Jorion Var 09

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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