Elices 2009 Bespoke model validation - 1er Seminaire parisien de...
Transcript of Elices 2009 Bespoke model validation - 1er Seminaire parisien de...
ModelModel ValidationValidation GroupGroupRisk Risk MethodologyMethodologyGrupo SantanderGrupo Santander
BespokeBespoke modelmodel validationvalidation
1er 1er SeminaireSeminaire parisienparisien de de validationvalidation de de modmod èèlesles financiersfinanciers ..InstitutInstitut Telecom, Telecom, amphiteatreamphiteatre TheveninThevenin ..Paris, Paris, MarchMarch 15th, 201015th, 2010
Alberto ElicesAlberto Elices
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Outline
� Introduction.� Model validation philosophy.� Reconcile FO and Risk interests: provisions.� Examples of bespoke validation:
� Double no touch, single barrier and portfolio of faders.� FX self-quanto options.� Effect of stochastic interest rates on callable products.
� Conclusions.
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Introduction
� After the crisis in the 2nd half of 2008, a big concernabout pricing models has been raised.� Risk management and model validation raise nowconsiderably more attention.� Model validation:
� Validation of model implementation is not enough.� Estimation of model risk.� Periodic reviews to identify adjustments and reflect changes
in market conditions.
� Risk management:� Apply reserves: provisions.� Limit model risk exposure (reduce volume of operations).
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Introduction
� Models are limited: they just approximate real prices.� Risk of prices out of market.� Risk of assumptions far from market practice.
� Taxonomy of model risk:� Not appropriate or incomplete theoretical framework.� Implementation errors.� Calibration errors.� Stability problems.� Market data inconsistencies.� User errors.
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Model Validation Philosophy
� Tests that could and sometimes should be performed:� Consistency with simpler products already validated.� Implementation of premium and greeks.� Stability of premium and greeks.� Convergence of premium and greeks.� Model robustness against inconsistent or illiquid market data
-> back testing vs real market data in fluctuating conditions.� Tests to estimate model risk.
� Model implementation:� Premium: compare with an independently developed model
for a representative set of deal inputs and market data.� Greeks: compare with simpler products already validated.
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Model Validation Philosophy
� Model risk estimation (premium based):� Premium sensitivity to non-calibrated (unobserved) or
innaccurately calibrated parameters: e.g. correlations, dividends.
� Comparison with other models with more accurate or simplydifferent hypothesis:
– Compare same product with different models available in FO.– Development of “toy” models:
Get sets of model parameters calibrated manually to market.Generate market and stressed market scenarios.Compare model under validation with “toy” model valuation.
� Simulation of hedging strategies: either back test with real or“toy” model market data.
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How to reconcile Front Office and Risk department inter ests?: provisions
� The provision should cover the expected hedging loss:� When hedging is done using a model with aggressive
prices, the expected hedging loss is around the fair priceminus the aggresive price: that difference is the provision.
� Provisions as a means to approve campaigns usinglimited models with controllable risk:
� A provision allows approving campaigns which would not be possible with a more sophisticated model (performance).
� Provisions to foster improvement of FO models:� Models with limitations should be given provisions which
should be released the more the model is improved.
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How to reconcile Front Office and Risk department inter ests?: provisions
� Provision calculation philosophy:� They should be transparent, easy to compute.� They should be dynamic stable, with smooth evolution
through time (they might ↑ or ↓ approaching expiry).� They should balance risk limitation and trading mitigation.� Front Office should be able to reproduce them.
– Use provision tables calculated from studies.– Use FO pricing models to calculate provisions:
Changing market or deal parameters (move correlations, barriers, coupons, leverages, volatilities etc).
Compare prices of deals valued with different FO models(better models might take too long on a daily basis).
� Other possible but more complicated approaches:– Simulate or back test portfolio hedging: sometimes impractical.– Compare FO versus Risk models.
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How to reconcile Front Office and Risk department inter ests?: provisions
� Market provisions.� Liquidity: inflation, credit spreads, volatility.� Movement of unobserved market parameters: correlations,
dividends.
� Model provisions: applied to specific models.� Digital jumps: varying call spread width depending on digital
jump and gamma sensitivity through tables.� Sensitivity to calibration set selection.� Movement of unobserved model parameters: mean
reversion, vol of vol, etc.� Model risk: tables from studies comparing different models.
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Example of provisions: liquidity of volatility surface.
� Liquidity provisions should allow unwinding a position.� Sensitivities are diffused through time (towards lower more
liquid maturities) and moneyness (towards closer to ATM).
Diffusion of offsetting sensitivities
for the 10 year tenor
with W=0.5
0
200
400
600
800
1,000
1,200
1 2 3 4 5 7 10 15 20 25 30 40 50 60 70
Tenor
Sensitivities (£k)
Original Sensitivity 1 tailed distribution
2 tailed distribution
Diffusion of sensitivities
for the 5% strike with W=0.05
ATM = 6%
0
200
400
600
800
1,000
1,200
2.50
%
3.00
%
3.50
%
4.00
%
4.50
%
5.00
%
5.50
%
6.00
%
6.50
%
7.00
%
7.50
%
8.00
%
Tenor
Sensitivities (£k)
2 tail diffusion 1 tailed diffusion
Original Sensitivities
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Example of provisions: liquidity of volatility surface.
� Liquidity provisions in practice:� Movements of implied volatility surface might be mainly
explained by parallel and slope (skew) shifts.� Parallel shifts: hedged with ATM options.� Slope shifts: hedged with risk reversals sensitive to slope.
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Example of provisions: uncertainty of volatility.
� A Heston model calibrated to market is considered:� Confidence intervals for total variance & vol are calculated.� Up and down increments of vol are calculated vs maturity.� Provision: max(|up & down movement * vega|).
0 2 4 6 8 100
0.02
0.04
0.06
0.08
0.1
0.12
Maturity in years
Tot
al v
aria
nce
mea
n &
qua
ntile
s; a
lpha
=0.
8 v0: 0.0076; Kappa: 1.9006; Theta: 0.0088; sigma: 0.1800
UpQtlMeanDownQtl
0 2 4 6 8 100.06
0.07
0.08
0.09
0.1
0.11
Maturity in years
Vol
atili
ty m
ean
& q
uant
iles;
alp
ha=
0.8
v0: 0.0076; Kappa: 1.9006; Theta: 0.0088; sigma: 0.1800
UpQtlMeanDownQtl
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Examples of bespoke validation: Double no touch (DNT).
� Methodology: simulation of hedging strategy.� Justification of expected hedging loss as premiumdifference according to different models.� Provisions as the sum of expected hedging loss plus uncertainty of hedging loss dispersion.
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Examples of bespoke validation: Double no touch (DNT).
� Double no Touch exotic options are often priced with a volga-vanna (heuristic) model.� Which model is better for hedging?
� The heuristic model.� The ATM model: Black Scholes with at-the-money volatility
� The criterion of comparison: P&L distribution at expiry.� Working hypothesis: market behaves as a Hestonmodel with certain parameters.
dtdYdW
dYdtvdv
dWvdtS
dS
tt
y
t
ρσθκ
µ
,
)(
=><
+−=
+=
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Examples of bespoke validation: Double no touch (DNT).
� At each point in time:� Heston’s spot and variance are simulated according to a set
of parameters calibrated to market.� Volatility surface is calculated with those parameters.� Heston’s two risk factors (underlying and variance) are
hedged with underlying and a 6m vanilla ATM call option.
� Pricing models are analytical.� This calculation is carried out with the aid of a grid ofcomputers.
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Examples of bespoke validation: Double no touch (DNT).
�Heston parameters: calibrated to 1y EUR/USD on August 2006 :� Kappa (κ) : 1.9006 Theta (θ) : 0.0088� Sigma (σ) : 0.1807 Rho (ρ) : 0.1289� Var0 (v0 ) : 0.0076
0
1
2
3
4
5
0
0.2
0.4
0.6
0.8
1
0.084
0.086
0.088
0.09
0.092
0.094
0.096
0.098
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.086
0.087
0.088
0.089
0.09
0.091
0.092
0.093
0.094
0.095Calibrated implied volatility for maturity 1y
Vol
atili
ty in
per
uni
t
Delta Delta
Maturity in years
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Examples of bespoke validation: Double no touch (DNT).
Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb07 Apr070.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8Van1y: Spot price paths; HedgeFreq: 0.25 days
Sp
ot
price
Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb07 Apr070
0.05
0.1
0.15
0.2
0.25
0.3
0.35Van1y: Volatility paths; HedgeFreq: 0.25 days
Vo
latility
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Examples of bespoke validation: Double no touch (DNT).
� A 1 year and 2 months double no touch is considered.� Hedging is performed once a day.� Pricing:
� Heuristic model (volga-vanna): 0.0839.� ATM model: 0.0466.� Heston’s model: 0.1327.
Price
Spot
2130.1 3622.1
2812.1=Spot
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Examples of bespoke validation: Double no touch (DNT).
Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−100
−50
0
50SndCDNTch−1y17usd: Hedged delta paths; HedgeFreq: 1.00 days
De
lta
Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−50
0
50
100
150
200
250SndCDNTch−1y17usd: Hedged vega paths; HedgeFreq: 1.00 days
Ve
ga
� Heuristic double no touch:
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Examples of bespoke validation: Double no touch (DNT).
� There is a consistent bias towards P&L losses.
Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15SndCDNTch−1y17usd: PnL Paths; HedgeFreq: 1.00 days
Pe
r u
nit
of
no
min
al
Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0SndCDNTchATM−1y17usd: Portfolio value; HedgeFreq: 1.00 days
Po
rtfo
lio p
rice
Heuristic model Black Scholes ATM model
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Examples of bespoke validation: Double no touch (DNT).
Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−0.035
−0.03
−0.025
−0.02
−0.015
−0.01
−0.005
0SndCDNTch−1y17usd: ExpVal of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov070
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05SndCDNTch−1y17usd: StdDev of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
� P&L distribution at expiry for heristic model.� ExpVal: -0.031.� StdDev: 0.045.
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Examples of bespoke validation: Double no touch (DNT).
Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02SndCDNTchATM−1y17: ExpVal of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov070
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1SndCDNTchATM−1y17: StdDev of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
� P&L distribution at expiry for ATM model.� ExpVal: -0.09.� StdDev: 0.093.
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Examples of bespoke validation: Double no touch (DNT).
Initial price of the option
Expected hedging losses0.03100.0839
0.09000.0466
0.1149 0.1366Option price with Heston model (Stdev=0.0279)0.1327
Premium + hedging costs
� The expected hedging cost plus the premium gives thefair price of the option.
0.1327
With smileSML
Without smileATM
[ ]LPEPP &modelHeston +≈
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Examples of bespoke validation: Double no touch (DNT).
� To calculate the precision of Heston’s price estimator, 100 estimations with 120 paths are simulated.� Heston’s price: Exp(Pr) = 0.1327 y Std(Pr)=0.0279.
0.1366: Total hedging cost ATMExp(Pr)=0.1327
Exp(Pr)+Std(Pr)=0.1606
Exp(Pr)-Std(Pr)=0.1048
0.1149: Total hedging cost SML
� Hedging cost difference between SML and ATM can be explained because only 120 paths were used.
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Examples of bespoke validation: Double no touch (DNT).
Initial price of the option
Expected hedging loss
A standard deviation of P&L is provisioned
With smile Without smile
Final price with provision
0.0450
0.0310
0.0839
0.0930
0.0900
0.0466
0.1599 0.2296
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Examples of bespoke validation: Double no touch. Conclu sions.
� The heuristic model is better than the ATM model:� Pricing: it provides a closer price to Heston´s model (the fair
value under the assumptions taken).� Hedging: it provides a lower P&L dispersion.
� The provision chosen in this situation is approximatelyequal to the expected hedging loss + deviation of P&L.
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Examples of bespoke validation: Simple barrier options.
� Two traded simple barriers considered:� 1y Up & In and 3 month Down & In.� Simple barrier options have signicantly less risk than DNT.
1.26850.33y1.22251.25Down & InPutBarr2
1.26851y1.421.27Up & InCallBarr1
SpotExpiryBarrier levelStrikeTypeCall/PutName
1.26851.22251.25Down & InPutBarr2
1.26851y1.421.27Up & InCallBarr1
SpotExpiryBarrier levelStrikeTypeCall/PutName
0.0025-0.00040.0112Barr2SML
0.0026-0.00050.0113Barr2ATM
0.00600.00000.0279Barr1SML
0.00850.00040.0274Barr1ATM
P&L Std DevExpected P&LPriceCase
0.0025-0.00040.0112Barr2SML
0.0026-0.00050.0113Barr2ATM
0.00600.00000.0279Barr1SML
0.00850.00040.0274Barr1ATM
P&L Std DevExpected P&LPriceCase
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Examples of bespoke validation: FX self-quanto options.
� Self quanto options are the worst case for correlation is 1.� Four valuation methods are compared:
� Black Scholes with at-the-money (ATM) volatility (left equation).� Black Scholes with volatility selection (VolSelection) to the strike
level of the non-quanto option (left equation).� Vega-volga-vanna (VVV) method.� Estimation of probability distribution of ST using Gatheral’s
parametrization of implied volatility surface (right equation)
( )[ ]),( TtPKSprice EURQ
EUR T
+−= E
tttEUR
tUSD
tQt
Qt dWdtrr
S
dS σσ ++−= )( 2
( )[ ]0
1),(
STtPSKSprice USDTTEUR
+−= E
ttEUR
tUSD
t
t
t dWdtrrS
dS σ+−= )(
0.39500.39900.37120.3704
VVVGatheralVolSelectionATM
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Examples of bespoke validation: FX self-quanto options.
� Gatheral’s parametrization is fitted to market data:� Left plot: market fitting (maximum difference is 5.65bp).� Right plot: implied probability distribution.
−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80.185
0.19
0.195
0.2
0.205
0.21
0.215
0.22
0.225
0.23
Moneyness: k = log(strike/forward)
Total
varia
nce:
varia
nce x
Matu
rity
0 50 100 150 200 2500
0.2
0.4
0.6
0.8
1
1.2x 10
−3
Pro
babi
lity
dens
ity fu
nctio
nUnderlying at expiry
33.8032.2231.4731.0330.7730.5930.5531.2531.6532.1332.7933.8035.87Vol
0.950.90.850.80.750.70.50.30.250.20.150.10.05Delta
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Examples of bespoke validation: FX self-quanto options.
� A set of a hundred random scenarios changing dates, interestrates and strikes preserving shape of volatility surface is gene rated.� Call option prices are compared.
-0.0600
-0.0400
-0.0200
0.0000
0.0200
0.0400
0.0600
0.0800
0 10 20 30 40 50 60 70 80 90 100
Gatheral-VVV
Gatheral-VolSelect
Gatheral-ATM
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Examples of bespoke validation: FX self-quanto options.
� Put option prices give a lot less differences for same scenari os.� Volatility selection or ATM methods can significantly di verge fromfair value. Best method: volga-vanna.
-0.0100
-0.0050
0.0000
0.0050
0.0100
0.0150
0.0200
0 10 20 30 40 50 60 70 80 90 100
Gatheral-VVV
Gatheral-VolSelect
Gatheral-ATM
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Examples: Effect of stochastic interest rates (IR) on cal lable products.
� Critical on long term products.� Consider a hybrid model with Hull White evolution forrates and Black Scholes for FX:
� Callable products are very related to digital options:� Products called when ST is outside a window is equivalent to
the sum of a digital OTM call and put.� Therefore, digital call options are studied.
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Examples: Effect of stochastic interest rates (IR) on cal lable products.
� Digitals with strike around Fwd have no correction.� As Fwd paths spread around mean, more paths end at
higher or lower ST and less around mean.� Probability of calling outside a window increases.
2 4 6 8 101
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Maturity in years
FX
forw
ard
Fwd Mean and quantiles with alpha=0.75
UpQtlMeanDownQtl
34
Examples: Effect of stochastic interest rates (IR) on cal lable products.
� Increasing correlation between interest rates makesdispersion lower => less correction.
35
Examples: Effect of stochastic interest rates (IR) on cal lable products.
� Higher correlation between local IR and FX:� ↑ Strikes => ↑ price; ↓ Strikes => ↓ price.� More probability to cancel at upper thresholds.� Opposite conclusion with correlation foreign IR - FX.
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Conclusions
� Model validation has raised considerable more attention in the past months due to the 2008 crisis.� Validation of implementation is no longer enough. Estimation of model risk has become a major issue:
� Sensitivity to unobserved parameters.� Compare models using hedging strategies.� Compare same product with different models.
� Provisions: between perfect & reasonable:� They should cover uncertainty of hedging costs.� They may allow campaigns with limited models.� They allow to foster improvement of FO models.
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Annexe: Examples of bespoke validation: a portfolio of F X Faders.
� Heston hedging strategy simulation is used.� This portfolio represents a real portfolio on EUR/USD.� All operations are forward contracts (long a fader calland short a fader put with same data).� All faders are “in” with a lower level (notionalincreases when spot is higher than lower level).� Most faders limit client losses with a continuous lowerbarrier (when trespassed no more notional isaccumulated).� Faders are valued with volga-vanna heuristic model.
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Annexe: Examples of bespoke validation: a portfolio of F X Faders.
� The portfolio of faders can be represented by thefollowing three positions:
175.1=DLEVEL
29.1=K
15.1=DBARR
2096.1=Spot
175.1=DLEVEL
29.1=K
2096.1=Spot
175.1=DLEVEL
29.1=K
15.1=DBARR
2096.1=Spot
yExpiry 5.1: yExpiry 8.0: yExpiry 5.0:
%54:Nominal %29:Nominal %17:Nominal
weeksfrequencyFading 4: weeksfrequencyFading 4: weeksfrequencyFading 2:
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Annexe: Examples of bespoke validation: a portfolio of F X Faders.
� Three delta periods are observed (0.5y, 1.3y y 1.5y).� Lower delta limits depend on notionals of still living options (-0.66, -0.83 y –0.54).
Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep07−1
−0.5
0
0.5
1
1.5FwdFader: Hedged delta paths; HedgeFreq: 1.00 days
Del
ta
Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep07−4
−3
−2
−1
0
1
2
3FwdFader: Hedged vega paths; HedgeFreq: 1.00 days
Veg
a
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Annexe: Examples of bespoke validation: a portfolio of F X Faders.
� Expected hedging gain at maturity is 25pb and thestandard deviation is 27bp.� No provision is needed.
Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep070
0.5
1
1.5
2
2.5
3x 10
−3FwdFader: Standard deviation of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep07−0.5
0
0.5
1
1.5
2
2.5x 10
−3 FwdFader: ExpVal of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
41
Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep070
0.5
1
1.5
2
2.5
3
3.5
4x 10
−3 FwdLloss−1y5: StdDev of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb070
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
−3 FwdLloss−0y8: StdDev of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
Mar06 Apr06 May06 May06 Jun06 Jul06 Jul06 Aug06 Aug06 Sep060
0.5
1
1.5
2
2.5
3
3.5
4x 10
−3 FwdSloss−0y5: StdDev of PnL; HedgeFreq: 1.00 days
Per
uni
t of n
omin
al
Annexe: Examples of bespoke validation: a portfolio of F X Faders.
� The standard deviation of each componenent is: 37pb, 42pb y 38pb.
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Annexe: Examples of bespoke validation: a portfolio of F X Faders.
� Theoretical limits of the portfolio standard deviation:� Compensación perfecta: 0 (posiciones largas y cortas del
mismo producto).� Maximum diversification (no compensation effect):
� Perfect correlation among standard deviations:
� The portfolio considered has 27pb. There exists somecompensation effect.
pbnomnomnom prodprodprodprodprodprod 39332211 =++ σσσ
pbnomnomnom prodprodprodprodprodprod 2423
23
22
22
21
21 =++ σσσ