1 Using VIX Futures as Option Market-Makers’ Short Hedge 10 December 2010 Yueh-Neng Lin, Anchor Y. Lin National Chung Hsing University

2 Volatility Risk Given the large and frequent shifts in volatility in the recent past, especially in periods like Black Monday of Stock Market Crash in 1987, Gulf War I in 1990, Asian Currency Crises in 1997, Russian Default and LTCM in 1998, Bursting of Dot-Com Bubble in 2000, 9-11 in 2001, Gulf War II in 2003, Subprime Mortgage Crises of 2007, and Leman Brothers bankruptcy in September 2008, there is a growing need for instruments to hedge volatility risk.

3 Volatility Risk Management Financial instruments to hedge volatility risk: 1. Traditional Call and Put options also have delta and gamma risk 2. Volatility/Variance Swaps A forward contract on realized or implied volatility/variance Only available at OTC 3. Strategies Options on a straddle (Carr and Madan, 1998) Options on at-the-money forward-start straddle (Brenner, Ou and Zhang, 2006) Forward-start strangle (Rebonato, 1999)

4 Volatility Risk Management (continued..) 4. Futures or options on a volatility index Brenner and Galai (1989) believe that different volatility measures that are highly correlated to the volatility exposures of all market participants are different. Hence, the volatility index may be based on standard deviation of historical data, implied volatility of options, or the combination of historical and option data. Then, the futures or options on alternate volatility index may satisfy the hedging demand of different participants. VIX futures and VIX options

5 VIX The Chicago Board Options Exchange (CBOE) introduces the VIX (volatility index) in 2003, which is the sentiment volatility implicit in S&P 500 index (SPX) option prices. VIX squared is the forward price of annualized total variance of S&P 500 index returns over 30 days (τ=30/365), implied in SPX option prices:

6 VIX & S&P 500 VIX tends to rise when the S&P 500 falls. VIX tends to decline or remain constant when the S&P 500 rises. Historically, VIX hits its highest levels during times of financial turmoil and investor fear. Hence, VIX is also known as the investor “Fear Index” (Whaley, 2000) in the marketplace.

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8 VIX & VIX Derivatives VIX futures are launched on March 26, VIX options are launched on February 24, VIX term structure are built in December Negative Correlation between S&P 500 Index and VIX VIX futures and VIX options could be a “catastrophe hedging” tool for stock portfolios.

9 VIX Futures Since VIX futures settle to the forward 30-day implied volatility of the S&P 500 ( forward VIX ), they are natural to hedge the “ forward vega” risk of S&P 500 options.

Hedging a Futures Call Option’s Forward Vega Risk A short position on S&P 500 futures call options is chosen as our target instrument, since it consists of the volatility risk between option’s expiry and its underlying futures’ expiry, or equivalently the “forward vega” risk. 10

11 The aim of this paper is to investigate whether and by how much VIX futures improve hedging a futures call option’s vega risk

12 Hedge a Call Option’s Spot Vega Risk Psychoyios and Skiadopoulos (2006) use the short position on a standard European-style call option as the target instrument. Two hedging schemes with two traditional option models and three volatility option models to investigate which combination gets the best hedging performance. They find that the traditional option is a more efficient instrument than the volatility option for hedging a traditional option.

American-Style Futures Option For options with early exercise potential, compute the BAW (Barone-Adesi and Whaley, 1987) early-exercise premium, treating it as if the stock volatility. Adding this early-exercise adjustment component to the European futures option price in the pricing formula should result in a reasonable approximation of the corresponding American futures option price. 13

14 Implicit Features of S&P 500 Index Prices 1. Stochastic volatility (SV) Heston (1993), Cox, Ross and Ingersoll (1985) The volatility is allowed to be mean-reverting and stochastic. 2. Price jumps (SVJ) Bates (1996), Bakshi, Cao and Chen (1997) To internalize more skewness and kurtosis.

American-Style Futures Options (continued..) For stochastic volatility and jump-diffusions, the constant-volatility analytic approximation results in Bates’ (1996) American currency futures options. 15

16 VIX Futures Valuation Formula

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19 VIX - VIX Futures What is the relationship? Futures prices can swing from a premium to a discount to VIX. VIX futures prices are based purely on expectations. No cost of carry between spot VIX and VIX futures.

20 Fair Price of VIX Futures (continued..) Carr and Wu (2006) present that the price of the VIX futures has a lower bound and an upper bound. Dupire (2006) finds the concavity adjustment which needs to be subtracted from the price of forward variance to arrive at the fair value of VIX futures. Zhang and Zhu (2006) choose a Heston-type stochastic- variance model and developed a numerical expression for VIX futures. Lin (2007) drives VIX futures price formulas under affine stochastic-volatility index processes with the convexity adjustment.

21 Fair Price of VIX Futures By Martingale pricing theory, the current VIX futures prices expiring at t can be computed under a risk-neutral probability measure Q as,

22 Strategies for Hedging Forward Volatility Risk

Hedging Strategies This study uses a short position on the futures option as the target portfolio at time t, i.e. TAR t = － C t (F) The expiration dates of SPX futures options and SPX futures are respectively T 1 and T 2, where t <T 1 <T 2. T 2 － T 1 ≡ 30 days The hedged portfolio consists of (1) N 1,t shares of underlying SPX futures F t, and (2) N 2,t shares of either forward-start strangle portfolios (HS1) or VIX futures (HS2), both denoted as INST t. 23

Hedging Strategies (continued..) The profit of this hedging portfolio is defined as Further add the constraints of delta-neutral and vega-neutral by 24

Two Hedging Schemes Hedging Scheme 1 (HS1): The hedged portfolio consists of N 1,t shares of underlying futures, and N 2,t shares of forward-start strangle portfolios. The forward-start strangle portfolio consists of (1) a short position of short-term strangle (the expiration date of the standard call and standard put is T 1 ; the strikes of the call and put are K 2 and K 1 with K 1 < K 2 ) ; and (2) a long position of long-term strangle (the expiration date of the standard call and standard put are T 2 ; the strikes of the standard call and put are K 2 and K 1 ). 25

Two Hedging Schemes (continued..) Hedging Scheme 1 (HS1): 26

Two Hedging Schemes (continued..) Hedging Scheme 2 (HS2): The hedged portfolio consists of N 1,t shares of underlying futures, and N 2,t shares of the VIX futures, i.e. 27

Hedging Error The hedging error is defined as the additional profit (loss) over the risk-free return : 28

Absolute Hedging Error The absolute hedging error through a hedging period (T 1 － t) is calculated as where 29

30 Empirical Data In order to assess the hedging performance of forward volatility risk, we use the SPX futures, the SPX options and the VIX futures to hedge a short position on the SPX futures call option. The data period: from July 3, 2006 to June 30, The daily prices for the SPX futures and the SPX futures option are obtained from the CME. The daily prices for the SPX options are obtained from CBOE, and the daily prices for the VIX futures are captured from CBOE. The interest rate data are daily annualized Treasury-bill rates obtained from DataStream Database. The daily dividend-yield ratio data are obtained from the S&P Corporation.

31 Empirical Data (continued..) 1. SPX futures that expire in March, June, September, and December. 2. SPX futures call options that expire in February, May, August, and November. 3. Forward-start strangle portfolio involves in two strangles. 1 ST strangle: SPX options that expire in February, May, August, and November. 2 nd strangle: SPX options that expire in March, June, September, and December. 4. VIX futures that expire in February, May, August, and November.

32 Empirical Results Our findings show that VIX futures outperform forward-start strangle strategies at most moneyness- maturity categories and the results are robust across models and hedging strategy. Overall, VIX futures are a better hedging instrument than standard options if the target option is a futures option, or equivalently if the risk exposure is the forward volatility risk. In particular, incorporating SV on average outperforms in OTM hedging. However, adding price jumps further enhances the hedging performance for SR options during the post- crash-relaxation period.

33 The End. Thank you.