The Market for Volatility Trading - Google Sites

The Market for Volatility Trading Jin Zhang Dept of Accountancy and Finance University of Otago Dunedin 9054, New Zealand

FINC405 Mathematical Finance

Jin Zhang (Otago)

The Market for Volatility Trading

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Introduction

Three major risk factors traded in financial markets Index, interest rate and volatility 1

Index/market risk is traded in stock market

2

Interest rate risk is traded in bond and interest derivatives market

3

Volatility risk is indirectly traded in options market

Options market also trades index risk. The interest rate risk could be a risk factor in the price of long-term options Volatility risk in an option portfolio is often contaminated by index risk and sometimes by interest rate risk Developing a financial market that trades volatility risk ONLY has been a central concern for both researchers and practitioners

Jin Zhang (Otago)


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History

Historical development of volatility market In 1977, Gastineau first created a volatility index based on option market prices. In 1979, Galai proposed indexes for traded options. In 1989, Brenner & Galai suggested again the idea of developing a volatility index. In 1993, Brenner & Galai introduced a volatility index based on implied volatilities from at-the-money (ATM) options. In 1993, the Chicago Board Options Exchange (CBOE) introduced a volatility index, named VIX. The VIX is computed from the implied volatilities of the eight near-the-money, nearby, and second nearby S&P 100 index (OEX) options based on Whaley’s (1993) design. It is a proxy of the implied volatility of 30-calendar-day ATM options. In April 1993, Reuters began reporting the VIX index. Jin Zhang (Otago)


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History

Historical development of volatility market (Cnt’d) In December 1994, the German Futures and Options Exchange (DTB) launched a volatility index called VDAX which tracks the three-month implied volatility of DAX index calls and puts. The DAX index is a value-weighted index of the 30 largest firms traded on the Frankfurt Stock Exchange. In 1995, the Austrian Futures and Options Exchange (OTOB) announced a volatility index on its Austrian Traded Index (ATX) calls and puts. In 1996, Volatility swaps began to trade in the over-the-counter (OTC) market.

Jin Zhang (Otago)


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History

Historical development of volatility market (Cnt’d) In 1996, the Wall Street Journal reported that the CBOE plans to unveil options on the VIX index shortly, but it never happened. In 1996, an issue of Futures reported that the American Stock Exchange was also considering developing volatility options on the U.S. stock market and that market regulators had privately endorsed the concept. In January 1997, the London based subsidiary of the Swedish exchange (OMLX) launched volatility futures. The trading volume was very low. In October 1997, the French exchange (MONEP) started publishing VX1/VX6, which represents the ATM implied volatility of the CAC 40 options with 31/185 calendar days to maturity. Jin Zhang (Otago)


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History

Historical development of volatility market (Cnt’d) The failure of VOLAX futures In January 1998, the German Futures and Options Exchange (DTB) launched the VOLAX future as a futures contract on the three-month implied volatility of an at-the-money DAX option. The underlying instrument for the new contract is a weighted average of the VDAX volatility subindices which are calculated by Deutsche Boerse AG since July 1997. It failed to attract significant volume. The peak-volume occurred in the second month of trading and was followed by a strong decline in volume reaching zero in September 1998 (Herrmann and Luedecke 2002).

Jin Zhang (Otago)


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History

Historical development of volatility market (Cnt’d)

In late 1998, variance swaps became very popular in the aftermath of the Long Term Capital Management meltdown when implied stock index volatility levels rose to unprecedented levels. Hedge funds took advantage of this by selling the realized volatility at high implied levels. In 1998 - 1999, Carr and Madan (1998), and Derman’s (1999) Quantitative Strategy Group in Goldman Sachs found independently a formula to determine the fair value of variance swap rate.

Jin Zhang (Otago)


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History

Historical development of volatility market (Cnt’d) In September 2003, the CBOE adopted Carr-Madan/Derman’s theory to design a new methodology to calculate the CBOE volatility index, VIX. The new VIX is based on the prices of a portfolio of 30-calendar-day out-of-the-money (OTM) S&P 500 (SPX) index call and put options. The new VIX squared represents the SPX 30-day variance swap rate. The old VIX has been renamed to be VXO. On March 26, 2004, the CBOE Futures Exchange (CFE) started trading the VIX futures, the first-ever volatility futures traded in the US. On May 18, 2004, the CFE listed the SPX three-month variance futures. On February 24, 2006, the CBOE listed the VIX options. On July 1, 2008, the CBOE listed the VIX binary options. Jin Zhang (Otago)


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History

Historical development of volatility market (Cnt’d) On January 29, 2009, S&P 500 VIX Short Term Futures Index (SPVXSTR) was created. It utilizes prices of the next two near-term VIX futures contracts to replicate a position that rolls the nearest month VIX futures to the next month on a daily basis in equal fractional amounts. This results in a constant one-month rolling long position in first and second month VIX futures contracts. Barclays Bank PLC’s iPath issued an exchang-traded note (ETN), VXX that tracks the SPVXSTR index. On January 30, 2009, VXX = 6693.12, Trading volume = 3,300 On April 14, 2014, VXX = 44.38, Trading volume = 42,727,000 More than 30 VIX exchange-traded products (ETPs) are now listed with an aggregate market investment value of nearly $4 billion, generating a daily trading volume in excess of $800 million. On May 28, 2010, the CBOE started offering VXX options. Jin Zhang (Otago)


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History

Historical development of volatility market (Cnt’d) Many volatility indexes have been launched by exchanges: Volatility indexes on international equity indexes: VIX, VXV (3-month) and VXST (9-day) on S&P 500, VXD on DJIA, VXN on NASDAQ-100, RVX on Russell 2000, VDAX on DAX 30, VSTOXX on Dow Jones EURO STOXX 50, VSMI on SMI, Volatility Indexes on AEX, BEl 20 and CAC 40, VFTSE on FTSE 100, MVX on XIU, SIVX on TOP40, AVIX on S&P/ASX 200, VHSI on HSI CBOE Equity Volatility Indexes: VXAZN, VXAPL, VXGS, VXGOG, VXIBM CBOE Volatility Indexes on ETPs: EVZ on CurrencyShares Euro Trust, GVZ on SPDR Gold Shares, OVX on US Oil Fund, VXEEM, VXSLV, VXFXI, VXGDX, VXEWZ, VXXLE, VXEFA CBOE Interest Rate Swap Rate Volatility Index: SRVX CBOE/CBOT 10-year U.S. Treasury Note Volatility Index: VXTYN Jin Zhang (Otago)


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OTC Variance Swaps

Over-the-counter variance swap market Volatility/variance has become an asset class in its own right. In late 1990s, Wall street firms started trading variance swaps, forward contracts written on the realized variance. These swaps are now the preferred route for many hedge fund managers and proprietary traders to make bets on market volatility. According to some estimates, the daily trading volume in equity index variance swaps reached USD 4-5 million vega notional in 2006. On an annual basis, this corresponds to payments of more than USD 1 billion, per percentage point of volatility (Carr and Lee 2007).

Jin Zhang (Otago)


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OTC Variance Swaps

Variance swap pricing Variance swaps are forward contracts written on realized variance. On maturity date, T , the party with a long variance position will have to pay a fixed amount, V0 that is called variance swap rate, in order to receive the realized variance, Vrealized , between current time, t = 0, and maturity date T . The net cash flow to the long party is Vrealized − V0 .

(1)

Since there is no cost entering into forward contracts, the value of the variance swaps at the time of inception, t = 0, must be zero. Then the variance swap rate can be determined by V0 = E0Q [Vrealized ]. Jin Zhang (Otago)

(2)


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OTC Variance Swaps

Variance swap pricing theory Suppose equity index level is modeled by dSt = µdt + σt dBt , St

or

1 d(ln St ) = (µ − σt2 )dt + σt dBt , 2

where the volatility, σt , follows a general stochastic process. This implies dSt 1 − d(ln St ) = σt2 dt. St 2 Integrating this equation gives the realized variance Z T Z 1 T 2 2 dSt ST Vrealized = σ dt = − ln . T 0 t T St S0 0

(3)

Combining equation (2) and (3) gives the variance swap rate as follows Z T dSt 2 Q ST 2 F0 ST Q V0 = E0 − ln = ln − E0 ln , (4) T St S0 T S0 S0 0 where F0 = S0 e (r −q)T is the index forward price. Jin Zhang (Otago)


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OTC Variance Swaps

Carr-Madan (1998) and Derman (1999) variance swap pricing theory The issue becomes pricing a derivative with payoff ln SST0 , a log contract. A general result from Calculus: for a twice differentiable function, f (ST ), we have following decomposition formula Z S∗ ∗ 0 ∗ ∗ f 00 (K ) max(K − ST , 0)dK f (ST ) = f (S ) + f (S )(ST − S ) + 0 Z +∞ + f 00 (K ) max(ST − K , 0)dK S∗

Applying this general result to a log contract gives ST S∗ ST ln = ln + ln ∗ S0 S0 S Z S∗ ∗ S ST − S ∗ 1 = ln + − max(K − ST , 0)dK ∗ S0 S K2 0 Z +∞ 1 − max(ST − K , 0)dK , K2 S∗ Jin Zhang (Otago)


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OTC Variance Swaps

Carr-Madan/Derman variance swap pricing theory (Cnt’d) Then the expected value of a log contract is ST S ∗ F0 − S ∗ E0Q ln = ln + S0 S0 S∗ ! Z S∗ Z +∞ 1 1 −e rT p0 (K )dK − c0 (K )dK , K2 K2 0 S∗ where c0 /p0 is European call/put price. The variance swap rate is 2 F0 ST 2 F0 F0 Q V0 = ln − E0 ln = ln ∗ − −1 T S0 S0 T S S∗ !# Z +∞ Z S∗ 1 1 rT +e p0 (K )dK + c0 (K )dK . K2 K2 0 S∗ ∗

If S = F0 ,

2 V0 = e rT T

Z 0

F0

1 p0 (K )dK + K2

Z

+∞

F0

(5)

1 c0 (K )dK . K2

Variance swap rate can be determined by all the OTM call and put prices. Jin Zhang (Otago)


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VIX and Listed Volatility Products

The CBOE Volatility Index, VIX In September 2003, the CBOE adopted Carr-Madan/Derman’s theory to design a new methodology to compute VIX. The new VIX is computed from the option quotes of all available OTM calls and puts on the S&P 500 (SPX) with a non-zero bid price using following formula VIX = 100 × σ,

2 1 F 2 X ∆Ki RT e Q(Ki ) − − 1 , (6) σ = T T K0 Ki2 2

i

T is 30 days, F is the implied forward index level derived from the nearest to the money option prices by using put-call parity Ki is the strike price of ith OTM options, ∆Ki is the interval between two strikes, K0 is the first strike that is below the forward index level Q(Ki ) is the midpoint of the bid-ask spread of each option with strike Ki VIX 2 represents the S&P 500 30-day variance swap rate Jin Zhang (Otago)


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The daily closing level of VIX and S&P 500 Indexes

The historical highest closing level was 80.86 on 20 November 2008. The intraday highest level was 89.53 on 24 October 2008. Jin Zhang (Otago)


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Exchange-listed volatility/variance products Table 1 Brief Summary of Exchange-listed Volatility/Variance Products There are twelve volatility/variance derivative products listed in the CBOE Futures Exchange (CFE) and the Eurex. In Panel A, we provide information on the CFE volatility futures; in Panel B, we provide the information on the CBOE volatility options. In Panel C, we provide information on the Eurex volatility futures. The information on trade volume and open interest is the number of contracts on a randomly-chosen date: November 21, 2007. Products Trade Volume Panel A: The CFE volatility/variance futures Nasdaq-100 Volatility Index (VXN) 5 Russell 2000 Volatility Index (RVX) 50 Volatility Index (VIX) 3,303 DJIA Volatility Index (VXD) 416 S&P 500 Three-Month Variance 76 S&P 500 Twelve-Month Variance 0 Panel B: The CBOE volatility options Nasdaq-100 volatility index (VXN) Russell 2000 volatility index (RVX) Volatility Index (VIX) Panel C: The Eurex volatility futures R Futures (FVDX) VDAX-NEW R Futures (FVSM) VSMI R Futures (FVSX) VSTOXX

Jin Zhang (Otago)

Open Interest

Listing Date

55 2,467 88,319 961 2,052 60

Jul. 6, 2007 Jul. 6, 2007 Mar. 26, 2004 Apr. 25, 2005 May 18, 2004 Mar. 23, 2006

50 220 87,349

2,156 11,543 2,240,946

Sep. 27, 2007 Sep. 27, 2007 Feb. 24, 2006

0 0 30

0 0 0

Sep. 19, 2005 Sep. 19, 2005 Sep. 19, 2005


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Market size of volatility futures and VIX options CBOE launched VIX options on Feb 24, 2006, Friday. On June 13, 2006, Tuesday, SPX = 1223.69, VIX = 23.81 Products 12-month Variance futures 3-month Variance futures DJ Volatility Index futures VIX futures VIX options SPX options

Trade Volume 0 28 502 2,012 19,485 1,040,141

Open Interest 210 879 1031 37,301 827,398 8,095,700

Market size VIX futures: 23.81 × 10 × 100 × 37, 301 = 888, 136, 810 USD VIX options: 23.81 × 100 × 827, 398 = 1, 970, 034, 638 USD SPX options: 1223.69 × 100 × 8, 095, 700 = 9.9 × 1011 USD Jin Zhang (Otago)


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Market size of volatility futures and VIX options (Cnt’d) On December 11, 2008, Thursday, SPX = 873.59, VIX = 55.78 Products 12-month Variance futures 3-month Variance futures DJ Volatility Index futures VIX futures VIX options SPX options

Trade Volume 0 0 10 3,182 87,609 736,490

Open Interest 55 78 105 27,287 1,052,427 18,720,920

Market size VIX futures: 55.78 × 10 × 100 × 27, 287 = 1, 522, 068, 860 USD VIX options: 55.78 × 100 × 1, 052, 427 = 5, 870, 437, 806 USD SPX options: 873.59 × 100 × 18, 720, 920 = 1.64 × 1012 USD Jin Zhang (Otago)


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Average daily trading volume of VIX futures

Jin Zhang (Otago)


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Average daily trading volume of VIX options

Jin Zhang (Otago)


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Literature

VIX futures and option, SPX Variance futures Zhang and Zhu (2006, JFuM): Modeling VIX and VIX futures by using Heston (1993) model for the instantaneous variance, testing VIX futures pricing model by using market data on March 1, 2005 Zhu and Zhang (2007, IJTAF): An enhanced version of Zhang and Zhu (2006) by allowing long-term mean level to be time dependent, testing VIX futures pricing model by using market data on March 10, 2005 Lin (2007, JFuM): Affine jump-diffusion model with jumps in index and volatility, study VIX futures prices between April 21, 2004 to April 18, 2006. Sepp (2008ab, Risk, JCF): Affine jump-diffusion model with jump in volatility, study VIX futures and VIX option pricing model, estimate model by using daily VIX levels from February 28, 2003 to February 29, 2008 Zhang and Huang (2010, JFuM): Modeling variance futures by using Heston (1993) model, study SPX three-month variance futures prices and variance risk premium between May 18, 2004 and August 17, 2007 Zhang, Shu and Brenner (2010, JFuM): Detailed version of Zhang and Zhu (2006), approximate analytical VIX futures price formula, study VIX futures market prices between March 26, 2004 and February 13, 2009 Jin Zhang (Otago)


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Literature

VIX futures and option Lu and Zhu (2010, JFuM) pricing VIX futures with three variance factors Lin and Chang (2009, JFuM) and (2010, JEDC) pricing VIX futures and option by modeling SPX and instantaneous volatility in an affine jump-diffusion framework Dupoyet, Daigler and Chen (2011, JFuM) pricing VIX futures by modeling VIX with a CEV jump-diffusion process Wang and Daigler (2011, JFuM) compare performance of four VIX option pricing models. Cheng et al (2012) show that Lin and Chang’s formula is not an exact solution as claimed Zhu and Lian (2012, JFuM) pricing VIX futures in Lin and Chang’s setup Lian and Zhu (2013, DEF) pricing VIX options in Lin and Chang’s setup Mencia and Sentana (2013, JFE) pricing VIX futures and options by modeling VIX directly Lin (2013, JBF) pricing VIX option by modeling the ratio between forward VIX squared and VIX futures directly. Jin Zhang (Otago)


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Literature

Other research on volatility derivatives market Albanese, Lo and Mijatovic (2009, QF) spectral methods for volatility derivatives Psychoyios, Dotsis and Markellos (2010, RQFA) a jump diffusion model for VIX volatility options and futures Hilal, Poon and Tawn (2011, JBF) Conditional heteroskedasticity and tail dependence in S&P500 and VIX Konstantinidi and Skiadopoulos (2011, IJoF) predicability of VIX futures price Chen, Chung and Ho (2011, JBF) diversification effects of volatility related assets Chung et al (2012, JFuM) the information content of SPX and VIX options Branger and Volkert (2012, wp) consistent pricing of VIX derivatives Song and Xiu (2012, wp) state-price densities implied from SPX and VIX option prices Luo and Zhang (2012, JFuM) term structure of VIX Cont and Kokholm (2013, MF) consistent pricing model for index options and volatility derivatives Huskaj and Nossman (2013, JFuM) a term structure model for VIX futures Jin Zhang (Otago)


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Literature

VXX ETN and VXX option

Bao, Li and Gong (2012, EJOR) pricing VXX option by modeling VXX directly Whaley (2013, JPM) document the large negative return of VXX ETN. Eraker and Wu (2017, JFE) develop an equilibrium model for VIX futures price and study VXX return

Jin Zhang (Otago)


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Literature

Other VIX related research Carr and Wu (2006, JoD): Propose a theory to calculate an upper bound for VIX futures price, and the variance of the VIX at future time by using current prices of European options with different strikes Dotsis, Psychoyios and Skiadopoulos (2007, JBF): Study the jump diffusion models for the volatility indices, such as VIX, VXO, VXD, VDAX and VX1/VX6 Jiang and Tian (2007, JoD): Examine the robustness of the CBOE procedure in calculating the new VIX Konstantinidi, Skiadopoulos and Tzagkaraki (2008, JBF): Address the question whether the evolution of implied volatility can be forecasted by studying a number of European and US implied volatility indices Carr and Wu (2009, RFS): Study the variance risk premiums of individual stock options market Duan and Yeh (2010, JEDC): Develop an estimation method for extracting the latent stochastic volatility from VIX Badshah, Frijns and Tourani-Rad (2013, JFuM): Contemporaneous spill-over among equity, gold, and exchange rate implied volatility indices Jin Zhang (Otago)


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Literature

VIX research for local market

Dowling and Muthuswamy (2005, RFuM): Introduce the Australian Volatility Index (AVIX) by using S&P/ASX200 index options Frijns, Tourani-Rad and Zhang (2008, NZ Economic Papers): The New Zealand implied volatility index Frijns, Tallau and Tourani-Rad (2010, JFuM): Construct Austrlian VIX (AVX) by using 3-month (66 trading days) at-the-money implied volatility of S&P/ASX 200 index options, study information content of the constructed AVX

Jin Zhang (Otago)


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VIX Futures Market

VIX and VIX futures On March 26, 2004, the newly created CBOE Futures Exchange (CFE) started trading the first-ever listed volatility product in the US: VIX futures, a futures contract written on the VIX index. It is cash settled with the VIXt multiplied by 1000 dollars. Since VIX is not a traded asset, one cannot replicate a VIX futures contract using the VIX and a risk free asset. Thus a cost-of-carry relationship between VIX futures and VIX cannot be established. FtT 6= VIXt e r (T −t) The relation between FtT and VIXt is an outstanding issue. Jin Zhang (Otago)


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VIX Futures Market

VXBt (10 × VIXt ) and VIX futures prices, Ftt+τ , τ fixed 300

Trading Volume VXB 30-day VIX Futures 60-day VIX Futures 90-day VIX Futures

250 200 150 100 50

2006-9-26

2006-7-26

2006-5-26

2006-3-26

2006-1-26

2005-11-26

2005-9-26

2005-7-26

2005-5-26

2005-3-26

2005-1-26

2004-11-26

2004-9-26

2004-7-26

2004-5-26

2004-3-26

0

Figure 1. VXB and VIX futures prices with three fixed time-to-maturities between March 26 2004 and November 21 2006. The VXB time series is from the CBOE. The

fixed maturity VIX futures prices are constructed by using the market data of available contracts with a linear interpolation technique. The bar chart shows the trading volume (normalized by 100 contracts) of futures of all maturities on each day. Jin Zhang (Otago)


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VIX Futures Market

Stochastic volatility model In the physical measure P, the SPX index, St , is assumed to follow Heston (1993) stochastic variance model p P , dSt = µSt dt + Vt St dB1t p P . dVt = κP (θP − Vt )dt + σV Vt dB2t Changing probability measure from P to Q as follows µ−r Q P dB1t = dB1t − √ dt, Vt

Q P dB2t = dB2t −

λ p Vt dt, σV

where λ is the market price of volatility risk. The risk-neutral dynamics of SPX is then given by p Q dSt = rSt dt + Vt St dB1t , p Q dVt = κ(θ − Vt )dt + σV Vt dB2t . The future variance VT conditional on Vt is non-central χ2 distributed. Jin Zhang (Otago)


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VIX Futures Market

A simple model for the VIX The relation between physical parameters, (κP , θP ) and risk-neutral parameters, (κ, θ) θ=

κP θP , κP + λ

κ = κP + λ.

VIXt2 is defined as the variance swap rate over the next 30 calendar days [t, t + 30/365], we have (Zhang and Zhu 2006) Z t+τ0 Z VIXt 2 1 1 t+τ0 Q Q = Et Vs ds = Et (Vs )ds 100 τ0 t τ0 t Z i 1 t+τ0 h −κ(s−t) = θ + (Vt − θ)e ds = (1 − B)θ + BVt , (7) τ0 t where B =

1 − e −κτ0 and τ0 = 30/365. κτ0

Jin Zhang (Otago)


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VIX Futures Market

A simple model for the VIX futures price The price of VIX futures with maturity T is (Zhang and Zhu 2006) FtT = EtQ (VXBT ) = EtQ (10 × VIXT ) i hp (1 − B)θ + BVT . = 1000 × EtQ

(8)

With asymptotic analysis, we have an approximate analytical formula (Zhang, Shu and Brenner 2010) FtT 1/2 = [θ(1 − BE ) + Vt BE ] 1000 σ2 1−E −3/2 2 − V [θ(1 − BE ) + Vt BE ] B Vt E 8 κ 4 2 (1 − E ) σ −5/2 3 3 + V [θ(1 − BE ) + Vt BE ] B Vt E + 16 2 κ2

(1 − E )2 2κ 1 (1 − E )3 θ , 2 κ2 +θ

1 − e −κτ0 , E = e −κ(T −t) . Combining (7) and (9) gives κτ0 = FtT (VIXt ; T − t, κ, θ, σV , λ).

(9)

where B = FtT

Jin Zhang (Otago)


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VIX Futures Market

Figure: Model and market prices of 30-day VIX futures

30-day VIX Futures and VXB

200

150

100

50

01Jun04

01Dec04

01Jun05

01Dec05

01Jun06

The upper solid line is the market price. The dots are model predicted prices based on the 200

tures and VXB

parameters calibrated from VIX futures prices term structure on the previous day. The lower solid line is VXBt − 90 for an easy comparison. The average prices of 30-day VIX futures is 150

144.8. The RMSE between model price and market price is 2.56. Relative error < 2%. Jin Zhang (Otago)


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Variance Risk Premium

Variance risk premium

The variance risk premium (VRP), measured by the return of a 30-day variance swap (Carr and Wu 2009), is given by Z t+τ0 Z t+τ0 1 1 Q P Vs ds − Et Vs ds VRP = Et τ0 t τ0 t = (1 − B P )θP + B P Vt − (1 − B)θ − BVt , where B P =

Pτ

1 − e −κ κP τ0

Jin Zhang (Otago)

(10)

0

.


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Variance risk premium (Cnt’d) Substituting κP θP = κθ and κP + λ = κ into equation (10) gives h κ i VRP = (1 − B P ) − (1 − B) θ + (B P − B)Vt κP " ! # κ 1 − e −(κ−λ)τ0 1 − e −κτ0 = 1− − 1− θ κ−λ (κ − λ)τ0 κτ0 # " 1 − e −κτ0 1 − e −(κ−λ)τ0 − Vt + (κ − λ)τ0 κτ0 1 − e −κτ0 − κτ0 e −κτ0 κτ0 (1 + e −κτ0 ) − 2(1 − e −κτ0 ) θ + V λτ0 + O(λ2 τ02 ) = t κ2 τ02 κ2 τ02 1 1 1 = κτ0 + O(κ2 τ02 ) θ + − κτ0 + O(κ2 τ02 ) Vt λτ0 + O(λ2 τ02 ), (11) 6 2 3

where the last two equalities are due to the Taylor expansion for small λτ0 and κτ0 . The variance risk premium is almost proportional to the market price of volatility risk, λ, if λτ0 is small. Jin Zhang (Otago)


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200

Figure. Variance risk premium 0 01Jul2004

03Jan2005

01Jul2005

03Jan2006

03Jul2006

03Jan2007

01Jul2005

03Jan2006 t

03Jul2006

03Jan2007

02Jul2007

400 VRP Mean of VRP

200

VRP

0

-200

-400

-600 01Jul2004

03Jan2005

02Jul2007

Figureshows 5: Thethe top variance graph shows 30-day historical HVtin , realized variance, RVt , This figure riskthepremium (VRP) variance, embedded the SPX options 2

and V IXt . The lower graph shows the variance risk premium (VRP), measured by the

2 2 market,difference measured by the between theand 30-day and VIX , i.e., between thedifference 30-day realized variance V IX 2 ,realized i.e., V RPvariance t = RVt − V IXt . The

average value variance risk value premium −66.0413. VRPt = RVt − VIXoft2 .the The average of isthe variance risk premium is −66.0413. Jin Zhang (Otago)


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Variance Risk Premium 200

The market price of volatility risk λ 0 01Jul2004

03Jan2005

01Jul2005

03Jan2006

03Jul2006

03Jan2007

02Jul2007

03Jul2006

03Jan2007

02Jul2007

t

40

λ Mean of λ

20

0

λ

-20

-40

-60

-80

-100 01Jul2004

03Jan2005

01Jul2005

03Jan2006 t

Figure 6: The top graph shows V IXt2 and the instantaneous variance, Vt , calculated from the observable V IXt . The lower graph shows the market price of volatility risk, λ, calcuThis figure price of floating volatility risk, λ, κcalculated from VRP t , Vof t and latedshows from Vthe RPt ,market Vt and calibrated θt with fixed = 1.2929. The average value λ is −19.1184. calibrated floating θ with fixed κ = 1.2929. The average value of λ is −19.1184. t

Jin Zhang (Otago)


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Parameter values in Heston (1993) model

Lin (2007) Duan and Yeh (2010) Zhang and Huang (2010)

Data period 21 Apr 2004 - 18 Apr 2006 2 Jan 2001 - 29 Dec 2006 18 May 2004 - 17 Aug 2007

κ 5.3500 -1.7986 1.2929

λ -0.3528 -7.5697 -19.1184

Why are the parameters estimated with different approaches so different? What is the most reliable way is to estimate the parameters in the volatility process?

Jin Zhang (Otago)


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The Term Stucture of VIX

Instantaneous VIX In the risk-neutral measure, Q, SPX is modeled by √ dSt Q = rdt + vt dB1t + (e x − 1)dNt − λt E Q (e x − 1)dt. St − Applying Ito’s Lemma gives √ 1 Q Q x d ln St = r − vt − λt E (e − 1) dt + vt dB1t + xdNt . 2 Luo and Zhang (2012) propose a new concept of instantaneous VIX as follows Vt =

dSt − d ln St = vt + 2λt E Q (e x − 1 − x). St −

Jin Zhang (Otago)


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The Term Stucture of VIX

The term structure of VIX Luo and Zhang (2010) model instantaneous VIX, Vt , by using a two-factor model as follows dVt dθt

Q = κ(θt − Vt )dt + dM1t , Q = dM2t ,

Q Q where dM1t and dM2t are increments of two martingale processes. The term structure of VIX, VIXt,τ , is then given by Z t+τ dSu 1 Q 2 E − d ln Su du VIXt,τ = τ t Su t Z t+τ 1 Q = E Vu du τ t t 1 − e −κτ = (1 − ω)θt + ωVt , ω= . κτ

Jin Zhang (Otago)


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Competing Models

Lin and Chang Lin (2007, JFuM), Lin and Chang (2009, JFuM) (2010, JEDC) model SPX futures price, Ft (T ), in an affine jump-diffusion framework √ 1 d ln Ft (T ) = − vt dt + vt dωS,t + ln(1 + Jt )dNt − κλt dt, 2 √ dvt = κv (θv − vt )dt + σv vt dωv ,t + zv dNt . They obtain a model for VIX VIXt2 =

ζ1 (aτ vt + bτ ) + ζ2 , τ

where ζ1 , ζ2 , aτ and aτ are functions of model parameters. They present formulas for VIX futures and option prices. Cheng et al (2012) point out that Lin and Chang’s solution is not exact as claimed. Jin Zhang (Otago)


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Competing Models

Lian and Zhu

In the same setup as Lin and Chang (2009, 2010),

Zhu and Lian (2012, JFuM) provide a formula for VIX futures price in terms of a single integration.

Lian and Zhu (2013, DEF) provide a formula for VIX option price in terms of a single integration as well.

Jin Zhang (Otago)


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Competing Models

Mencia and Sentana Grunbichler and Longstaff (1996, JBF) square root process (SQR) p dV (t) = κ[θ − V (t)]dt + σ V (t)dW Q (t) Detemple and Osakwe (2000, EFR) log-normal OU (LOU) process d ln V (t) = κ[θ − ln V (t)]dt + σdW Q (t) Mencia and Sentana (2013, JFE) model VIX directly with a concatenated SQR (CSQR) process p dV (t) = κ[θ − V (t)]dt + σ V (t)dWvQ (t), √ dθ(t) = κ[θ − θ(t)]dt + σ θdWθQ (t), dWvQ (t)dWθQ (t) = 0, and a few others based on LOU with time-varying mean, jumps and stochastic volatility in isolation and combination. VIX option is priced in an affine jump-diffusion framework. Jin Zhang (Otago)


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Competing Models

Lin

Lin (2013, JBF) defines z(t, Tj ) =

VIX 2 (t, Tj ; τ ) ∼ Q Fj − martingale. FtVIX (Tj )

She then model z-ratio as d

dz(t, Tj ) X F = σk (t, Tj ) · dωk j (t). z(t, Tj ) k=1

VIX option is priced in a Black-Scholes type of formula.

Jin Zhang (Otago)


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Competing Models

Bao, Li and Gong Bao, Li and Gong (2012, EJOR) model VXX directly by p SR dVXXt = κ(θ − VXXt )dt + σ VXXt dWt , p SRJ dVXXt = κ(θ − VXXt )dt + σ VXXt dWt + ydNt , LR

LRJ

d ln VXXt = κ(θ − ln VXXt )dt + σdWt ,

d ln VXXt = κ(θ − ln VXXt )dt + σdWt + ydNt .

They also model VXX with default risk by JDLRJ, JDLRSV and JDLRJSV d ln VXXt∆ = κ(θ∗ − ln VXXt∆ )dt + σdWt + ydNt + ln(1 − L)dHt , p d ln VXXt∆ = κ(θ∗ − ln VXXt∆ )dt + Vt dWt + ln(1 − L)dHt , p d ln VXXt∆ = κ(θ∗ − ln VXXt∆ )dt + Vt dWt + ydNt + ln(1 − L)dHt VXX option is then priced in an affine jump diffusion framework. Jin Zhang (Otago)


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Outstanding Issues

Outstanding issues

1

Consistent modeling for the SPX, VIX and VXX

2

Pricing SPX, VIX and VXX options

3

Equilibrium model for the variance risk premium

Jin Zhang (Otago)


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