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Option Risk Management
Copyright © 2000 – 2006Investment Analytics
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 2
Agenda
Option sensitivity factorsDelta
Delta hedging
Option time valueGamma and leverageVolatility sensitivityGamma and Vega hedging
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 3
Option Sensitivity Factor
What affects the price of an optionthe asset price, Sthe volatility, σthe interest rate, rthe time to maturity, tthe strike price, X
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 4
Black-Scholes Equation
Consider delta-hedged option portfolioMust grow at risk free rate, else arbitrage
Leads to following relationship:
rVSVrS
SVS
tV
=∂∂
+∂∂
+∂∂
2
222
21 σ
DeltaGammaTheta
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 5
Option Greeks
Delta (“price sensitivity”)change in option price due to change in stock price
Gamma (“leverage”)change in delta due to change in stock price
Vega (“volatility sensitivity”)change in option price due to change in volatility
Theta (“time decay”)change in option value due to change in time to maturity
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 6
Option DeltaKey sensitivity dV/dS
Change in option value for infinitely small moveIn reality use: ∆V/∆SBetter still:
⎟⎠⎞
⎜⎝⎛
∆∆
+∆∆
= +− SV
SVDelta
21
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 7
The Delta of a Call OptionDelta changes as stock price changes
Gamma measures rate of change of delta
X Stock Price
Cal
l Val
ue
1
0.50
Out of the money
At the money
In the money
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 8
IBM Option Delta
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 9
A New Look at Delta
Pro
babi
lity
Delta = Probability of Option Finishing in-the-money
S X.e-rt
Out-of-the-Money
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 10
Delta: At-the-MoneyP
roba
bilit
y
S = X.e-rt
At-the-Money
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 11
Delta: In-The-MoneyP
roba
bilit
y
SX.e-rt
Delta 1
In-the-Money
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 12
Delta and Derivatives
Forward FX ContractsF = e(r-rf)t SSo delta of forward is e-rft
Stock index with dividend yield dDelta = e-dt
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 13
Delta and Volatility
How does (future) volatility affect delta?Example, Stock $100, 25% vol
$100 1-year Call, delta = 0.63$100 1-year Put, delta = -0.37
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 14
Call Delta and Volatility
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 15
Put Delta and Volatility
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 16
Delta and Volatility
ITM/ATM Call OptionDelta will fall with rising volatilityDue to fatter tails, probability of finishing OTM increases
OTM Call OptionDelta will increase with volatilityDue to fatter tails, probability of finishing ITM increases
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 17
Option Value
Intrinsic Value
Time Value
• Asset Price
• Strike Price
• Interest rates
• Time toExpiration
• Volatility
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 18
How Time Value DecaysP
rem
ium
Stock price
9 months
6months3 months
Expiration
Strike Price
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 19
Time Decay (Theta)Decay Acceleration
Pre
miu
m
0
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 20
Time Value of IBM Option
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 21
IBM Option Theta
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 22
Theta AdjustmentsModified Theta
Takes account of rolldown on volatility curveDaily loss of value assuming volatility is at level with 1-day shorter expiration
Shadow ThetaLosses from decay are often compounded by drop in implied vol in quiet marketsShadow theta factors in expected changes in IV.
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 23
Option Gamma
Change in Delta for small change in stockSecond derivative: δ2V/δS2
“Rate of Acceleration” of option value with price
Call and put options have positive gammaOption delta becomes larger as stock appreciatesBecomes smaller (more negative) as stock declines
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 24
Gamma Formula
tSdN
σ)( 1′
=Γ
2/2
21)( xexN −=′π
ttXSd
σσ )2/()/ln( 2
1+
=
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 25
IBM Option Gamma
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 26
Lab: Leverage
An Experiment:Assume stock $100Risk free rate 10%, volatility 25%Call option, strike price 100 (at the money)
Leverage:If stock moves by $5, how much does option value change?
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 27
Solution: Leverage
NOTES: Col 2 is option value with stock price = 100Col 3 is option value with stock price = 105Col 4 is ‘Leverage’: (C1 - C2)/C1
Call Prices ReturnMaturity S=100 S=105 (%) Gamma1.00 12.34 15.66 27% 0.0150.50 8.26 11.48 39% 0.0220.25 5.60 8.80 57% 0.0320.10 3.40 6.69 97% 0.0500.01 1.02 5.07 396% 0.160
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 28
Time vs Leverage
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 29
Gamma Characteristics
For ATM options Gamma is max nearer to expirationFor OTM options Gamma is max further away from expiration
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 30
Up- and Down- Gamma
Gamma is not symmetricUp-Gamma: change in delta for small gain in stock priceDown-Gamma: change in delta for small loss in stock price
Beware averaging!Some high risk positions have large +veup-gamma and large –ve down-gamma
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 31
Gamma and Volatility
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 32
Shadow Gamma
Volatility adjusted GammaUp- Shadow Gamma
Down Shadow-Gamma
[ ] [ ])(
,)(,SS
SdS−
∆−+∆=Γ +
++ σσσ
[ ] [ ])(
,)(,SS
SdS−
∆−+∆=Γ −
−− σσσ
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 33
Lab: Shadow GammaShadow Gamma
-1500
-1000
-500
0
500
1000
1500
80 84 88 92 96 100 104 108 112 116 120
Stock Price
P/L
GammaShadow Gamma
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 34
Delta and Gamma BleedDelta Bleed
Change in Delta per dayOTM (ITM) options move further OTM (ITM) with timeReduces (increases) delta
Gamma Bleed Change in Gamma per day
Increases for ITM/ATM optionsDecreases for OTM options
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 35
Delta Bleed – ATM
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 36
Delta Bleed - ITM
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 37
Gamma Bleed ITM
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 38
Gamma Bleed OTM
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 39
Alpha: Gamma Rent
Theta per Gamma ratioLow alpha means taking little theta risk for the gammaAlpha = (modifed Theta) / (shadow) GammaTheta = -½ Γσ2S2
Alpha = -½ σ2S2
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 40
Volatility Greeks
Vega - sensitivity to implied volatilityGamma - sensitivity to actual volatilityExample: Weather
People carrying umbrellas (implied risk of rain) = Vega Rain (the wet stuff) = Gamma
Implied volatility estimated s.d. implied by option prices by B-S model“market’s” estimate of current volatility
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 41
Vega
Measures change in option value for small change in implied volatility
)( 1dNtSV ′=∂∂σ
2/2
21)( xexN −=′π
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 42
Vega CharacteristicsCall and put options have positive Vega
Increase in value as implied volatility increases
Vega changes with underlying stockHighest for ATM options
Most sensitive to changes in implied volatility
Most Vegas decrease with time(except knock-outs and other exotics)
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 43
IBM Option Vega
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 44
Vega and Time
60 80
100
120
140
160
0.1
0.4
0.7
1.0
0
10
20
30
40
50
Vega
Asset
Time
Vega and Time
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 45
Vega Convexity
Nonlinearity of payoffVega convexity δ2V /δσ2
Vega is a non-linear function of σPrice
Linear for ATM optionsNon-linear for ITM/OTM options
Payoff like an option on volatility
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 46
Vega Convexity
Call Option Price and Vega
0
10
20
30
40
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Volatility
Pric
e
0
10
20
30
40
50
Veg
a
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 47
Lab: Option Sensitivities
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 48
Solution: Option Sensitivities
Theta
Gamma
Vega
Delta
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 49
Solution: Option Sensitivities
DeltaShort-dated, ATM options on less volatile stock are more sensitive
GammaGreatest for short-dated ATM options on less volatile stock
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 50
Solution: Option Sensitivities
VegaLong-dated, ATM options on less volatile stock more volatility sensitive
ThetaShort-dated ATM options on more volatile stock experience greatest rate of decay
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 51
Volatility Smiles & SurfacesImplied volatilities of options with different strikes varies
Inconsistent with Black-ScholesImplies volatilities of OTM options typically greater than ATM options
Smile: Plot IV vs. StrikeShows “smile” effect
Surface: Plot IV vs. Strike & Maturity
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 52
Volatility Smile – Example
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 53
Volatility Surface – Example
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 54
Forward Volatility
T0 T1 T2
σ 12 σ F
2
σ 22
σ σ σF T T T T T T= − − − −[( ) ( ) ] / ( )2 1 22
1 0 12
2 1
Market focus on spot volatilityCreates arbitrage opportunities in forward market
Minor fluctuations in spot volatility create large swings in forward volatility
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 55
Forward Volatility Curve
Volatility Curve
0%
10%
20%
30%
40%
50%
60%
70%
30 60 90 120 150 180 210 240 270 300 330 360
SpotForward
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 56
Curve Movement
ParallelSteepening/FlatteningConvexity changes
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 57
Lab: YAHOO
Construct implied volatility smile & surface
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 58
Solution: YAHOO
95
100
105
110
115 0.07
0.15
0.32
0.57
70%
71%
71%
72%
72%
73%
73%
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 59
Solution: YAHOO
95
100
105
110
1150.07
0.15
0.32
0.57
70%
71%
71%
72%
72%
73%
73%
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 60
Simple Gamma Hedging
Suppose unhedged position = one callYou can make it delta neutralBut this only works for small changesNeed to eliminate gamma risk too
How do you make it Delta & Gamma neutral?Can’t use stock: gamma is zeroMust use other options, O1 and O2
Solve: mδ1 + nδ2 = 0mΓ1 + nΓ2 = 0
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 61
Simple Vega Hedging
Make a portfolio Delta-Vega neutralAgain, use other optionsSolve:
mδ1 + nδ2 = 0mV1 + nV2 = 0
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 62
Lab: Greek Hedging
Excel: WorkbookExcel: WorkbookWorksheet: Greek hedging
Sensitivity:Check how position value, delta changes with stock priceCheck how position value changes with volatility
Hedging:Gamma hedge the given portfolioCheck sensitivity of position to stock price
Use Solver
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 63
Greek Hedging –Using SOLVER
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 64
Solution: Greek HedgingPosition Stock 45
Value Delta Gamma Vega Theta Volatility 14.0%Portfolio -493.9 9.2 -15.9 -1,169.8 282.2 Risk Free 5.0%
Hedge 493.9 -9.2 15.9 1,168.1 -281.6Net Position -0.0 -0.1 -0.0 -1.7 0.6
Target 0.0 0.0 0.0 1,169.8 282.2Abs. Position 0.0 0.1 0.0 1.7 0.6
Quantity Type Strike Maturity Price Delta Gamma Vega Theta0.0 C 35 0.25 10.44 1.00 0.000 0.01 -1.730.0 C 40 0.25 5.53 0.97 0.020 1.49 -2.33
99.5 C 45 0.25 1.55 0.59 0.119 8.77 -3.702.0 C 50 0.25 0.14 0.10 0.053 3.90 -1.310.1 C 55 0.25 0.00 0.00 0.004 0.27 -0.080.0 P 35 0.25 0.00 0.00 0.000 0.01 0.00
-0.1 P 40 0.25 0.04 -0.03 0.020 1.49 -0.35-0.7 P 45 0.25 0.99 -0.42 0.119 8.77 -1.4775.3 P 50 0.25 4.52 -0.90 0.053 3.90 1.160.0 P 55 0.25 9.32 -1.00 0.004 0.27 2.63
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 65
Modified Vega
Measures sensitivity to nonparallel changes in volatility curve
Vi are vegas in maturity buckets i = 1 , . . ., nωi are weights to be determined
∑=
=n
iiiVV
1* ω
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 66
Theoretical WeightingSquare root time ruleStart with e.g. 30-day exposureWeight exposures at other maturities by (30/t)1/2
Example: 120 day Vega is weighted by (30/120)^0.5 = 0.5So $1M Vega risk in one month is equivalent to $2M exposure at 120 days
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 67
Forward Volatility Analysis
Separate Vega risk into forward bucketsExample:
0 – 30 days30 – 60 days60 – 90 days90 – 180 days180 – 360 days
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 68
Forward Volatility Analysis
Bucket volatility & correlationEstimate volatility of volatility for each forward bucketEstimate volatility correlations between forward buckets
Use these to weight the exposures in each forward bucket and compute P/L
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 69
Value at Risk Models
Delta-NormalSimple, linear model uses derivatives deltaIgnores high-order effects
Non-Linear ModelMakes adjustments for non-linear effects (Gamma risk)Important for derivatives portfolios
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 70
The Delta-Normal Model
VAR = Market Value x Confidence Factor x Volatility x Delta
Same as standard model, just incorporating deltaNB: stock portfolio: delta = 1
A simple linear function of deltaAssumes that returns are normally distributedIf necessary, adjust volatility by T1/2 for appropriate holding period, as before
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 71
Derivative Portfolio VaRExample
Long S&P100 OEX index callsMarket value $9.45MMDaily volatility 1%Option delta 0.5Confidence level 99% (CF = 2.33)
VAR = $9.45 x 2.33 x .01 x 0.5 = $110,000There is a 1% chance that the call portfolio will lose more that $110K in a day
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 72
Options & VAR
A deep In-the-Money optionHas approximately same VAR as underlying stock
Assuming equal $ amounts invested in each Only true for short-term holding periods -Why?Answer:
Delta of ITM option ~ 1For long holding period, delta bleed
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 73
VaR & Delta
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
150
146
142
138
134
130
126
122
118
114
110
106
102 98 94 90 86 82 78 74 70 66 62 58 54 50
Moneyness
VA
R
At the Money
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 74
VaR and Delta
VAR increases with DeltaMinimum for OTM options, delta ~ 0Maximum for ITM options, delta ~ 1Changes most rapidly for
ATM optionsShort maturity options
because of Gamma
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 75
Delta-Normal VaR: Equivalent Formulation
VAR = S x CF x σ x ∆p
• S = underlying (stock) price• CF = confidence factor• σ = volatility• ∆p = Delta of portfolio
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 76
VAR and Gamma
Gamma adjustment required forATM optionsShort-dated options
Gamma risk is minor for Deep ATM/OTM long dated optionsShort holding periods
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 77
VAR Formula –Gamma Adjusted
VaR = S x CF x σ x [∆p2 + 1/2 (S x σ x Γ )2]1/2
S = underlying stock priceCF = confidence factorσ = volatilityΓ = Gamma∆p = Delta of portfolio
Note: same as Delta-Normal model when Γ = 0
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 78
VAR & Gamma Risk ExampleLong 1000 ATM calls
Stock price is $50Daily volatility 1.57% (25% annual)Portfolio delta 700Gamma is 27.8Confidence level 95% (CF = 1.65)
VAR = $50 x 1.65 x 0.0157 x [7002 + 1/2(50 x 0.0157 x 27.8)2]1/2 = $907
There is a 5% chance that the call portfolio will lose more that $907 in a day
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 79
Limitations of Delta-Gamma Normal Model
Major sources of error:Gamma changes over time (bleed)
Gamma increases as maturity approachesHigh-order effect produces very rapid changes in VaR
Gamma-adjusted VaR non-NormalRight-skewed distributionOvertstates true VaR
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 80
Zangari’s Moment Adjustment
Adjust Normal confidence parameterCorrection for skewness & kurtosis
Zα is the distribution’s lower α-percentileτ is the skewnessκ is the kurtosis
2332 ))(32)(36/1()3)(24/1()1)(6/1( τκτζ ααααα ZZZZZ −−−+−≈
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 81
Adjusting for Fat TailsStandardized residual process
εt ~ Normal(0, 1)Zangari’s Normal Mixture Approach
δt is binary variable, usually 0, sometimes 1ε2,t ~ Normal (µt, σ2t)
ttrR εσ =/
ttttrR 2,1,/ εδεσ +=
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 82
Zenari’s GEDGeneralized Error Distribution
Normal distribution when v = 2Probability of extreme event rises as v gets smaller
)(1/2)( )1/(1
|/(1/2)|
υυε υ
λε υ
Γ= +
− tef t
1/2)(2/ )/3](1/[2 υυλ υ Γ= −
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 83
Model Testing
Back testing recommended by Basle CommitteeCheck the failure rate
Proportion of times VaR is exceeded in given sampleCompare proportion p with confidence level
Problem: Hard to verify VAR for small confidence intervals
Need very many sample periods to obtain adequate test
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 84
Failure Rates
Test Statistic: see Kupiec 19951 + 2Ln[(1-p)T-NpN] - 2Ln[(1-(N/T))T-N (N/T)N]
ChiSq distribution, 1d.f.N = # failures; T = Total sample size
Example: Confidence level = 5%Expected # failures = 0.05 x 255 = 13Rejection region is 6 < N < 21
If # failures lies in this range, model is adequateIf N > 21, model underestimates large loss riskIf N < 6, model overestimates large loss risk
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 85
Additional TestsChristofferson (1996)
Interval test for VARVery general approach
Zangari Excessive Loss Test (1995)Calculates expected losses in “tail” event
f and F are standard Normal density / Distribution fnsUse t-test to compare sample mean losses against expected
)(/)(]|[ αασασ FfRRE tttt −=<
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 86
Lopez Probability Forecasting ApproachMost tests have low power
Likely to misclassify a bad model as goodEspecially when data set is small
Lopez ApproachUses forecasting loss function
e.g Brier Quadratic Probability Score
ptf is forecast probability of event taking place in interval t
It takes value 1 if event takes place, zero otherwiseIdentifies true model in most simulated cases
TIpQPS t
T
t
ft /)(2 2
1−= ∑
=
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 87
Empirical TestsZengari (1996)
Tested Standard Normal, Normal Mixture and GED VAR
12 FX and equity time series
All performed well at the 95% confidence levelNormal Mixture ad GED performed considerably better at 99% confidence level
ConclusionBoth NM and GED improve on Normal VAR
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 88
Other Approaches to VAR
Historical SimulationStress TestingMonte-Carlo Simulation
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 89
Historical Simulation
CalculationCalculate return on portfolio over past periodCalculate historical return distributionLook at -1.65σ point, as before
Pros & ConsDoes not rely on normal distributionOnly one path (could be unrepresentative)
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 90
Stress Testing
Scenario approach:Simulates effect of large movements in key financial variables
E.g. Derivatives Policy Group GuidelinesParallel yield curve shifts +/ 100 bpYield curve twist +/ 25 bpEquity index values change +/ 10%Currency movements +/ 10%Volatilities change +/ 20% of current values
Pros & ConsMore than one scenarioValidity of scenarios is crucialHandles correlations poorly
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 91
Monte Carlo Simulation
Sometimes called “full valuation”methodWidely applicable
Does not assume Normal distributionHandles all types of securities
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 92
Monte-Carlo Methodology
Simulate movement in asset valueRepeat 10,000 times, get 10,000 future valuesCreate histogram
Find cutoff value such that 95% of calculated values exceed cutoff
Cutoff value is the portfolio VARFor given confidence level (95%)For given holding period
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 93
Generating Simulated Values
∆S = S0 x (µ + σε)∆S is change in valueS0 is initial value µ is average daily returnσ is daily volatilityε is random variable
Procedure:Generate ε (random)Compute change in portfolio valueRepeat many times (10,000+)Create a histogram of portfolio values
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 94
Example
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 95
CrashMetricsDifferent Approach
Assumes worst case scenarioExamines behavior of assets in market crashesFinds optimal hedging strategy
Advantages:Distribution-freeRobustWorks with complex derivative positionsStatic hedging at known cost
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 96
Lab: Implementing a Simple VaR ModelImplementing a VaR model:
Delta normalDelta-GammaMonte-Carlo simulation
HedgingDelta neutralDelta-Gamma
Worksheet: Risk Management
Copyright © 2000-2006 Investment Analytics Advanced Option Risk Management Slide: 97
Summary: Option Risk ManagementOption Greeks
Importance of interaction effectsBucketing techniques
Value at RiskSevere limitations for option books
CrashmetricsFree of most unrealistic assumptions