following p. jorion, value at risk, mcgraw-hill chapter …yats.free.fr/doc/jorion-var-09-en.pdf ·...

1

Daniel HERLEMONT

Financial Risk Management

Following P. Jorion, Value at Risk, McGraw-Hill

Chapter 9

VaR Methods

Daniel HERLEMONT

VaR Methods

�Local Valuation Methods

�valuing the portfolio once, using local derivatives :

�delta normal method

�delta-gamma ("Greeks") method

�Most appropriate to portfolios with with limited sources

of risk.

�Full Valuation Methods

�re-pricing the portfolios over a range of scenarios,

including:

�Historical

�Monte Carlo

2

Daniel HERLEMONT

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:

� αααα is the standard normal deviate corresponding to the

confidence level, e.g. 1.645 for a 95% confidence level

Daniel HERLEMONT

Delta Normal - Fixed Income Portfolio

The price-yield relationship:

where D* is the (modified) Duration

where σσσσ is the volatility in of change in level of yield

3

Daniel HERLEMONT

Distribution with linear exposure

Daniel HERLEMONT

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

4

Daniel HERLEMONT

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 methods

�Monte Carlo: sampling from a distribution (e.g. normal)

�Historical Simulations: sampling from historical data

Daniel HERLEMONT

Full Valuation

�The portfolio is priced for each draw

�VAR 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

5

Daniel HERLEMONT

Daniel HERLEMONT

Delta Gamma Approximations

�Extends the delta normal method with higher

moments

Γ Γ Γ Γ Γ second derivative of portfolio value

Γ Θ Θ Θ Θ is the time drift

6

Daniel HERLEMONT

Delta Gamma - Examples

�Fixed Income

�D is the Duration, C is the convexity

�Vanilla Call Options:

�valid for long (Γ>0) Γ>0) Γ>0) Γ>0) or short (Γ<0) Γ<0) Γ<0) Γ<0)

�The second term decrease the linear VAR.

Daniel HERLEMONT

Delta Gamma for a long call

� the downside risk for the option is less than given by deltaapproximation .... this is the "raison d'être" of option ...

7

Daniel HERLEMONT

Delta Gamma for complex portfolios

� taking the variance at both side:

� then, under normal hypothesis:

0),cov( and )](variance[2)(variance 222 == dSdSdSdS

Daniel HERLEMONT

Delta Gamma - Cornish Fisher Expansion

ξ is the Skewness

ξ Negative Skewness increases VAR

ξ the same applies for positive Excess Kurtosis

8

Daniel HERLEMONT

Skewness

Daniel HERLEMONT

Kurtosis

9

Daniel HERLEMONT

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

Daniel HERLEMONT

Delta Gamma - Multiple risk factors

�∆∆∆∆ and dS are vectors

�computationally intensive

�requires estimates of:

�Gamma (implicit correlations)

�Covariance matrix

10

Daniel HERLEMONT

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

Daniel HERLEMONT

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 time

�daily 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 ...

11

Daniel HERLEMONT

Example: Leeson's Straddle

Daniel HERLEMONT

Sell Straddle payoff

Sell Straddle = sell call + sell put

Strike = at the money

Successful, only if the spot remains stable

Delta = 0

12

Daniel HERLEMONT


Daniel HERLEMONT


13

Daniel HERLEMONT


VaR Analysis could have prevented bankruptcy

if positions were known

Daniel HERLEMONT


14

Daniel HERLEMONT


Daniel HERLEMONT


15

Daniel HERLEMONT


Daniel HERLEMONT

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 factors

�estimating the covariance matrix of risk exposure

16

Daniel HERLEMONT

Delta Method Implementation

Daniel HERLEMONT

Delta Normal Implementation

�Advantages

�easy to implement (matrix computation)

�fast

� simple to explain

�adequate in many situations

�Problems

� fat tails �� under estimate risks

� inadequate for non linear instrument

17

Daniel HERLEMONT

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

Daniel HERLEMONT


�Advantages

� simple to implement (brute force)

� if historical data are available ...

� no need to estimate covariance matrix, etc ...

� model free method

�allow non linearities, capturing gamma, vega,

correlations risks

�account for fat tails

18

Daniel HERLEMONT


�Problems

�assume 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, ...

Daniel HERLEMONT

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

19

Daniel HERLEMONT

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

Daniel HERLEMONT

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

20

Daniel HERLEMONT

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

Daniel HERLEMONT

Comparison of approaches to VAR

21

Daniel HERLEMONT

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

following p. jorion, value at risk, mcgraw-hill chapter …yats.free.fr/doc/jorion-var-09-en.pdf ·...

Documents