jorion var 09
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Risk MgmtTRANSCRIPT
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Daniel HERLEMONT
Financial Risk Management
Following P. Jorion, Value at Risk, McGraw-HillChapter 9
VaR Methods
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VaR Methods
Local Valuation Methodsvaluing the portfolio once, using local derivatives :delta normal methoddelta-gamma ("Greeks") methodMost appropriate to portfolios with with limited sources
of risk.
Full Valuation Methodsre-pricing the portfolios over a range of scenarios,
including:HistoricalMonte Carlo
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Delta Normal Methods
Usually rely on normality assumption
Worst loss for V is attained for extreme values of S If dS/S is normal, the portfolio VaR is:
is the standard normal deviate corresponding to the confidence level, e.g. 1.645 for a 95% confidence level
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Delta Normal - Fixed Income Portfolio
The price-yield relationship:
where D* is the (modified) Duration
where is the volatility in of change in level of yield
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Distribution with linear exposure
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Approximation depends on the optionality of the portfolio and the horizon
For options (as well as bonds) non linearities exist, However, they don't necessarily invalidate the delta normal method for
small changes and/or short term horizons
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Full Valuation
Delta Normal may become inadequate: when the worst loss may not be obtained for extremes realizations
of the underlying options are near expiration and at-the-money with unstable deltas
(straddle, barriers, ...) The Full Valuation considers the portfolio for a wide range
of price levels:
The new values can be generated by simulation methodsMonte Carlo: sampling from a distribution (e.g. normal)Historical Simulations: sampling from historical data
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Full Valuation
The portfolio is priced for each drawVAR is then calculated from the percentiles of the
full distribution of payoffs. it accounts for
non linearities income payments time decay
potentially: the most accurate method but the most computationally demanding
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Delta Gamma Approximations
Extends the delta normal method with higher moments
second derivative of portfolio value
is the time drift
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Delta Gamma - Examples
Fixed Income
D is the Duration, C is the convexity
Vanilla Call Options:
valid for long (>0) >0) >0) >0) or short (
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Delta Gamma for complex portfolios
taking the variance at both side:
then, under normal hypothesis:0),cov( and )](variance[2)(variance 222 == dSdSdSdS
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Delta Gamma - Cornish Fisher Expansion
is the Skewness
Negative Skewness increases VAR
the same applies for positive Excess Kurtosis
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Skewness
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Kurtosis
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Delta Gamma Monte Carlo
also known as the partial simulation method:
Create random simulation for risk factors
then uses Taylor expansion (delta gamma) to create simulated movements in option value
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Delta Gamma - Multiple risk factors
and dS are vectorscomputationally intensive requires estimates of:
Gamma (implicit correlations)
Covariance matrix
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Comparison of methods
For lager portfolios where optionality is not dominant, the delta normal method provides a fast and efficient method for measuring VAR
For portfolios exposed to few sources of risk and with substantial option components, the Greeks (delta-gamma) provides increase precision at low computational cost
For portfolios with substantial option components or longer horizons, a full valuation method may be required
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Note on the "Root Squared Time" rule
Normally daily VAR can be adjusted to other period by scaling by a square root of time factor
However, this adjustment assume: position is constant during the full period of timedaily returns are independent and identically
distributed
Hence, the time adjustment is not valid for options positions (that can be replicated by dynamically changing positions in underlying)
For portfolios with large options components, the full valuation must be implemented over the desired horizon ...
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Example: Leeson's Straddle
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Sell Straddle payoff
Sell Straddle = sell call + sell putStrike = at the money
Successful, only if the spot remains stableDelta = 0
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Example: Leeson's Straddle
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Example: Leeson's Straddle
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Example: Leeson's Straddle
VaR Analysis could have prevented bankruptcy
if positions were known
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Example: Leeson's Straddle
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Example: Leeson's Straddle
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Example: Leeson's Straddle
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Example: Leeson's Straddle
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Delta Normal Implementation
Simple porfolios
More complex portfolios / instruments specifying a list of risk factors mapping the linear exposure of all instruments onto
these risk factorsestimating the covariance matrix of risk exposure
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Delta Method Implementation
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Delta Normal Implementation
Advantageseasy to implement (matrix computation)fast simple to explainadequate in many situations
Problems fat tails under estimate risks inadequate for non linear instrument
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Historical Simulation Implementation
Consist in going back in time (say 250 days), and apply historical returns
Hypothetical prices for scenario k provide a new portfolio value
Then VAR is estimated from the full sample
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Historical Simulation Implementation
Advantages simple to implement (brute force) if historical data are available ... no need to estimate covariance matrix, etc ... model free methodallow non linearities, capturing gamma, vega,
correlations risksaccount for fat tails
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Historical Simulation Implementation
Problemsassume we have sufficient historical data only one sample path is used assume that past data is representative of the future
the window may omit important data or n the other hand, may include not relevant data
simple historical simulation may miss some dynamic aspects (time varying volatility and clustering, ...)
put the same weight on all observations, including old data
quickly become cumbersome for large portfolios
note: most of the problems can be mitigated by time varying models like GARCH, RiskMetrics, ...
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Monte Carlo Implementation
2 steps procedure specifying stochastic processes for financial variables then simulate price paths
At each horizon considered, the portfolio is evaluated VAR is estimated from simulated portfolio values similar to historical simulation, except that hypothetical price changes is created by random draws
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Monte Carlo Implementation - Advantages
by far the most powerful method to compute VAR account for a wide range of risk and features, including non linear price risk time varying volatility fat tails extreme scenarios can also be used to estimate expected loss beyond the VAR time decay of options effect of pre defined trading or hedging dynamic strategies
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Monte Carlo Implementation - Problems
Major drawback: computation time ex: 10000 sample path for 100 assets => 1 million full
valuations in addition, each valuation may require inner
simulation to price derivatives, for example ! (Monte Carlo of Monte Carlo)
too heavy to implement on a regular day to day basis require strong skills and infrastructure (Software &
Hardware) Model Risk
in case the stochastic processes and pricing formulas are wrong sensitivity analysis
Subject to (Small) Sample Variation Effects
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Empirical Comparisons
Foreign currency portfolio Delta Normal is
at 99% confidence level, slightly underestimate actual VAR the fatest method
Full Monte Carlo most accurate slowest method
for lage portfolios, bank still prefer the delta normal, however, this method may dangerously underestimate actual losses in case of optionality features
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Comparison of approaches to VAR
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Aactual Uses of Methods
In practice all methods are used by bank:
42% delta normal and simple covariance approach
31% use historical simulation
23% Monte Carlo
source Britain's FSA survey