Derivatives > Advanced Option Risk Management

1

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


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


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


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


Option DeltaKey sensitivity dV/dS

Change in option value for infinitely small moveIn reality use: ∆V/∆SBetter still:

⎟⎠⎞

⎜⎝⎛

∆∆

+∆∆

= +− SV

SVDelta

21


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


IBM Option Delta


A New Look at Delta

Pro

babi

lity

Delta = Probability of Option Finishing in-the-money

S X.e-rt

Out-of-the-Money


Delta: At-the-MoneyP

roba

bilit

y

S = X.e-rt

At-the-Money


Delta: In-The-MoneyP

roba

bilit

y

SX.e-rt

Delta 1

In-the-Money


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


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


Call Delta and Volatility


Put Delta and Volatility


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


Option Value

Intrinsic Value

Time Value

• Asset Price

• Strike Price

• Interest rates

• Time toExpiration

• Volatility


How Time Value DecaysP

rem

ium

Stock price

9 months

6months3 months

Expiration

Strike Price


Time Decay (Theta)Decay Acceleration

Pre

miu

m

0


Time Value of IBM Option


IBM Option Theta


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.


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


Gamma Formula

tSdN

σ)( 1′

=Γ

2/2

21)( xexN −=′π

ttXSd

σσ )2/()/ln( 2

1+

=


IBM Option Gamma


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?


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


Time vs Leverage


Gamma Characteristics

For ATM options Gamma is max nearer to expirationFor OTM options Gamma is max further away from expiration


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


Gamma and Volatility


Shadow Gamma

Volatility adjusted GammaUp- Shadow Gamma

Down Shadow-Gamma

[ ] [ ])(

,)(,SS

SdS−

∆−+∆=Γ +

++ σσσ

[ ] [ ])(

,)(,SS

SdS−

∆−+∆=Γ −

−− σσσ


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


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


Delta Bleed – ATM


Delta Bleed - ITM


Gamma Bleed ITM


Gamma Bleed OTM


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


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


Vega

Measures change in option value for small change in implied volatility

)( 1dNtSV ′=∂∂σ

2/2

21)( xexN −=′π


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)


IBM Option Vega


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


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


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


Lab: Option Sensitivities


Solution: Option Sensitivities

Theta

Gamma

Vega

Delta



DeltaShort-dated, ATM options on less volatile stock are more sensitive

GammaGreatest for short-dated ATM options on less volatile stock



VegaLong-dated, ATM options on less volatile stock more volatility sensitive

ThetaShort-dated ATM options on more volatile stock experience greatest rate of decay


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


Volatility Smile – Example


Volatility Surface – Example


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


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


Curve Movement

ParallelSteepening/FlatteningConvexity changes


Lab: YAHOO

Construct implied volatility smile & surface


Solution: YAHOO

95

100

105

110

115 0.07

0.15

0.32

0.57

70%

71%

71%

72%

72%

73%

73%


Solution: YAHOO

95

100

105

110

1150.07

0.15

0.32

0.57

70%

71%

71%

72%

72%

73%

73%


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


Simple Vega Hedging

Make a portfolio Delta-Vega neutralAgain, use other optionsSolve:

mδ1 + nδ2 = 0mV1 + nV2 = 0


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


Greek Hedging –Using SOLVER


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


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


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


Forward Volatility Analysis

Separate Vega risk into forward bucketsExample:

0 – 30 days30 – 60 days60 – 90 days90 – 180 days180 – 360 days


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


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


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


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


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


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


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


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


VAR and Gamma

Gamma adjustment required forATM optionsShort-dated options

Gamma risk is minor for Deep ATM/OTM long dated optionsShort holding periods


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


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


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


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 −−−+−≈


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 +=


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= −


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


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


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 −=<


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−= ∑

=


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


Other Approaches to VAR

Historical SimulationStress TestingMonte-Carlo Simulation


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)


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


Monte Carlo Simulation

Sometimes called “full valuation”methodWidely applicable

Does not assume Normal distributionHandles all types of securities


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


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


Example


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


Lab: Implementing a Simple VaR ModelImplementing a VaR model:

Delta normalDelta-GammaMonte-Carlo simulation

HedgingDelta neutralDelta-Gamma

Worksheet: Risk Management


Summary: Option Risk ManagementOption Greeks

Importance of interaction effectsBucketing techniques

Value at RiskSevere limitations for option books

CrashmetricsFree of most unrealistic assumptions

Derivatives > Advanced Option Risk Management

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