Volatility Dynamics and Option Prices · Spring School and Workshop on Volatility Dynamics and...

Volatility Dynamics and Option Prices

Loriano Mancini

Swiss Finance Institute andEcole Polytechnique Federale de Lausanne

Spring School and Workshop on Volatility Dynamics and OptionPrices, and Econometrics of Intraday Data, 10–13 April 2017

Outline

Volatility DynamicsDiscrete-time ModelsContinuous-time Models

Option PricingChange of MeasureGARCH Model

Volatility DerivativesVariance SwapsTerm Structure ModelsOptimal InvestmentVIX

References

2/69

Outline




References

Volatility Dynamics 3/69

Asset Returns

I Many asset classes

I Equities, bonds, commodities, currencies, etc.

I Significantly different economic features

I Empirical regularities of asset returns or stylized facts

I Surprisingly similar

I Model-free phenomena (essentially)

I Impose constraints on models of asset returns

I Relevant for asset and risk management, derivative pricing, etc.


Market Returns: S&P500 Index

1982 1987 1993 1998 2004 2009 2015−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Figure: Daily log-returns, log(St/St−1), of the S&P500 equity marketindex St . Note: i) Stochastic volatility, ii) volatility clustering, and iii)large market returns, are evident. Other asset returns exhibit similarphenomena.


Asset Returns: Stylized facts

I Time variation of asset returns or volatility

I Stochastic (i.e., changes randomly over time)

I Persistent (i.e., temporal dependence or volatility clustering)

I Price discontinuities (i.e., jumps)

I Heavy tails of return distributions

I Leverage effect, i.e., Cov[asset return, volatility change] < 0

I . . .


Two Modeling Frameworks for Asset Returns and Volatility

I Discrete-time models

I Inference :)

I Prototype model: GARCH model

I Engle (1982), Bollerslev (1986)

I Continuous-time models

I “Closed-forms” :)

I Prototype model: Square-root model

I Heston (1993)


Outline




References


GARCH model: Classic Specification

I Asset log-return rt = log(St/St−1)

rt = µ+ εt

where εt = σt zt , and zt is an i.i.d. shock D(0, 1)

I GARCH(1,1) conditional variance σ2t = Var[rt |Ft−1]

σ2t = β0 + β1σ

2t−1 + β2ε

2t−1

where Ft−1 is the information set at time t − 1

I Plus other conditions to ensure σ2t > 0, stationary, etc.

⇒ σ2t is stochastic, persistent, and known at time t − 1


GARCH model: Maximum Likelihood Estimation

I Joint density of log-returns r1, . . . , rT from p.d.f. f (rt |Ft−1)

I Log-likelihood function, assuming rt |Ft−1 ∼ N (µ, σ2t )

log f (r1, . . . , rT ) =T∑

t=1

log f (rt |Ft−1)

= −1

2

T∑t=1

[log(2π) + log σ2

t + (rt − µ)2/σ2t

]

I Maximize log-likelihood w.r.t. µ, β0, β1, β2


Market Returns and GARCH Volatility

1982 1987 1993 1998 2004 2009 2015

−0.2

−0.1

0

0.1

1982 1987 1993 1998 2004 2009 20150

0.2

0.4

0.6

0.8

1

Figure: Upper graph: Daily S&P500 log-returns. Lower graph: EstimatedGARCH volatility σt annualized. Note: GARCH volatility tracks marketreturn variability.


Market Returns and GARCH Innovations

1982 1987 1993 1998 2004 2009 2015

−0.2

−0.1

0

0.1

1982 1987 1993 1998 2004 2009 2015−10

−5

0

5

Figure: Upper graph: Daily S&P500 log-returns. Lower graph: EstimatedGARCH innovations zt = (rt − µ)/σt . Note: GARCH innovations exhibitmuch less volatility clustering than market returns.


Temporal Dependence

0 5 10 15 20 25 30 35 40 45−0.1

0

0.1

0.2

0.3

Lag

AC

F r t2

0 5 10 15 20 25 30 35 40 45−0.1

0

0.1

0.2

0.3

Lag

AC

F z

t2

Figure: Upper graph: Sample autocorrelations of squared daily S&P500log-returns. Lower graph: Sample autocorrelations of squared GARCHinnovations (zt)2, where zt = (rt − µ)/σt . Note: Squared market returnsexhibit strong temporal dependence (volatility clustering), squaredGARCH innovations do not.


GARCH Models: Some Common Specifications

I Return innovation zt (driving rt = µ+ σt zt) non-normal

I e.g., Bollerslev (1987), Engle and Gonzalez-Rivera (1991)

I Asymmetric GARCH (to capture “leverage” effects)

σ2t = β0 + β1σ

2t−1 + β2ε

2t−1 + β3It−1ε

2t−1

where It−1 = 1 when εt−1 < 0, and It−1 = 0 otherwise

I Glosten, Jagannathan, and Runkle (1993)

I Long-run and short-run component GARCH

σ2t = qt + β1(σ2

t−1 − qt−1) + β2 η1,t−1

qt = γ0 + γ1qt−1 + γ2 η2,t−1

where η1,t−1 and η2,t−1 are zero-mean innovations

I Christoffersen et al. (2008)Volatility Dynamics 14/69

GARCH Models Nowadays

I Widely used to estimate and forecast conditional variances

I Largely applied for risk management, derivative pricing, etc.

I Multivariate GARCH models well-developed

I e.g., Bauwens, Laurent, and Rombouts (2006)


Outline




References


Continuous-time Stochastic Volatility Models

I Instantaneous variance vt of asset returns is unobservable

I Call for filtering, inference not straightforward

I Heston model, or square-root diffusion, prototype model

dSt/St = µ dt +√

vt dW1,t

dvt = κ(θ − vt)dt + σ√

vt dW2,t

where dW1,t and dW2,t Brownian increments (correlation ρ),speed of mean reversion κ > 0, long run mean θ >0, etc.

I Exercise: Dynamics of√

vt (use Ito’s Lemma)?

I Affine models, drift and variance of vt are affine in vt

I Characteristic function of log(ST/St) in closed-form

E[e iu log(ST /St )|Ft ] = eC(T−t)+D(T−t)vt


Exercise: Moment Generating Function of Log-returnsI Given the Heston model in the previous page

I Use Ito’s Lemma to get the dynamics of xt = log(St)

Goal: Derive the moment generating function E[eu log(ST /St )|Ft ]I Define the martingale process ζt = E[euxT |Ft ]I Guess the functional form ζt = euxt +C(T−t)+D(T−t)vt

I Use Ito’s Lemma to get the dynamics of ζt

dζt =∂ζt

∂tdt+

∂ζt

∂xtdxt+

1

2

∂2ζt

∂x2t

(dxt)2+∂ζt

∂vtdvt+

1

2

∂2ζt

∂v 2t

(dvt)2+∂2ζt

∂xt∂vt(dxt dvt)

I Plug in all partial derivatives, and dxt and dvt termsI Drift(ζt) = 0, ∀dt and ∀vt dt, because ζt is a martingaleI Setting dt terms = 0 and vt dt terms = 0 gives two ODEs,

one for D(T − t) and one for C (T − t)

D ′ =1

2σ2D2 + (uρσ − κ)D +

1

2u2 − 1

2u

C ′ = uµ+ Dκθ

with initial conditions D(0) = 0 and C (0) = 0Volatility Dynamics 18/69

Continuous-time Model: Common Specification

I Remaining in the class of affine models

I Two-factor affine jump diffusion (to increase “vol of vol”):

dSt/St− = µ dt +√

vt dW1,t + (eJt − 1)dNt − νt dt

dvt = κv (mt − vt)dt + σv√

vt dW2,t

dmt = κm(θm −mt)dt + σm√

mt dW3,t

where jump intensity νt is affine, plus other conditions

I From Transform Analysis of Duffie, Pan, and Singleton (2000)

E[

exp

(−∫ T

tγ · Xs ds

)eu·XT |Ft

]= eA(T−t)+B(T−t)·Xt

where Xt = [log(St), vt ,mt ] is the state vector, andA(T − t) and B(T − t) solve ODEs


Volatility Dynamics from Two-factor Affine Model

97 98 99 00 01 02 03 04 05 06 07 08 09 100

10

20

30

40

50

60

70

80

v1/2

m1/2

Figure: Estimated instantaneous volatility,√

vt , and its stochastic longrun mean,

√mt , from S&P500 data, using a two-factor affine jump

diffusion model. Volatility annualized and in percentage units.


Continuous-time Stochastic Volatility Models Nowadays

I Multi-factor models

I e.g., Gruber, Tebaldi, and Trojani (2015)

I “Unspanned” risks

I e.g., Andersen, Fusari, and Todorov (2015)

I Non-affine models

I e.g., Filipovic, Gourier, and Mancini (2016)

I . . .


Outline




References

Option Pricing 22/69

DerivativesI No doubt things can go wrong with derivatives (see subprime)

I Derivative markets serve the fundamental purpose of risksharing by meeting hedging and speculative demands

I Option markets are large and fast growingI e.g., S&P500 options is a multi-trillion market

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Num

ber

of options

×104

0

2

4

6

8

10

12Number of option contracts by year

Calls

Puts

Figure: Number of out-of-the-money puts and calls on S&P500.Option Pricing 23/69

European Call and Put Options

I Options are financial instruments

I Call / Put option

I Written on an underlying asset S (e.g., S&P500 index)

I Gives the right, but not the obligation, to buy / sell theunderlying asset at the strike price K

I European options can only be exercised at maturity T

I Option price determined at time t < TI Paid by the option buyer to the option seller (or writer)


What determines an option price?

I Put option on market index pays out when market drops, i.e.,an insurance contract against market drop

80 85 90 95 100 105 110 115 1200

2

4

6

8

10

12

14

16

18

20

(K −

ST)+

ST

Figure: Put option payoff (K − ST )+ with strike price K = 100.

I Put option price at time t < T depends on

I Objective probability of market drop, and

I Investors’ preferences for protection, risk aversion, etc.⇒ Investors’ preferences lead to risk premiaOption Pricing 25/69

Outline




References


Equilibrium Model: Intuition

I One period exchange economy with representative agent

I One consumption good, one risky asset ψ

I One representative agent has endowment wt , can buy ξ unitsof the risky asset, consumes ct , discounts future utility u(cT )at β, and at terminal time T > t receives the payoff of the riskasset and consumes everything

I Representative agent maximizes the expected utility

maxξ

u(ct) + EP[βu(cT )|Ft ]

subject to ct = wt − ξψt , with cT = ξψT

I First order condition

0 = u′(ct)(−ψt) + EP[βu′(cT )ψT |Ft ]Option Pricing 27/69

Fundamental Asset Pricing EquationI First order condition ⇒ Fundamental Asset Pricing Equation

ψt = EP[β

u′(cT )

u′(ct)ψT |Ft

]= EP [Mt,T ψT |Ft ]

I When ψ is the risk free bond, ψt = e−r(T−t) and ψT = 1

e−r(T−t) = EP[β

u′(cT )

u′(ct)|Ft

]I When ψ is the risky asset, taking the ratio, and solving for ψt

ψt = e−r(T−t)EP[

βu′(cT )/u′(ct)

EP [βu′(cT )/u′(ct)|Ft ]ψT |Ft

]= e−r(T−t)

∫dPt,T

βu′(cT )/u′(ct)∫dPt,Tβu′(cT )/u′(ct)

ψT

= e−r(T−t)

∫dQt,T ψT = EQ

[e−r(T−t)ψT |Ft

]Option Pricing 28/69

Risk Neutral Pricing

I Risk neutral measure, or pricing measure

dQt,T = dPt,Tβu′(cT )/u′(ct)

EP[βu′(cT )/u′(ct)|Ft ]= dPt,T

Mt,T

EP[Mt,T |Ft ]

I Risky assets can be priced as if investors were risk neutral

ψt = EQ[e−r(T−t)ψT |Ft

]I Stochastic discount factor Mt,T

Mt,T = βu′(cT )

u′(ct)= e−r(T−t) dQt,T

dPt,T

I Exercise: Derive Mt,T when u(c) = c1−γ/(1− γ) has CRRA γ

I How does Mt,T look like “in practice”?Option Pricing 29/69

Estimated Stochastic Discount Factor from SPX data

2002/10

2004/02

2005/07

2006/11

2008/04

2009/08

0.5

1

1.5

0

1

2

3

4

5

6

ST / S

t

SD

F

Figure: Empirical pricing kernel projected on one-year S&P500 grossreturns ST/St , Mt,T = βu′(cT )/u′(ct) = e−r(T−t)dQt,T/dPt,T , basedon Barone-Adesi, Engle, and Mancini (2008). Note: Mt,T as a functionof ST/St , estimated each day t, changes shape over time.


Equilibrium Model: Recap

I Three key quantities

I Objective probability measure Pt,T

I Risk neutral measure Qt,T

I Investors’ preferences βu′(cT )/u′(ct) = e−r(T−t)dQt,T/dPt,T

Note: Any two determine the third

I Objective probability P needs to be risk-adjusted for optionpricing, which is under Q ⇒ Change of measure from P to Q

I Active research on which variables enter the SDF

Mt,T = βu′(cT )/u′(ct)

i.e., on priced risk factors


Outline




References


GARCH Option Pricing

I Asset return dynamic under P

log(St/St−1) = r + λσt −1

2σ2

t + εt

σ2t = β0 + β1σ

2t−1 + β2ε

2t−1

where εt |Ft−1 ∼ N (0, σ2t ), and risk premium λ > 0

I Time varying excess return, as EP[St/St−1|Ft−1] = er+λσt

I How do we change measure from P to Q?

One approach is Locally Risk Neutral Valuation RelationshipI Duan (1995)

There are (infinitely many) other ways of changing measure


Locally Risk Neutral Valuation Relationship

I Definition: Risk neutral measure Q satisfies LRNVR when

I Q is absolutely continuous with respect to PI EQ[St/St−1|Ft−1] = er

I VarP[log(St/St−1)|Ft−1] = VarQ[log(St/St−1)|Ft−1]

I log(St/St−1)|Ft−1 ∼ N under Q

I Comments on the LRNVR

X To avoid arbitrage, Q must be absolutely continuous withrespect to P, i.e., for any event A, P(A) = 0⇒ Q(A) = 0

X Under Q, any traded asset (stock, bond, option, etc.) earnsthe risk free rate in expectation (i.e., risk neutral pricing)

Same GARCH variance under P and Q from t − 1 to t

Same class of log-normal return distributions under P and QOption Pricing 34/69

Risk Neutral GARCH Dynamic

I Asset return dynamic under Q, from LRNVR

log(St/St−1) = r − 1

2σ2

t + (εt + λσt)

σ2t = β0 + β1σ

2t−1 + β2ε

2t−1

where εt |Ft−1 ∼ N (−λσt , σ2t ), due to change of measure

I From P to Q, p.d.f. f (εt |Ft−1) “rigidly shifts to left”

I Under Q, (εt + λσt) is the zero-mean return shock

I Change of measure in continuous time model is “similar”,e.g., P-Brownian motion gets some drift from P to Q

I Exercise: If P-dynamic is dSt/St = (r + λ√

vt)dt +√

vtdW Pt ,

what is the Q-dynamic of St?


Pricing Options: Two Methods

I Using risk neutral GARCH dynamics, there are two methodsto price options:

I Monte Carlo simulationI Simulate many price paths from t to T , S (i)

T , i = 1, . . . ,mI Compute, say, time-t put price, e−r(T−t)

∑mi=1(K − S

(i)T )+/m

I e.g., Duan (1995), Barone-Adesi, Engle, and Mancini (2008)

I Fourier inversionI Specify (affine) GARCH modelI Derive moment generating function, EQ[eu log(ST/St )|Ft ]I Price options via Fourier inversion

I e.g., Fang and Oosterlee (2008)

I e.g., Heston and Nandi (2000), Majewski et al. (2015)


Risk Neutral Parameters

I Calibration (classic approach): Minimize the distance betweenmodel-based and market-based option prices by changingmodel parameters

I Given N market option prices Pmarket(Kj ,Tj ), j = 1, . . . ,NI For a given set of model parameters θ = β0, β1, β2, λ,

compute model-based prices Pmodel (Kj ,Tj ; θ), j = 1, . . . ,NI Change model parameters θ as to

minθ

N∑j=1

(Pmodel (Kj ,Tj ; θ)− Pmarket(Kj ,Tj )

weightj

)2

I Likelihood based method (recent approach): Maximize theweighted sum of log-likelihood of asset returns andlog-likelihood of option pricing errors

I e.g., Badescu, Cui, and Ortega (2016)


Implied Volatility Surface from SPX Options

0.40.5

0.60.7

0.80.9

11.1

1.21.3

0

50

100

150

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

moneyness = Strike/Futures

Impl

ied

vola

tility

Figure: Implied volatilities σIV (K ,T ) for various strike prices K andmaturities T from SPX options on 31-Dec-2004. Recall, σIV (K ,T ) issuch that PBlack-Scholes(K ,T ;σIV (K ,T )) = Pmarket(K ,T ). Note: Twofactors in volatility help fitting term structure of σIV ; large negative pricejumps help fitting short-term skew of σIV .


Outline




References

Volatility Derivatives 39/69

Volatility Derivatives

I “Volatility has become an asset class”— (Any) investment bank’s brochure, 1990s

I “The next financial crisis will come from volatility markets”— (Anonymous) variance swap trader, last year



I “Volatility has become an asset class”— (Any) investment bank’s brochure, 1990s

I “The next financial crisis will come from volatility markets”— (Anonymous) variance swap trader, last year



I Many volatility derivatives

I Volatility swaps, futures/options on VIX, VIX ETF, etc.

I Variance Swaps

I Basic contract


Outline




References


Variance Swap (time 0)

0 50 100 150 200 250−30

−20

−10

0

10

20

30

Time (days)

Pay

off

0 50 100 150 200 250

1000

1050

1100

1150

1200

Time (days)

Sto

ck P

rice


Variance Swap (time T > 0)

0 50 100 150 200 250−30

−20

−10

0

10

20

30

Time (days)

Pay

off

0 50 100 150 200 250

1000

1050

1100

1150

1200

Time (days)

Sto

ck P

rice


Variance Swaps Term Structure

I Variance Swaps

I actively traded over-the-counter

I for different time horizons ⇒ term structureI previous slide: only one time horizon of 250 days

I Develop term structure models for variance swaps


Variance Swap Payoff

Variance Swap payoff = Floating leg − Fix leg

= QVt,t+τ −VSt,t+τ

I Floating leg: Quadratic variation over horizon [t, t + τ ]

I Fix leg: Variance Swap rate, fixed at time t

VSt,t+τ = EQ[QVt,t+τ |Ft ]

depend on time horizon τ ⇒ term structure

I Because variance swap is a forward contractEQ[e−r(T−t)(QVt,t+τ −VSt,t+τ )|Ft ] = 0


Variance Swap Term Structure: Data

97 98 99 00 01 02 03 04 05 06 07 08 09 10 236

12

24

10

20

30

40

50

60

70

80

Maturity months

Year

Var

ianc

e S

wap

Rat

e %

Figure: Variance swap rates,√VSt,t+τ × 100, on the S&P500 index

from 4-Jan-1996 to 2-Sep-2010, daily quotes. Source: Bloomberg.


Summary Statistics

Variance Swap ratesτ Mean Std Skew Kurt

2 22.14 8.18 1.53 7.083 22.32 7.81 1.32 6.056 22.87 7.40 1.10 4.97

12 23.44 6.88 0.80 3.7724 23.93 6.48 0.57 2.92

Realized Variances2 18.90 12.40 4.31 28.403 19.06 12.04 3.80 21.816 19.46 11.33 2.93 13.17

12 20.13 10.47 1.97 6.8624 20.60 8.81 1.09 3.48

Table: Daily data from 4-Jan-1996 to 2-Sep-2010, volatility percentageunits. Note: Variance risk premium tends to increase with time horizon τ .


Outline




References


Modelling Variance Swaps: SV Approach

I From stochastic volatility (SV) literature

I Specify price index dynamics (judiciously)

dSt

St−= rt dt + σt dWt +

∫Rξ (χ(dt, dξ)− νt(dξ)dt)

where σt diffusive volatility, χ jump measure, νt compensator

I Quadratic variation over time horizon [t, t + τ ]

QVt,t+τ =1

τ

(∫ t+τ

tσ2

s ds +

∫ t+τ

t

∫R

(log(1 + ξ))2χ(ds, dξ)

)

I Variance swap rate (possibly in closed-form)

VSt,t+τ = EQ [QVt,t+τ |Ft

]Volatility Derivatives 50/69

Modelling Variance Swaps: TS Approach

I From term structure (TS) literature on bond yields

I Observed term structure: Bond yield curve

y(t,T ) = − 1

T − tlogEQ

[exp

(−∫ T

trs ds

)|Ft

]⇒ Spot interest rate rt = y(t, t), often affine in latent factors

I Observed term structure: Variance swap curve

VS(t,T ) =1

T − tEQ[∫ T

tvs ds|Ft

]⇒ Spot variance vt = VS(t, t), flexible function of latent factors

I Link to SV approach: vt = σ2t +

∫R(log(1 + ξ))2νt(dξ)


Quadratic Variance Swap Model

I Spot variance vt quadratic in Xt (subsumes affine models)

vt = φ+ ψ>Xt + X>t πXt

from Filipovic, Gourier, and Mancini (2016)

I State process Xt is m-dimensional quadratic diffusion:

dXt = µ(Xt)dt + Σ(Xt)dWt

where µ(x) linear in x , and Σ(x)Σ(x)′ quadratic in x

I Quadratic term structure model for variance swap

VS(t,T ) = Φ(T − t) + Ψ(T − t)>Xt + X>t Π(T − t)Xt

where Φ, Ψ and Π satisfy a linear system of ODEsVolatility Derivatives 52/69

Model Estimation

I State process Xt is not observable: Require filtering

I Maximum likelihood with (unscented) Kalman filter

I Quadratic model in state-space form:

Xt = Φ0 + ΦX Xt−1 + ωt , ωt ∼ N (0,Qt) (Transition)

Zt = H(Xt) + ut , ut ∼ N (0,Ωt) (Measurement)

I Daily observations Zt = [VSt,T1 , . . . ,VSt,T5 ]′, t = 1, . . . ,T

I Measurement equation nonlinear (quadratic)I Standard Kalman filter not applicable


Unscented Kalman Filter (UKF)KF: If measurement equation were linear H(X ) = H0 + H1 X

I Prediction step of Xt and Zt :

Xt|t−1 = Φ0 + ΦX Xt−1, with Pt|t−1 = Cov[Xt − Xt|t−1]

Zt|t−1 = H0 + H1Xt|t−1, with Ft|t−1 = Cov[Zt − Zt|t−1]

I Update step of Xt :

Xt = Xt|t−1 + Wt(Zt − Zt|t−1), with Pt = Cov[Xt − Xt ]

Kalman gain Wt such that ∂trace(Pt)/∂Wt = 0

UKF: Because measurement equation is nonlinear H(X )

Approximate p.d.f. of X with “sigma points” χ(i)t|t−1, ωi

I Zt|t−1 =∑

i ωi H(χ(i)t|t−1), and Ft|t−1, using “sigma points”

I Log-likelihood based on (unscented) Kalman filter

T∑t=1

−1

2

[5 log(2π) + log |Ft|t−1|+ (Zt − Zt|t−1)′F−1

t|t−1(Zt − Zt|t−1)]


Filtered Trajectory from Bivariate Quadratic Model

1998 2000 2002 2004 2006 2008 2010−1

0

1

2

3

4

5

6

X1−(λ0+β12X2)

λ1+β11

Figure: Filtered trajectory of bivariate state vector Xt . Quadratic modelfitted to variance swap rates on the S&P500.

I X1t more volatile, mimics trajectories of short-term VS rates

I X2t more persistent, mimics long-term movements in VS rates


Fitting Variance Swap Rates

1998 2000 2002 2004 2006 2008 201010

20

30

40

50

60

70

80T = 2 months

Model−based VS ratesActual VS rates

1998 2000 2002 2004 2006 2008 201010

15

20

25

30

35

40

45

50T = 24 months

Model−based VS ratesActual VS rates

Figure: Model-based and actual (i.e., observed) variance swap rates onthe S&P500. Bivariate quadratic model fitted to variance swap rates.


Outline




References


Optimal Portfolio Problem

Maximize expected utility from terminal wealth VT of a powerutility investor with constant relative risk aversion (CRRA) η

maxnt ,wt ,φt ,0≤t≤T

EP

[V 1−η

T

1− η

]

By dynamically and optimally investing:

I nt = (n1t , n2t) relative notional exposures to each on-the-runτi -variance swap, i = 1, 2

I wt fraction of wealth invested in stock index

I φt fraction of wealth invested in index option

I and risk-free bond

Solution and implementation via HJB equation (a lot of work)

I Filipovic, Gourier, and Mancini (2016)Volatility Derivatives 58/69

Optimal Investment in VS: Short-Long Strategy

1998 2000 2002 2004 2006 2008 2010−5

−4

−3

−2

−1

0

1

2

n

1t

n2t

I Short position in 2-year VS (earn variance risk premium),long position in 3-month VS (hedge volatility risk)

I Periodic patterns in nt

I Based on bivariate quadratic model


Optimal Investment in Stock Index and Put Option

1998 2000 2002 2004 2006 2008 20100.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

wt

1998 2000 2002 2004 2006 2008 2010−5

0

5

10

15

20x 10

−3

φ t

I Positive optimal weight wt in stock index

I Positive, tiny optimal weight φt in put optionVolatility Derivatives 60/69

Wealth Trajectory with Optimal Investment

1998 2000 2002 2004 2006 2008 201050

100

150

200

250

300

Optimal portfolioProxy portfolioS&P500

I Smooth wealth growth with little volatilityI Merely a diversification effect? Not really. . .

I Suited for risk-averse investors (CRRA η = 5)


Wealth Trajectory with Optimal Investment: Log-investor

1998 2000 2002 2004 2006 2008 201050

100

150

200

250

300

350

400

Optimal portfolioProxy portfolioS&P500

I Wealth fluctuations larger than S&P500 indexI To seek risk premiums (increased exposures to risk factors)

I Suited for less risk-averse investors (CRRA η = 1)


Outline




References


VIX Index and Variance Swap RateI VIX: CBOE 30 days (“fear”) volatility index, highly watched

I Assuming no jumps in the index price S , using theQ-dynamic of the forward price1 Fs = EQ[ST |Fs ]

dFs/Fs =√

vs dW Qs

d log Fs = −1

2vs ds +

√vsdW Q

s

1

T − t

∫ T

tvs ds = − 2

T − tlog

FT

Ft+

2

T − t

∫ T

tdFs/Fs

1

T − tEQ[∫ T

tvs ds|Ft

]=

2

T − tEQ[− log

FT

Ft|Ft

]= VIX 2

t

I VIX 2t is (T − t) days VS rate when S does not jump

1And using the identity

− logFT

Ft= 1− FT

Ft+

∫ Ft

0

(K − FT )+

K 2dK +

∫ ∞

Ft

(FT − K)+

K 2dK


Difference between VS rates and VIX-type indices

97 98 99 00 01 02 03 04 05 06 07 08 09 10−5

0

5

10

15

Year

VS

min

us V

IX %

2−month3−month6−month

Figure: VSt,t+τ −VIXt,t+τ for τ = 2, 3, 6 months, daily. VIX-typeindices calculated using SPX options and revised VIX methodology fromCBOE. Volatility percentage units. Note: VSt,t+τ −VIXt,t+τ can beinterpreted as profit/loss from selling VS and hedging with SPX optionsand futures, in frictionless markets, when index price S jumps.


Wall Street Journal, 22 October 2008


Outline




References

References 67/69

ReferencesI Andersen, Fusari, and Todorov (2015), “The Risk Premia Embedded in Index

Options,” Journal of Financial Economics, 117, 558–584.

I Barone-Adesi, Engle, and Mancini (2008), “A GARCH Option Pricing Modelwith Filtered Historical Simulation,” Review of Financial Studies, 21, 1223–1258.

I Badescu, Cui, and Ortega (2016), “Non-Affine GARCH Option Pricing Models,Variance Dependent Kernels, and Diffusion Limits,” working paper, SSRN.

I Bauwens, Laurent, and Rombouts (2006), “Multivariate GARCH Models: ASurvey,” Journal of Applied Econometrics, 21, 79–109.

I Bollerslev (1986), “Generalized Autoregressive Conditional Heteroskedasticity,”Journal of Econometrics, 31, 307–327.

I Bollerslev (1987), “A conditionally Heteroskedastic Time Series Model forSpeculative Prices and Rates of Return,” Review of Economics and Statistics,69, 542–547.

I Christoffersen, Jacobs, Ornthanalai, and Wang (2008), “Option Valuation withLong-Run and Short-Run Volatility Components,” Journal of FinancialEconomics, 90, 272–297.

I Duan (1995), “The GARCH Option Pricing Model,” Mathematical Finance, 5,13–32.

I Duffie, Pan, and Singleton (2000), “Transform Analysis and Asset Pricing forAffine-jump Diffusions,” Econometrica, 68, 1343–1376.

I Engle (1982), “Autoregressive Conditional Heteroscedasticity with Estimates ofthe Variance of United Kingdom Inflation,” Econometrica, 50, 987–1007.References 68/69

References (Continued)

I Engle and Gonzalez-Rivera (1991), “Semiparametric ARCH models,” Journal ofBusiness and Economic Statistics, 9, 345–359.

I Fang and Oosterlee (2008), “A Novel Pricing Method for European OptionsBased on Fourier-Cosine Series Expansions,” SIAM Journal on ScientificComputing, 31, 826–848.

I Filipovic, Gourier, and Mancini (2016), “Quadratic Variance Swap Models,”Journal of Financial Economics, 119, 44–68.

I Glosten, Jagannathan, and Runkle (1993), “On the Relation Between theExpected Value and the Volatility of the Nominal Excess Returns on Stocks,”Journal of Finance, 48, 1779–1801.

I Gruber, Tebaldi, and Trojani (2015), “The Price of the Smile and Variance RiskPremia,” working paper, SSRN.

I Heston (1993), “A closed-form solution for options with stochastic volatility,with applications to bond and currency options,” Review of Financial Studies, 6,327–343.

I Heston and Nandi (2000), “A Closed-form GARCH Option Valuation Model,”Review of Financial Studies, 13, 585–625.

I Majewski, Bormetti, and Corsi (2015), “Smile From The Past: A GeneralOption Pricing Framework with Multiple Volatility and Leverage Components,”Journal of Econometrics, 187, 521–531.

References 69/69

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