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HyperVolatility – End of the Year Report 2013

The HyperVolatility End of the Year Report 2013 has been completed.

The report has an interactive Table of Contents, therefore, you can simply click on the asset class you are interested in and jump straight to the analysis.

The first copy is read–only while the second file is a printer–friendly version of the research.

The HyperVolatility End of the Year Report 2013 can be downloaded FOR FREE at the following link (no registration required):

HyperVolatility End of the Year Report 2013

HyperVolatility End of the Year Report 2013 (PRINT)

The 1st part of the report examines the performances of the most important asset classes in the world (equities, currencies, bonds and commodities) in 2013.

The asset classes that have been object of our annual research are the following

Equity futures: DAX, E–Mini S&P500, FTSE/MIB

Treasury Bonds futures: German Bund, American Treasury Bonds

Currency futures: Euro, Japanese Yen

Commodity futures: WTI Crude Oil, Gold

Volatility Indices: VIX  

The 2nd part, instead, presents the macroeconomic scenario in USA, Europe and BRICS economies (Brazil, Russia, India, China and South Africa). The analysis focuses on important indicators such as GDP growth, inflation, Debt–to–GDP ratio, unemployment rate, inflation rate and credit rating.

A Chinese and an Italian version of the aforementioned research will be uploaded in the upcoming hours.

Option Greeks and Hedging Strategies

The aims of the actual research are, firstly, to present some of the most efficient methods to hedge option positions and, secondly, to show how important option Greeks are in volatility trading. It is worth mentioning that the present study has been completely developed by Liying Zhao (Quantitative Analyst at HyperVolatility) and all the simulations have been performed via the HyperVolatility Option Tool–Box. If you are interested in learning about the fundamentals of the various option Greeks please read the following studies Options Greeks: Delta, Gamma, Vega, Theta, Rho and Options Greeks: Vanna, Charm, Vomma, DvegaDtime .

In this research, we will assume that the implied volatilityσis not stochastic, which means that volatility is neither a function of time nor a function of the underlying price. As a practical matter, this is not true, since volatility constantly change over time and can hardly be explicitly forecasted. However, doing researches under the static–volatility framework, namely, the Generalized Black–Scholes–Merton (GBSM) framework, we can easily grasp the basic theories and then naturally extend them to stochastic volatility models.

Recall that the Generalized Black–Scholes–Merton formula for pricing European options is:

     C = Se (b-r)T N(d1) – Xe -rT N(d2)

P = Xe -rT N(–d2) – Se (b-r)T N(–d1)

Where

d1

d2

And N ( ) is the cumulative distribution function of the univariate standard normal distribution. C = Call Price, P = Put Price, S = Underlying Price, X = Strike Price, T = Time to Maturity, r = Risk–free Interest Rate, b = Cost of Carry Rate, σ= Implied Volatility.

Accordingly, first order GBSM option Greeks can be defined as sensitivities of the option price to one unit change in the input variables. Consequentially, second or third–order Greeks are the sensitivities of first or second–order Greeks to unit movements in various inputs. They can also be treated as various dimensions of risk exposures in an option position.

1. Risk Exposures

Differently from other papers on volatility trading, we will initially look at the Vega exposure of an option position.

1.1 Vega Exposure

Some of the variables in the option pricing formula, including the underlying price S, risk–free interest rate r and cost of carry rate b, can be directly collected from market sources. Strike price X and time to maturity T are agreed with the counterparties. However, the implied volatility σ, which is the market expectation towards the magnitude of the future underlying price fluctuations, cannot be explicitly derived from any market source. Hence, a number of trading opportunities arise. Likewise directional trading, if a trader believes that the future volatility will rise she should buy it while, if she has a downward bias on future volatility, she should sell it. How can a trader buy or sell volatility?

We already know that Vega measures option’s sensitivity to small movements in the implied volatility and it is identical and positive for both call and put options, therefore, a rise in volatility will lead to an increase in the option value and vice versa. As a result, options on the same underlying asset with the same strike price and expiry date may be priced differently by each trader since everyone can input her own implied volatility into the BSM pricing formula. Therefore, trading volatility could be, for simplicity, achieved by simply buying under–priced or selling over–priced options. For finding out if your implied volatility is higher or lower than the market one, you can refer to this research  that we previously posted.

Let’s assume an option trader is holding a so–called ‘naked’ short option position, where she has sold 1,000 out–of–the–money (OTM) call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ=30%, which is currently valued at $434.3. Suppose the market agreed implied volatility decreases to 20%, other things being equal, the option position is now valued at $70.6. Clearly, there is a mark–to–market profit of $363.7 (434.3-70.6) for this trader. This is a typical example of Vega exposure.

Figure 1 shows the Vega exposure of above the option position. It can be easily observed that the Vega exposure may augment or erode the position value in a non–linear manner:

option vega exposure

(Figure 1. Source: HyperVolatility Option Tool – Box)

A trader can achieve a given Vega exposure by buying or selling options and can make a profit from a better volatility forecast. However, the value of an option is not affected solely by the implied volatility because when exposed to Vega risk, the trader will simultaneously be exposed to other types of risks.

1.2 Theta Exposure

Theta is the change in option price with respect to the passage of time. It is also called ‘time decay’ because Theta is considered to be ‘always’ negative for long option positions. Given that all other variables are constants, option value declines over time, so Theta can be generally referred to as the ‘price’ one has to pay when buying options or the ‘reward’ one receives from selling options. However, this is not always true. It is worth noting that some researchers have reported that Theta can be positive for deep ITM put options on non–dividend–paying stocks. Nevertheless, according to our research, which is displayed in Figure 2, where X=$100, T=30 days, r=0.5%, b=0, σ =30%, the condition for positive Theta is not that rigorous:

option theta

(Figure 2. Source: HyperVolatility Option Tool – Box)

For deep in–the–money (ITM) options (with no other restrictions) Theta can be slightly greater than 0. In this case, Theta cannot be called ‘time decay’ any longer because the time passage, instead, adds value to bought options. This could be thought of as the compensation for option buyers who decide to ‘give up’ the opportunity to invest the premiums in risk–free assets.

Regarding the aforementioned instance, if an option trader has sold 1,000 OTM call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ=30%, the Theta exposure of her option position will be shaped like in Figure 3:

option theta exposure

(Figure 3. Source: HyperVolatility Option Tool – Box)

In the real world, none can stop time from elapsing so Theta risk is foreseeable and can hardly be neutralized. We should take Theta exposure into account but do not need to hedge it.

1.3 Interest Rate/Cost of Carry Exposure (Rho/Cost of Carry Rho)

The cost of carry rate b is equal to 0 for options on commodity futures and equals r–q for options on other underlying assets (for currency options, r is the risk–free interest rate of the domestic currency while q is the foreign currency’s interest rate; for stock options, r is the risk–free interest rate and q is the proportional dividend rate). The existence of r, q and b does have an influence on the value of the option. However, these variables are relatively determinated in a given period of time and their change in value has rather insignificant effects on the option price. Consequently, we will not go too deep into these parameters.

1.4 Delta Exposure

Delta is the sensitivity of the option price with regard to changes in the underlying price. If we recall the aforementioned scenario (where a trader has sold 1,000 OTM call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ =30%, valued at $434.3) if the underlying price moves downwards, the position can still be able to make profits because the options will have no intrinsic value and the seller can keep the premiums. However, if the underlying asset is traded at, say, $105, all other things being equal, the option value becomes $6,563.7, which leads to a notable mark–to–market loss of $6,129.4 (6,563.7-434.3) for the option writer. This is a typical example of Delta risk one has to face when trading volatility. Figure 4 depicts the Delta exposure of the above mentioned option position, where we can see that the change in the underlying asset price has significant influences on the value of an option position:

option delta exposure

(Figure 4. Source: HyperVolatility Option Tool – Box)

Compared to Theta, Rho and cost of carry exposures, Delta risk is definitely much more dominating in volatility trading and it should be hedged in order to isolate volatility exposure. Consequentially, the rest of this paper will focus on the introduction to various approaches for hedging the risk with respect to the movements of the underlying price.

2. Hedging Methods

At the beginning of this section, we should clearly define two confusable terms: hedging costs and transaction costs. Generally, hedging costs could consist of transaction costs and the losses caused by ‘buy high’ and ‘sell low’ transactions. Transaction costs can be broken down into commissions (paid to brokers, etc.) and the bid/ask spread. These two terms are usually confounded because both of them have positive relationships with hedging frequency.  Mixing these two terms up may be acceptable, but we should keep them clear in mind.

2.1 ‘Covered’ Positions

A ‘covered’ position is a static hedging method. To illustrate it, let us assume that an option trader has sold 1,000 OTM call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ=30%, and has gained $434.3 premium. Compared to the ‘naked’ position, this time, these options are sold simultaneously with some purchased underlying assets, say, 1,000 stocks at $90. In this case, if the stock price increases to a value above strike price (e.g. $105) at maturity, the counterparty will have the motivation to exercise these options at $100. Since the option writer has enough amount of stocks on hand to meet the exercise demand, ignoring any commission, she can still get a net profit of 1,000*(100-90) + 434.3= $10,434.3. On the contrary, if the stock price stays under the strike price (e.g. $85) upon expiry date, the seller’s premium is safe but has to suffer a loss in stock position which in total makes the trader a negative ‘profit’ of 1,000*(85-90) + 434.4=-$4565.7. The ‘covered’ position can offer some degrees of protection but also induces extra risks in the meantime. Thus, it is not a desirable hedging method.

2.2 ‘Stop-Loss’ Strategy

To avoid the risks incurred by stock prices’ downward trends in the previous instance the option seller could defer the purchase of stocks and monitor the movements of the stock market. If the stock price is higher than the strike price, 1,000 stocks will be bought as soon as possible and the trader will keep this position until the stock price will fall below the strike. This strategy seems like a combination of a ‘covered’ position and a ‘naked’ position, where the trader is ‘naked’ when the position is safe and he is ‘covered’ when the position is risky.

The ‘stop-loss’ strategy provides some degrees of guarantee for the trader to make profits from option position, regardless of the movements of the stock price. However, in reality, since this strategy involves ‘buy high’ and ‘sell low’ types of transactions, it can induce considerable hedging costs if the stock price fluctuates around the strike.

2.3 Delta–Hedging

A smarter method to hedge the risks from the movements of the underlying price is to directly link the amount of bought (sold) underlying asset to the Delta value of the option position in order to form a Delta– neutral portfolio. This approach is referred to as Delta hedging. How to set up a Delta–neutral position?

Again, if a trader has sold 1,000 call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ=30% the Delta of her position will be -$119 (-1,000*0.119), which means that if the underlying increases by $1, the value of this position will accordingly decrease by $119. In order to offset this loss, the trader can buy 119 units of underlying, say, stocks. This stock position will give the trader $119 profit if the underlying increases by $1. On the other hand, if the stock price decreases by $1, the loss on the stock position will then be covered by the gain in the option position. This combined position seems to make the trader immunized to the movements of the underlying price.

However, in the case the underlying trades at $91, we can estimate that the new position Delta will be -$146. Obviously, 119 units of stocks can no longer offer full protection to the option position. As a result, the trader should rebalance her position by buying 27 more stocks to make it Delta–neutral again.

By doing this continuously, the trader can have her option position well protected and will enjoy the profit deriving from an improved volatility forecasting. Nevertheless, it should be noted that Delta–hedging also involves ‘buy high’ and ‘sell low’ operations which could cause a loss for every transaction related to the stock position. If the price of the underlying is considerably volatile, the Delta of the option position would change frequently, meaning the option trader has to adjust her stock position accordingly with a very high frequency. As a result, the cumulative hedging costs can reach an unaffordable level within a short period of time.

The aforementioned instance show that increasing hedge frequency is effective for eliminating Delta exposure but counterproductive as long as hedging costs are concerned. To reach a compromise between hedge frequency and hedging costs, the following strategies can be taken into considerations.

2.4 Delta–Gamma Hedging

In the last section, we have found that Delta hedging needs to be rebalanced along with the movements of the underlying. In fact, if we can make our Delta immune to changes in the underlying price, we would not need to re–hedge. Gamma hedging techniques can help us accomplishing this goal (recall that Gamma is the speed at which the Delta changes with respect to movements in the underlying price).

The previously reported example, where a trader has sold 1,000 call options priced with S=$90,X=$100,T=30days,r=0.5%,b=0 ,

σ =30% had a position Delta equal to -$119 and a Gamma of -$26. In order to make this position Gamma–neutral, the trader needs to buy some options that can offer a Gamma of $26. This can be easily done by buying 1,000 call or put options priced with the same parameters as the sold options. However, buying 1,000 call options would erode all the premiums the trader has gained while buying 1,000 put options would cost the trader more, since put options would be much more expensive in this instance. A positive net premium can be achieved by finding some cheaper options.

Let us assume that the trader has decided to choose, as a hedging tool, the call option priced with S=$90, X=$110, T=30 days, r=0.5%, b=0, σ=30%, with 0.011 Delta and 0.00374 Gamma. To offset his sold Gamma, the trader needs to buy 26/0.00374 = 6,952 units of this option which cost him $197.3 which leads to an extra Delta of 6,952*0.011=$76. At this point, the trader has a Gamma–neutral position with a net premium of $237 (434.3-197.3) and a new Delta of -$43(-119+76). Therefore, buying 43 units of underlying will provide the trader with Delta neutrality. Now, let’s suppose that the underlying trades at $91, the Delta of this position would become -$32 but since the trader had already purchased 43 units of stocks, she only needs to sell 12 units to make this position Delta–neutral. This is definitely a better practice than buying 27 units of stocks as explained in section 2.3 where the trader had only Delta neutralized but was still running a non–zero Gamma position.

However, Delta–Gamma Hedging is not as good as we expected. In order to explain this, let us look at Figure-5 which shows the Gamma curve for an option with S=$90, X=$110, T=30 days, r=0.5%, b=0, σ=30%:

option gamma

(Figure 5. Source: HyperVolatility Option Tool – Box)

We can see that Gamma is also changing along with the underlying. As the underlying comes closer to $91, Gamma increases to 0.01531 (it was 0.00374 when the underlying was at $90) which means that, at this point, the trader would need $107 (6,954*0.01531) Gamma and not $26 to offset her Gamma risk. Hence, she would have to buy more options. In other words, Gamma–hedging needs to be rebalanced as much as delta – hedging.

Delta–Gamma Hedging cannot offer full protection to the option position, but it can be deemed as a correction of the Delta–hedging error because it can reduce the size of each re–hedge and thus minimize costs.

From section 2.3 and 2.4, we can conclude that if Gamma is very small, we can solely use Delta hedging, or else we could adopt Delta–Gamma hedging. However, we should bear in mind that Delta–Gamma hedging is good only when Speed is small. Speed is the curvature of Gamma in terms of underlying price, which is shown in Figure – 6:

option speed

(Figure 6. Source: HyperVolatility Option Tool – Box)

Using the basic knowledge of Calculus or Taylor’s Series Expansion, we can prove that:

 Δ Option Value Delta * ΔS + ½ Gamma * ΔS + 1/6 Speed * ΔS

We can see that Delta hedging is good if Gamma and Speed are negligible while Delta–Gamma hedging is better when Speed is small enough. If any of the last two terms is significant, we should seek to find other hedging methods.

2.5 Hedging Based on Underlying Price Changes / Regular Time Intervals

To avoid infinite hedging costs, a trader can rebalance her Delta after the underlying price has moved by a certain amount. This method is based on the knowledge that the Delta risk in an option position is due to the underlying movements.

Another alternative to avoid over–frequent Delta hedging is to hedge at regular time intervals, where hedging frequency is reduced to a fixed level. This approach is sometimes employed by large financial institutions that may have option positions in several hundred underlying assets.

However, both the suitable ‘underlying price changes’ and ‘regular time intervals’ are relatively arbitrary. We know that choosing good values for these two parameters is important but so far we have not found any good method to find them.

2.6 Hedging by a Delta Band

There exist more advanced strategies involving hedging strategies based on Delta bands. They are effective for finding the best trade–off between risks and costs. Among those strategies, the Zakamouline band is the most feasible one. The Zakamouline band hedging rule is quite simple: when the Delta of our position moves outside of the band, we need to re–hedge and just pull it back to the edge of the band. However, the theory behind it and the derivation of it are not simple. We are going to address these issues in the next research report.

Figure-7 provides an example of the Zakamouline bands for hedging a short position consisting of 1,000 European call options priced with S=$90, X=$110, T=30 days, r=0.5%, b=0, σ=30%:

zakamouline bands

(Figure 7. Source: HyperVolatility Option Tool – Box)

In the next report, we will see how the Zakamouline band is derived, how to implement it, and will also see the comparison of Zakamouline band to other Delta bands in a quantitative manner.

The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial

The VIX Index: step by step

The VIX Index has been introduced by the CBOE in January 1993 and since then it has become the most popular and well known volatility index in the world. The VIX is an index extracted from S&P500 options and its calculation has been changed in September 2003 in order to obtain observations not linked or dependant from any model idiosyncrasy. The present study, inspired by the paper written and compiled by the CBOE, will show how to calculate the VIX Index in a step–by–step fashion. There will be ample explanations about how the index works and we will break down the formula in order to provide a better understanding of the function and weight of every component. Let’s start with the general formula for the VIX Index:

\dpi{120} \bg_white \large \sigma ^2 = \frac{2}{T}\sum_\imath \frac{\Delta K_\imath}{K_\imath ^2} e^ R^T Q (K_\imath )- \frac{1}{T} \left [ \frac{F}{K_0}-1 \right ]^2                                     (1)

Where

\dpi{120} \bg_white \large \sigma = \dpi{120} \bg_white \frac{VIX}{100}\Rightarrow VIX = \sigma x 100

T = Time to expiration

F = Forward Index price (the ATM strike price)

K0=  First strike price below the Forward price F

Ki= Strike of the ith out–of–the–money option. A call option strike will be considered if Ki>K0 while a put option strike will be used if Ki=K0

ΔKi = it is the midpoint between the strike prices. Specifically, the strikes used will be on both the sides of Ki. In other words, ΔK for the lowest strike is just the difference between the lowest strike and the penultimate strike of the option chain while ΔK for the highest strike is equivalent to the difference between the highest strike and the strike immediately below the top one. The formula is the following \dpi{120} \bg_white \Delta K_\imath = \frac{K_\imath_+_1 - K_\imath_-_1}{2}

R = Risk free interest rate (the interest rate used is usually the Treasury Bond yield that more closely matches the option expiration date)

Q (Ki) = The midpoint of the spread between bid price and ask price for each option with a strike Ki

This is the formula that it is currently employed to calculate the most famous risk management index in the world. Before we provide any real example it is necessary to clarify some points. First of all, the VIX index is calculated using 2 expirations: the front month and the second front month contracts. In our study we will be using S&P500 option prices recorded on the 10th of October 2013 and therefore the front month options will be those expiring on the 18th of October 2013 (which will provide us with information about the volatility in the near term) while the second front month options will expire on the 15th of November 2013 and they will provide us with information about the volatility in the medium term. It is important to point out that near term options must have at least 7 days to expiration and when this criteria is no longer met the model automatically rolls to the next available contract. Let’s give an example, our front month options expire on the 18th of October while the medium term options will expire on the 15th of November, hence, on Friday the 12th the VIX will mechanically roll. At this point, the front month options used will be the ones expiring on the 15th of November while the second front month options employed in the calculation will become those expiring in December. This measure has to be adopted in order to avoid mispricing issues that commonly happen when options are about to expire. Another important detail to mention is that T (time to expiration) is calculated in minutes using calendar days, consequently, the time to expiration will be given by

\dpi{120} \bg_white T = \frac {M_c_u_r_r_e_n_t_ \hspace {1 mm}_d_a_y + M _s_e_t_t_l_e_m_e_n_t \hspace {1 mm}_d_a_y + M_o_t_h_e_r \hspace {1 mm}_d_a_y_s }{Minutes\hspace {1 mm}in \hspace {1 mm}a\hspace {1 mm} year}                                  (2)

M current day= minutes remaining until midnight of the same day

M settlement day= minutes remaining from midnight until 08:30 am on SPX settlement day

M other days= total minutes in the days between current day and settlement day

If we assume that the data have been recorded exactly at 08:30 am on the 10th of October and that options will expire at 08:30 am of the 18th of October:

\dpi{120} \bg_white T_1 =\frac {930+510+1,920}{525,600} = 0.0246575

\dpi{120} \bg_white T_2 =\frac {930+510+3,540}{525,600} = 0.1013699

Needless to say that T1 and T2 refer to near term and medium term options respectively. The yield for 1 month Treasury Bills as of the 9th of October 2013 was 0.27% and therefore, given that our October options expire in 9 days while the November options will expire in 37 days, we will use this figure for both near and medium terms.

We now need to determine the F price of the index and in order to do so we take the strike price at which puts and calls have the smallest difference in absolute terms. The below reported table displays call and put prices with their absolute differential:

 S&P500 options

The red back–grounded figures are the strike prices at which calls and puts have the lowest difference, however, the two expiration dates have 2 different strike prices: ATM for October options is at 1,650 while ATM for November options is 1,655. In order to simplify calculations, and given the fact that the difference is extremely small, we will take 1,650 as F price because it is the front month contract. We can calculate F using the following formula:

\dpi{120} \bg_white F = Strike \hspace {1 mm}Price + e^R^T x ( Call \hspace {1 mm}Price - Put \hspace {1 mm} Price)                                 (3)

Obviously, there will be two forward index values, F1 and F2 , the first for near term and the second one for medium term options respectively:

\dpi{120} \bg_white F_1: 1,650+ e^(^0^.^0^0^2^7 \hspace {1mm}^x \hspace {1mm}^0^.^0^2^4^6^5^7^5^)x (23.75 - 22.7) = 1,651.05

\dpi{120} \bg_white F_2: 1,650+ e^(^0^.^0^0^2^7 \hspace {1mm}^x \hspace {1mm}^0^.^1^0^1^3^6^9^9^)x (39.75 - 37.3) = 1,652.45

Now we have both  F1 and F2 , so we can identify K0 which is the strike price immediately below the forward prices. In our case, the closest strike right below F1 and F2 is 1,650. Consequentially, K0,1= 1,650 and K0,2 = 1,650.

The next step is to select all put options whose strike is lower than K0 and all call options whose strike is higher than K0. The selection excludes every option with a bid equal to 0 and it terminates when there are 2 consecutive strike prices that equal zero:

s&p500 options

The above reported table explains very well what stated before. In the month of October the lowest puts are at 790 and 795 but the two red back–grounded options cannot be accepted in our calculation because there are two consecutive 0 in their bid prices and therefore they are our stopping point. In other words, no option with a lower than 800 strike price will be considered into the calculation of the VIX. The same principle applies to calls and in fact 2,095 and 2,100 are the highest call option strike prices that will be considered. The same procedure will be applied to November options. It is important to point out that the option chains will rarely have the same amount of strikes available because according to volatility fluctuations the number of strikes that will be priced by market makers will vary. It goes without saying that high volatility explosions will obligate market makers to price even Far–Away–From–The–Money options on both sides because the demand for these instruments will rise. Consequentially, the number of strikes that will be used for the purpose of calculating the VIX will vary according to volatility fluctuations and swings in S&P500 futures. The following table lists the options that will be used for calculating the volatility index:

ATM options

The ATM strike is highlighted in blue. The put options in the near term contract that will be used start from strike 800 until strike 1,645 while the call options range from strike 1,655 until strike 2,100. The medium term strikes, instead, goes from 985 until 1,645 for put options and from strike 1,655 until strike 2,100 for calls. We now calculate the specific weight that every single strike will have in the calculation by using the following formula and we will take the 800 put as an example:

\dpi{120} \bg_white \frac{\Delta K_8_0_0 \hspace {1 mm}Put} {K^2_8_0_0 \hspace {1 mm}Put} e ^R^T^1 Q(800 Put)                        (4)

Please bear in mind that Q is the midpoint between bid and ask while ΔK is the difference between the last option’s strike and the closest next strike, hence, in our case ΔK is (805 – 800) = 5. Let’s proceed with the calculation:

\dpi{120} \bg_white \frac{5}{(800)^2}\hspace {2 mm}2.71828 ^(^0^.^0^0^2^7 \hspace {1 mm}^x \hspace {1 mm} ^0^.^0^2^4^6^5^7^5^) (0.05)= 0.000000390651

The next table summarizes the contribution of each option strike to the overall computation of the VIX Index:

Option Strike contribution

Now, the weights need to be added up and multiplied by 2/T1 for the near term and by 2/T2for the medium term and by performing this calculation we would have 0.074319519775 for the near term and 0.041553088 for the medium term.

The equation (1) that we presented at the beginning of this study is almost completed, in fact, the final part is the only one yet to be estimated. Let’s proceed:

\dpi{120} \bg_white \frac{1}{T_1} \left [ \frac{F_1}{K_0}-1 \right ]^2 = \frac {1}{0.0246575} \left [\frac {1,651.05}{1,650}-1 \right ]^2 = 0.000016423

\dpi{120} \bg_white \frac{1}{T_2} \left [ \frac{F_2}{K_0}-1 \right ]^2 = \frac {1}{0.1013699} \left [\frac {1,652.45}{1,650}-1 \right ]^2 = 0.000021750

We can now complete the calculation by subtracting the two members of equation (1):

Near Term σ21 : 0.074319519775 – 0.000016423 = 0.074303096

Medium Term σ22 : 0.041553088 – 0.000021750 = 0.041531338

The final VIX computation is given by the following formula:

\dpi{120} \bg_white VIX = 100 \hspace {1 mm} x \hspace {1 mm}\sqrt\left \left \{ T_1 \sigma_1^2 \left [ \frac{N_T_2-N_3_0}{N_T_2 - N_T_1}\right] - T_2\sigma _2^2 \left [ \frac{N_3_0 - N_T_1}{N_T_2 - N_T_1} \right ]\right \} x \frac {N_3_6_5}{N_3_0}

The second term of the equation is nothing but the computation of the square root of the 30 day average of σ21 and σ22which are subsequently multiplied by the first term 100.  The weights of σ21 and σ22 are less than 1 or almost equal to 1 when the near term options have less 30 days and the medium term options have more than 30 days to expiration. However, it is worth noting that when the VIX rolls both near and medium term options have more than 30 days to expiration. Let’s now conclude the VIX estimation:

NT1= minutes left before the settlement of near – term options

NT2= minutes left before the settlement of medium – term options

N30= number of minutes in 30 days

N365=  number of minutes in a year composed by 365 days

If we plug in the numbers we have the following outcome:

\dpi{120} \bg_white \sqrt\left \{ 0.0246575 \hspace {1 mm}x \hspace {1 mm}0.074303096 \hspace {1 mm} x \left [ \frac{53,280 - 43,200}{53,280 - 12,960}\right] + 0.1013699 \hspace {1 mm} x \hspace {1 mm} 0.041531338 \hspace {1 mm}x \hspace {1 mm}\left [ \frac{43,200 - 12,960}{53,280 - 12,960} \right ]\right \} \hspace {1 mm} x \hspace {1 mm} \frac {525,600}{43,200}

This leads to the very last step: VIX = 100 x 0.209736087 = 20.9%

Now, the number we came out with is fairly accurate but it is limited to the moment in which the option prices have been “frozen”. Consequentially, the aforementioned procedure and calculation, in order to be precise, has to be automated and repeated every instant because any changes in option premiums will inevitable affect the final assessment of the VIX.

The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial.

The Volatility Smile

Option markets are multidimensional, in fact, options spreads can be created using different strikes, different maturities or different type of options (calls / puts). Besides, option traders cannot track instantaneous price changes on the entire option chain, hence, it is easy to derive that the most important factor in option trading is not the price of the option itself. The variable that has to be accurately tracked and monitored at all times is, in fact, what drives and determines the price of the option: volatility. In reality, there are different types of volatilities but the one extracted from option premiums is called implied volatility (the volatility extracted from futures prices is instead referred to as realized volatility) and its shapes and fluctuations are crucial to any market player involved in options trading. The present research will try to describe the dynamics of the implied volatility shape and to analyze its most common evolutions: Smile, Smirk and Forward Skew. In particular, the implied volatility figures used in the present examination have been extracted from front month WTI option premiums traded on the 30th of August 2013 which expire in October. All calculations and charts have been respectively performed and created with the HyperVolatility Option Toolbox. The following chart displays the so–called volatility smile:

Volatility Smile

(Source: HyperVolatility Option Toolbox)

As we can see from the above reported graph, the curve is higher at the extremes but rather low in the middle (bear in mind that the At–the–Money strike was $107.5). This is the typical shape for a front month implied volatility curve where the high demand for ITM calls and OTM puts as well as for ITM puts and OTM calls drives the volatility higher. In many cases, you will hear that the volatility for Out–of–the–Money options is higher but such statement is clearly incorrect because without further specification it implies that only OTM puts and calls experience such high volatility. The chart evidently shows that the volatility for ITM and OTM options is higher but, for obvious reasons, the volatility for the strike where a call option is In–the–Money will be almost as high as the volatility for the Out–of–the–Money put option and vice versa. Consequentially, saying that Away–from–the–Money options (both calls and puts) have a higher implied volatility than At–the–Money options is without a doubt the most correct statement. The most natural questions at this point would be: Why? What does a smile–shaped curve tell us?

The most obvious thing to say is that the volatility on AFTM options is higher because investors tend to trade them more often and they consequently push the volatility on the upside. The reason why investors buy wings is that ITM options have more intrinsic value than the ATM ones while OTM options have more extrinsic value than an option struck At–the–Money. Consequentially, the presence of an implied volatility smile–shaped curve is typical of more speculative markets. The smile suggests that, when large volatility shifts happen, many market players rush to buy OTM options for speculative reasons while ITM options are primarily purchased to stabilize portfolio gains. The next chart shows how the curve moves:

Volatility Smile - Dynamics

(Source: HyperVolatility Option Toolbox)

Many researches on implied volatility curve dynamics showed that the 75% of volatility changes can be defined by a total shift, up or down, of the entire curve as you can see from the chart. A further 15% of the movements consist of curve twisting (the right hand side of the curve goes deeper down while the left side gets pulled on the right or vice versa) while the remaining 10% is the product of a change in convexity (wings getting wider or tighter).

Smile–shaped curves are frequently found in equity index options, stock options and popular commodities / currencies (Euro, WTI, Gold, etc). Nevertheless, it is worth noting that the shape of the curve can even evolve over time. This concept can be better explained by looking at the following chart:

Volatility Smirk

(Source: HyperVolatility Option Toolbox)

The smirk is a particular volatility profile where ITM calls and OTM puts are priced with a much higher implied volatility. This phenomenon is commonly found in equity markets and risky assets. In this simulation the ATM strike is 116 and its volatility is 13.1% but OTM puts and ITM calls are much more expensive because they are trading above the 70% level. As previously mentioned, the implied volatility curve can change and evolve and a volatility smile can turn into a smirk if investors, traders and market players are expecting a market crash or if the plunge in price has already happened. Smirks are simply telling us that lower strikes are more traded than higher strikes and OTM puts as well as ITM calls are being heavily traded. If the market is heading south, a smile would easily evolve into a smirk because of the great buying pressure generated by market players rushing to buy OTM puts to protect their portfolios. The purchase of ITM calls, even during market crashes, makes sense because ITM call options have already an established intrinsic value and they have the highest probability to expire In–the–Money; in other words they are safer. However, in the event of market downtrends the evolution of a volatility smile into a smirk would predominantly be caused by the large buying volume on OTM puts.

The Volatility smile, nevertheless, can go through another metamorphosis whose final output is the so–called forward skew:

Volatility Forward Skew

(Source: HyperVolatility Option Toolbox)

The forward skew is nothing but a reversed form of smirk, in fact, the volatility here tends to become higher for ITM puts and OTM calls. This type of curve is more frequently found in commodity markets, particularly agricultural products, than equity indices or stock options. Even in this case the increase in volatility is provoked by an augment in demand for these options. However, the strong buying pressure concentrated on OTM calls is often the main cause of such shape. Let us break it down. Many players in commodity markets are commercials (mining and energy companies, grain / wheat / sugar / coffee producers) and therefore a disruption in the supply chain of a particular commodity can generate serious problems. A shortage in oil supply due to geopolitical variables, a disappointing crop due to a frost or to challenging meteorological conditions, continuous strikes in a particularly large mine are all factors that would force companies to buy as quickly as possible the commodity they need in order to lock in the order. Consequentially, the remarkable buying pressure on OTM calls would inevitably drive their price up and that is why the implied volatility of higher strikes is more elevated than others.

Let us now summarize the main concepts in order to avoid confusion:

1) An implied volatility smile means that Away–from–the–Money options have a higher implied volatility than At–the–Money options

2) Implied volatility smile–shaped curves are typical of highly speculative markets

3) Many researches on implied volatility curve dynamics showed that the 75% of all volatility changes consist of a shift, up or down, of the entire curve

4) Smile–shaped curves are frequently found in equity index options, stock options and the most popular commodities / currencies

5) The smirk is a particular volatility profile where ITM calls and OTM puts are priced with a much higher implied volatility

6) Volatility smile curves can turn into a smirk if investors, traders and market players are expecting a market crash or if the plunge in price has already happened

7) The forward skew is a reversed form of smirk, in fact, the volatility here tends to become higher for ITM puts and OTM calls

8) The forward skew curve is more frequently found in commodity markets (particularly in agricultural products)

9) The formation of a forward skew curve is often the consequence of a shortage in the supply chain due to transportation issues, geopolitical problems, adverse meteorological conditions, etc

The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial.

Extracting Implied Volatility: Newton-Raphson, Secant and Bisection Approaches

The aim of the present research is to identify an efficient method to extract implied volatility from options prices. It is worth mentioning that the present study has been completely developed by Liying Zhao (Options Engineer at HyperVolatility) and all the simulations have been performed via the HyperVolatility Option Tool – Box. Options traders, and by “traders” we obviously mean market makers too, are more concerned with volatility than premium itself because they know perfectly well that the latter is a consequence of the former. Therefore, it is essential to find out which implied volatility level has been used to price options in order to determine which trading strategy should be adopted. It goes without saying that if an option trader “discovers” some options priced with a lower implied volatility level he would immediately consider buying strategies while the opposite scenario would apply when there is a case of overpricing. At a first glance, things seem quite simple but what would happen if the implied volatility that you have extracted from market prices does not match the one employed by your counterparty? Chances are that if you extract a ‘wrong’ implied volatility figure from the ‘right’ market price you could probably be misled by it and eventually lose money.

There are many ways to “recuperate” the implied volatility from market prices and among them the Newton–Raphson (NR) method is undoubtedly one of the most popular employed by option traders. The question is, can the Newton–Raphson approach be trusted? In order to answer this question we first need to examine the NR method. The formula is the following:

formula1a

Where  CodeCogsEqn(3) if (i=0) is the initial guess for the implied volatility, c-sigma I is the option price derived from the initial guess, cm is the market price of the option, deltac-deltasigma is obviously the Vega in terms of the initial guess, and finally sigma I+1 would be the updated implied volatility. While the absolute value of cm-csigma falls into a desired degree, sigma I+1 would be the desired estimation of the implied volatility used to price the option. It is worth noting that the initial guessed values are figures that can be obtained by using other formulas, past experience but they can even be chosen randomly.

The NR method is very fast and, if the initial guesses are good the efficiency of the model is enhanced which is why many option traders prefer to use it. However, in order to use this method the value of Vega is needed. Fortunately, Vega can be easily derived following an analytical approach at least for European style options. What happens if we are trading American style options? Well, Vega can be numerically computed for American options; nevertheless, in this case the advantage of the NR method (that is high efficiency) is, more or less, sacrificed.

Let’s omit the aforementioned issue and let us focus on the accuracy of the NR method. We will now present a simulation where we will price an entire options chain written on a hypothetical commodity using the BSM model and then we will extract the implied volatility via the NR approach. Let us assume that we are trading European style commodity options where the underlying price equals $100, the risk–free interest rate is 0.5%, the cost of carry is 0, the implied volatility is 20% and there are 20 days left before expiration. Given those parameters, we can price an option chain with strike prices ranging from $50 to $150 by using the Black–Scholes–Merton (BSM) formula. Bear in mind we need to assume that the theoretical prices of the option chain are perfectly matched by market prices. Now, we can finally use market prices inversely with the NR method and we will theoretically get an implied volatility equal to 20% for each instrument in the option chain. The following chart help clarifying the concept:

implied  volatility

(Source: HyperVolatility Option Tool – Box)

The chart displays only 1 curve (which shows the implied volatility for put options) because the volatility curve for call options moves in the exact same way so there is a problem with overlapping. Nevertheless, we have got a ‘volatility skew’. Do not worry. This is not a real volatility skew. Volatility skews exist because Out-of-The-Money (OTM) and In-The-Money (ITM) options are more heavily traded. Nonetheless, we are not dealing with real trading conditions; instead, we just want to get back to the implied volatility we used to price these options. Thus, we can infer that this ‘volatility skew’ originates from the inaccuracy of the NR method. According to some deeper researches the Newton–Raphson method becomes very inaccurate when the strike of the option is more than 20% Away-From-The-Money (AFTM).

If we use the biased implied volatility that we previously obtained and plug it into the BSM formula to price the same option chain we would get a range of new option prices. Luckily, the correlation coefficient between the prices of the option chain with biased and unbiased implied volatility is equal to 1, which means, the bias of implied volatility for AFTM options has little effect on the pricing. Why? To explain this, let us consider the following Vega chart:

options vega

(Source: HyperVolatility Option Tool – Box)

It is quite clear from the chart that the implied volatility is only influential for slightly In-The-Money (ITM), or slightly Out-of-The-Money (OTM) options. In other words, shifts in the implied volatility level would significantly influence options whose strike is close to the ATM area but would not impact that much Away–From–The–Money options. Therefore, when the strike price is far away from the underlying price the change in implied volatility becomes less important when pricing an option. On the other hand, Near-The-Money (NTM) options are very sensitive to changes in implied volatility. For instance, using 10% or 30% implied volatility while we are pricing an Away–From–The–Money option ($100 underlying, $60 strike) would not show much difference in terms of premiums. For the same reason, when we take the reversed BSM formula to extract implied volatility from market prices for AFTM options, it is normal that the resulting implied volatility does not match the volatility level we used to price the option chain with in the first place. From a deeper angle, it is worth saying that Vega approaches zero when the underlying price is far away from strike price. Furthermore, small errors in the estimation of Vega, being it the NR formula’s denominator, can lead to large errors in the implied volatility.

Another approach that can be used to extract implied volatility from options prices is the so-called Secant Method. Some ‘mathematicians’, 3,000 years before Newton, developed a root–finding algorithm called Secant Method (SM) that uses a succession of roots of secant lines to approximate a root of a function. This method works similarly to the NR method and consequently it is just as fast as the NR approach. Moreover, the Secant Method does not require the knowledge and computation of Vega. Can we assert that the Secant Method is the perfect approach? Before answering this question, let’s see what the Secant Method is about:

formula2a

where cm is the market price,  sigma n is the result for each iteration, c sigma n is the theoretical price for each result. With two initial guesses sigma 0 and sigma 1 we can get sigma 2, but we are going to repeat this process until we get sigma n and sigma n-1 (which are close enough) and then sigma n that would be the desired answer.

To evaluate the reliability of this method we are going to let the statistics speak. Suppose we are trading a European style commodity options with underlying price equal to $100, risk–free interest rate 0.5%, cost of carry set to 0, implied volatility 20% and 20 days to maturity. Given the aforementioned parameters, we now price an option chain with strike prices ranging from $50 to $150 by using the BSM formula. However, this time we use the theoretical prices inversely but with the Secant Method and try to extract the implied volatility which should equal 20% for each option.

implied volatility

(Source: HyperVolatility Option Tool – Box)

The above reported chart clearly shows that we can perfectly recover the implied volatility from market prices for Near–The–Money options. Nevertheless, Away–From–The–Money options present a more complex scenario because at the ‘tails’ of the strike distribution we have an increased deal of inaccuracy. The main reason for the existence of these irregular observations is that this algorithm may not converge if the first derivative of the function in terms of the argument, Vega in this instance, is close to 0 (remember the Vega chart for AFTM options: Vega approaches 0 and here the convergence of the Secant Method is not guaranteed). It is also worth noting that in order to execute the Secant Method we need two initial guesses but in this case both the guessed values must be very close to the target value. Thus, this approach requires a great deal of preparatory work before the start of iterations otherwise one would run the risk to get worse–than–expected results.

Alternatively, we can try other ways to extract the implied volatility from market prices. One of them is the Bisection Method (BM), where we also need two initial volatility estimates: a ‘low’ estimation of the implied volatility sigma L , which would generate the option value c L and a ‘high’ volatility estimation, sigma H,generating the option value c H. The option market price, cm , then should fluctuate between c L and c H. The Bisection computation of the implied volatility is given by:

formula3a

where c sigma MINOR we replace sigma L with sigma I+1 while if c sigma MAJOR we replace sigma H with sigma I+1. When sigma H-L falls into the desired interval, sigma I+1 would be the desired implied volatility.

Let us now assess the accuracy of this method by using the same approach adopted for the Newton–Raphson formula. Again, suppose we are trading European style commodity options with underlying price equal to $100, risk-free interest rate 0.5%, cost of carry set to 0, implied volatility 20% and 20 days to maturity. We will now price an option chain with strike prices ranging from $50 to $150 by using the BSM formula. However, this time we employ the theoretical prices inversely and we will try to find the implied volatility via the Bisection Method:

chart4

(Source: HyperVolatility Option Tool – Box)

The chart highlights that the implied volatility become slightly inaccurate only for very deep ITM options and in this case the ‘confident interval’ is significantly enlarged. The advantages of the Bisection Method are that it can be implemented without the knowledge of Vega and it can be used to find the implied volatility when dealing with American options. On the other hand, it may still not be too attractive extracting implied volatility with this method because it is not as efficient as the Newton – Raphson formula. Furthermore, the Bisection Method is computationally expensive, hence, when dealing with a great amount of options the capacity of the computer adopted becomes crucial.

There is no perfect method to extract implied volatility from market prices without errors and with high speed. Nevertheless, the Newton–Raphson approach, despite its drawbacks, is the dominating method in this area. In fact, with good initial guesses the NR formula can be a very ‘smart’ method which provides accurate results particularly for Near–The–Money options with high efficiency. The Bisection Method is considered to be ‘dull’ but it provides larger ‘confident intervals’. Besides, the Secant Method is as fast as NR one but it is very hard to implement. Finally, it is worth mentioning that, in spite of what the majority of option traders believe, there actually are analytical expressions to extract the implied volatility from market prices. The problem with these closed formulas is that they are too complicated. In fact, each function can even have more than thirty parameters and each parameter, in turn, will have to be calibrated with a great deal of accuracy. Luckily, there is a lot of room for improvement in this field and perhaps future researches will propose an analytical solution to this problem that will manage to replace the Newton–Raphson approach.

In order to summarise the findings of the present research, please have a look at the following table where all the models and characteristics are not only listed but also ranked. The way to interpret the results is very simple: the higher the mark, the better. Even the section called “Difficulty” has been quantified in the same way, hence, an higher mark means that it is easier to implement:

chart5

It is clear enough that the Newton–Raphson method and the Bisection Method are the ones showing the highest marks. However, they are recommended not in all cases. In particular, the NR approach is more suitable when dealing with a large set of options while the Bisection Method has to be preferred when the data set available is rather small.

If you are interested in options trading or pricing you may want to read some HyperVolatility researches related to this topic:

“The Pricing of Commodity Options”

“Options Greeks: Delta, Gamma, Vega, Theta, Rho”

“Options Greeks: Vanna, Charm, Vomma, DvegaDtime”

The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial

 

Commodity Volatility Indices: OVX and GVZ

The present research will examine the most popular commodity volatility indices proposed by the CBOE: the OVX and the GVZ. It is important to point out that this study is the second component of a bigger research whose first part is entitled “Equity Volatility Indices: VIX, VXN, VXD, RVX” . The analysis will follow the structure presented in the first half and it is aimed to provide pivotal volatility levels. (The HyperVolatility Forecast Service provides market projections for the VIX, Gold, WTI crude oil and many other asset classes. Send an email to info@hypervolatility.com and get a free 14 days trial). Before getting started it is worth reminding that the OVX is a volatility index based on the performance of the options written on the USO (United States Oil Fund) while the GVZ is calculated on options written on the SPDR Gold Shares Trust. The next chart displays the ranking of both commodity volatility indices:

Commodity Volatilities: Distribution Ranking

The most important levels to observe are the 25%, the median and the 75% ones. The OVX has a median value fluctuating around 32.8% while the GVZ is much lower and, on average, it does not get higher than 18.1%. In case of low volatility environments the OVX remains around the range 29% – 30% while the GVZ tends to oscillate around 16%. On the other hand, high volatility days would probably see the OVX moving around 37% while the GVZ does not usually surpass the 21.3%. These levels are crucial to anyone who wants to trade oil or gold volatilities because they will provide buy / sell signals for options traders. Let’s list some of them (remember these are not recommendations but only a general rule):

1) Long Gold or WTI Oil volatility when the OVX is below 30% and the GVZ is around 16%

2) Range trading strategies, such as condors or butterflies, should be adopted when the OVX is higher than 35% and the GVZ is above 19%

3) If the OVX or the GVZ are fluctuating around their median levels, it implies that some major movement is about to occur. Therefore long volatility strategies could be implemented when the OVX is around 30% – 31% and the GVZ is in the 17% – 18% interval.

The fact that a volatility index has a high value does not mean that it is the most volatile one. We already tried to address this issue when dealing with equity volatility indices so now we will attempt to explain such a phenomenon for commodity volatilities too. The next chart plots the volatilities for the OVX and GVZ indices:

Volatility of Commodity Volatilities

The chart clearly displays the volatility of both commodity volatility indices. The relationship between the OVX and the GVZ is positive, in fact, the correlation between the 2 indices is +0.77 while the correlation between the volatility of the OVX and the volatility of the GVZ is +0.62. It is interesting to notice that the 10% – 13% interval is a key mean reverting level in both markets because all the time the volatility of the OVX or GVZ touched this level both asset classes experienced a major volatility explosion. The next chart shows the distribution ranking for the volatility of commodity volatility indices:

Volatility of Commodity Volatility: Distribution Ranking

The distribution ranking demonstrates that the GVZ, although lower than the OVX in volatility points, is more volatile than the OVX index. Specifically, GVZ’s volatility oscillates on average around the 18% while the OVX’s oscillation rate does not go higher than 13.3%. In low volatility environments the oil volatility index’s fluctuations rate is not lower than 10.6% while gold’s index oscillations rarely gets lower than 13.9% (these are the mean reverting points for the volatility of commodity volatilities). On the other hand, volatility explosions are not higher than 20.7% for the volatility of the OVX and 22.5% for the GVZ. Now we can improve on the list of buy/sell signals we previously mentioned (again these are not trading recommendations but only a general guide):

1) Long Gold or WTI Oil volatility when OVX’s volatility is around 10% – 11% or when GVZ’s volatility is within 13% – 14%

2) Condors or butterflies, should be entered when OVX’s volatility is higher than 19% – 20% and the GVZ’s one is above the 20% – 21% interval

3) If OVX and GVZ volatilities are oscillating around their median values some long volatility strategies are definitely safer than selling straddles or naked options. Hence, when the volatility of the OVX is around 13% – 14% and the volatility of the GVZ is in the 18% – 19% interval, it is a good risk management practice to hedge the portfolio against a potentially higher degree of market fluctuations

If you are interested in trading gold futures or options you might want to read our research “Trading Gold and Silver: A Realized Volatility Approach”

Instead, if you are looking to trade WTI oil futures or options you will find the research “The Oil Arbitrage: Brent vs WTI” very helpful

Equity Volatility Indices: VIX, VXN, VXD, RVX

The present research will go through the most popular equity volatility indices proposed by the CBOE. The aim of the research is to identify how they fluctuate and to quantify the magnitude of their movements. The actual study will be followed by a second one entirely focused on commodity volatilities and the examined asset classes will be Gold and WTI Crude Oil (the HyperVolatility Forecast Service provides market projections for the VIX, Gold, WTI crude oil and many other asset classes. Send an email to info@hypervolatility.com and get a free 14 days trial). The great attention towards volatility indices and the remarkable popularity gained by VIX futures are surely undeniable but what is the risk involved in these markets? What is the volatility index that best fits your trading style and portfolio needs? Do you know how the VIX performs compared to other volatility indices? The HyperVolatility team trades and analyzes volatility movements on a daily basis on different asset classes and the present study is a summary of some of our findings. The examined data sample goes from January 2011 until April 2013 and the present research will examine 4 equity volatility indices: the VXN (Nasdaq 100), the VXD (Dow Jones), the RVX (Russell Index) and the VIX (S&P500). The first chart displays the distribution ranking of each index and it is a good approximation of how volatile the aforementioned indices can be:

Volatility Indices - Distribution Ranking

The graph clearly indicates that, on average, almost all indices move within the 18% – 22% interval (the RVX is the only one above this range as its median is 23.43%). The lowest fluctuations, for all indices, are concentrated within the 9% – 12% range (the VXD is the only index showing a figure lower than 10%) while the wildest volatility explosions tend to group around the 45% – 48% area (the RVX again is the only outsider with a 58.84% peak). Statistically speaking, the most important intervals to consider are the 25%, the median and the 75% ones because they represent the most common spectrum of oscillation. Consequently, in case of low volatility most of the indices will tend to move within the 15% – 18% range (the RVX does not go below 20% though) while in high volatility environments equity volatility indices will tend to oscillate between 20% and 24% (the RVX is again the only outsider with 28.1%). The RVX is definitely the index showing the highest figures, which is normal given the fact that it tracks the volatility of less liquid stocks, but is it really the most volatile one? The answer is no because the fact that the RVX has an average value of 23.43% does not mean that it experiences the highest degree of fluctuations; it simply implies that the RVX is constantly higher than the others. In order to clarify this issue it is necessary to quantify the magnitude of the equity volatility indices’ movements. The next chart displays the volatility of the volatility indices. This time, however, the oscillations are expressed in monthly volatility because the CBOE indices are already annualized:

Volatility of Volatility

The chart evidently shows that the most volatile equity index, amongst those proposed by the CBOE since 2011 so far, is the VIX and not the RVX. The second most volatile is the VXD and the least volatile is the RVX. How is this possible? Wasn’t the RVX the index showing the highest values? The explanation is simple. The VIX and VXD indices have a higher variability rate because a fairly good amount of moves tend to be large, fast and powerful. On the other hand, the RVX has a lower variability because it tends to fluctuate more often around its mean value and consequently the dispersion is lower. The next graph, which plots the ranking of the dispersion rate for the volatility of CBOE equity volatility indices, will help to expand more this concept:

Volatility of Volatility - Distribution Ranking

The graph confirms what previously stated: the VIX and the VXD are, on average, the most volatile indices. The 3rd index is the VXN while the RVX is the one that experiences the lowest degree of diffusion. The dispersion of the VIX and VXD is the largest even for small and large moves indicating that these 2 volatility indices move a lot more quickly than the others. Almost all indices do not go over the 30% threshold but when large market moves happen the monthly volatilities reach 50% and in some cases, like the VIX, 60%. To sum up, the VIX and VXD are the most volatile indices while the VXN and RVX ranked 3rd and 4th respectively (it is worth reminding that the sample analyzed goes from January 2011 until April 2013). Financial markets have different risk dimensions and equity volatility indices make no exception. The next table tries to provide empirical evidence to quantify 1 of those dimensions: the intraday risk

Intraday Risk

Before getting started it is important to point out that the numbers reported in the table are volatility points. The intraday risk, on average, is around 1.35% for all equity volatility indices but the RVX reaches 1.5%. On the other hand, the table suggests that very large market moves can push equity volatility indices up or down by 13%, 14% or 15% in one single day.

However, one question, given the above mentioned results, would be particularly appropriate: wasn’t the RVX the least volatile index? If that is true, how come the intraday risk is higher?

The RVX index is more “dangerous” than other volatility indices as far as intraday risk is concerned. Nevertheless, even if the RVX fluctuates more wildly during trading hours its mean reverting pressure is higher in the medium term. In other words, the RVX is fairly volatile on a daily basis but it tends to mean revert towards its mean much more quickly than other indices. Furthermore, violent volatility bursts or aggressive drops tend to counterbalance themselves more effectively and in the medium term they result in a lower dispersion of the RVX with respect to its mean. The aforementioned phenomena explain why the RVX has a higher value in terms of volatility points but it is generally less volatile than the VIX or the VXD.

Conclusions

1) On average all indices tend to oscillate within the 18% – 22% interval

2) In case of low volatility, indices will tend to move within the 15% – 18% range while in high volatility environments the equity volatility indices will likely oscillate between 20% and 24%

3) The RVX is definitely the index showing the highest value in terms of volatility points but it is the less volatile because the mean reverting pressure is higher in the medium term

4) The most volatile equity indices, amongst those proposed by the CBOE, are the VIX and the VXD

5) The dispersion rate for the VIX and VXD indices is the largest even for small and large moves which means that they oscillate more quickly and frequently than others

6) Intraday risk is around 1.35% for all equity volatility indices but the RVX reaches 1.5%

 The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

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Options Greeks: Vanna, Charm, Vomma, DvegaDtime

The present article deals with second order Options Greeks and it constitutes the second part of a previously published article entitled “Options Greeks: Delta,Gamma,Vega,Theta,Rho”. Before getting started it is important to highlight the great contribution that Liying Zhao (Options Analyst at HyperVolatility) gave to this report. All the calculations and numerical simulations that will be shown and commented are entirely provided by Mr Zhao.

Second-order Greeks are sensitivities of first-order Greeks to small changes in different parameters. Mathematically, second-order Greeks are nothing else but the second-order partial derivatives of option prices with respect to different variables. In practical terms, they measure how fast first order options Greeks (Delta, Vega, Theta, Rho) are going to change with respect to underlying price fluctuations, volatility, interest rate changes and time decay. Specifically, we will go through Vanna, Charm (otherwise known as Delta Bleed), Vomma and DvegaDtime. It is important to point out that all charts have been produced by assuming that the underlying asset is a futures contract on WTI crude oil, the ATM strike (X) is 100, risk-free interest rate (r) is 0.5%, implied volatility is 10% while the cost of carry (b) is 0 (which is the case when dealing with commodity options).

Vanna: Vanna measures the movements of the delta with respect to small changes in implied volatility (1% change in implied volatility to be precise). Alternatively, it can also be interpreted as the fluctuations of vega with respect to small changes in the underlying price. The following chart shows how vanna oscillates with respect to changes in the underlying asset S:

Options VannaThe above reported chart clearly shows that vanna has positive values when the underlying price is higher than strike (in our case S>$100) and it has negative values when the underlying moves just below it (S<$100). What does that imply? The graph highlights the fact that vega moves much more when the underlying asset approaches the ATM strike ($100 in our case) but it tends to approximate 0 for OTM options. Consequently, the delta is very sensitive to changes in implied volatility when the ATM area is approached. However, it is important to point out that delta will not always increase if the underlying moves from, say, $80 to $100 because in many risky assets (stocks, equity indices, some currencies and commodities) the implied volatility is inversely correlated to the price action. As a result, if WTI futures go from $80 to $100 the implied volatility will probably head south and such a phenomenon would decrease vanna which, in turn, would diminish the value of delta.

Charm (or Delta Bleed): Charm measures delta’s sensitivity to a small movement in time to maturity (T). In practical terms, it shows how the delta is going to change with the passage of time. The next chart displays graphically the relationship between the aforementioned variables:

Options Charm

The chart suggests that, like in the case of vanna, the charm achieves its highest absolute values when the options are around the ATM area. Therefore, slightly in-the-money or out-of-the-money options will have the highest charm values. This makes sense because the greatest impact of time decay is precisely on options “floating” around the ATM zone. In fact, deep ITM options will behave almost like the underlying asset while OTM options with the passage of time will approach 0. Consequently, the deltas of slightly ITM or OTM options will be the most eroded by time. Charm is very important to options traders because if today the delta of your position or portfolio is 0.2 and charm is, for instance, 0.05 tomorrow your position will have a delta equal to 0.25. As we can clearly see, knowing the value of charm is crucial when hedging a position in order to keep it delta – neutral or minimize portfolio risk.

Vomma: Vomma measures how Vega is going to change with respect to implied volatility and it is normally expressed in order to quantify the influence on vega should the volatility oscillate by 1 point. The fluctuations of vomma with respect to S are shown in the next chart:

Options VommaAs displayed in the above reported chart out-of-the-money options have the highest vomma, while at-the-money options have a low vomma which means that vega remains almost constant with respect to volatility. The shape of vomma is something that every options trader should bear in mind while trading because it clearly confirms that the vega that will be influenced the most by a change in volatility will be the one of OTM options while the relationship with ATM options will be almost constant. This makes sense because a change in implied volatility would increase the probability of an OTM options to expire in-the-money and this is precisely why vomma is the highest around the OTM area.

DvegaDtime: DvegaDtime is the negative value of the partial derivative of vega in terms of time to maturity and it measures how fast vega is going to change with respect to the time decay. The next chart is a visual representation of its fluctuations with respect to the underlying asset S:

Options DvegaDtime

The above reported graph clearly displays that the influence of time decay on volatility exposure measured by vega is mostly felt in the ATM area especially for options with short time to maturity. The fact that DvegaDtime is mathematically expressed as negative derivatives makes sense because time decay is clearly a price that every options holder has to pay. In order to make things easier have a look at the plots of vega and theta because you will immediately realize that both volatility and time decay have their highest and lowest values in the ATM area. It goes without saying that ATM options have the highest volatility potential and therefore vega will be effected the most by the passage of time when the strike of our hypothetical options and the underlying price gets very close.

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Options Greeks: Delta,Gamma,Vega,Theta,Rho

First of all I would like to give credit to Liying Zhao (Options Analyst at HyperVolatility) for helping me to conceptualize this article and provide the quantitative analysis necessary to develop it. The present report will be followed by a second one dealing with second order Greeks and how they work.

Options are way older than one might imagine. Aristotele mentioned options for the first time in the “Thales of Miletus” (624 to 527 B.C.), Dutch tulip traders began trading options at the beginning of 1600 while in 1968 stock options have been traded for the first time at the Chicago Board Options Exchange (CBOE). The pricing of options has always attracted academics and mathematicians but the first breakthrough in this field was pioneered at the beginning of the 1900 by Bachelier. He literally discovered a new way to look at option valuation, however, the real shift between academia and business occurred in 1973 when Black, Scholes and Merton developed the most popular and used option pricing model. Such a discovery opened an entire new era for both academics and market players. Being one of the most crucial financial derivatives in the global market, options are now widely adopted as an effective tool to leverage assets or control portfolio risk. Nowadays, it is easy to find articles, researches and studies on option pricing models but this article will instead focus on options Greeks and in particular first-order Greeks (derived in the BSM world). Options Greeks are important indicators for assessing the degree of risk coming from exogenous variables, in fact, they measure option premium’s sensitivities to small changes in different parameters. Mathematically, Greeks are the partial derivatives of the option price with respect to different factors such as volatility, interest rate and time decay.

The purpose of this article is to explain, as clearly as possible, how Options Greeks work but we will concentrate only on the most popular ones: Delta, Gamma, Vega (or Kappa), Theta and Rho. It is worth mentioning that all the charts that will be presented have been extrapolated by assuming that the underlying is a WTI futures contract, that the options have a strike price (X) of 100, that the risk-free interest rate (r) is 0.5%, that the cost of carry (b) is 0 while implied volatility is 10%.

Delta: Delta measures the sensitiveness of the option’s price to a $1 fluctuation in the underlying asset price. The chart displays how the Delta moves in respect to the underlying price S and time to maturity T:

options delta

The chart clearly shows that in-the-money call options have much higher Delta values than out-of-the-money options while ATM options have a Delta which oscillates around 0.5. Call options have a Delta which ranges between 0 and 1 and it gets higher as the underlying approaches the strike price of the option which means that out-of-the-money call options will have a Delta close to 0 while ITM options will have a Delta fluctuating around 1. Many traders think of Delta as the probability of an option expiring in-the-money but this interpretation is not correct because the N(d) term in its formula expresses the probability of the option expiring ITM but only in a risk-neutral world.  In real trading conditions higher Delta calls do have a higher probability to expire ITM than lower Delta ones, however, the number itself does not provide a reliable source of information because everything depends on the underlying. The Delta simply expresses the exposure of the options premium to the underlying: a positive Delta tells you that the premium will rise if the underlying asset will trend higher and it will decrease in the opposite scenario. Put options, instead, have a negative Delta which ranges between -1 and 0 and the below reported chart displays its fluctuation in respect to the underlying asset.

put option delta

It is easy to notice that as the underlying asset moves below the $100 threshold (the strike price of our hypothetical put option) the Delta approaches -1, which implies that ITM put options have a negative Delta close to -1, while OTM options have a Delta oscillating around 0. In practical trading the value of the Delta is very important because it tells you how the options premium is going to change in the case the underlying moves by $1. Let’s assume you purchase a 100 call options on crude oil with a Delta of +0.5 and the premium was $1,000.  If the option is at-the-money the WTI (the underlying asset) will be at $100 but if oil futures go up by $1 dollar to $101 the premium of your long call will move to $1,500. The same applies to put options but in this case the ATM Delta will be -0.5 and your long put option position will generate a profit if WTI futures move from $100 to $99.

Gamma: Gamma measures Delta’s sensitivity to a $1 movement in the underlying asset price and it is identical for both call and put options. Gamma reaches its maximum when the underlying price is a little bit smaller, not exactly equal, to the strike of the option and the chart shows quite evidently that for ATM option Gamma is significantly higher than for OTM and ITM options.

option gamma

The fact that Gamma is higher for ATM options makes sense because it is nothing but the quantification of how fast the Delta is going to change and an ATM option will have a very sensitive Delta because every single oscillation in the underlying asset will alter it.

How Gamma can help us in trading? How do we interpret it?

Again, the value of Gamma is simply telling you how fast the Delta will move in the case the underlying asset experience a $1 oscillation. Let’s assume we have an ATM call option on WTI with a Delta of +0.5 while futures prices are moving around $100 and Gamma is 0.08, what does that imply? The interpretation is rather simple: a 0.08 gamma is telling us that our ATM call, in the case the underlying moves by $1 to $101, will see its Delta increasing to +0.58 from +0.5

Vega (or Kappa): Vega is the option’s sensitivity to a 1% movement in implied volatility and it is identical for both call and put options. The below reported 3-D chart displays Vega as a function of the asset price and time to maturity for a WTI options with strike at 100, interest rate at 0.5% and implied volatility at 10% (the cost of carry is set to 0 because we are dealing with commodity options).

option vega

The chart clearly highlights the fact that Vega is much higher for ATM options than for ITM and OTM options. The shape of Vega as a function of the underlying asset price makes sense because ATM options have by far the highest volatility potential but what does Vega really tell us in real trading conditions? Again, Vega (or Kappa) measures the dollar change in case of a 1% shift in implied volatility, therefore, an at-the-money WTI options whose value is $1,000 with a Vega of , say, 100 will be worth $1,100 if the implied volatility moves from 20% to 21%. Vega is a very important risk measure for options traders because it estimates how your P/L is going to change as a function of implied volatility. Implied volatility is the key factor in options pricing because the price of a single options will vary according to this number and this is precisely why implied volatility and Vega are essential to options trading (the HyperVolatility Forecast service provides analytical, easy to understand projections and analysis on volatility and price action for traders and investors).

Theta: Theta measures option’s sensitivity to a small change in time to maturity (T). As time to maturity is always decreasing it is normal to express Theta as negative partial derivatives of the option price with respect to T.  Theta represents the time decay of option prices in terms of a 1 year move in time to maturity and to view the value of Theta for a 1 day move we should divide it by 365 or 252 (the number of trading days in one year). The below reported chart shows how Theta moves:

option theta

Theta is evidently negative for at-the-money options and the reason behind this phenomenon is that ATM options have the highest volatility potential, therefore, the impact of time decay is higher. Think of an option like an air balloon which loses a bit of air every day. The at-the-money options are right in the middle because they could become ITM or they could get back into the OTM “limbo” and therefore they contain a lot of air, consequently, if they have got more air than all other balloons they will lose more than others when the time passes. Let’s look at a practical example. Let’s assume we are long an ATM call option whose value is $1,000 and has a Theta equals to -25, if the day after both the underlying price and volatility are still where they were 1 day before our long call position will lose $25.

Rho: Rho is the option’s sensitivity to a change in the risk-free interest rate and the next chart summarises how it fluctuates with respect to the underlying asset:

option rho

ITM options are more influenced by changes in interest rates (negative Rho) because the premium of these options is higher and therefore a fluctuation in the cost of money (interest rate) would inevitably cause a higher impact on high-premium instruments. Furthermore, it is rather clear that long dated options are much more affected by changes in interest rates than short-dated derivatives. The below reported chart displays how Rho oscillates when dealing with put options:

put option rho

The Rho graph for put options mirrors what it has been stated for calls: ITM have a larger exposure than ATM and OTM put options to interest rate changes and long term derivatives are much more affected by Rho than in the short term (even in this case the 3-D graph displays negative values). As previously mentioned Rho measures how much the option’s premium is going to change when interest rates move by 1%. Hence, an increase in interest rates will augment the value of a hypothetical call option and the rise will be equal to Rho. In other words, the value of the call option will increase by $50 if interest rates move from 5% to 6% and our WTI call option has a premium of $1,000 but Rho equals 50.

As stated at the beginning of the present report this is only the first part and a second article dealing with second order Greeks will be posted soon.

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