Presentation PGR9 (2)

Three Essays about Options on Leveraged ExchangeTraded Funds

Yahua Xu

Auckland University of Technology

[email protected]

July 1, 2015

Yahua Xu (AUT) PGR9 July 1, 2015 1 / 53

Background Knowledge

Exchange Traded Fund (ETF): An ETF is an investment fundtraded on stock exchanges. It has advantages such as low costs, taxefficiency and stock-like features.

Leveraged ETF (LETF): A LETF is a special type of ETF anddesigned to track a multiple daily return of a target asset.

LETFs tracking two important indices:S&P 500 Index.VIX Short Term Futures Index.



Option: A call (put) option gives the holder the right to buy (sell) anunderlying asset at a specified strike price on a specified date:

Payoff of a Call Option: max{st k , 0}.Payoff of a Put Option: max{k st , 0}.

LETF Option: The underlying asset is a LETF.

LETF Option Market:The option on equity LETF was launched in 2011.The option on volatility LETF was launched in 2012.

In my research, I am going to analyze dynamics of the two specified LETFs

and price options written on them.



It is from a regulatory notice about the LETFs of the Financial IndustryRegulatory Authority (FIRA) published in June 2009:

While such products may be useful in some sophisticated tradingstrategies, they are highly complex financial instruments that aretypically designed to achieve their stated objectives on a daily basis.Due to the effects of compounding, their performance over longerperiods of time can differ significantly from their stated dailyobjective.


Analyzing Objectives

S&P 500 Index

Options

VIX Index

LETFs

UPRO(+3)

Options

SSO(+2)

Options

SPY(+1)

Options

SH(1)

Options

SDS(2)

Options

SPXU(3)

Options


Analyzing Objectives

VIX Index

VIX Futures

VIX Short Term Futures Index

LETFs

UVXY(+2)

Options

VIXY(+1)

Options

SVXY(1)

Options


Equity LETF Prices

Figure: Equity LETF Prices

0

10

20

30

40

50

60

70

LETFs Price

SPXU (-3) SDS (-2) SH (-1) SPY (+1) SSO (+2) UPRO (+3)

Note. Times series for equity LETF from 2011/06/01 to 2012/06/01.


Research Questions (RQs)

RQ1

How is the empirical performance of the Heston model in regard ofpricing options on equity LETFs?

RQ2

What is the proper dynamic model for volatility LETFs? How is theempirical performance of the model in regard of pricing options onvolatility LETFs?

RQ3

What are the option pricing formulas when jump risk is added intostochastic volatility framework?


Thesis Structure

Stochastic Volatility Framework

Chapter2

Logarithmic Model with StochasticVolatility (LRSV) for Volatility LETFs

Proposing LRSV for the volatilityLETFs;

Empirical analysis about theperformance of LRSV

Chapter1

Heston Model for Equity LETFs

Empirical analysis about theperformance of Heston


Thesis Structure

Stochastic Volatility + Jump Framework

Chapter 3

LRSV + Jump (Volatility LETFs)

LRSV +Stochastic Jump

LRSV +Constant Jump

Heston + Jump (Equity LETFs)

Heston +Stochastic Jump

Heston +Constant Jump


Volatility Skew

Volatility Skew/Smile is the phenomena of options impliedvolatilities varying with moneyness.

It is one of the most stylized facts observed in the option market:

Equity options exhibit negative volatility skew.Volatility options exhibit positive volatility skew.

The stochastic volatility models are capable of capturing the volatilityskew observed in option market, as documented in the literature suchas Heston(1993) and Bates(1996).


Volatility Skew

Figure: Negative Volatility Skew for Equity Option

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

SPY(+1)

Note. Smile for the maturity 0.14 for the LETF SPY(+1) for the day 2011/10/24.


Volatility Skew

Figure: Positive Skew for Volatility Option

0.00

0.50

1.00

1.50

2.00

2.50

VIXY(+1)

Note. Smile for the maturity 0.14 for the LETF VIXY(+1) for the day 2011/10/24.


Chapter I

Chapter I:

Options on Equity LETFs, Calibration and Error

Analysis


Chapter I: Literature Review

Avellaneda and Zhang (2010) is the first to study the dynamics of theequity LETF returns from a theoretical perspective:

The equity LETFs fail to reproduce the multiple return of theunderlying over a long period of time.

The long-run return of a LETF is path-dependent.

They proposed the Heston model for the dynamics of equity LETFs.



The other studies such as Ahn et al. (2012) and Deng et al. (2014) alsoemploy the Heston model for pricing options on the equity LETFs:

Zhang (2010) only utilizes one days underlying ETF option data tocalibrate the Heston model. The information of model fitting is onlyshowed by figures.

Ahn et al. (2012) only utilizes the option data of June, 14, 2013. Inthe calibration, they fix the correlation coefficient without givingthe reason.

Deng et al. (2014) also only uses one days option data, withoutreporting any pricing error.



How to systematically evaluate the performance of an option pricingmodel?

The empirical study of Bakshi et al. (1997) on the performance ofseveral S&P 500 option pricing models provides a good reference:

The daily calibration is rolled based on a set of time series data.Estimated parameters and pricing errors are both provided.


Chapter I: Model and Option Pricing

Assume the price of the underlying asset st follows the Heston model:

dstst

= (r q)dt +vtd1t ,dvt = ( vt)dt + vtd2t ,

with initial value s0 known.

t =(1t ,

2t

)t0 is a Brownian motion with correlation coefficient .



For a LETF with leverage ratio m, the price lt is related to st as follows:

dltlt

= mdstst

+ (1m)rdt,dvt = ( vt)dt + vtd2t .

Moreover, the log return of lt can be expressed in terms of st , vt and m:

d ln lt = md ln st m2 m

2vtdt

volatility bias

+(1m)rdt,

When |m| > 1, the volatility bias is negative.



The characteristic function of ln lt can be written as:

X (z) = E[eiz ln lt

l0

]= E

[eiz(m ln st

s0m2m

2

t0 vudu+(1m)rt)

]= e i(1m)zrtE

[eizm ln st

s0iz m2m

2

t0 vudu

]= e i(1m)zrt

(t, zm, z

[m

2 m2

])= e i(1m)zrt+A(t)+B(t)v0+izmx .

where the expressions of A(t) and B(t) can be computed and their valuesare based on the values of z , m, r , q, , , , .



The option pricing formula can be obtained by inverting the characteristicfunction via Fourier Transform:

c(t, k,m) =kert

pi

0

Re

[X (z)e

iz ln k 1iz(iz 1)

]dzR ,


Chapter I: Data

We have a rich dataset which contains prices of options on a sextet ofequity LETFs. The dataset spans from March 24, 2011 to February 16,2015. The equity LETFs offered by Proshares are as follows:

Table 1: Equity LETFs

Fund Name Ticker Name Leverage Ratio

UltraPro Short S&P 500 ETF SPXU -3UltraShort S&P 500 ETF SDS -2Short S&P 500 ETF SH -1SPDR S&P 500 ETF SPY +1Ultra S&P 500 ETF SSO +2UltraPro S&P 500 ETF UPRO +3


Chapter I: Calibration

We adopt the norm-in-price criterion. At day t, we separately calibrateoptions on the LETFs with different leverage ratios. The objectivefunction is:

minvt ,,,,

1

Nj

i

(Vmarket(ti ,Ki ,mj) Vmodel(ti ,Ki ,mj)

)2,



For a given day t:

If the estimated parameters are the same, the model is good!

If different, analyze the distribution of pricing errors corresponding todifferent leverage ratios. It is quite likely the larger the absolute valueof the leverage, the greater the pricing errors are.



A rolling calibration using several days option data:

We make a comparison of parameters (, , , ) obtained from eachdays calibration to see if they are stable over time.

If the estimated parameters are stable over time, then the pricingmodel is a robust.

If not, the jump risk may be needed to improve the modelperformance.


Chapter I: Contribution

It is the first to provide a systematic and comprehensive analysis forthe fit of the Heston model in pricing options on equity LETFs.


Chapter II

Chapter II:

Options on Volatility LETFs with Positive

Volatility Skew


Chapter II: Options on Volatility LETFs with PositiveVolatility Skew

The underlying index for the volatility LETFs is the VIX Short-TermFutures Index.

The VIX futures index has a close relationship with the VIX index.

The modeling of VIX dynamics can serve as a good reference for thedynamic model of volatility LETFs.


Chapter II: Literature Review

Detemple and Osakwe (2000) proposed a mean-reverting logarithmicprocess for the VIX dynamics.

Sepp (2008) suggests incorporating stochastic volatility into the VIXdynamics to model the feature of positive skew in VIX options.

Kaeck and Alexander (2010) developed a stochastic volatility ofvolatility model for the logarithmic VIX dynamics.


Chapter II: Literature Review

Bao et al. (2012) adopts the model of Kaeck and Alexander (2010)for the dynamics of VXX (an ETN similar to VIXY (+1) tracking thedaily return of VIX Short Term Futures Index):

Bao et al. (2012) only focuses on the dynamics of the unleveragedvolatility ETF.No research about the volatility Leveraged ETFs has been carried out.

Based on Bao et al.(2012), a logarithmic stochastic volatility modelfor the volatility LETFs is proposed, which is the first model of thiskind.


Chapter II: Model and Option Pricing

Assume the spot price of the VIX Short Term Future Index Vt follows thelogarithmic model with stochastic volatility (LRSV):

d lnVt = ( lnVt)dt +vtd1t ,dvt = v (v vt)dt + vvtd2t ,

with initial value V0 known.

t =(1t ,

2t

)t0 is a Brownian motion with correlation coefficient .



For a LETF with leverage ratio m, the return xt is based on the underlyingasset return Vt :

dxtxt

= mdVtVt

+ (1m)rdt,dvt = v (v vt)dt + vvtd2t ,

with the initial value x0 = x known.

Moreover, the log return of xt can be expressed in terms of Vt , vt and m:

d ln xt = md lnVt m2 m

2vtdt

volatility bias

+(1m)rdt,

When |m| > 1, the volatility bias is negative.



The characteristic function of ln xt can be written as:

X (z) = E[eiz ln xt

x0

]= E

[eiz(m ln Vt

V0m2m

2

t0 vudu+(1m)rt)

]= e i(1m)zrtE

[eizm ln Vt

V0iz m2m

2

t0 vudu

]= e i(1m)zrt

(t, zm, z

[m

2 m2

])= eA(t)+B(t)v+C(t)x .

The expressions of A(t), B(t) and C (t) can be computed and they arebased on the values of z , m, r , q, , , , v , v and v .




c(t, k,m) =kert

pi

0

Re

[X (z)e

iz ln k 1iz(iz 1)

]dzR ,


Chapter II: Data

We have a rich dataset which contains prices of options on a set ofvolatility LETFs. The dataset spans from March 19, 2012 to October 31,2014. The volatility LETFs offered by Proshares are as follows:

Table 1: Volatility LETFs

Fund Name Ticker Name Leverage Ratio

Short VIX Short-Term Futures ETF SVXY -1VIX Short-Term Futures ETF VIXY +1Ultra VIX Short-Term Futures ETF UVXY +2


Chapter II: Calibration

The calibration procedure is similar to the previous part. Firstly, thenorm-in-price criterion is adopted. We implement calibration separately foroptions written on different LETFs on the same day t. The objectivefunction is:

minvt ,,,,v ,v ,v

1

Nj

i

(Vmarket(ti ,Ki ,mj) Vmodel(ti ,Ki ,mj)

)2,



For a given day t:

If the estimated parameters are the same, the model is good!

If different, analyze the distribution of pricing errors corresponding todifferent leverage ratios. It is quite likely the larger the absolute valueof the leverage, the greater the pricing errors are.



A rolling calibration using several days option data:

We make a comparison of parameters (vt , , , , v , v , v ) obtainedfrom each days calibration to see if they are stable over time.

If the estimated parameters are stable over time, then the pricingmodel is a robust.

If not, the jump risk may be needed to improve the modelperformance.


Chapter II: Contribution

It is the first model proposed for dynamics of volatility LETFs.

An extended calibration on a time series of option data will be carriedout to assess the models performance.


Chapter III

Chapter III:

Analytical Extensions of Pricing Options on Equity

and Volatility LETFs


Chapter III: Analytical Extensions of Pricing Options onEquity and Volatility LETFs

In the first two chapters, we focus on utilizing the stochastic volatilitymodels addressing the characteristic of volatility skew observed in theLETF option market:

The Heston model is adopted for the dynamics of an equity LETF toincorporate the negative volatility skew.The LRSV model is adopted for the dynamics of a volatility LETF toincorporate the positive volatility skew.


Chapter III: Literature Review

Some research declares that the inclusion of jumps into the stochasticvolatility model may better explain the volatility skew:

Bates (1996) is the first to introduce a jump component into thestochastic volatility model to analyze the Deutsche Mark Optionmarket.Bakshi et al. (1997) conjecture that jumps in volatility may benecessary to fully explain the volatility smile observed in the S&P 500option market.Duffie et al. (2000) also found that the inclusion of jumps improves themodel fitting.Broadie et al. (2007) utilize an extensive data sample of S&P 500futures option. Both time series and cross section test show that jumpin volatility can improve the model fit.


Chapter III: Literature Review

The specification of jumps is furthermore important:

Typical jump models such as Bates (1996) usually assume the jumpintensity is constant.Bates (2000) found that the time-varying jump risk and stochasticvolatility work together to better explain the negative skewness.Eraker (2004) also assumes that the jump arrival intensity isstate-dependent.

However, evidence supporting the specification of jumps is mixed.Therefore, we are going to analyze both cases.


Chapter III: Heston Model Extension with Constant JumpIntensity

The dynamics of the underlying asset price st is specified as follows:

dstst

= (r q)dt +vtd1t ,dvt = ( vt)dt + vtd2t + JdNt ,


1t and 2t has correlation coefficient , Nt is a Poisson Process with

constant intensity . The magnitude J of jumps is an exponential randomvariable, i.e. J exp ( 1J ).


Chapter III: Heston Model Extension with Stochastic JumpIntensity

The dynamics of the underlying asset price st is specified as follows:

dstst

= (r q)dt +vtd1t ,dvt = ( vt)dt + vtd2t + JdNt ,dt = ( t)dt +

td

3t ,


1t and 2t has correlation coefficient , and

1t and

3t has correlation

coefficient 0. Nt is a Poisson Process with stochastic intensity t . Themagnitude J of jumps is an exponential random variable, i.e. J exp ( 1J ).


Chapter III: Heston Model Extension with Stochastic JumpIntensity

We denote by lt the spot price of the LETF. The characteristic function ofln ltl0 can be written as:

X (z) = E[eiz ln lt

l0

]= E

[eiz(m ln st

s0m2m

2

t0 vudu+(1m)rt)

]= e i(1m)zrt

(t, zm, z

[m

2 m2

]).


Chapter III: LRSV Extension with Constant Jump Intensity

We assume the underlying asset price Vt follows the logarithmic modelwith stochastic volatility (LRSV), and is specified as follows:

d lnVt = ( lnVt)dt +vtd1t ,dvt = v (v vt)dt + vvtd2t + JdNt ,


1t and 2t has correlation coefficient . Nt is a Poisson Process with

constant intensity . The magnitude J of jumps is an exponential randomvariable, i.e. J exp ( 1J ).


Chapter III: LRSV Extension with Stochastic JumpIntensity

We assume the price Vt follows the logarithmic model with stochasticvolatility (LRSV), and is specified as follows:

d lnVt = ( lnVt)dt +vtd1t ,dvt = v (v vt)dt + vvtd2t + JdNt ,dt = ( t)dt +

td

3t ,


1t and 2t has correlation coefficient , and

1t and

3t has correlation

coefficient 0. Nt is a Poisson Process with stochastic intensity t . Themagnitude J of jumps is an exponential random variable, i.e. J exp ( 1J ).


Chapter III: LRSV Extension with Stochastic JumpIntensity

We denote by lt the spot price of the LETF. The characteristic function ofln ltl0 can be written as:

X (z) = E[eiz ln lt

l0

]= E

[eiz(m ln st

s0m2m

2

t0 vudu+(1m)rt)

]= e i(1m)zrt

(t, zm, z

[m

2 m2

]).


Chapter III: Option Pricing


c(t, k,m) =kert

pi

0

Re

[X (z)e

iz ln k 1iz(iz 1)

]dzR ,


Chapter III: Sensitivity Analysis

We will carry out sensitivity analysis for the models to see the impactof the parameters.


Chapter III: Contribution

It is the first to introduce jump risk to the stochastic volatilityframework for the dynamics of equity and volatility LETFs.


Thank You


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