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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.

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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

 

The Pricing of Commodity Options

The present research will prove particularly useful to option traders. The analysis proposed by the HyperVolatility Team will explain, in a few bullet points, how the most popular commodity options pricing models behave and what the practical divergences in terms of prices are. The present study is very valuable to anyone interested in trading options because most trading platforms allow the trader to choose the model via which the theoretical value of the options will be calculated and consequently shown (the HyperVolatility Forecast Service provides market projections for many asset classes. Send an email to info@hypervolatility.com and get a free 14 days trial). The pricing models that will be analyzed are the Barone–Adesi –Whaley, the Bjerksund & Stensland (the 2002 version), the Black–76, the Binomial Tree and the classic Black–Scholes–Merton one. The models have been tested against each other and the following charts graphically show the divergence of 1 pricing model with respect to all others. The research has been performed assuming that the underlying asset (S) is a WTI crude oil futures contract, that the volatility (σ) is 20%, that the interest rate (r) is 0.5% and that the Cost of Carry is 0 (which is normal when dealing with commodity options).

As previously mentioned, the study will examine 1 pricing model at the time and, in order to avoid confusion and make things simpler, we decided to list the most important aspects below each graph:

Barone Adesi Whaley model

1) The Barone-Adesi-Whaley model overprices options when compared to other formulas. The pricing spread with respect to other models is on average between 0.06% and 0.08%

2) The Barone-Adesi-Whaley prices tend to get closer to other models as the expiration increases

3) The Barone-Adesi-Whaley model, on average, tends to overprice options with respect to the Binomial Tree (~ 0.16% higher) for short maturities. The trend is higher for out-of-the-money options and particularly for put options

4) The prices derived from the Bjerksund & Stensland model are always lower than Barone-Adesi-Whaley prices. The difference is bigger for 1 month options (~ 0.16%)

5) The Black-76 performs as well as the Black–Scholes–Merton model, however, their results overlap and that is why the Black-76 curve is not visible

6) The difference with the Black–Scholes–Merton model becomes larger as the expiration increases but it is not higher than 0.1%

 

Bjerksund & Stensland model

1) The Bjerksund & Stensland model under–prices options in respect to other models. On average the difference ranges between 0.05% – 0.06%

2) The under–pricing tends to reduce as the expiration increases

3) The Bjerksund & Stensland  model produces prices which are lower than the Barone–Adesi–Whaley one for any expiration

4) The Black–Scholes–Merton model approximates to the Bjerksund & Stensland one from the 8th month onwards

5) The Black–76 performed as well as the Black–Scholes–Merton model and that is why the overlapped curve cannot be seen  in the chart

6) The Binomial Tree approach shows the highest differential with respect to the Bjerksund & Stensland model. The divergence in pricing oscillates around 0.15%

 

Black-76 model

1) The Black–76 model over–prices options only with respect to the Bjerksund & Stensland one (almost 0.05%)

2) The divergence between Black–76 and Bjerksund & Stensland attenuates when longer expirations are approached

3) The Barone–Adesi–Whaley model prices are slightly higher than Black–76 ones and the discrepancy augments with the passage of time (between 0.08% and 0.1% for 10 months and 1 year expiring options respectively)

4) The Binomial Tree approach, if we exclude the short term, delivers higher prices than the Black–76 model but the divergence oscillates around the interval 0.03% – 0.04%

5) The Black–76 model performed as well as the Black–Scholes–Merton one and that is why the BSM curve is flat to 0. Needless to say that the Black–Scholes–Merton curve suggests that there is no difference in pricing

 

Binomial Tree model

1) The Binomial Tree under–prices options with respect to other models in the short term (around 2.5%) but the divergence is much lower for longer dated derivatives

2) The Barone–Adesi–Whaley model and the BSM model perform as well as the Black–76 one therefore their curves are hidden in the chart

3) The Bjerksund & Stensland model provided higher prices for short dated options but in the long term the Binomial Tree approach shows a slight over–pricing tendency with respect to the former. However, the spread is no higher than 0.04% – 0.05%

 

Black Scholes Merton model

1) The performances of the Black–Scholes–Merton formula with respect to other pricing models match perfectly well with the outcome generated by the Black–76 model

2) The above reported chart is identical to the graph extrapolated for the Black–76 model, in fact, the green curve does not move from the 0 axis

If you are interested in trading options you might want to read also the HyperVolatility researches entitled “Options Greeks: Delta, Gamma, Vega, Theta, Rho” and “Options Greeks: Vanna, Charm, Vomma, DvegaDtime”

The HyperVolatility Forecast Service enables you to receive the 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.

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.

The HyperVolatility Forecast Service enables you to receive the 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.

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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