1. Market Crashes and Modeling Volatile Markets Prof. Svetlozar (Zari) T
Market Crashes and Modeling Volatile Markets Prof. Svetlozar (Zari) T.Rachev Chief-Scientist, FinAnalytica Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering University of Karlsruhe Department of Statistics and Applied Probability University of California, Santa Barbara Thalesian seminar , London - December 2, 2009
Today, my goal is to demonstrate the existence of a solid framework for quantitative modeling for volatile markets, or markets exhibiting extreme events.
The recent market events provoked hot discussions and even dismay towards the appropriateness of quantitative approaches towards portfolio construction and risk management.
Many established theories, concepts, approaches and measures have been strongly criticized. The concepts of diversification measured by correlation, the notion of a hedge portfolio, etc have all been discredited. Still, it is clear to everyone, investment decision making process today cannot exist without a quantitative insight. FinAnalytica has been working since its inception to provide more reliable tools that can be adequate in normal market conditions, but also during market stress.
The first part of my presentation will consider the theoretical foundations. In the second one, we will see some applications for portfolio construction and risk management.
In the first part we will go through the market phenomena of a factor (or risk driver)level as well as on multivariate, we will then discuss the models to handle them and finally what risk measures shall we use in the extended class of models that now accommodate for extreme events.
So, let’s start by making sure we understand the notion of extreme events and it’s place in the structure of the market phenomena

2. Post Modern Portfolio Theory

3. MPT “translation” for Volatile Markets
Real World
Old World
Normal (Gaussian)
Distributions
Correlation
Sigmas
Sharpe Ratios
BS Option Pricing
Markowitz Optimal Portfolios
Fat-tailed Distributions
Tail & Asymmetric Dependence
Expected Tail Loss
STARR Performance
Tempered-Stable Option Pricing
Fat-tail ETL Optimal Portfolios
60 years ago the Normal distribution became the underlying model for financial application and theory because it is possible to relatively easy solve all main problems: risk measurement, portfolio optimization, hedging….
Today, we see after dozen years of research and development we can build a similar consistent framework based on realistic market assumptions… 3

4. Models Map

5. The Fat-tailed Framework
Agenda
The Fat-tailed Framework
Univariate model (single asset)
Subordinated models
Stable model
Dependence
Risk and Performance measures
Applications
Option pricing - Some extension of the main fat-tailed model: Tempered Stable models
Modeling market crashes
Risk monitoring
Portfolio management and optimization

6. Fat-tail Modeling Framework

7. Phenomena of Primary Market Drivers - 1
Univariate level
Time-varying volatility
Fat-tails
Asymmetry
Long-range dependence (intra-day)
DJ Daily returns
When observing the return process of primary market drivers such as stocks, interest rates, exchange rates, indices we face 3 pronounced phenomena – time-varying volatility, fat-tails and asymmetry.… Other phenomena such as jumps (due to external events such as regulation, …), illiquidity, long-range dependence, etc. can be observed in certain financial time-series and to a different degree can be accommodated for in a quantitative framework. But let’s (at least for now), concentrate on those three aspects which we constantly observe across all markets…
Time-varying volatility stands for the fact that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes.“
Fat-tails stand for the fact that from time to time we observe very large returns which probability is practically zero if normal or Gaussian distribution is used. This mathematically is expressed by having a power decay of the probability distribution function when the return (its argument) goes to plus or minus infinity. For us, it means observing market crashes or at least severe negative returns on the markets of say minus 8% - 10% daily not once in the life of the universe as the normal model would suggest, but every 5 to 10 years…
Finally asymmetry means that we usually observe larger in magnitude negative extreme events, though not so frequently compared to the extreme positive returns or in some cases vice versa.
So, how do we usually visualize those aspects of the returns. The first approach is to plot the probability density function. ….
The second way is to construct QQ plot which depicts the theoretical quantile vs empirical ones. If the distribution is indeed normal all dot’s will be on the same line… Deviation in the tail in this direction means ,……

8. Subordinator (g(W)) < 1 Subordinator (g(W)) < 1 g <0
Fat-tailed
Fat-tailed
Subordinator (g(W)) < 1
“On the days when no new information is available, trading is slow and the price process evolves slowly. On days when new information violates old expectations, trading is brisk, and the price process evolves much faster”.
Clark (1973)
Subordinator (g(W)) < 1
g <0
We start with the normal model… how to make it better to fit the data and model the excess curtosis (the concentartion around the center ), fat-tailes and the skewness?...Well, its appear it is sufficinet to introduce a new random variable which we call market time intensity. If we look the price process what we do is to say – Ok… the return process is a Brownian motion but under a different, unobservable time – the marker time. If we look at a point of time, we see that this model actually results in randomizing the volatility ….We can incorporate skewness and a shift very naturally as well….So, what happens when we draw random numbers in a MC expiriment from such a model is the following: we sample togther a normal variable and a market time intensity variabl, which we call a subordinator… IF the subordinator is smaller than 1, the resulting simulation from the return will become smaller and thus we will introduce peakness. … on opposite, if it is larger – we introduce extreme events or fat-tailes… Subordinated models, it turns out have very nice additional features…
Emp.

9. Subordinator > 1

10. Stable Family Rich history in probability theory
Kolmogorov and Levy ( ), Feller (1960’s)
Long known to be useful model for heavy-tailed returns
Mandelbrot (1963) and Fama (1965)

11. Fat Tails Study: 17,000+ factors
85%, 95%, 97.5%, and 99% VaR tested

12. Fat Tails Study: Factors Breakdown
Factors Tested
Number
Percentage
Equities
8346
48.5%
CDS Spreads
7803
45.3%
Interest Rates
528
3.1%
Implied Volatilities
518
3.0%
Currencies 12
0.1%
Total
17207
100.00%

13. Alpha Tail Parameter: Varies Across Assets & Time
Important to:
Distinguish tail risk contributors and diversifiers
Changes in the market extreme risk
after removing GARCH

14. There is NO universal tail index!
Tail parameter Alpha for 41 indices after removing GARCH effect
/May 15th 2009/
There is NO universal tail index!

15. Phenomena of Primary Market Drivers - 2
Tail Dependence
Zero tail dependence
Gaussian copula
Now, suppose we have extreme events for our market drivers… say Russell 2000 and Nasdaq. Will they appear on this way? No, this is what the world would be if there was zero tail dependence. But this is not what our world is … The way we know they appear is this!

16. Dependence Models Asymmetric Dependence
There is more to say about the dependence – it is also assymetric – positive extreme events have lower tail dependence..

17. correct investment decisions
Dependence Models
Modeling of Extreme Dependency in market crashes is critical for taking
correct investment decisions
Bi-variate Normal
Fat-tailed indices
Gaussian Copula
Fat-tailed copula
Observed returns in Q3 1987
Explain the charts
Copulas – flexible and non-linear functions describing dependence structure
Gaussian copula
it is more or less equivalent to simple correlation matrix
BUT – other copulas exist
Non-Gaussian copulas
address directly questions such tail dependence and asymmetry
Objective way to define-stress-tests: Scanning for “Black Swans” (unknown unknowns)
Challenges
Developing efficient numerical algorithms.
Choosing an appropriate parametric structure
F is the multivariate cdf, C is the copula function and Fi are the one-dimensional cdf.

18. Risk & Performance Measures
Symmetric risk penalty
Downside risk penalty
Downside risk penalty
and upside reward
In the modern portoflio theory, based on the Gaussina distribution the natural risk measure is the sigma and the natural perf. Measure is the Sharpe ration. Those masures howeve does not work in a fat-tailed word. We need something different . The risk measure should account for the assymetric nature of the risk, skewed and fat-tailed distibution. Such coice is ETL or also know as….The can form performance measures based on it …Advantages of ETL: looks in the tail, coherent, good for interpretation, interpretation… semi-st-deviation, omega, … 18

19. Why not Normal ETL?

20. Fat-tailed world is a complex one:
Summary
Fat-tailed world is a complex one:
GARCH is not enough
Fat-tails are not enough
Copula choice is important
Fat-tails change across assets and across time
Beware of pseudo-fat-tailed models
Fat-tailed ETL as a risk measure is important

21. Application 1 – Option pricing Stable and Tempered Stable Distributions

22. Tempered Stable Models Introduction
The stable model does not allow for unique equivalent martingale measure
Take a stable model and make the very end of the tails lighter (still much heavier than the Gaussian)
All moments exist
No-arbitrage option pricing exists

23. Tempered Stable

24. Tempered Stable

25. Map of Tempered Stable Distributions
Kim-Rachev
(KR)
Modified
Tempered
Stable
(MTS)
Normal
Tempered
Stable
(NTS)
Smoothly
Truncated
Stable
(STS)
Rapidly
Decreasing
Tempered Stable
(RDTS)
Classical
Tempered
Stable(CTS)

26. Incorporating GARCH Effect
other tempered stable models

27. Is GARCH Enough? … No!
QQ plots between the empirical residual and innovation distributions for daily return /data for IBM/

28. Option Prices and GARCH Models
SPX Call Prices (April 12, 2006)
where N is the number of observation, is the n-th price determined by the
simulation, and is the n-th observed price.

29. Model Universe
We studied the full spectrum of tractable (infinitely divisible) models
We see that Stable ARMA-GARCH is the best choice to model primary risk drivers
We propose a form of tempered stable (RDTS) for option pricing

30. The Option Pricing Models Universe

31. References

32. Application 2 - Modeling Market Crashes

33. Daily Returns: S&P 500 Index

34. Crash Probability: Black Monday
On October 19 (Monday), 1987 the S&P 500 index dropped by 23%. Fitting the models to a data series of 2490 daily observations ending with October 16 (Friday), 1987 yields the following results:

35. Crash Probability: U.S. Financial Crisis
On the September 29 (Monday),2008 the S&P 500 index dropped by 9%. Fitting the models to a data series of 2505 daily observations ending with the September 26 (Friday), 2008 yields the following results:

36. S&P Backtest

37. References

38. Application 3 – Risk Monitoring

39. Backtest Example Long-short stock portfolio
99% VaR backtest was run from 8/1/2007 to 5/15/2008 (206 days)
250 rolling window used to fit the models
Models:
Historical method
Normal method
Constant Volatility
EWMA for Cov matrix
Asymmetric Stable with Copula
Volatility Clustering

40. Model Comparison Quantitative - Number of exceedances Qualitative
Average - must be on average 2
Number of exceedances above 4
/95% CI is 0-4/
Checked on portfolio and industry level
Qualitative
Visual check of VaR evolution vs returns
Historical
3.25
24%
Normal 99
7.03
76%
Normal 99 EWMA
3.90
42%
Asym Stable Fat-tail Copula
1.64 3%
Asym Stable Fat-tail Copula Volatility Clustering
2.27 6%
Av. # of
exceedances
% Industries
VaR rejected

41. Risk Backtest Both are important!
Fat-tailed VaR with constant volatility provides long-term equilibrium VaR
Fat-tailed VaR with volatility clustering provides dynamic short-term view of the tail risk (VaR)
Both are important!
Returns
Normal 99
Normal 99 EWMA
Asym Stable Fat-tail Copula
Asym Stable Fat-tail Copula Vol Clustering

42. Application 4 – Portfolio Management and Optimization

43. Portfolio Risk Budgeting
Marginal Contribution to Risk
Standard Approach: St Dev
ETL:
The expression for marginal contribution to ETL is
and the resulting risk decomposition:

44. Portfolio Optimization
Flexibility in problem types, a very general formulation is
where the first ETL is of a tracking-error type, the second one measures absolute risk and l ≤ Aw ≤ u generalizes all possible linear weight constraints
If future scenarios are generated, there are two choices:
Linearize the sample ETL function and solve as a LP
Solve as a convex problem

45. Portfolio Optimization References

46. Modeling Fat-tailed world is a complex task BUT crucial for:
Summary
Modeling Fat-tailed world is a complex task
BUT crucial for:
Option pricing
Explaining volatility smile
Identifying statistical arbitrage opportunities
Crash warning indicators
Helps identify changes in the market structure faster
Risk monitoring
Realistic understanding of risk and its evolution
Portfolio construction and optimization
Achieve higher risk-adjusted returns

47. Q&A…
Thank you!