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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

 

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