1. Numerical Software, Market Data and Extreme Events Robert Tong

2. Outline Market data Pre-processing Software components Extreme events
Example: wavelet analysis of FX spot prices
Implications for software design

3. Market data minute, hour, day, … – low-high price
Tick – as transactions occur,
high frequency, irregular in time
quote/price with time stamp
Sample tick data at regular times –
minute, hour, day, … – low-high price
Bid-ask pairs – FX spot market
Time series – construct from sampled and processed data

4. FX spot market prices - USD-CHF
ticks (e.g. see
minutes
hours
From:

5. Data cleaning Required to remove errors in data –
inputting errors
test ticks to check system response
repeated ticks
copying and re-sending of ticks
scaling errors
How can false values be reliably identified and rejected ?
what assumptions must be imposed?
elimination of outliers based on an assumed probability distribution

6. Pre-processing Must not introduce spurious structures to data
Tick data irregular in time – construct homogeneous
time series by interpolation: linear, repeated value
Bid-ask spread – use relative spread
Remove seasonality
Account for holidays
Must not introduce spurious structures to data

7. Software components

8. Implementation issues
Algorithm design –
Stability
Accuracy
Exception handling
Portability
Error indicators
Documentation
These are independent of the problem being solved

9. Extreme events Software – Weather – storm Warfare – explosion
Markets – crash
Software –
How should it respond to the unpredictable?
What is the role of software when its modelling assumptions break down?

10. An illustration – another type of bubble
Underwater explosions are used to destroy ships –
the initial shock is expected and often not as damaging as the later gas bubble collapse.
Left: raw data from sensitive, but un-calibrated pressure gauge
Right: calibrated gauge uses averaging to produce smooth curve
Use of averaging obscures critical event in this case.

11. Example: wavelet analysis of FX spot prices
Wavelet transforms provide localisation in time and frequency for analysis of financial time series.
This is achieved by scaling and translation of wavelet basis.
Decompose time series, by convolution with dilated and translated mother wavelet, or filter,
Discrete (DWT) Orthogonal Filter pair:
H – high pass, G – low pass
followed by down-sampling

12. Wavelet filters Family of filters by scaling Daubechies D(4) wavelet
filters result from sampling a continuous function

13. Multi-Resolution Analysis
Discrete Wavelet Transform (DWT) d1 d2

14. DWT implementation Orthogonal wavelet transform uses
filters defined by sequences: ,
satisfying: , ,
This allows for a number of variants in implementation
numerical output from different software providers
is not identical

15. Discrete Wavelet Transform – Multi-Resolution Analysis
For input data , length ,
produces representation in terms of ‘detail’ and ‘smooth’ wavelet coefficients of length
Uses
Data compression – discard coefficients
De-noising
Disadvantages
Difficult to relate coefficients to position in original input
Not translation invariant – shifting starting position produces different coefficients

16. Maximal Overlap Wavelet Transform (MODWT) (Stationary Wavelet Transform)
Convolution: wavelet filters as in DWT
No down-sampling
MRA produces N coefficients at each level
Requires more storage and computation
Not orthonormal
Advantages
Translation invariant
Can relate to time scale of original data
Does not require length(x) =

17. Choice of wavelet filter
Short
can introduce ‘blocking’ or other features which
obscure analysis of data
Long
increases number of coefficients affected by ends
of data set
Basis Pursuit
seeks to optimise choice of wavelet at each level
but requires more computation

18. FX: USD, GBP, EUR – NZD 12 noon buying rates, Jan – Jul 2007

19. FX: JPY, USD, GBP, EUR – NZD 12 noon buying rates, Jan – Jul 2007 (from www.x-rates.com)

20. JPY-NZD, LA(8), MODWT (includes boundary effects)
x(t) d1 d2 d3 d4

21. JPY-NZD, LA(8) MODWT (includes boundary effects)
x(t) d5 d6 s6

22. Boundary conditions – end extension
Wavelet transform applies circular convolution to data
What happens at the ends of the data set?
End extension techniques –
periodic
reflection – whole/half-point
pad with zeros
Boundary effects contaminate wavelet coefficients
software should indicate where output is
influenced by end extension

23. End extension
Periodic Whole-point reflection

24. Periodic end extension
USD-NZD, Haar, MODWT
Periodic end extension
Level 1 detail coefficients Level 2 detail coefficients

25. USD-NZD, Haar, MODWT (end effects removed)
x(t) d1 d2 d3
USD-NZD, Haar wavelet, MODWT
The MODWT (Maximal Overlap Discrete Wavelet Transform, also called the Stationary Wavelet Transform) is applied to a time series of the daily average spot prices for USD-NZD. The simplest basis, the Haar wavelet, is used in this example. The wavelet transform is computed by convolution with a filter pair. In the Haar case, the wavelet filter is defined as (1,-1)/sqrt(2) and the scaling filter as (1,1)/sqrt(2). The application of the wavelet transform can be viewed as a combination of differencing and averaging with a dilation of the filter pair in passing from one level to the next in the multi-resolution analysis.
Unlike the commonly used DWT (Discrete Wavelet Transform), the MODWT does not discard even (or odd) coefficients and thus stores more than the number of coefficients of the original time series. The MODWT is chosen because it is translation invariant and is better suited to statistical analysis. The DWT is used for data compression and for de-noising.
The detail coefficients resulting from the application of the wavelet filter, d1, d2, …, d6 are stored along with the final smooth coefficients, s6, produced by the scaling filter. The coefficients are plotted as lines with those representing the smallest time scale given by d1. The smooth coefficients, s6, give an overall averaging and here show an upward trend.
Boundary effects are confined to the left ends as shown in the plots for this particular implementation and those coefficients affected have been deleted.
The data for USD-NZD shown cover a period when the NZ central bank intervened in the market to try to halt the upward rise of the NZD. Interventions were made on 11th and 19th June. These interventions did not have any effect on the underlying trend.

26. USD-NZD, Haar MODWT (end effects removed)
x(t) d4 d5 d6 s6

27. Wavelet analysis for prediction
Extrapolation from present to near future is useful
Apply wavelet filters to for
avoiding boundary effect
Select wavelet scales to identify trend and stochastic parts of data set
Use wavelet coefficients to compute prediction
(see Renaud et al., 2002)

28. Implications for software development
Reproducibility is desirable –
algorithms precisely defined to allow independent implementations
to produce identical results
Edge effects –
contaminate ends of transform for finite signals –
software must indicate coefficients affected
Smoothing/averaging –
software should indicate when underlying assumptions likely to be invalid
Pre-processing –
ensure that structure is not introduced by interpolation to give homogeneous data set

29. Implications for software development
For extreme events –
must not obscure or remove data relevant to critical events by averaging, smoothing, filtering.