what have rough paths got to do with finance? - ccfz

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


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


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500 Ticks

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Last Traded Price

Source: QuantHouse, 2012 (www. quanthouse.com)

05:44 GMT on the 10th November 10th 2009 New York Crude Oil


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

500tickbidoffer.mpeg


Data as a control

inputs

order book

news

strategy

returns

Order Book

Hedge Fund Returns

Extra Information


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)


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.


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


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.


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


Seeing the signature


Regression


Generalise vol

S_1995U.mpeg


The third term is very visible e.g. in classic futures markets


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


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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what have rough paths got to do with finance? - ccfz

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