Financial market

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Contents

Introduction

Literature review

1. Sampling procedure

2.Implied volatility forecast

2.1Theoretical base for estimation

2.2Possible drawbacks and assumptions of the BS formula

3.Comparison of historical and IV volatility forecasts

3.1Forecast based on the implied volatility

3.2Forecast based on GARCH(1,1)

3.3Mixed strategies

3.4Comparative Statistics

Conclusion

References



Introduction

There is no doubt that nowadays processes in the financial markets are characterized by the existence of a high level of uncertainty and complexity. This is the reason why economists during several decades try to analyze and create some tools in order to decrease this extent of uncertainty and try to predict future movements of the market.

One of these tools, which is widely used nowadays, is to use market of derivatives, in particular, options market, in order to infer the degree of uncertainty about future prices of the underlying asset. In particular, in this paper the author will concentrate on the option-implied volatility and its' ability to predict future movements.

So this paper is dedicated to the subject of financial market efficiency and, namely, to the topic of option-implied volatility for the prediction of the underlying asset volatility. The thesis of the given paper is that implied volatility helps to predict ex-post realized volatility and can serve as an indicator of important economic events.

As for the motivation of the paper, the concept of capital market efficiency is central to the sphere of finance. Market efficiency represents one of the most crucial issues in finance because it implies that prices fully reflect all available information on the market.

The fact that some of the algorithms for predicting the volatility of the underlying asset were developed by the author and also the fact that there is no previous research in academic literature dedicated to the implied volatility on the RTS Index show the novelty of the research.

In view of the aforesaid, the practical applicability of the research is that the result obtained in this paper can be used in forecasting and analyzing the volatility of one of the most traded Russian indexes.

The purpose of this paper is to verify whether option-priced volatility is a good tool for forecasting. In order to achieve this purpose, particular goals of the research have been formulated:

· To verify whether implied volatility predicts future volatility of the underlying asset in statistical sense;

· To test whether it predicts efficiently and without bias;

· To test IV forecast in combination with autoregressive conditional heteroscedasticity (GARCH) models;

· To compare the results among different methods and choose the best one.

Finally, the finding of the paper will represent the reasonability to use implied volatility in the forecating of the volatility of the underlying asset.

The rest of this paper is organized as follows. Firstly, there will be a literature review where the author highlights the main articles dedicated to the predictive efficiency of the implied volatility for different financial instruments. Then the main part of the research will be divided into 3 components. In the first part, the data sampling procedure will be described. The second part will be dedicated to the implied volatility estimation process. And the last part will be devoted to different methods of forecasting of actual volatility and the examination of the statistical accuracy among them. Finally, the author will discuss the obtained results and choose the best model for predicting volatility of the RTS Index.



Literature review

Option-implied volatility for several decades remains subject to long-standing controversy. Fischer Black and Myron Scholes (1972) were the first economists who decided to analyze the efficiency of options market. They derived a formula for the estimation of the European option price, which is widely applied in all subsequent research dedicated to this topic.

Depending on different sampling bases and various forecasting techniques academicians have received mixed results. At one end of the spectrum, some researchers suppose that due to a number of restrictions assumed by Black and Scholes model historical volatility performs better than implied volatility. To illustrate, C. Doidge and J.Z. Wei (1998) received this result for the Canadian stock market. They concluded that nonsimultaneity of prices, unfamiliarity of the market, and a noncompetitive trading environment leaded to bad performance of IV estimator. Moreover, the similar research was done for the American market. Canina and Figlewski (1993) and Day and Lewis (1992) showed that for S&P 100 stock index implied volatility does not contain any valuable information. They found out that possible reasons may include model misspecification and expiration day effects. In addition to this, Treasury bill rate may not be a good proxy for the relevant rate facing by an options arbitrageur.

In contrast, some of the economists are sure that implied volatility forecast is superior to a time series volatility forecast. For example, the research done for the same S&P 100 stock index by B.J. Christensen and N.R. Prabhala (1998) revealed that the implied volatility outperforms historical volatility. They explained efficiency and unbiasedness of their results by using nonoverlapping sample. Another paper dedicated to the American market, C.J. Neely (2005), revealed a strong coincidence for three-month eurodollar interest rates between changes in implied volatility and major economic events.

In order to avoid the inaccuracy in the estimation of the model based on the implied volatility, the author takes into account the experience of the previous researchers, namely:

· the data is accumulated on the nonoverlapping basis;

· only the closest expiration term of the option series is used in the analysis;

· the prices of option contracts and the underlying index are taken at the end of each trading day.



1. Sampling procedure

In this paper the RTS Index is chosen as the underlying asset because of the number of reasons. First of all, the majority of researchers, who studied the predictive efficiency of the implied volatility, concentrated on the American and European markets, while the author has not found similar studies for Russian financial instruments. Secondly, the RTS Index is one of the most representative indexes of Russian market. It is a free-float capitalization-weighted composite index of Russian stocks of the largest and dynamically developing issuers traded on the Moscow Exchange. The stocks included in the RTS Index are the 50 most liquid stocks, which are reviewed on a quarterly basis [Moscow Exchange website]. Another reason of that choice for the underlying asset is that options on RTS Index futures contract are plain vanilla, which is one of the assumption of the Black-Scholes model.

The RTS Index was launched on September 01, 1995. The Index is calculated on a real time basis and denominated in US dollars. Speaking about the options on the RTS Index it is important to highlight several characteristics:

· They are futures-style options on RTS Index futures contract;

· Expiration occurs on the third Thursday of every quarter (March, June, September, December). It is important to say that on the Moscow Exchange there are RTS options that expiry quarterly, monthly and even weekly, however, in the current research only options with quarter expiration are taken into account because of their high liquidity;

· They have American style meaning that they allow option holders to exercise this instrument at any time prior to its' maturity date.

For the given research the author has chosen daily data for the RTS option prices for the 3-year period from the January 5, 2014 till December 28, 2017. This chosen interval is large enough in order to estimate and test the estimated results, and moreover, it covers the financial crisis of 2014-2015 in Russia, which is interested for testing the predictive power of the implied volatility.

Moreover, there is need for measure of the risk-free rate that investors face. In this paper the usual technique will be used. As a proxy for the risk-free rate author will use daily values for 1-month MosPrime Rate. This rate is calculated by the National Foreign Exchange Association together with Thomson Reuters based on the offer rates of deposits denominated in rubles as quoted the leading money market participants to the superior financial institutions [The Central Bank of the Russian Federation website].

To mitigate the possible bias in further estimation related to nonsynchronous trading the limitations will be chosen for the option closing prices and days remaining to maturity.

2. Implied volatility forecast

2.1 Theoretical base for the estimation

According to the definition, a European style call (put) option is a right, but not an obligation, to purchase (sell) an asset at a strike price on option maturity date [Poon and Clive W. J. Granger (2003), p.485]. As already been mentioned, for pricing such options the well-known Black-Scholes formula derived from the partial differential equation will be used [F.Black and M.Scholes (1973)]. The main idea of the Black-Scholes partial differential equation, which describes the option price over time is that one can totally avoid risk by hedging the option by trading the underlying asset [J.C. Hull (2008), pp. 287-288]:

, where(1)

V (S,t) - the price of an option as a function of time and stock price (C is the notation for a call option price, P is the notation for a put option price);

- the time in years;

S - the price of the underlying asset;

r - the annualized continuously compounded risk-free interest rate;

- the volatility of the underlying asset.

According to the Black and Scholes model and using Itф's lemma, logarithm of the price of the underlying asset should follow this dynamic:

(2)

In other way dynamic can be described by the following formula:



, where(3)

lnS_t - natural logarithm of price level of the underlying asset;

lnS₀ - natural logarithm of initial price level of the underlying asset;

м - mean value;

у - standard deviation;

W_t - the standard Wiener process (Brownian motion).

It means that the logarithm of price of the underlying asset follows a normal distribution with parameters and . Therefore, in further research this point should be taken into account in order to modify data from price levels of the underlying asset to natural logarithms of them.

However, turning back to the financial instrument chosen for this research, namely, Futures-style options on RTS Index futures contract, one should point out several necessary extensions to the standard Black and Scholes model.

The first moment that should be mentioned is that in this paper the author considers American style option. The main difference of such a type is that it can be exercised on the maturity date or at any time t prior to maturity of the contract . Therefore, the aforesaid equality becomes an inequality of the following form:

, where(4)

- the payoff of the option when the price of the underlying asset is S.

Another extension to the Black and Scholes model is the futures contract as the underlying asset.

This modification was presented a bit later by Fischer Black, who replaced the spot price of the underlying asset by the discounted futures price F [F. Black (1976)]:

(5)

The model discussed above allows to determine the option price as a function of a number of variables:

· F_t - the current price of the futures contract;

· X - the strike price of the option;

· r - the risk-free interest rate;

· T-t - time remaining to maturity;

· - volatility of the underlying asset over the time remaining to maturity.

One can notice that F_t , X, r and (T-t) are observable on the market. Moreover, the market price for this option is also a known value: it does not matter whether it is just a quote or whether there was a transaction. Nevertheless, in this research the author uses the prices of the real trades in order to obtain more relevant results. Using a backward induction, one can derive the volatility introduced in this formula that market took as an input for the quote. So this market's expectation volatility is called the implied volatility and this is the parameter that the author of the current paper is interested in.

According to the Black and Scholes model with the previously mentioned extensions, the formulas for the call and put option prices are as follows:

(6)

, where(7)

and



- call option price;

- put option price;

N(d) - standard normal distribution function.

The author assumes informationally efficient market, consequently the market price and the calculated according the formulas theoretical price should be equal. This assumption allows to derive the implied volatility:

and .

On the one hand, it seems that the implied volatility should contain much valuable information about the future volatility as it reflects market participants' expectations and investors' sentiments. Therefore, it is expected to be an efficient predictor of the actual index proce movements. Nevertheless, forecast based on any theoretical model requires a number of assumptions to hold, which is difficult to provide in reality. Moreover, one should draw his attention to the fact that market driven pricing irregularities may lead to some inaccuracies in the modeling procedure. These drawbacks of using model-based implied volatility will be explained in the next subchapter.

2.2 Possible drawbacks and main assumptions of the Black and Scholes formula

The reason why the forecast built on the implied volatility may lead to irrelevant results is a number of assumptions of the Black-Scholes model that should hold [F.Black and M.Scholes (1973), p.640]. The "ideal conditions" mentioned in this article include:

· volatility of the underlying asset, у, is constant;

· risk-free interest rate, r, is known and constant over time;

· underlying price follows the log-normal distribution;

· index pays no dividends;

· there are no transaction costs or taxes;

· underlying asset should be divisible;

· there are no restrictions for short-selling;

· there is continuous trading without arbitrage.

These are "ideal conditions" and violation of any of them will result in some inaccuracy in the estimated theoretical price. However, these assumptions are strict enough, and some of them seem unrealistic and are rarely met in reality.

To illustrate, the author will concentrate on the first assumption about constant volatility. It implies each underlying asset can have only one volatility.

Nevertheless, what one could usually see in reality is so-called "volatility smile". This phenomenon holds when options with different strikes but with the same time to maturity give different values for implied volatility for the same underlying asset. Options either deep in-the-money or out-of-the-money tend to produce higher values for the implied volatility.

Based on the data sample for RTS options it is possible to demonstrate how "volatility smile" looks like in reality. As mentioned earlier, RTS options expiry on the quarter basis (on the 3-d Thursday of March, June, September). So, it is interesting to see how the volatility behaves for different strikes, for example, a month prior to expiration. The graphs below show the dependency between the implied volatility and strikes. On the horizontal axis there are shifts from the central strike, or in other words, the strike closest to the settlement price of RTS futures. The tick size for the RTS option contract is 2 500, so shift +1 means the central strike + 2 500.

The "volatility smile" for call RTS options:



Picture 1. Volatility smile of the call RTS option on the 16-th of November, 2017

Picture 2. Volatility smile of the call RTS option on the 15-th of May, 2017

The "volatility smile" for put RTS options:

Picture 3. Volatility smile of the put RTS option on the 16-th of November, 2017

Picture 4. Volatility smile of the put RTS option on the 15-th of May, 2017

Poon and Clive W. J. Granger (2003) and B.J.Christensen and N.R.Prabhala (1998) highlight 4 possible reasons why this puzzle may take place.

Distributional assumption. According to Black and Scholes model, price of the underlying asset should follow the lognormal distribution (3). However, there is a wide range of academic papers that show that in real world there are usually leptokurtic tails [R.C.Blattberg and N.J. Gonedes (1974) and Fama (1965)]. This results in overestimating of the implied volatility at very low and very high strikes.

Stochastic volatility. Another possible situation is when the volatility of the underlying asset has its' own dynamics and its' own volatility. So in order to avoid this type of problem researchers tend to use at-the money options for forecasting.

Market Microstructure and measurement errors. The lack of ideal trading environment, which include no-arbitrage, zero transaction cost and continuous trading conditions, leads to the situation when options are traded with some deviation from their theoretical price. So, this is situation when the assumption about informationally efficient market does not hold.

Investor risk preference. In the Black and Scholes model investor risk is irrelevant in option pricing. However, risk preference influences option prices, which in its' turn, influence the volatility.



3. Comparison of historical and IV volatility forecasts

3.1 Forecast based on the implied volatility

Forecast based on the implied volatility is market based volatility forecast as it reflects expectations of market participants. It means that it tend to represent an ex-ante measure. In order to compute the ex-post volatility over remaining time to maturity the sample standard deviation for daily index returns will be calculated using the standard way:

, where(8)

- the mean RTS index return;

N - the total number of observations.

In the academic literature there are different estimates that are supposed to be optimal for volatility calculation. For instance, according to Canina and Figlewski (1993), suggested forecasting horizon is equal to 35 days. Nevertheless, in the current research the number of observation equals 75 trading days. This time horizon covers a quarter period which is assumed to be interesting for an investor or a trading person.

Having the estimation of the ex-post volatility using the current values of the index, one could try to estimate it using previous values in order to apply this algorithm in forecasting. The first thing the author can do is to estimate the informational content of the implied volatility. In order to see whether the implied volatility shows some information about the actual volatility the conventional analysis using the following regression is used:

(9)



This regression equation (9) of a parsimonious form allows to verify at least 3 hypotheses. The first one is whether implied volatility is informative for the prediction of future volatility. In this case the coefficient before the independent variable should be non-zero:

H₀:

The second hypothesis that is be checked by this simple regression is whether forecasting using implied volatility is unbiased. This implies that coefficient for should be close to 1, while the constant term should tend to 0:

H₀:

And finally, one is needed to verify whether the implied volatility is an efficient forecasting measure. In that case, the error term should follow the white noise and be uncorrelated with the independent variable of interest :

H₀:

For reliable results all three hypotheses should hold and, moreover, it is needed to verify the significance of all coefficients and the absense of serial autocorreletion in the error term. After the fullfillment of all these conditions the author saves the residuals and the obtained coefficients in order to use the mixed strategies for forecasting the actual index volatility.



3.2 Forecast based on GARCH(1,1)

This subchapter will be devoted to the forecast based on the historical data. For this purpose the generalized autoregressive conditional heteroscedasticity (GARCH) model described by Engle (1982) and Bollerslev (1986) was chosen. According to the majority of the previous reseerch, a parsimonious GARCH(1,1) specification is assumed to be sufficient to capture the stochastic nature of the index returns movements.

The GARCH(1,1) model represents 2 estimated equations, one of which is a conditional mean equation (10) and the other - a conditional variance equation (11):

(10)

(11)

, where

Ш _t-1 - information set that contains all information which is available and known at time t-1;

N(0, _t ) - normal distribution function.

In order to obtain a well-defined process this regression will be restricted by 2 conditions. The first one is needed to avoid non-stationarity, so the sum of coefficients should be inside the unit circle: . The second restriction is used to ensure non-negativity of volatility, so all coefficients should not be less than 0: .

After obtaining the estimates for volatility based on GARCH(1,1) one uses the same conventional analysis in order to estimate the informational content of the historical volatility for the prediction of the ex-post volatility:



(12)

Again, the aforesaid hypotheses are verified for this specification of the regression equation and obtaining well-defined and significant estimates resiaduals and coefficients are saved for constructing the mixed variations of different approaches.

3.3 Mixed strategies

Previous research showed that neither implied volatility by itself nor the historical volatility on its' own represent good measures for forecasting future volatility. The possible reason is that they contain different pieces of information about the future movements. In this case the appropriate strategy is to combine the aforesaid approaches and construct some variations of mixed strategies. It is assumed that in combination the option-based method and GARCH(1,1) produce more reliable results.

To begin with, the simplest method with equal weights developed by Vasilellis and Meade (1996) is applied. For each of these volatility measures one assigns an equal weight:

, where(13)

- volatility estimated by GARCH(1,1) method at time t;

- volatility estimated by the Black and Scholes model at time t.

However, it is a simple average of two approaches that does not take into account how well each of them manages to predict the future volatility value. Moreover, these weights are constant in time that does not seem to reflect the reality. It is possible to assume that there are periods when the implied volatility works better, or on the contrary, when the historical volatility predicts more informatively. This is why the author modifies the weights assigned to each estimation procedure in such a way that a procedure with lower error term () and from(12)) will have larger weight.

One way of doing it is to find the weight of inverse error of each method in cumulative inverse error:

(14)

(15)

Another way of assigning weights suggested by the author is not to minimize error effect, but on the contrary, maximize the informative effect. This is realized by using estimated significant standardized coefficients a₁ and a₃ from equations (6) and (9), respectively. In this case, however, the weights again will not depend on time:

(16)

(17)

3.4 Comparative Statistics

In this subchapter the usual technique in forecasting is applied: division of the sample into 2 parts. The first part of the sample period is used to give the in-sample estimation of the parameters of the time-series models. This in-sample period is used on a rolling basis to evaluate out-of-sample forecasting performance with rolling window. The out-of-sample estimation will be compared to the calculated actual volatility measures. The proportion of the in-sample period to the out-of-sample period is 3 to 1: having data from January 05, 2014 till December 28, 2017 the first three years are used for the estimation of regressions.

In order to compare different forecasting strategies among themselves the author uses the technique of minimization of forecast errors. Three main measures are analyzed in order to choose which strategy performs better.

· The mean absolute error:;

· The mean absolute percent error: ;

· The root mean square error: , where

n - the number of predicted values: n=N(total number of observations) - rolling window.

In the current research both call and put options are analyzed in order to produce the implied volatility component of the forecast. But in order to calculate concrete values different option types are analyzed separately. The obtained results in terms of forecast errors are as follows:

· based on the implied volatility form call options:

	MAE	MAPE	RMSE
Implied Volatility	0. 0319	30.87	0.0316
GARCH(1,1)	0.0317	30.24	0. 0323
Simple average	0.0284	26.05	0.0295
Weighted average (1)	0.0278	25.48	0.0299
Weighted average (2)	0.0292	28.51	0.0303

· based on the implied volatility form put options:

	MAE	MAPE	RMSE
Implied Volatility	0.0253	23.94	0.0264
GARCH(1,1)	0.0317	30.24	0. 0323
Simple average	0.0237	21.67	0.0259
Weighted average (1)	0.0241	22.31	0.0268
Weighted average (2)	0.0250	22.85	0.0256

The given results show that in most cases the models containing both methods (the implied volatility and time series) perform better. However, there is no unique conclusion concerning the weighting scheme: which method is more informative and, as a consequence, should have higher weight.



Conclusion

Currently, the rapid development of financial markets and, as a consequence, high level of uncertainty enforce investors to develop some techniques which allow them to estimate and predict to some extent the future movements of prices. The academic community and scientific researchers, in their turn, prepare and analyze the theoretical base for such forecasting tools.

One of these tools applicable for prediction is the implied volatility. In the academic literature there are many papers dedicated to the informational content of the option-based volatility. This issue was raised even in the last century but it does not lose its' relevance nowadays. The author concentrates the attention on the current Russian market, namely, the RTS Index which is less studied than, for example, American or European ones. Moreover, in the current paper the author suggests some ways how the implied volatility can be included in the prediction process and add some valuable information for forecasting based on the classical algorithm using historical data.

To draw the conclusion, it is necessary to say that market efficiency remains one of the most disputable issues nowadays. Many academicians devote their papers to this topic verifying whether the market prices can reveal some information about future changes or not. The author compares predictive power of different strategies using the technique of minimization of errors and maximization of the informative base.

The results of the current research show that the combination of the implied volatility with GARCH model predicts better ex-post realized volatility than the implied volatility or GARCH model independently. It means that although the implied volatility by itself can be not reliable instrument, it has some useful informational content that could be complement to the information contained in the historical data.

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