Market Crashes and Modeling Market Crashes and Modeling Volatile Markets Volatile Markets Prof. Svetlozar (Zari) T.Rachev Prof. Svetlozar (Zari) T.Rachev Chief-Scientist, FinAnalytica Chief-Scientist, FinAnalytica Chair of Econometrics, Statistics and Mathematical Finance Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering School of Economics and Business Engineering University of Karlsruhe University of Karlsruhe Department of Statistics and Applied Probability Department of Statistics and Applied Probability University of California, Santa Barbara University of California, Santa Barbara Thalesian seminar , London - December 2, 2009 Thalesian seminar , London - December 2, 2009
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Market Crashes and Modeling Volatile Market Crashes and Modeling Volatile MarketsMarkets
Chair of Econometrics, Statistics and Mathematical Finance Chair of Econometrics, Statistics and Mathematical Finance School of Economics and Business Engineering School of Economics and Business Engineering University of Karlsruhe University of Karlsruhe Department of Statistics and Applied Probability Department of Statistics and Applied Probability University of California, Santa Barbara University of California, Santa Barbara
Thalesian seminar , London - December 2, 2009Thalesian seminar , London - December 2, 2009
Post Modern Portfolio TheoryPost Modern Portfolio Theory
MPT “translation” for Volatile MarketsMPT “translation” for Volatile Markets
1% STABLE ETL vs. NORMAL VAR AND ETL: $1M OVERNIGHT
Normal VaR = $47K
Normal ETL = $51K
Stable ETL = $147K
STABLE DENSITYNORMAL DENSITY
SummarySummary
• 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
Application 1 – Option pricing Application 1 – Option pricing Stable and Tempered Stable DistributionsStable and Tempered Stable Distributions
• QQ plots between the empirical residual and innovation distributions for daily return /data for IBM/
Option Prices and GARCH Models
where N is the number of observation, is the n-th price determined by thesimulation, and is the n-th observed price.
SPX Call Prices (April 12, 2006)
Model UniverseModel 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
The Option Pricing Models UniverseThe Option Pricing Models Universe
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:
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:
S&P BacktestS&P Backtest
-0.3
-0.25
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Return normal GARCH VaR 99% fat tail GARCH VaR 99% fat tail GARCH ETL 99%
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
Risk BacktestRisk Backtest
Application 4 – Portfolio Management and Application 4 – Portfolio Management and OptimizationOptimization
Portfolio Risk BudgetingPortfolio Risk Budgeting
• Marginal Contribution to RiskStandard Approach: St Dev
( ) cov( , )i i Pi
P P
r rMCTR
Ωw
Pi i P
i P
w MCTR
w Ωw
ww
ppii
i rVaRrrEw
ETLMCETL
|
The expression for marginal contribution to ETL is
and the resulting risk decomposition:
pi
ppiii
ii rETLrVaRrrEwMCETLw |
ETL:
Portfolio OptimizationPortfolio 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
uAwl
ew
ts
Erwrwrrw
T
TTb
T
w
1
..
ETLETL min 21
SummarySummary
• Modeling Fat-tailed world is a complex taskBUT crucial for: