A Cointegration Approach to Identifying Systemic Risk in Markets
Michael Kane
Bryan Lewis
Who am I?
Currently an Assistant Professor at Yale
Research Area is in scalable statistical computing and machine learning
I was a Co-PI in DARPA's XDATA program
Premise: What can we do when we can't predict signal (in this case stock price)?
Outline
PART 1 - THE FLASH CRASH AS A MOTIVATING EXAMPLE OF SYSTEMIC RISK
PART 2 - QUANTIFYING SYSTEMIC RISK USING COINTEGRATION
PART 3 - SCALING TO REAL-TIME MARKET DATA
The Flash Crash
On May 6, 2010 the stock market losses about 1 trillion dollars in value in about 15 minutes
What happened afterward?
A compelling explanation was not found but conspiracies abound.
Single-stock circuit breakers (and collars) are instituted.
Do Single-Stock CB's Work?
When a large-cap stock's price goes up (or down) 5% in 5 minutes trading stops.
Drop for the Dow was 9.2%
... but it's worse than that because stocks are stopped when they shouldn't be.
... and even worse than that
A halt is meant to allow traders to "take a step back" and assess the market
They do not.
What is the underlying problem?
Single-stock trading halts manage single-stock volatility
They don't manage systemic risk or risk caused by an event severe enough to cause instability in the financial system
Should we regulate volatility?
Characterizing Systemic Risk
Dynamically spanning sectors over time
Makes markets vulnerable to events that result in high-volatility
We may not be able to forecast events but we may be able to estimate vulnerability
Assume stock that "behave similarly" are susceptible to similar price-moving factors
Quantify "similar behavior" using Cointegration
y(t) - B x(t) = u(t)
Two time series y(t) and x(t) are cointegrated if
where B is a real-valued constant and u(t) is a stationary process.
It is OLS with noise that doesn't diverge.
Who uses it?
Economists - model macroeconomic trends
Financial Engineers - Provides a generalization to pairs trading
Me - to find stocks "behaving similarly"
Preliminary Study
May 6, 2010, S&P 500 stocks, sliding window
124,750 comparisons for each second of a 23,400 second day
3 days and about 60 machines at the Oak Ridge National Labratories
Off-the shelf R packages
Is this really "massive"?
Results for the Entire Market
Results by Sector
A NORAD for Markets
We can tell when stocks behave similarly
They react similarly after a shock
We may not be able to predict shocks but we can definitely detect when a shock will be system-wide
Now if only we could run this in real-time...
Scaling the Solution to Higher Frequency
Euclidean distance is related to correlation
Correlation is necessary for Cointegration
Idea behind the algorithm
Project the data onto a low-dimensional subspace
If you are far away on the subspace you are at least as far away in the original space
That distance tells you something about the correlation
A distance approximation using the SVD
|| x_i - x_j ||^2 = \sum_{l=1}^k \sigma_l (e_i^T v_l - e_j v_l)^2
x_i
X
\sigma_l
v_l
e_i
- standardized matrix
- ith column of x
- lth column of x
- lth right singular vector of X
- unit basis, all zeroes except 1 at position i
A distance approximation using the SVD
The truncated SVD can be calculated very efficiently using the Implicitly Restarted Lanczos Bidiagonalization Algorithm
Then we can use the following relationship to get highly correlated pairs
cor(x_i, x_j) = 1 - ||x_i - x_j||^2/2
Result
Computational complexity was reduced by about one and a half orders of magnitude
What used to take 3 days on a cluster now takes a few hours on a laptop.
This is now scaled for real-time and allows us to try new approaches that weren't previously feasible
R Packages Used
foreach [Calaway, Weston and Revolution Analytics, https://cran.r-project.org/web/packages/foreach/index.html]
nxcore [Hafen and Kane, https://github.com/hafen/nxcore]
tcor [Lewis, https://github.com/bwlewis/tcor]
rthreejs [Lewis, https://github.com/bwlewis/rthreejs]
Thanks
A Cointegration Approach to Identifying Systemic Risk in Markets
By Michael Kane
