what have rough paths got to do with finance? - ccfz
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Oxford-Man Institute of Quantitative Finance
What have rough paths got to do with finance?
Terry LyonsNi Hao
Greg Gyurko….
With research support from ERC,SRC,EPSRC, and the Oxford Man Institutedata via the Oxford Man Institute, Quanthouse and Pinnacle
Oxford-Man Institute of Quantitative Finance
Two interrelated applications
Describing complex data streams
Extending Itô’s integral
Better and more quantitative understanding of the key features in a market that affect performance of automated trading strategies
Oxford-Man Institute of Quantitative Finance
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Tick
500 Ticks
Bid
Ask
Last Traded Price
Source: QuantHouse, 2012 (www. quanthouse.com)
05:44 GMT on the 10th November 10th 2009 New York Crude Oil
Oxford-Man Institute of Quantitative Finance
Data as a path
How do we talk about or discuss that time series?It is vector valued (3, 5,.. dimensions).The order in which thing happen really matters
Bid Offer already has a lot of structure
Clue: We care about it for its effect on the trading strategy
How do we describe it – well certainly not as a markovprocess unless we understand better the state variables
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500 Ticks on the 10th Novemebr 2009 New York Crude
500 Ticks on the 10th Novemebr 2009 New York Crude
Oxford-Man Institute of Quantitative Finance
Data as a control
inputs
order book
news
strategy
returns
Order Book
Hedge Fund Returns
Extra Information
Oxford-Man Institute of Quantitative Finance
What are we trying to do?
How should one describe these data streams
How should one describe their randomness if one believes the markovian model is too simplistic
How can one learn this randomness from data
How can one introduce feedback effects
Does any of the approaches help understanding and development of investment strategies
Order Book(t)
Extra Information(t)
Oxford-Man Institute of Quantitative Finance
The signature of a path
There is a natural transform of a path in ‘stream space’ to a series of co-efficients that are graded according to their effect
For the signature the main reference is:109-167(171) (2010) Ben Hambly, Terry Lyons, Annals of Mathematics
There is a wide literature on rough paths; one basic mathematical text is :Differential Equations Driven by Rough Paths, Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004 Lyons, Terry J., Caruana, Michael J., Lévy, Thierry 2007.
Although the mathematics is very abstract compared with the potential for applications, software now exists in python that makes the transformation from time series to first few terms in the signature routine.
Oxford-Man Institute of Quantitative Finance
We can describe this data
Classical approach
one minute returns…
daily returns …
No theoretical basis
Signature is Information one can actually trade
The signature of a path is a transform – it does not need a model
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0.00045 0.00135 0.0003 -0.00045 0.00045 -0.0003 0.0015 0.0021 0.0018
Oxford-Man Institute of Quantitative Finance
Simple application – Validate a trading platform simulation
Real data500 ticks -> first few terms
Simulated dataSim-ticks -> first few terms
Compute the empirical expectations and compare
Reject the model (or at least understand the biases it introduces into the outputs of any system)
Basic factsOne can always compute the
first few terms in the signature of the process.
When it comes to understanding the outcome of an investment strategy this information is all that matters
The full expected signature completely describes the law of the (non markov) process.
Oxford-Man Institute of Quantitative Finance
Use in other contexts
Anastasia Papasiviliou, Christophe Ladroue, Annals of Statistics 2011(39) 2047-2073
Develop an expected signature matching estimator (ESME)
Use it to create models from empirical data
(model hedge fund strategies from their returns)
Taking GMME (Hansen, Mykland) beyond the Markov setting.
Parameterised
Fractional Noise with unknown
parameters
Unknown
Non-linear
equation
Observed
response
Oxford-Man Institute of Quantitative Finance
The third term is very visible e.g. in classic futures markets
Oxford-Man Institute of Quantitative Finance
The third term is very visible e.g. in classic futures markets
Oxford-Man Institute of Quantitative Finance
The future: Tracking Event CascadesThe first non-obvious tern in the
signature of a multidimensional time series is the “area” and it is a pathwise measure of lead lag relationships. If a shock hits one asset then another there will be a big area generated between the assets.
After a shock, get a graph given by the signs of the areas, indicating the flow of information across the assets.
A practise datasetseismic information from 100 microphones in a Chilean copper mine for a dayRegular controlled explosionsLead and lag are effective distances from the source
https://wiki.cs.umd.edu/cmsc734_11/index.php?title=Analysis_of_Patent_Citation_Networks#The_Followers_and_The_Followed
Oxford-Man Institute of Quantitative Finance
The future: Quantifying changing markets
Do markets change11905 sequences of volnormalised prices for the 59 last days of futures contract (from the OMI data).
Did the markets change after 2008?
The higher terms in the signatures seem to produce interesting changes but need to be investigated further.
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