Hedging with VIX

The VIX is the ticker symbol for the volatility index that the Chicago Board Options Exchange (CBOE) created to measure the implied volatility of options on the S&P 500 index (SPX) over the next 30 calendar days. The formal name of the VIX is the CBOE Volatility Index. Originally, it was pioneered by Professor Robert Whaley in 1993. In the same year CBOE introduced Volatility Index that measured market’s expectation of 30 day volatility implied by eight at-the money S&P 100 (OEX) put and call option prices. Throughout a decade VIX gained popularity and recognition in the capital market especially in the US. In 2003, CBOE together with Goldman Sachs reviewed and updated VIX which is based on a broader index S&P 500 (SPX) one of the main indexes of U.S. equities.

The new VIX estimates the expected volatility for next 30 days by averaging weighed price of SPX puts and calls option over wide range of strike price. VIX is commonly referred as Fear Gauge or Fear Index because it is said to be a good indicator of the level of fear and greed of investors in the U.S. on the equity market. When there is a fear among investors, the VIX level is notably higher than normal. The ratio of VIX to S&P 500 during our economic crisis of 2008 clearly underlines this property as demonstrated in graph 1. This strong negative correlation will be examined later in the report and brings the second use of this product, hedging. (CBOE, 2009) Graph 1:

Today VIX has established itself from abstract concept to standard use that many market participants find it practical. CBOE introduced the first exchange-traded VIX futures contract in March 24, 2004. After two years time, CBOE also launched VIX options. These financial products are very successful. (CBOE, 2009) Within 5 years volumes of VIX options and futures has grown to more than 100,000 contracts a day as supported by graph 2. They have become highly liquid and both have also received award for most innovative index derivative product. Graph 2:

Methodology
VIX is a Volatility Index consists of options rather than stocks. The original VIX which was introduced in 1993 (which was based on Black Scholes Formula) is calculated slightly different than the new VIX that is currently in use. New calculation (not based on Black Scholes) differs from the former by two main respects: it is based on S&P 500 rather than S&P 100 (S&P 500 gives more accurate view of investors’ expectations on future market volatility); and it utilizes broad range of strike prices rather than at-the-money option prices. Calculation of VIX involves rules for selecting component options and a formula to calculate the index values. (Whaley, 2008)

Generalized formula used in the VIX calculation

σ^2=2/T ∑_i▒(〖∆K〗_i/(K_i^2 ) e^RT Q(K_i)) -1/T [F/K_0 -1]^2 Where
σVIX/100 => VIX= σ * 100
T Time to expiration
F Forward index level derived from index option prices
K_0 First strike below the forward index level, F
K_i Strike price of i^(th) out-of-the-money option; a call if K_i >K_0 and a putif K_i < K_0; both put and call if K_i =K_0 〖∆K〗_i Interval between strike prices – half the difference between the strike on either side of K_i 〖∆K〗_i=(K_(i+1)-K_(i-1))/2

(Note: ∆K for the lowest strike is simply the difference between the lowest strike and the next higher strike. Likewise, ∆K for the highest strike is
the difference between the highest strike and the next lower strike.)

R Risk free interest rate
Q(K_i) The midpoint of the bid-ask spread for each option with strike K_i

VIX is a measure of expected volatility of SPX for the next 30 days. Calculation uses two options: near term and next term put and call options usually in the first and second SPX contract months. Requirements: Near term option must have at least 7 days to expiration with the intention to minimize pricing anomalies that could potentially occur close to expiration. If the near term option has less than week to expiration, VIX rolls to the second and third SPX contract. For instance, VIX will be calculated using SPX options expiring in March and April on the second Friday in March. On the following Monday April will replace near-term and May will replace next-term. Example: the near-term and next-term options have 9 days and 37 days to expiration To replicate the precision commonly used by option and volatility traders, VIX calculation measure T divides each day into minutes.

T={M_(Current day)+M_(Settlement day)+M_(Other days) }/(Minutes in a year)

Where
M_(Current day) Minutes remaining until the midnight of the current day M_(Settlement day) Minutes remaining from midnight until 8:30 a.m. SPX settlement day M_(Other days) Total minutes in days between current day and settlement day Example: the near-term and next-term options have 9 days and 37 days to expiration T_1 ={930 + 510 + 11,520}/525,600 = 0.0246575

T_2 ={930 + 510 + 51,840}/525,500 = 0.1013699
Bond-equivalent yield of U.S Treasury Bill maturing closest to the expiration date of SPX option is the risk free interest rate R that is used in VIX calculation. In our example, we assume it is R=0.38% for both sets of options After calculating the T, VIX involves selecting the option to be calculated. Only, options with non zero bid price is selected. Out-of-the money SPX puts and out-of-the money SPX calls centered around at-the money exercise price are chosen for the calculation. Near-TermNext-Term

Strike PriceCallPutAbsolute DifferenceStrike PriceCallPutAbsolute Difference 90048.9527.2521.6090073.652.820.80
90546.1529.7516.4090570.3554.715.65
91042.5531.7010.8591067.3556.7510.60
91540.0533.556.5091564.7558.95.85
92037.1536.650.5092061.5560.551.00
92533.3037.704.4092558.9563.054.10
93032.4540.157.7093055.7564.49.65
93528.7542.7013.9593553.0567.3514.30
94027.5045.3017.8094050.1569.819.65

To find out the forward SPX level, F, the strike price with the smallest absolute difference between call and put prices is chosen. In our case, this is 920 F=Strike Price× e^RT (Call Price-Put Price)

Therefore, forward price for near-term and next-term options are F_1=920+e^((0.0038×0.0246575) )×(37.15-36.65)=920.50005
F_2=920+e^((0.0038×0.1013699) )×(61.55-60.55)=921.00039

Where: F_1is near-term forward index and F_2is next-term forward index price. Next step is to determine the strike price K_0 which is 920 for both near-term and next-term options. K_(0,1)= 920 and K_(0,2)= 920

Now we need to pick out of the money put options with strike price smaller than K0 starting with the put strike immediately lower than K0 and move on to the following lower strike prices. Any put option that has zero bid price is excluded. Once two puts with consecutive strike prices are found to have zero bids, no puts with lower strikes are considered. Put StrikeBidAskInclude?

20000.05Not considered following two zero bids
25000.05
30000.05
35000.05No
37500.10No
4000.050.20Yes
4250.050.20Yes
4500.050.20Yes

Following step, pick out of the money call options with strike price greater than K0 beginning with the call strike immediately greater than K0 and move to the following higher strike prices. Call options with zero bid is also excluded. Once we find two consecutive call options with zero bid prices, no calls with higher strike prices are considered.

Call StrikeBidAskInclude?
12150.050.50Yes
12200.051.00Yes
122501.00No
123001.00No
123500.75Not considered following two zero bids
124000.50
124500.15
12500.50.10
12550.001.00

VIX calculation uses average of quoted bid and ask prices for each selected options. K0 put and call are averaged to give single value. Price used for the 920 strike, next term is (61.55+50.55)/2=61.05 and near term is (37.15+36.65)/2=36.90

Next-term StrikeOption TypeMid-quote PriceNext-term StrikeOption TypeMid-quote Price 400Put0.125400Put0.325
425Put0.125425Put0.30
450Put0.125450Put0.50

910Put31.70910Put56.75
915Put33.55915Put58.90
920Put/Call Average36.90920Put/Call Average61.05
925Call33.30925Call58.95
930Call32.45930Call55.75

1210Call0.2751210Call0.825
1215Call0.2751215Call0.725
1220Call0.5251220Call0.60

Step 2 Application of VIX formula for each near and next term options

VIX is a blend of information mirrored in the prices of all options selected. Each option’s contribution to the VIX value is proportional to ∆K and its price, and inversely proportional to the square of the option’s strike price. In general, 〖∆K〗_i is half the difference between the strike prices on both sides ofK_i. At the upper and lower edges of any given strips, 〖∆K〗_i is simply the difference between K_i and the adjacent strike price. In our example, 400 put is the lowest in the strip of near-term options and 425 is the adjacent. Thus, 〖∆K〗_(400 Put)= (425 – 400) = 25.

(∆K_(400 Put))/(K_(400 Put)^2 ) e^(RT_1 ) Q(400 Put)=25/〖400〗^2 e^.0038(0.0246575) (0.125)=0.0000195

In the next table, same calculation is carried out for each option. The contribution values are then summed and multiplied by 2/T_1 for near-term and 2/T_2 for next-term options

Next-term StrikeOption TypeMid-quote PriceContribution by StrikeNext-term StrikeOption TypeMid-quote PriceContribution by Strike 400Put0.1250.0000195400Put0.3250.0008182
425Put0.1250.0000173425Put0.300.0002501
450Put0.1250.0000139450Put0.500.0001531

910Put31.700.0001914910Put56.750.0003428
915Put33.550.0002004915Put58.900.0003519
920Put/Call Average36.900.0002180920Put/Call Average61.050.0003608 925Call33.300.0001946925Call58.950.0003446
930Call32.450.0001876930Call55.750.0003224

1210Call0.2750.00000091210Call0.8250.0000031
1215Call0.2750.00000091215Call0.7250.0000027
1220Call0.5250.00000181220Call0.6000.0000022

2/T ∑_i(〖∆K〗_i/(K_i^2 ) e^RT Q(K_i))
0.47277990.3668297

Now, the following step is to calculate 1/T [F/K_0 -1]^2for each term T_1and T_2:

1/T_1 [F_1/K_0 -1]^2=1/0.0246575 [920.50005/920-1]^2=0.0000120 1/T_2 [F_2/K_0 -1]^2=1/0.1013699 [920.00039/920-1]^2=0.0000117 Now we can calculate σ_1^2 and σ_2^2

σ_1^2= 2/T_1 ∑_i(〖∆K〗_i/(K_i^2 ) e^(RT_1 ) Q(K_i)) -1/T_1 [F_1/K_0 -1]^2=0.4727799-0.0000120=0.4727679 σ_2^2= 2/T_2 ∑_i(〖∆K〗_i/(K_i^2 ) e^(RT_2 ) Q(K_i)) -1/T_2 [F_2/K_0 -1]^2=0.3668297-0.0000117=0.3668180 The final step is to calculate the 30 day weighed average of σ_1^2 and σ_2^2 and then take the square root of that value and multiply by 100 to get VIX. VIX=100×√({T_1 σ_1^2 [(N_(T_2 )-N_30)/(N_(T_2 )-N_(T_1 ) )]+T_2 σ_2^2 [(N_30-N_(T_1 ))/(N_(T_2 )-N_(T_1 ) )]}×N_365/N_30 ) When near-term options have less than 30 days to expiration and next-term options have more than 30 days to expiration, the result of VIX value reflects an interpolation of σ_1^2and σ_2^2. Each individual weight is equal to or less than 1 and sum of the weights equals to 1. When the VIX rolls, both near-term and next-term options have more than 30 days to maturity. The same formula is used to calculate the 30 day weighted average but the result will be an extrapolation of σ_1^2and σ_2^2 meaning sum of weights is still 1. (CBOE, 2009) N_(T_1 ) = number of minutes to settlement of the near-term options (12,960) N_(T_2 ) = number of minutes to settlement of the next-term options (53,280) N_30 = number of minutes in 30 days (30 × 1,440 = 43,200)

N_365 = number of minutes in a 365-day year (365 ×1,440 = 525,600) Therefore,
VIX=100×√({0.0246575×0.4727679[(53,280-43,200)/(53,280-12,960)]+0.1013699×0.3668180[(43,200-12,960)/(53,280-12,960)]}×525,600/43,200)

VIX= 100×0.612179986=61.22

Uses of VIX
Investor Fear Gauge:
VIX has received the title of “fear gauge” while volatility, in technical terms, is the standard deviation of the daily logarithmic return. Hedgers have dominated SPX index option market. Their main motive of participating in this market is to secure themselves from a potential drop in the stock market. Hedgers do this by buying index puts. This is no different from a person buying car insurance. If the accident rate in a city increases, the insurance price will also increase. The same is true for portfolio insurance. The more the investor demand, the higher the price. Therefore, VIX is a good indicator that reflects the price of portfolio insurance. The ratio between VIX and S&P 500 can be observed to reflect the investors’ sentiment of the equity market. Graph 3, once more, illustrates the market confidence reaches its lowest level after the financial crisis. Graph 3:

Hedging with VIX:
Many previous researches underline the strong negative correlation between VIX and S&P500 index. In the past 7 years the correlation between the daily logarithmic changes of VIX and S&P 500 was a healthy -.74. Nevertheless the volatility of the VIX is in average 4 times greater. (Standard & Poor’s, 2008) Graph 4:

As a result of the strong negative correlation and high volatility, small drops in S&P 500 will be reflected by high increase in VIX. S&P 500 Daily Return smaller than <Probability of rise in VIXAverage VIX Return 0.00%79.52%4.70%

-0.50%93.56%7.78%
-1.00%95.37%10.51%
-1.50%95.89%12.54%
Numbers are calculated on the past 7 years closing prices.
These properties on the VIX make it an extremely good hedging tool for S&P 500 correlated portfolio or US equity fund. Nonetheless the VIX in itself cannot be traded. Market participant for hedging or speculation purposes can either use Options and Futures traded on the VIX since 2004 and 2006 respectively or trade the ETN (VXX & VXZ) iPath S&P 500 VIX Short and Mid Term Futures. This ETN track the VIX performance but offer smaller volatility. (Standard & Poor’s, 2009) S&P 500 Daily Return smaller than <Probability of rise in VIXProbability of rise in VXX 0.00%81.36%71.47%

-0.50%93.88%90.31%
-1.00%96.72%95.08%
-1.50%97.44%96.15%

3 Year correlations:
S&P 500 VIX Short-Term Futures IndexS&P 500 VIX Mid-Term Futures IndexVIXS&P 500 S&P 500 VIX Short-Term Futures Index1.0000.87420.8691-0.7626 S&P 500 VIX Mid-Term Futures Index1.0000.7692-0.7446

VIX1.000-0.7367
S&P 5001.000

Numbers are calculated using data from 2005 to 2008, closing prices. This strong correlation makes these ETNs useful hedging tool, but their low volatility diminish their hedging performance relative to the VIX.

Hedging with VIX Options and Futures:
As previously stated VIX Option and Future have become highly liquid enabling market participant to use them as strong hedging instrument to protect against market drops. To hedge a portfolio using VIX options, first of all the portfolio must be highly correlated with SPX index. Commonly practiced strategy called “The Reverse Collar” is when an investor purchases near-term slightly out-of-the money VIX calls continuously. In order to reduce the cost of the hedge investor sells slightly out-of-the money VIX puts of the same expiration date. It is clearly highlighted in the call to put VIX ratio that market participant hedge or speculate against volatility increase using call. (Dennis et al, 2005) (Dong, 2007) Graph 6:

VIX futures can be used to hedge equity returns and spread against market volatility. For instance, long volatility is a good way to hedge long equity position. Similarly, VIX future can be used without the downside of long volatility that can be very expensive to maintain over long period of time.

Risk management case study application:
From the previous section, the ability of the VIX and its sub products to hedge against large equity market drops, has been demonstrated. To frame its use in a real life application, we will consider a manager of a fixed pre-selected US equity portfolio. These are represented by Unit Investment Trust, they offer a low management fee but do not offer readjustment to risk as they are “fixed” and are only allowed to take long position. A well-diversified portfolio of stock will be strongly correlated to the S&P 500 and will probably have a beta of 1. The strategy recognized is a Benchmark Asset Allocation. Return and risk need to drift to the lower left corner of efficient frontier to offer the best risk reward ratio. Sharpe ratio is used to measure this. Futures on the VIX index are the most suitable for this hedging purpose. They are cost effective, high product availability, high liquidity, no premium, low transaction cost, low margin requirements and transparent pricing compare to option trading. We will place our manager back as of 1 of January 2004, before the financial crisis to evaluate the hedging strategy. To optimize the hedge 13 years historical data of the S&P 500 and VIX index are used. Since future contracts have only been traded since 2004 a hypothetical VIX future index will be used. The question is how much weight allocation should be used to optimize the Sharp ratio? Optimizing the Sharpe ratio:

Sharpe Ratio=(w×R_p+(1-w)×R_v-R_f)/o_(p,v)
Where:
R_p: Annual return of the original portfolio
R_v: Annual return of VIX index
R_f : Risk free rate
w: asset weight of the original portfolio
o_(p,v): Annual standard deviation of combined portfolio
R_v=o_v/o_0
o_v : Annual standard deviation of VIX index
o_0: Annual standard deviation of VIX index of the previous year o_(p,v)=√(w^2×o_p^2+〖(1-w)〗^2×o_v^2+2×w×(1-w)×o_p×o_v×ρ_(p,v) ) o_p : Annual standard deviation of original portfolio

Combining these the Sharpe ratio is given by:
S=(w×R_p+(1-w)×o_v/o_0 -R_f)/√(w^2×o_p^2+〖(1-w)〗^2×o_v^2+2×w×(1-w)×o_p×o_v×ρ_(p,v) )

The first derivative of the Sharpe ratio in respect to w is considered: ∂S/∂w=(R_p o_v (wρo_p-(w-1) o_v )-R_v o_p (wo_p-(w-1)ρo_v )+R_f (wo_p^2+(1-2w)ρo_p o_v+(w-1) o_v^2))/(w^2 o_p^2-2(w-1)wρo_p o_v+〖(w-1)〗^2 o_v^2 )^(3⁄2)

Let this derivative equal 0 and solve the equation for w when it is at its maximum or minimum then: w=(o_v (R_f o_0 (o_v-o_p ρ_(p,v) )+o_v (o_p ρ_(p,v)-R_p o_0)))/(R_f o_0 〖(o〗_p^2+o_v^2-2o_p o_v ρ_(p,v))+o_v (o_p (R_p o_0+o_v ) ρ_(p,v)-o_p^2-R_p o_0 o_v ) Since R_v=o_v/o_0

w=(o_v (〖ρR〗_v o_p-R_p o_v+R_f (o_v-ρo_p)))/(R_p (〖ρo〗_p-o_v ) o_v+R_v o_p (ρo_v-o_p )+〖R_f (o〗_p^2-2ρo_p o_v+o_v^2))

This equation enables us to find the optimizing asset allocation. Using data from 1990 to 2003 the optimum asset allocation was found. Asset allocation optimization with sharp ratio
YearRisk freeS&P volatilityS&P ReturnVIX VolatilityVIX ReturnW 19906.82%15.90%-8.55%116.56%42.54%96.52%
19914.12%14.27%23.35%99.44%-31.20%95.30%
19923.61%9.70%4.37%70.04%-42.93%76.84%
19933.63%8.60%6.82%84.86%-7.51%98.11%
19947.20%9.84%-2.43%99.87%23.27%94.10%
19955.18%7.80%29.35%74.98%-5.29%95.88%
19965.51%11.80%17.68%91.40%54.01%89.72%
19975.51%18.20%25.23%82.50%13.78%97.96%
19984.53%20.42%26.43%100.84%1.69%94.72%
19995.98%18.08%17.68%90.45%10.52%96.87%
20005.32%22.16%-9.64%87.51%8.59%93.60%
20012.17%21.77%-13.84%86.61%-12.06%96.97%
20021.32%25.94%-27.90%91.40%18.44%93.25%
20031.26%17.01%23.92%60.52%-29.85%96.52%

Average94.02%
The risk free interest rate comes from the 1-year Treasury bill. It is highlighted that the average optimized portfolio is constituted of 5.98% of the hypothetical security linking to VIX index and the remaining in the S&P 500 portfolio. Our manager applies this weighted ratio to his fixed portfolio has of 1 of January 2004. The performance is evaluated over the next 5 following years. The graph below represents the hedged portfolio against the S&P 500 performance readjusted as of 1 January 2004. Graph 7:

This graph clearly underlines the result of this hedged portfolio. In booming environment the returns are slightly decreased, but in a downturn the drawdown is minimized. Volatility is reduced. YearHedged Portfolio Standard DeviationS&P 500 Standard DeviationHedged Portfolio Return S&P 500 ReturnChange in Standard deviationChange in Return 20047.83%11.11%6.91%8.92%-3.28%-2.01%

20057.22%10.28%2.47%2.96%-3.06%-0.48%
20066.82%10.01%11.87%12.28%-3.20%-0.41%
200710.09%16.00%5.84%3.02%-5.91%2.82%
200820.79%41.08%-37.33%-47.14%-20.29%9.81%
200915.48%27.27%12.51%21.07%-11.78%-8.56%
Average11.37%19.29%0.38%0.18%-7.92%0.19%
Total2.26%1.11%1.16%

The table above summarises the 5-year performance of this hedged portfolio compare to the S&P 500. Over 5 year in average the standard deviation is reduced by nearly 8% every year. Its risk is reduced. On the other hand, because of the economic downturn, the returns are actually higher by 0.19% average. As a result our Sharpe ratio is improved significantly. YearHedged Sharpe ratioS&P 500 Sharpe ratio

20040.880.80
20050.340.29
20061.741.23
20070.580.19
2008-1.80-1.15
20090.810.77

Over the five-year period the hedged portfolio has outperform with its Sharpe ratio from .20 compare to the S&P 500 low 0.06.
Conclusion:
Volatility Index is an innovative instrument and good measure of expected volatility of stock index. Since calculation shift from S&P 100 to S&P 500, it has proven to be a good indicator of investors’ sentiment in relation to future confidence. Its strong negative correlation with S&P 500 over the past years supported with its high volatility underlines its ability to hedge an US equity position. Futures and Options volume augmentation since their introduction in 2004 and 2006 respectively, emphasize the trend of increased market participation. Its higher volume makes it a liquid and available instrument. As a result it has received various award for most innovative derivative product. The fixed portfolio scenario proves the previous mentioned fact. Over an economic downturn, such as the financial crisis of 2008, the max draw down is reduced significantly along with volatility, while at the same time returns are nearly unaffected. Optimising a risk-adjusted return for this fixed portfolio. The risk reward ratio, underlined through the Sharpe or other measures, is optimized.

Bibliography:
Web source:

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