)

We define the forecasting model as t=x

twhere for a well specified predictor=||.

The absolute value is used so that the information content of the predictive model can

be negative. If|||| then the predictor is biased.We have now defined a data generating process for well-specified forecasting

models with a variable degree of predictability.

The variance of the predictions is y=

2and the covariance between predicted and

asset price returns is y

= . The correlation coefficient is given by y

= and isconsidered to be the measure of the prediction accuracy of the forecasting model. The

R2of the model is then equal to the variance of the predictions as expected for a well-

specified model.

Next we devise a parameterised decision rule that converts a predicted return into a

trading position. The difference between the new and previous trading position defines

the amount of the asset that must be either bought or sold. A set of decision rules D,

can be investigated by defining two parameters k and mas follows:

Dt= m |t|k

sign(t ) (4)

wheret is the forecast return andD

tthe trading position at time t.

When k=1the decision rule is a linear function with slope m; when k=0the rule is

a step function between two states +mand -m1. Other positive values of kand mgive

additional non-linear decision functions. Figure 3 shows the parameterised decision

rule for 4 values of k. The parameter mis equivalent to a leverage factor with no upper

limit. This parameterised decision rule is a generalisation of the optimal solution of the

risk averse investor with no leverage constraint. In this case k=1 and m= /2,

where is the risk aversion parameter for the utility function, is the expected return

and the expected standard deviation of returns.

1For

t and k=0,D assumes 0

0=0 using Lopitals theorem.


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Parameterised Decision Rule

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

forecast

position

k=0 m=1

k=0.5 m=1

k=1 m=1

k=2 m=1

Fig. 3.shows the decision rule for 4 values of parameters k and m.

The expected investment return for a decision epoch is defined to be the average of

the trading position multiplied by the asset price return over n time periods . The

expected mean and variance of the investment returns can then be used to form a risk

adjusted performance measure for the trading objective. In this paper we consider theratio of average return divided by standard deviation of return, defined by Sharpe in

[4]. The trading performance will be some function of both prediction accuracy, andthe decision rule parameter k. In the next section we investigate this relationship by

varying these parameters.

4 Simulation Results

The investment returns were simulated for different values of the decision rule

parameters and levels of prediction accuracy. The mean and variance of investment

returns are shown for three values of k in figure 4. From these results it is clear that

the mean and variance are sensitive to both prediction accuracy and the decision ruleparameter k.

Expected return

0

0.2

0.4

0.6

0.8

1

1.2

0.05 0.2 0.35 0.5 0.65 0.8 0 .95

prediction accuracy

return

k=1

k=0

k=0.5

Expected Variance

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.05 0.2 0.35 0.5 0.65 0.8 0.95

prediction accuracy

variance k=1

k=0

k=0.5

Fig. 4. shows mean and variance of expected investment returns per decision epoch for three

values of the decision parameter, k, and varying prediction accuracy.


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The Sharpe ratio is now calculated for varying levels of prediction accuracy and

values of decision parameter,k. Note that the trading rule parameter, m, is a common

multiplying factor of both return and variance and so does not effect the value of the

Sharpe Ratio. It is therefore ignored for this performance metric.

The left hand graph in Figure 5 shows how the optimal trading rule parameter, k,

varies with the prediction accuracy of the model. The graph shows that the optimal

value of kdecays exponentially as the prediction accuracy of the model increases. The

right hand graph in figure 5 shows trajectories of expected position in return-risk space

for three different values of trading rule parameter for increasing prediction accuracy.

This graph shows how the significance of the choice of trading rule parameter, k,

effects the mean and variance of returns.

Optimal value of Trading parameter, k

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0. 4 0 .5 0. 6 0 .7 0.8 0.9 1

prediction accuracy

optimalk

Return/Risk Plot for varying prediction accuracy

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5

variance

mean

k=1

k=0

k=0.5

Fig. 5.The graph on the left shows the estimated optimal value of the trading rule parameter, k,

for different levels of prediction accuracy. The graph on the right shows the trajectories of

expected position in return-risk space for different values of trading rule parameter, k, by

increasing the prediction accuracy.

The key finding of these results is that the Sharpe Ratio, for a given level of

predictability, varies significantly as a function of the trading parameter, k. Therefore

maximisation of this parameter can significantly improve the performance of a trading

strategy. In the next section we apply the decision making methodology to a realapplication within statistical arbitrage.

5 Application within Statistical Arbitrage

We use the term statistical arbitrage as a generalisation of traditional zero risk

arbitrage where the price dynamics of a related group of assets are known and

exploited in order to achieve a profit. In this wider sense the relationships are

statistical rather than known and so statistical arbitrage trading is no longer risk

free and so risk should be controlled in the trading strategy.

Hourly price data was collected from 15 May 1998 to 7 July 1998 for a group of

equity index futures (FTSE, S&P, DAX, CAC, FIB and SSMI). A methodology usingprincipal component analysis developed in [5] was applied to the data to estimate

relative weightings of the Equity indices and so generate a relative mispricing, p as

follows:


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p = 0.58 FTSE + 0.17 S&P 0.15 DAX 0.76 FIB + 0.19 SSMI (5)

The left hand graph of figure 6 shows the value of the mispricing through time and the

right hand graph is the variance ratio profile for the mispricing. It shows that the

variance ratio of the mispricing is below one and so has significant cyclical or mean

reverting behaviour.

Relative mispricing of Equity Indices

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0 100 200 300

time

value

0

0.5

1

1.5

2

2.5

3

1 6 11 16 21 26 31 36 41 46

VR1

VR2

VR5

VR10

Fig. 6. The left hand graph shows the estimated mispricing, p between the prices of FTSE,

S&P, DAX, FIB and SSMI futures indices. The right hand graph shows the Variance Ratio plotfor the mispricing.

The mean reversion in the mispricing allows us to define a predictive model as the

negative of the relative mispricing. For this forecasting model the prediction accuracy

(i.e. the correlation between the forecasts and the change in the mispricing) was

measured to be 0.12.

The trading objective was specified as maximising Sharpe Ratio and so for this

level of prediction accuracy we expect, from the simulation results in section 4, that

the optimal value of k will be 1.2. Using the sample data to optimise k results in a

value of 1.8 as shown in the left hand graph in figure 7.

Anualised Sharpe Ratio

0

1

2

3

4

5

6

0 1 2 3 4

parameter, k

sharperatio

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

sample

synthetic

Equity Curve

-0.1

-0.05

0

0.05

0.1

0.15

0.2

116

31

46

61

76

91

106

121

136

151

166

181

196

211

226

241

256

271

time

profit

k=1.2

k=1.8

Fig. 7. The graph on the left shows the Sharpe Ratio for different values of parameter kfor the

synthetic and sample data. The graph on the right shows the equity curves for the two optimal k

values.

It is proposed that the difference between the two optimal k values is due toassumptions of normality and an unbiased forecasting model in the synthetic data

generating process which is violated in the statistical arbitrage forecasting model.


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The right hand graph in figure 7 shows the equity curves for the two optimal kvalues

assuming no transaction costs.

5 Conclusions

We have described a methodology which is intended to optimise the objective of

the trading strategy using a parameterise decision rule. We apply this methodology to

a forecasting model, based upon principal component analysis, of relative mispricing

between Equity indices. The prediction accuracy of the forecasting model is measured

and the parameterised decision rule is optimised for the synthetic data and the sample

data. The methodology improves the trading performance from an annualised Sharpe

ratio of 3.14, for the naive decision rule (k=0), to 5.31 for the optimal decision rule, a

factor of 1.7. To gain this level of trading performance would approximately equate to

improving the prediction accuracy of the forecasting model from 0.12 to 0.165, a

factor of over 35%.

6 References

1. J. Moody, Optimisation of Trading Systems and Portfolios, Proceedings from

Fourth International Conference on Neural Networks in the Capital Markets, 1996.

2. M. Choey, A. S. Weigend, Nonlinear Trading Models through Sharpe Ratio

Maximization, Proceedings from Fourth International Conference on Neural

Networks in the Capital Markets, 1996.

3. A. N. Burgess, Controlling nonstationary in statistical arbitrage using a portfolio

of cointegration models, Proceeding from Fifth International Conference on

Computational Finance, 1997.

4. W. F. Sharpe, Mutual fund performance, Journal of Business pp. 119-138, 1966.

5. A. N. Burgess, N. Towers, Statistical arbitrage models in Equity and FixedIncome markets, Technical Report, London Business School, 1998.

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