I have hinted at Z-transforms and Hilbert transforms and how they are applied to financial time series. But generally speaking, there are a number of other mathematical transforms which find their use in financial modeling problems.
Mathematical Transforms are important because they help convert functions into other domains thereby making them easier to understand and solve. Here I give a qualitative description of common transforms and some examples of how they are applied to problems in finance.
Obviously the choice for invoking a particular transform depends on the problem being solved.
1. Fischer Transform
The Fischer Transform converts any data set into a modified data set whose probability density function is approximately gaussian. An immediate benefit is that one can then analyze the transformed data set in terms of its deviation from the mean - something which might not have been possible prior to the transformation. Consider for instance absolute prices on a bar chart. Are they normally distributed (i.e. bell shaped)? No. The returns density of the time series are usually near bell shaped, but the actual prices are no where close to bell shaped. This means attaching Gaussian confidence intervals to absolute price data is impossible, but Fischer transformed data will modify the data so that they become bell shaped hence allowing the majority of Gaussian statistical functions to be applied. The benefit is seen in modified technical indicators that are similar to, but more responsive than conventional oscillators such as the Commodity Channel Index (CCI) or Moving Average Convergence Divergence (MACD).
2. Fourier Transform
The Fourier Transform enables conversion of functions or data sets from the time-domain to the frequency domain and vice-versa. In Signal Processing this is essentially what describes the relationship between the time domain and the frequency domain. In terms of option pricing, the Fourier Transform provides a framework for fast price calculation compared to Monte Carlo methods. As shown in [Szymon et all] the Fast Fourier Transform (FFT) algorithm is about 3000 times faster than Monte Carlo simulation.
It is worth noting that FFTs are inappropriate for direct time series analysis because they deliver poor resolution in terms of cycle length. They are only capable of recording integer number of cycles, thereby missing any cycles that have a length falling in between two integer boundaries. A good algorithm which performs better than the FFT for measuring cycle period in financial time series is the Pisarenko harmonic decomposition.
3. Hilbert Transform
The Hilbert Transform is a procedure to create complex signals from the real price data that is plotted on the bar chart. With complex signals available one can compute more accurate and responsive indicators as well as create other indicators which cannot be calculated without the Hilbert Transform. Computations such as Signal-to-Noise ratio, Power Spectral Density and Cycle Period measurement can only be achieved by calculating the Hilbert Transform.
4. Laplace Transform
Laplace Transforms are useful for solving transient state systems that are described by differential equations. The transform helps simplify complex differential equations; notably in the case of partial differential equations used in option pricing formulas. Simplified equations in the Laplace domain are solved algebraically before the inverse Laplace transform is applied to return the solution to the original domain.
5. Wavelet Transform
The Wavelet Transform is similar to the Fourier Transform in that it converts time domain data into frequency domain representations. The main difference is that wavelets provide a number of ways for doing the conversion through a series of wavelet families. Each wavelet family has its own property which is exploited for doing more specific decomposition tasks such feature extraction, noise removal or cycle period estimation of time series. Decomposed time series can be analyzed in several ways to provide more detain into it’s intrinsic properties.
6. Z - Transform
The Z Transform converts discrete time-domain data points into a complex frequency-domain representation. Unlike the Laplace and Fourier Transforms which are applicable to differential equations, the Z Transform provides an excellent framework for working with difference equations in the frequency domain. One can go about describing filters in terms of their transfer functions and apply a host of algebraic techniques found in Digital Signal Processing to design better filters for time series analysis.
Summary
- Transforms are performed to make problems easier to understand and solve.
- Any Transform has an accompanying Inverse Transform algorithm so that a dual relationship is maintained between two domains.
- The Z, Wavelet, Fourier and Laplace transforms are related in that they maintain a relationship between the time domain and the frequency domain.
- Laplace Transforms are useful for analyzing transient systems where as Fourier Transforms are more suitable for steady state systems.
- aiQUANT thinks technical indicators and oscillators should be designed in the Z domain. This gives greater understanding of the frequency characteristics - something which is not obvious with just the time domain representation of the indicator.
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