The Market for Volatility Trading

Transcription

1 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 1 / 47

2 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) The Market for Volatility Trading 2 / 47

3 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) The Market for Volatility Trading 3 / 47

4 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) The Market for Volatility Trading 4 / 47

5 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) The Market for Volatility Trading 5 / 47

6 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 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) The Market for Volatility Trading 6 / 47

7 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 , 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) The Market for Volatility Trading 7 / 47

8 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) The Market for Volatility Trading 8 / 47

9 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 = , 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) The Market for Volatility Trading 9 / 47

10 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) The Market for Volatility Trading 10 / 47

11 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 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) The Market for Volatility Trading 11 / 47

12 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, V 0 that is called variance swap rate, in order to receive the realized variance, V realized, between current time, t = 0, and maturity date T. The net cash flow to the long party is V realized V 0. (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 V 0 = E Q 0 [V realized]. (2) Jin Zhang (Otago) The Market for Volatility Trading 12 / 47

13 OTC Variance Swaps Variance swap pricing theory Suppose equity index level is modeled by ds t S t = µdt + σ t db t, or d(ln S t ) = (µ 1 2 σ2 t )dt + σ t db t, where the volatility, σ t, follows a general stochastic process. This implies ds t S t d(ln S t ) = 1 2 σ2 t dt. Integrating this equation gives the realized variance V realized = 1 T σt 2 dt = 2 ( T ds t ln S ) T. (3) T T S t S Combining equation (2) and (3) gives the variance swap rate as follows V 0 = 2 [ T T E Q ds t 0 ln S ] T = 2 ( ln F [ 0 E Q 0 ln S ]) T, (4) S t S 0 T S 0 S 0 0 where F 0 = S 0 e (r q)t is the index forward price. Jin Zhang (Otago) The Market for Volatility Trading 13 / 47

14 OTC Variance Swaps Carr-Madan (1998) and Derman (1999) variance swap pricing theory The issue becomes pricing a derivative with payoff ln S T S0, a log contract. A general result from Calculus: for a twice differentiable function, f (S T ), we have following decomposition formula f (S T ) = f (S ) + f (S )(S T S ) + + S 0 + Applying this general result to a log contract gives ln S T = ln S + ln S T S 0 S 0 S = ln S + S T S S 0 S + S 0 S S 1 K 2 max(s T K, 0)dK, f (K) max(k S T, 0)dK f (K) max(s T K, 0)dK 1 K 2 max(k S T, 0)dK Jin Zhang (Otago) The Market for Volatility Trading 14 / 47

15 OTC Variance Swaps Carr-Madan/Derman variance swap pricing theory (Cnt d) Then the expected value of a log contract is [ E Q 0 ln S ] T = ln S + F 0 S S 0 S 0 S e rt ( S K 2 p 0(K)dK ) + 1 S K 2 c 0(K)dK, where c 0 /p 0 is European call/put price. The variance swap rate is V 0 = 2 ( ln F [ 0 E Q 0 ln S ]) T = 2 [ ln F ( ) 0 T S 0 S 0 T S F0 S 1 ( S )] + e rt 1 + K 2 p 1 0(K)dK + S K 2 c 0(K)dK. (5) If S = F 0, V 0 = 2 [ F0 1 + ] T ert 0 K 2 p 1 0(K)dK + F 0 K 2 c 0(K)dK. Variance swap rate can be determined by all the OTM call and put prices. Jin Zhang (Otago) The Market for Volatility Trading 15 / 47

16 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, σ 2 = 2 T i K i Ki 2 e RT Q(K i ) 1 ( ) F 2 1, (6) T K 0 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 K i is the strike price of ith OTM options, K i is the interval between two strikes, K 0 is the first strike that is below the forward index level Q(K i ) is the midpoint of the bid-ask spread of each option with strike K i VIX 2 represents the S&P day variance swap rate Jin Zhang (Otago) The Market for Volatility Trading 16 / 47

17 VIX and Listed Volatility Products The daily closing level of VIX and S&P 500 Indexes The historical highest closing level was on 20 November The intraday highest level was on 24 October Jin Zhang (Otago) The Market for Volatility Trading 17 / 47

18 VIX and Listed Volatility Products 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, Products Trade Volume Open Interest Listing Date Panel A: The CFE volatility/variance futures Nasdaq-100 Volatility Index (VXN) 5 55 Jul. 6, 2007 Russell 2000 Volatility Index (RVX) 50 2,467 Jul. 6, 2007 Volatility Index (VIX) 3,303 88,319 Mar. 26, 2004 DJIA Volatility Index (VXD) Apr. 25, 2005 S&P 500 Three-Month Variance 76 2,052 May 18, 2004 S&P 500 Twelve-Month Variance 0 60 Mar. 23, 2006 Panel B: The CBOE volatility options Nasdaq-100 volatility index (VXN) 50 2,156 Sep. 27, 2007 Russell 2000 volatility index (RVX) ,543 Sep. 27, 2007 Volatility Index (VIX) 87,349 2,240,946 Feb. 24, 2006 Panel C: The Eurex volatility futures VDAX-NEW R Futures (FVDX) 0 0 Sep. 19, 2005 VSMI R Futures (FVSM) 0 0 Sep. 19, 2005 VSTOXX R Futures (FVSX) 30 0 Sep. 19, 2005 Jin Zhang (Otago) The Market for Volatility Trading 18 / 47

19 VIX and Listed Volatility Products Market size of volatility futures and VIX options CBOE launched VIX options on Feb 24, 2006, Friday. On June 13, 2006, Tuesday, SPX = , VIX = Products Trade Volume Open Interest 12-month Variance futures month Variance futures DJ Volatility Index futures VIX futures 2,012 37,301 VIX options 19, ,398 SPX options 1,040,141 8,095,700 Market size VIX futures: , 301 = 888, 136, 810 USD VIX options: , 398 = 1, 970, 034, 638 USD SPX options: , 095, 700 = USD Jin Zhang (Otago) The Market for Volatility Trading 19 / 47

20 VIX and Listed Volatility Products Market size of volatility futures and VIX options (Cnt d) On December 11, 2008, Thursday, SPX = , VIX = Products Trade Volume Open Interest 12-month Variance futures month Variance futures 0 78 DJ Volatility Index futures VIX futures 3,182 27,287 VIX options 87,609 1,052,427 SPX options 736,490 18,720,920 Market size VIX futures: , 287 = 1, 522, 068, 860 USD VIX options: , 052, 427 = 5, 870, 437, 806 USD SPX options: , 720, 920 = USD Jin Zhang (Otago) The Market for Volatility Trading 20 / 47

21 VIX and Listed Volatility Products Average daily trading volume of VIX futures Jin Zhang (Otago) The Market for Volatility Trading 21 / 47

22 VIX and Listed Volatility Products Average daily trading volume of VIX options Jin Zhang (Otago) The Market for Volatility Trading 22 / 47

23 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, 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) The Market for Volatility Trading 23 / 47

24 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) The Market for Volatility Trading 24 / 47

25 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) The Market for Volatility Trading 25 / 47

26 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) The Market for Volatility Trading 26 / 47

27 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) The Market for Volatility Trading 27 / 47

28 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) The Market for Volatility Trading 28 / 47

29 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 VIX t 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. F T t r(t t) VIX t e The relation between F T t and VIX t is an outstanding issue. Jin Zhang (Otago) The Market for Volatility Trading 29 / 47

30 VIX Futures Market VXB t (10 VIX t ) and VIX futures prices, Ft t+τ, τ fixed Trading Volume VXB 30-day VIX Futures 60-day VIX Futures 90-day VIX Futures Figure 1. VXB and VIX futures prices with three fixed time-to-maturities between March and November 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) The Market for Volatility Trading 30 / 47

31 VIX Futures Market Stochastic volatility model In the physical measure P, the SPX index, S t, is assumed to follow Heston (1993) stochastic variance model ds t = µs t dt + V t S t db P 1t, dv t = κ P (θ P V t )dt + σ V Vt db P 2t. Changing probability measure from P to Q as follows db P 1t = db Q 1t µ r Vt dt, where λ is the market price of volatility risk. db P 2t = db Q 2t λ σ V Vt dt, The risk-neutral dynamics of SPX is then given by ds t = rs t dt + V t S t db Q 1t, dv t = κ(θ V t )dt + σ V Vt db Q 2t. The future variance V T conditional on V t is non-central χ 2 distributed. Jin Zhang (Otago) The Market for Volatility Trading 31 / 47

32 VIX Futures Market A simple model for the VIX The relation between physical parameters, (κ P, θ P ) and risk-neutral parameters, (κ, θ) θ = κp θ P κ P + λ, κ = κp + λ. VIXt 2 is defined as the variance swap rate over the next 30 calendar days [t, t + 30/365], we have (Zhang and Zhu 2006) ( ) 2 ( VIXt 1 t+τ0 ) = Et Q V s ds = 1 t+τ0 Et Q (V s )ds 100 τ 0 t τ 0 t = 1 t+τ0 [ θ + (V t θ)e κ(s t)] ds = (1 B)θ + BV t, (7) τ 0 t where B = 1 e κτ 0 κτ 0 and τ 0 = 30/365. Jin Zhang (Otago) The Market for Volatility Trading 32 / 47

33 VIX Futures Market A simple model for the VIX futures price The price of VIX futures with maturity T is (Zhang and Zhu 2006) Ft T = Et Q (VXB T ) = Et Q (10 VIX T ) [ ] = 1000 Et Q (1 B)θ + BVT. (8) With asymptotic analysis, we have an approximate analytical formula (Zhang, Shu and Brenner 2010) F T t 1000 = [θ(1 BE) + V tbe] 1/2 σ2 V 8 [θ(1 BE) + V tbe] 3/2 B 2 + σ4 V 16 [θ(1 BE) + V tbe] 5/2 B 3 [ V t E 1 E ] (1 E)2 + θ κ 2κ [ 3 2 V (1 E)2 te κ (1 E)3 θ 2 κ 2 where B = 1 e κτ 0, E = e κ(t t). Combining (7) and (9) gives κτ 0 = Ft T (VIX t ; T t, κ, θ, σ V, λ). F T t ], (9) Jin Zhang (Otago) The Market for Volatility Trading 33 / 47

34 VIX Futures Market Figure: Model and market prices of 30-day VIX futures da y VI XFuture san dvx B Jun04 01Dec04 01Jun05 01 Dec05 01Jun06 The upper solid line is the market price. The dots are model predicted prices based on the 200 parameters calibrated from VIX futures prices term structure on the previous day. The lower t uresan dvx B solid line is VXB t 90 for an easy comparison. The average prices of 30-day VIX futures is The RMSE between model price and market price is Relative error < 2%. Jin Zhang (Otago) The Market for Volatility Trading 34 / 47

35 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 ( 1 t+τ0 ) ( 1 t+τ0 ) VRP = Et P V s ds Et Q V s ds τ 0 t = (1 B P )θ P + B P V t (1 B)θ BV t, (10) where B P = 1 e κp τ 0 κ P τ 0. τ 0 t Jin Zhang (Otago) The Market for Volatility Trading 35 / 47

36 Variance Risk Premium Variance risk premium (Cnt d) Substituting κ P θ P = κθ and κ P + λ = κ into equation (10) gives [ κ ] VRP = κ P (1 BP ) (1 B) θ + (B P B)V t [ ( ) κ = 1 1 ( e (κ λ)τ ) ] e κτ 0 θ κ λ (κ λ)τ 0 κτ 0 [ ] 1 e (κ λ)τ e κτ0 V t (κ λ)τ 0 κτ 0 [ κτ0 (1 + e κτ 0 ) 2(1 e κτ 0 ) = κ 2 τ0 2 θ + 1 e κτ 0 κτ 0 e κτ ] 0 κ 2 τ0 2 V t λτ 0 + O(λ 2 τ0 2 ) [( ) ( 1 1 = 6 κτ 0 + O(κ 2 τ0 2 ) θ ) ] 3 κτ 0 + O(κ 2 τ0 2 ) V t λτ 0 + O(λ 2 τ0 2 ), (11) where the last two equalities are due to the Taylor expansion for small λτ 0 and κτ 0. The variance risk premium is almost proportional to the market price of volatility risk, λ, if λτ 0 is small. Jin Zhang (Otago) The Market for Volatility Trading 36 / 47

37 Variance Risk Premium 200 Figure. Variance risk premium 0 01Jul Jan Jul Jan Jul Jan Jul VRP Mean of VRP VRP Jul Jan Jul Jan Jul Jan Jul2007 t This figure Figureshows 5: Thethe topvariance graph shows riskthe premium 30-day historical (VRP) variance, embedded HV t in, realized the SPX variance, options RV t, and V IXt 2. The lower graph shows the variance risk premium (VRP), measured by the market, difference measured between by the thedifference 30-day realized between variance theand 30-day V IX 2, realized i.e., V RPvariance t = RV t and V IXVIX t 2. The 2, i.e., VRP t = average RV t value VIXof 2 the variance risk premium is t. The average value of the variance risk premium is Jin Zhang (Otago) The Market for Volatility Trading 37 / 47

38 200 Variance Risk Premium The market price of volatility risk λ 0 01Jul Jan Jul Jan Jul Jan Jul2007 t 40 λ Mean of λ λ Jul Jan Jul Jan Jul Jan Jul2007 t Figure 6: The top graph shows V IXt 2 and the instantaneous variance, V t, calculated from the observable V IX t. The lower graph shows the market price of volatility risk, λ, calculatedshows from Vthe RP t, market V t and calibrated price offloating volatility θ t with risk, fixed λ, κ calculated = The from average VRPvalue t, Vof t and This figure calibratedλ is floating θ t with fixed κ = The average value of λ is Jin Zhang (Otago) The Market for Volatility Trading 38 / 47

39 Variance Risk Premium Parameter values in Heston (1993) model Data period κ λ Lin (2007) 21 Apr Apr Duan and Yeh (2010) 2 Jan Dec Zhang and Huang (2010) 18 May Aug 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) The Market for Volatility Trading 39 / 47

40 The Term Stucture of VIX Instantaneous VIX In the risk-neutral measure, Q, SPX is modeled by ds t S t = rdt + v t db Q 1t + (ex 1)dN t λ t E Q (e x 1)dt. Applying Ito s Lemma gives [ d ln S t = r 1 ] 2 v t λ t E Q (e x 1) dt + v t db Q 1t + xdn t. Luo and Zhang (2012) propose a new concept of instantaneous VIX as follows V t = ds t S t d ln S t = v t + 2λ t E Q (e x 1 x). Jin Zhang (Otago) The Market for Volatility Trading 40 / 47

41 The Term Stucture of VIX The term structure of VIX Luo and Zhang (2010) model instantaneous VIX, V t, by using a two-factor model as follows dv t = κ(θ t V t )dt + dm Q 1t, dθ t = dm Q 2t, where dm Q 1t and dmq 2t are increments of two martingale processes. The term structure of VIX, VIX t,τ, is then given by VIXt,τ 2 = 1 [ t+τ ( ) ] τ E t Q dsu d ln S u du t S u = 1 ( t+τ ) τ E t Q V u du t = (1 ω)θ t + ωv t, ω = 1 e κτ κτ. Jin Zhang (Otago) The Market for Volatility Trading 41 / 47

42 Competing Models Lin and Chang Lin (2007, JFuM), Lin and Chang (2009, JFuM) (2010, JEDC) model SPX futures price, F t (T ), in an affine jump-diffusion framework d ln F t (T ) = 1 2 v tdt + v t dω S,t + ln(1 + J t )dn t κλ t dt, dv t = κ v (θ v v t )dt + σ v vt dω v,t + z v dn t. They obtain a model for VIX VIX 2 t = ζ 1 τ (a τ v t + b τ ) + ζ 2, where ζ 1, ζ 2, a τ and a τ 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) The Market for Volatility Trading 42 / 47

43 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) The Market for Volatility Trading 43 / 47

44 Competing Models Mencia and Sentana Grunbichler and Longstaff (1996, JBF) square root process (SQR) 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 dv (t) = κ[θ V (t)]dt + σ V (t)dw Q v (t), dθ(t) = κ[θ θ(t)]dt + σ θdw Q θ (t), dw Q v (t)dw Q θ (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) The Market for Volatility Trading 44 / 47

45 Competing Models Lin Lin (2013, JBF) defines z(t, T j ) = VIX 2 (t, T j ; τ) F VIX t (T j ) Q F j martingale. She then model z-ratio as dz(t, T j ) z(t, T j ) = d k=1 σ k (t, T j ) dω F j k (t). VIX option is priced in a Black-Scholes type of formula. Jin Zhang (Otago) The Market for Volatility Trading 45 / 47

46 Competing Models Bao, Li and Gong Bao, Li and Gong (2012, EJOR) model VXX directly by SR dvxx t = κ(θ VXX t )dt + σ VXX t dw t, SRJ dvxx t = κ(θ VXX t )dt + σ VXX t dw t + ydn t, LR d ln VXX t = κ(θ ln VXX t )dt + σdw t, LRJ d ln VXX t = κ(θ ln VXX t )dt + σdw t + ydn t. They also model VXX with default risk by JDLRJ, JDLRSV and JDLRJSV d ln VXXt = κ(θ ln VXXt )dt + σdw t + ydn t + ln(1 L)dH t, d ln VXXt = κ(θ ln VXXt )dt + V t dw t + ln(1 L)dH t, d ln VXXt = κ(θ ln VXXt )dt + V t dw t + ydn t + ln(1 L)dH t VXX option is then priced in an affine jump diffusion framework. Jin Zhang (Otago) The Market for Volatility Trading 46 / 47

47 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) The Market for Volatility Trading 47 / 47