In the previous part of the overview we have discussed Greek coefficients for options (the Greeks) and explained why they are useful for successful trading. We have shown, however, that different trading programs calculate these indicators in different ways. In the best cases, the values differ by several percent fractions, while in the worst cases they may differ by times. Why do these divergences appear, and which trading terminal calculates everything correctly? To answer this question, we should first of all turn to the mathematics of the option models and understand the general idea behind them.

Trading software usually calculates the Greeks be the Black-Scholes model (later referred to as BS, link to Wikipedia.) It is based on a differential equation that allows deriving the dynamics of an option price given that we know several other values: the strike price, the price of the underlying asset, the future volatility of the underlying asset, risk-free interest rate and time to expiration. Originally, this model was developed for the European options on non-dividend equities, later being adapted for dividend stocks.

1. How to find necessary parameters to calculate Greeks? Historical and implied data

If we plug all necessary values into the Black-Scholes equation correctly, we get the price of the option at any given moment of time. It means, this equation is supposed to show the current fair, or theoretical, price of the option, so we could compare it to the current market price.

To get a hands-on experience, let’s use a ready-made BS calculator found here: https://www.mystockoptions.com/black-scholes.cfm. It allows defining the theoretical price of an option based on five values: the current price of the underlying asset, strike price, time to expiration, risk-free rate and the implied volatility (IV.)

This all is not that easy, however. If it were, we could just calculate the ‘fair’ price of the options, buy the underestimated ones and get profit — just as simple as that. There are two issues that make it more complicated, though.

The BS model is highly idealized. First of all, it is implied that the swings of the underlying asset price are purely accidental resembling, say, the Brownian movement. In the reality it does not work that way: the price reflects the current economic state which is, in its turn, affected by scientific discoveries, political issues and natural factors. The recent Brexit referendum, for example, is by no means factored in the BS model.
Even if the BS model were absolutely precise, it would be very difficult to find the exact data for its calculation. Some of the included values are pretty clear, like the strike price and the expiration date, but it also requires more specific data like the risk-free interest rate, dividends (if they are taken into account) and the future volatility. These data are quite ambiguous, and this ambiguity leads to the problems in calculating the BS equation result.

The first issue is fundamental, and it can be resolved only if we develop a detailed and precise behavioral pattern for every person, which sounds, as you might agree, more like a utopia. We can, though, gradually increase the precision of the models we have. Mathematics and economics are constantly developing, and more and precise (and complicated) models for price prediction appear. The BS model was developed in the 1970s, and today there are some more accurate alternatives. They are used, however, only by the most keen option traders capable of programming. In the future these models might as well be implemented in the regular trading software, but today trading platforms use only the Black-Scholes as the branch standard.

The second problem is much more contradictory. How to obtain the necessary data? There are two polar approaches to solving this problem.

The first approach (let’s call it historical) is to get all data in the natural form ‘as is’ — from official sources, financial websites and other sources of the historical market data. The interest rate, for example, can be derived from the state bonds interest rate or currency forwards. The information about dividends is available on, say, finance.google.com. The volatility can be calculated upon the statistics of the underlying asset quotes collected during several months by finding the standard deviation of the price and dividing it by the average price during the period. This is the way we have calculated the volatility in the previous part to estimate the projected profits. It was very rough, though.

The historical approach has several serious drawbacks. The volatility does not have to remain the same throughout the times, so using the historical volatility for forecasts can be compared with assuming that the revenues of a company grow at the same rate from year to year. This method is good to use only when there are no better alternatives. The same can be said about the dividend prediction: we know only historical data. Of course, there are companies that pay constant dividends every year, but most of them still do not. This is not to mention estimating dividends of an index that depends on a number of companies. The risk-free interest rate is also not that easy to assess, while it can hinge on different factors and change with the time. These uncertainties all lead to the ambiguous estimates of the Greeks.

The second approach (we call it evaluative) implies that the data are not just taken from financial websites, but are calculated using other, more reliable data — like the current option prices. We apply the following logic here: if all market players buy and sell the options at the given prices, they have some certain assumptions. There are several mathematical methods that provide an insight into the market players’ assumptions concerning the volatility and other factors with the help of market data analysis. These insights may be more precise than the clumsy attempts to extrapolate historical data.

Let’s take a closer look at this approach. We will calculate the volatility which will be called ‘implied.’

2. What is the implied volatility?

As we have said before, the BS model allows calculating a ‘fair’ option price if we know the volatility and other figures. If the volatility is unknown, we can, however, suppose that the current market price of the option is ‘fair’, plug it into our equation and find the volatility. This is the way the implied volatility (IV) is calculated.

Let’s take a practical example. Open the BS calculator: https://www.mystockoptions.com/black-scholes.cfm. Online calculator for option prices based on the BS model (https://www.mystockoptions.com/black-scholes.cfm)

It was created to calculate the theoretical price of a call option (Black-Scholes Value) given several values including the volatility. Let’s suppose, however, that we do not know the volatility, but the market price of this option is known. What if we assume that the market price is equal to the theoretical price and try to find the volatility to make up the right equation?

Let’s take one of the screenshots from the previous part. Options on S&P 500 index with expiration in July, 2016 on EXANTE Option board

Take a look at the call option with strike price 2,180 and suppose that its market price (\$0.65, the average between Bid and Ask prices) is fair. Let’s enter these data into the calculator:

• Stock Price: 2,068.
• Exercise Price: 2,180.
• Time to maturity: 0.082.
• Annual risk-free rate: 0.5% (this is a random value, but it is not significant for such short time periods.)

We will try to guess the volatility that will give us the price at the level of \$0.675. First, let’s try 10%. The Black-Scholes Value will be \$0.809 – it is too high. The market players probably expect a lower volatility, e.g. 9% With 9% we get \$0.428 – it is too low. 9.69% is a perfect match: the Black-Scholes Value equals to \$0.675. The Black-Scholes option price calculator deriving the implied volatility. We know the market price of the option (\$0.675,) and we have guessed that the correct volatility is 9.69%: this is the value that makes the Black-Scholes Value equal to the market price.

So, we have calculated the volatility. What is the economic sense of this value? This is the volatility expected by the market players, including those who use the most complicated predicting methods. By calculating the implied volatility, we act as if we were requesting the expert estimates of the asset’s future volatility from the trading community.

Calculating the implied volatility instead of the historic value is not a unique approach. This method is widely used in different trading terminals, including EXANTE, Think or Swim and Interactive Brokers. Evidently, we cannot get a fair price of an option with its help, as we suppose it is equal to the market price. However, now we can calculate the Greeks by plugging the new value into the Black-Scholes equation as σ (sigma.)

3. Okay, it is clear now how to get the volatility. What about other unknown values?

We have shown how to calculate the most problematic parameter of the Black-Scholes model, the future volatility of the underlying asset price. It becomes more of a psychological value than statistical or forecasting, though, and this fact should always be kept in mind (we will come back to it later.) Another question is where to get all other data.

One of the major problems, the volatility, is solved. There are, however, some other issues that can also spoil our calculations. These issues include finding the dividends, interest rate, and sometimes even the current market price of the underlying asset. Wrong calculation of these parameters will lead to mistakes in the estimation of the volatility and Greeks. Just reminding you of the divergence between the VIX (implied volatility index) value in different trading platforms:

 Type (C or P) and strike price Delta in Think or Swim Delta in Interactive Brokers Delta in EXANTE Volatility in Think or Swim Volatility in Interactive Brokers Volatility in EXANTE P16 -0.53 -0.34 -0.33 45% 73% 71% P14 -0.26 -0.17 -0.17 44% 61% 64% C30 0.16 0.16 0.17 128% 110% 108% C24 0.27 0.28 0.28 122% 100% 97% C20 0.39 0.42 0.42 116% 89% 85% C16 0.57 0.66 0.67 115% 73% 71% C14 0.67 0.83 0.83 123% 61% 64%

Probably, the difference in the results was caused by using non-identical methods to calculate the values listed above.

It is clear that estimating the dividends is a problem. As we have mentioned above, many companies do not stick to a strict schedule when paying the dividends out, while the size of the payments is even harder to predict. Formally, we could use the Dividend yield column on the finance.google.com portal, but the data stated there sometimes do not correspond with the reality. Moreover, these data show only the past, not saying anything about the future.

Another issue is the risk-free interest rate. What is it? Well, roughly speaking, it is the guaranteed interest you could get on your savings if you made a deposit in the most reliable bank ever, or bought some absolutely risk-free bonds. When an investor assesses the profitability of this or that option, he does not compare it to zero profitability: he compares it to the risk-free interest rate. The question here is, what is better, a risk-free bond or an option?

Absolutely reliable banks and bonds exist, unfortunately, only in the perfect world. If we choose from what we really dispose of, we will find out that not all variants are equally accessible by all investors. Those that are accessible are constantly changing the interest rate during the life-time of the option.

Moreover, different investors have different reliability requirements. Some imply that the amount of the invested funds should not change under any conditions. It is often said that the reference reliability is available at central banks, but they provide very low, if not negative, profitability. E.g., the European Central Bank offers –0.75% (!) per annum. Less meticulous investors can find regular commercial banks’ 1-2% per annum satisfactory. If an investor is aimed at very long-term deposits, like retirement funds, he could orient on the profitability of the major market indices (especially the US ones), like the S&P 500 — on average, it comprises 8-11%, but every single year it may rise or fall, so such assets are not usually considered risk-free. Our goal is to understand the real behavior of investors, though — in particular, to understand what they really call a risk-free interest rate. As we can see, they all think differently.

The calculator does not actually have a ‘Dividends’ field, but it has an ‘Interest rate’ field. For the options with short time to expiration it is not critical. However, if we take, say, the option on the S&P 500 index with expiration in December, 2018 and play with the interest rate, we will get significantly different results.

Let’s take a look at the screenshot from the first part again. There we can find the implied volatility (IV) column. This is the value from the EXANTE terminal. Take into account that it is different for different strike prices, notwithstanding the fact that we are talking about the volatility of one and the same underlying asset. Different market players asses it in different ways, and it is very important: we will discuss it a bit later. For now, let’s just study the option with strike price 2,100. Options on the S&P 500 index with expiration in December, 2018 in EXANTE trading platform. The IV column shows the implied volatility.

Let’s get back to the Black-Scholes calculator and enter all values we know except the volatility and the interest rate. The index is now 2,161, the strike price is 2,100, and the time to expiration is 2.42 years. Let’s find such volatility and interest rate that will produce the market price equal to \$241. Then let’s see how the volatility will change depending on different interest rates.

• If the interest rate is 8% (S&P 500), the volatility should be less than 1%. It is evident that it cannot be true.
• If the interest rate is 2% (typical banks), the volatility is around 11%. It is more likely, but still too far from the EXANTE’s result (18.59%.)
• If the interest rate is 0.5% (very moderate), the volatility will be around 15%.
• If the interest rate equals 0.01% (which is very low), the volatility will be 16%.
• If we try to take a negative interest rate, –0.75% (like the one provided by the European Central Bank), we will fail: the calculator refuses to count the value, as ‘all field values must be valid numbers greater than zero.’

It seems that the creators of this calculator could not foresee the market conditions we are currently observing. The same cannot be said about EXANTE developers, however: they have adopted the new reality and provisioned for the negative interest rates. Nevertheless, we have seen that a wrong interest rate used in calculations can lead to serious mistakes in the results.

The last issue we can encounter when calculating the volatility and Greeks is while calculating the current market price of the underlying asset. In some cases it is absolutely clear: take, say, S&P 500. However, it is not the same, for example, for the oil. The current oil price reflected in the mass media is represented not by the spot contract prices, but by the futures. The price of the oil here and now is more of an ephemeral value, while it is impossible to deliver oil barrels on the same day. Moreover, in some cases even the futures price is unknown: you can find an option on some asset, while the futures with the same expiration date may not exist.

There are different ways to avoid these issues when calculating the analytical values in trading software. As we have seen before, all these ways lead to different results of the calculation. In our opinion, the truly elegant and coherent approach is adopted by the mathematicians and programmers of EXANTE, and it’s worth being discussed in detail.

4. Rigorously market-oriented approach: ‘Everything is implied!’

As we have already mentioned before, using the implied volatility instead of the historical one is a well-known and widely used method. We can take a step forward and apply this approach not only to the volatility, but to all other values. And this is the way EXANTE does it.

Mathematicians from EXANTE have developed an elegant and almost ‘telepathic’ methodology based on the alternative Black-Scholes model that allows extracting all necessary data ‘from traders’ minds’. All these data are calculated as implied: implied forward price, implied volatility, implied risk-free rate, etc. The only input values are the ones that are unconditionally clear: prices of put and call options, their strike prices and the time to expiration. As a result, the Greeks they obtain are as close to the real market values as it is possible in comparison with other calculation methodologies that work with ready-to-use data about the dividends, etc.

Why is the alternative Black-Scholes model convenient? First of all, it helps getting rid of the dividends, even if they are paid. Secondly, it does not require the current price of the underlying asses, using the forward price instead, which usually coincides with the futures price. The only problematic parameter used is the interest rate. In EXANTE, they found a way to obtain all these values from the market data without using government websites or financial portals like Bloomberg.  They imply that there is only the market, and there are methods that allow deriving the implied interest rate and the forward price (if the real value is not available) from the current market data, and then they use these data to find the implied volatility and the Greeks.

Those who are acquainted with the methodological issues may now have two objections, so we have to explain the approach in more detail.

• Firstly, it is not clear how the algorithm can calculate so many unknown parameters based only on the few input values. The Black-Scholes calculator, for example, contains five parameters, one of which can be calculated if the other five are given. If we do not know two parameters, say, the volatility and interest rate, we will be unable to calculate anything.
• Secondly, if we assume that the option price is fair (is equal to the theoretic price) and calculate not only the volatility, but all other parameters on its basis, the real value of such data is questionable. First of all, we lose the opportunity to compare the market and the fair prices. Secondly, we lose cannot use the Greeks to estimate the profitability of the option on expiration: if the current price of options is fair, all options become equally profitable — or, to be more specific, their profitability is equal to zero. Moreover, as we can see from the screenshot above, traders estimate the volatility of one asset in different ways. Those who are fond of high strike prices think that the volatility will be lower than in the case of low strikes. Even for the volatility this approach is quite contradictory, isn’t it? The situation becomes even worse when we start calculating everything in the described way.

Both issues are quite serious and we cannot just dismiss them.

The first issue is quite simple. This contradiction could be right if we were talking about a single option with a given strike price and expiration date. In this case we would know only four parameters (call price, put price, strike price and the time to expiration,) and it would be impossible to find the volatility and Greeks. However, we analyze the whole set of options, and this fact is crucial.

What we dispose of is not only a selection of four values, but an enormous array of input data — option prices for different strike prices and expiration dates. This all comprises the market data. At the same time, there are not so many unknown values: the interest rate is the same for all options, the asset price is the same as well. From the formal mathematic point of view, there is enough data to achieve more or less precise results: the number of the known parameters is significantly larger than the number of the unknown ones.

The second issue is closely tied with an important peculiarity that is often hard to understand for those who are trying to learn how the Black-Scholes model works. The BS model is anything but a prognostic wonder. If the current option price is fair, distant forecasts do become senseless, as is the traders’ communal intellect deficient, which is shown by the dispersion of the volatilities they imply. When we calculate the implied values, we do not get any real forecasts of the future; these implied values give us only an idea about traders’ opinions — not even facts!

Traders’ actions are, however, based on their opinions, and this is the reason why they are so important to us!

The Greeks calculated with the implied data indicate the mood if the market. It may be irrational or even dull, but they are real. That’s why they are useful for us.

• The wrong approach is to calculate Greeks using the implied values first, and then to calculate the probability that the options will be executed in the money.
• The correct approach is to calculate the Greeks first, and then to predict the reaction of traders to the changing market conditions.
• For example, if we use the Delta from EXANTE Option board to calculate the probability that the option will be executed in the money, we are very likely to get a wrong result.
• If we use Delta as intended — to calculate the change in the option price depending on the change of the underlying price, however, the obtained value will be much more reliable.

• For example, we can ask, ‘What do you think, will this option be executed in the money?’ He will answer what he thinks, but he, obviously, cannot know for sure. The forecast we get is just a guess.
• We can ask, however, ‘At what price will you sell the option if the price of the underlying asset increases by \$10?’, and the answer to this question will be much more useful, as it becomes not just a pure guess, but quite a solid plan which is likely to be followed.

The second approach is exactly what EXANTE is doing. It does not just make predictions, but analyzes the traders’ psychology in the least controversial way. As a result, it obtains values that reflect the intentions of traders to set a particular price depending on the market conditions.

1前街，伦敦
EC2Y 9DT