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Signals From Implied Volatility Index Changes Finance Essay

An investor requires a prediction of the direction of the underlying asset price movements to devise a profitable trading strategy. In most of the techniques used for forecasting a point estimate of the expected return or its volatility is calculated. The point estimate, though useful, does not give the extremes of the estimate, which can be useful in evaluating the possible losses or gains for the asset. In this paper, we use machine learning based methods for quantile regressions to calculate an extreme interval estimate for the expected volatility in index return and use the inverse relationship between volatilities and index levels to generate a directional signal. We use the interval estimate to devise a trading strategy based on the expected direction of change in the interval to trade the underlying price index and compare the results obtained from linear quantile regressions to machine learning based methods.

1. Introduction

This paper investigates the profitability of a strategy based on short selling indices. It is constructed on the principle of the importance of leverage effects. This phenomenon relates to the fact that increases in volatility are linked with falls in stock prices and vice-versa for decreases in volatility. One way of extracting information about market consensus views of changes in volatility is to appeal to the implied volatility changes implicit in option prices. In this paper we employ changes in implied volatility derived from options on two indices: the S&P500 and the FTSE100. We utilise the history of the implied volatility series with quantile regression and machine learning techniques to forecast changes in the quantile intervals of volatility. We use these forecast changes in volatility quantile intervals to set up two profitable short selling strategies featuring the two indices. Short selling has attracted quite a lot of attention in the context of the recent financial crisis. The authors find some of the debate about short selling quite puzzling as we explain below.

The process of investing involves two basic positions: buying and selling. If you purchase a security or asset in the belief that its price will subsequently rise you have a long position in that security. To realise any ensuing gains from any price rise, you would subsequently have to sell the security to achieve the gains. You would then have a zero net position as both trades would have cancelled out.

Short-selling is the reverse of going long. If you thought that a security was likely to fall in price you could sell it, or go short, in the belief that the price would fall so that you could subsequently close out your position and realise a gain equal to the difference between the two reference prices; that at which you sold and that at which you subsequently purchase to close out the position. The main difference between the two strategies; which are reverse sides of the same coin, is that if you purchase a security, you have to come up with the money for the purchase immediately, or it will not be delivered to you. If you sell something, you could perhaps already have a position in the security, and in this case that would be engaging in ‘covered’ short selling, or you could sell a security you do not possess, in the hope that you could subsequently purchase it for delivery at a lower price; in this case this would be undertaking ‘naked’ short selling. In the case of ‘naked’ short selling there is a potential risk that you might fail to deliver if market circumstances changed and the price went up, or if your circumstances changed and you did not have the wherewithal to purchase the security you had sold for delivery.

Typically, markets have institutional frameworks and architectures which help guarantee delivery. Securities markets have margin calls if you have borrowed money to trade, and futures markets have both margin calls and marking to market to guarantee delivery. This leads on to the next important point: short selling can be done in the cash market for the basic instrument, or in derivative or futures markets based on the same underlying instrument. More recently, you could sell short by taking the appropriate position in contracts for the difference (CFDs) based on the same instrument. Thus, there are many different ways in which you can profit from the projected change in the price of an asset or security by taking appropriate positions in cash or derivative markets.

Short-selling came in for much misguided criticism at the height of the GFC. On February 24th 2010 the SEC adopted a new rule that places certain restrictions on short selling when a stock is experiencing significant downward price pressure. This Rule 201 imposes the following, amongst other conditions:

Short Sale-Related Circuit Breaker: The circuit breaker would be triggered for a security any day in which the price declines by 10 percent or more from the prior day's closing price. The duration of the Price Test Restriction is as follows: once the circuit breaker has been triggered, the alternative uptick rule would apply to short sale orders in that security for the remainder of the day as well as the following day. It was argued that once the circuit breaker is triggered it will enable long sellers to stand in the front of the line, and sell their shares before any short sellers. The uptick rule will permit short selling in a security if the price is above the current national best bid.

A 2009 IOSCO report on the regulation of short selling noted that: in some jurisdictions such as Hong Kong short selling is only permitted in stocks that meet certain eligibility criteria; whilst in others there is a requirement to pre-borrow the stocks before they can be short sold, while other jurisdictions such as the United States, previously had a ‘locate’ requirement. Short selling in Canada, Hong Kong and Japan is subject to trading controls such as price restriction rules. Furthermore, some jurisdictions like Australia, Canada, Japan, Hong Kong and the United States require the ‘flagging’ of short sales when orders are submitted to the exchange markets for execution. Another tool employed is the use of margin requirements to control short selling as is the case in Japan. Finally, in most jurisdictions, for transactions where stocks are not delivered within the standard settlement cycle, there is some form of mandatory buy-in or close-out requirement designed to cover the failed delivery of the stocks. IOSCO (2009) recommended the implementation of four broad principles in relation to short selling:

“The First Principle: Short selling should be subject to appropriate controls to reduce or minimise the potential risks that could affect the orderly and efficient functioning and stability of financial markets.

The Second Principle: Short selling should be subject to a reporting regime that provides timely information to the market or to market authorities.

The Third Principle: Short selling should be subject to an effective compliance and enforcement system.

The Fourth Principle: Short selling regulation should allow appropriate exceptions for certain types of transactions for efficient market functioning and development”.

These above recommendations appear sensible and non-controversial but in the authors’ view it is ironic that the report tries to side step the issues involving derivatives, stating that: “the Technical Committee understands that the reporting of short positions might not provide a full picture if the data excludes derivatives.” However, it leaves this up to the discretion of the local market authorities and in the process fails to heed Fischer Black’s observation that net positions in derivatives must balance out to zero: given that for each writer there must be a counter-party. This leaves one wondering why the position in derivatives would matter?

A further dilemma is related to the following observation about market expectations: if stock market prices are modelled as a strict random walk; then up or downward movements are equally likely. To impose short selling restrictions imposes restrictions on one side of the expected price path. However, if stock market prices are modelled as a random walk with drift, then short selling restrictions impose a condition on the price path that is in the reverse direction to the long-term trend. However, prices are stochastic and presumably price paths can take marked deviations from the long-term trend. Short-selling restrictions appear to sit awkwardly with considerations of the processes involved in price discovery and the stochastic nature of stock price changes.

However, we shall not dwell further on this controversy, but focus instead on two related synthetic products for two markets: a pair of major indices and a short trading strategy constructed around the implied volatility of option contracts written on these underlying index contracts.

The rest of the chapter is constructed as follows; a brief literature review and introduction to quantile regression, kernel quantile regression, and quantile regression forests follows in section two. The data and research method and modelling utilised to construct a trading strategy is introduced in section three, followed by the presentation of the results in section four, whilst a brief conclusion follows in section five.

2. Literature Review

2.1 Quantile Regression

In simple linear regression a bivariate normal distribution is assumed between the dependent and independent variables. The simple assumption of bivariate normality in a regression model may not be an appropriate assumption when the variables have some arbitrary joint distribution, in which case linear regression fails to describe the conditional distribution of the dependent variable.

Quantile Regression (Koenker and Basset, 1978), is a alternative technique which can be used as a substitute for simple linear regression as characterised by the Ordinary Least Squares (OLS) method. Through quantile regression we can get different quantile relationships for different quantiles of the conditional distribution of the dependent variable. Quantile regression is less susceptible to the influence of outliers and hence is useful in quantifying the relationship between dependent and independent variables across the distribution in the domain of the desired desired quantiles. Koenker (2005) discusses the asymptotics of quantile regression methods and comments that score tests for quantile regressions are in effect a class of generalized rank tests. He notes that the works of Spearman (1904), Hotelling and Pabst (1936), Friedman (1937) and Kendall (1938), are generally given credit for initiating the rank-based approach to statistical inference.

Equation-(1) gives the expression for simple linear (OLS) regression with X as independent variable, Y as dependent variable, α is the intercept, β the coefficient and e is an iid error term.

(1)

The above regression model works on the assumption of bivariate normality of the variables, but if the variables are not bivariate normal then we need more sophisticated regression method to model the conditional distribution F(Y|X) (Alexandar, 2008).

Quantile regression is modelled as an extension of classical ordinary least squares (OLS) estimates of conditional mean models to the estimation of quantile functions for a distribution (Koenker and Bassett, 1978). The central case in quantile regression is the median regression estimator that minimized a sum of absolute errors as opposed to OLS which minimizes the sum of squared errors. Other quantiles are estimated by minimizing an asymmetrically weighted sum of absolute errors. Taken together the ensemble of estimated conditional quantile functions offers a much more complete view of the effect of covariates on the location, scale and shape of the distribution of the response variable.

In quantile regression and (q is the quantile of interest) can be estimated as a solution to following optimization problem (Alexander, 2008)

(2)

where

(3)

See the Koenker (2005) monograph, or Alexander (2008) for a more comprehensive discussion of mathematical details.

As quantile regression provides inference in the quantiles, it can be used to build an interval prediction using extreme quantiles as the boundary intervals. If we predict the two boundary quantiles using quantile regression e.g, 1% and 99%, it gives us an interval estimate for our prediction and the value is expected to lie between these two boundary estimates. Instead of using point estimates from OLS which are around the mean we can use quantile regression to get an interval estimate which gives us the expected extreme loss or expected extreme gain in stock returns, based on this estimate a trading strategy can be constructed. The strategy here is to predict the next day quantile interval estimate using quantile regression and position the direction of the trade based on it. We predict [1%-99%] and [5%-95%] estimates for next day using a moving window of the last 250 days observations. The last six days returns are used as independent or predictor variables and the present day return as the dependent variable. The choice of intervals is based on the mostly commonly used value at risk quantile levels.

2.2 Kernel Quantile Regression

Kernel Quantile Regression, is an evolving quantile regression (Takeuchi I, et. al., 2006; Ll Youjuan, et. al., 2007) technique in the field of non linear quantile regressions. As kernel quantile regressions are capable of modelling the non linear behaviour of time series data, they prove to be more efficient in forecasting risk than other methods, including linear quantile regression. Kernel quantile regression is more efficient over non linear quantile regression as proposed in Koenker’s (2005) monograph on quantile regression (Takeuchi et al., 2006).

Youjuan LI et. al (2007) also did some work on kernel quantile regression in developing an efficient algorithm for their computation. The obvious advantage of kernel quantile regression is the use of kernel functions (weighting functions) to model the dependence, which allows modelling of both gaussian and non gaussian data. Kernel quantile regression can be used to forecast value at risk, using past return levels as a training set (Wang, 2009).

For a training data set where input and output . Assuming the mapping function to be , the general formula for calculating the quantile is given by

(4)

where is a Hilbert space (Youjuan LI et. al, 2007) and ρ(⋅) is defined as

(5)

Point x1 in the input space is mapped a point in the feature space by mapping function . An optimal linear quantile regression function in the feature space can be located by the following

(6)

The quantile hyperplane reproduced in kernel Hilbert space will be nonlinear in original space.

The quantile regression is given by the following optimization problem

(7)

where C is the regularization parameter. A larger C gives greater emphasis on the empirical error term. The above minimization can be transformed into a quadratic programming problem (see Wang, 2009;Youjuan LI et. al, 2007 and Takeuchi I, et. al., 2006 for more details).

We use KQR to predict the interval estimate using the last 250 days moving window of training data as a sample consisting of the last six days daily returns as the input and present day’s return as the output. This will be compared to linear quantile regression based estimates and the final returns based on the trading strategy (as described in section 3).

2.3 Quantile Regression Forests

Random forests (Breiman, 2001) are one of the many (Support Vector Machines, Neural Networks etc) popular machine learning tools for regression and classification based problems. Random forests provide inference about the conditional mean of the distribution in a random forest regression. Meinshausen (2006) generalized the random forests and showed that they can provide information about the full conditional distribution of the response variable, and can be used to forecast interval estimates. According to Meinshausen (2006) quantile regression forests give a nonparametric and accurate way of estimating conditional quantiles of high dimensional predictor variables.

Random forests, as the name suggests, is a ensemble of trees of n independent variables For each tree and each node in a large number of trees generated in random forests variables are spit at random. Random forests has a single tuning parameter which is the size of the random subset (subset of predictor variables at each node used for split point selection). In regression random forests they give the average of all tree responses as a prediction for a new data point (Breiman, 2001).

The Random forests method approximates the conditional mean by a weighted mean over the observations of the response variable,

(8)

where wi(x) is given by

(9)

where is a random parameter vector which determines how a tree is generated.

In Quantile Regression Forests trees are created with the same algorithm as in Random Forests. The weighted distribution (not the mean) of observed response variables gives the conditional distribution for quantile regression forests.

The conditional distribution of Y, given X=x is given by

(10)

Drawing analogies with the Random Forest approximation of the conditional mean, an approximation to can be defined by the weighted mean over the observations of

(11)

This approximation acts as the basis for the Quantile Regression Forests algorithm. See Meinshausen (2006) for further theoretical and mathematical details.

3. Data & Methodology

The basic intuition in this study is to use the values or levels (of implied volatilities) changes transformed into returns from the FTSE-100 and S&P-500 volatility indices as the basis for the decision on the position of a directional trade (short or long) in their respective underlying price indices (FTSE-100 and S&P-500). We use the last 4 years daily logarithmic returns for these four indices starting from January-2007 to October 2010. These are forward looking estimates of volatility; for example the S&P500 Volatility is an average of the expected 30 day variance of the index estimated from a strip of options on the index.

Simple linear regression methods, which forecast the conditional mean give a point estimate for the future value of the dependent variable, which may or may not be significantly close to the actual value. The predictions can be lower or higher than the actual value or can lie in an interval of lower and upper quantiles. A prediction interval for the future value can prove to be a better estimate when dealing with extreme value prediction like predicting the extreme low and extreme high next day return levels for a volatility index return as in the present case. We can form a prediction interval by forecasting the two extreme quantiles e.g., a 90% prediction interval for the value of Y is given by

There is a high probability of a future prediction lying within this prediction interval. These interval estimates can be forecasted using quantile regression, kernel quantile regression and quantile regression forests.

When forecasting financial time series return quantiles, lower and upper quantile estimates are equivalent to predicting value at risk for short and long positions in the market. Based on two extreme quantile estimates we can device a trading strategy which changes our directional position; whether we go long or short, depending on the degree of change in the estimated extreme values.

We predict two interval estimates [1%,99%] and [5%,95%], using linear quantile regression, kernel quantile regression and quantile regression random forests. We use the last six days of returns calculated from FTSE 100 Volatility Index and S&P 500 Volatility Index as independent variables and the present day return as the dependent variable. A moving window data set of the last 250 days is taken as a training sample to predict the next day’s observation. We trade the underlying FTSE-100 price index and S&P-500 price index based on the directional signal generated from the trading strategy. We predict daily interval estimates for year 2008, 2009 and 2010 (till 26/10/2010) using the previous years (250 days) data as training sample in a daily moving window.

We use the Quantreg, Kernlab and Quantregforest packages in R to do the empirical exercise, the data is collected from the Thomson Reuters Datastream database. We have used all the default optimization settings in calculating the interval estimates. The KQR estimates are obtained by means of a Radial basis function kernel (which is useful when no prior information is available about the training data) which has two parameters viz., cost (C) and sigma (s), C is taken to be 1 (default) and s is optimized by the automatic optimization provided in the Kernlab package. Quantile regression forests results are obtained using all the default parameters as in QuantregForest package.

Trading Strategy

We use two sets of interval estimates to predict the direction of return, i.e., 1 if the return is increasing and -1 if it is decreasing which are used to decide and to predict when to short the underlying price index. If (lt,ut) represents an interval estimate for time t where lt is the estimated lower quantile and ut­ is the estimated upper quantile, the direction of returns can be decided based on the following algorithm;

If lt+1lt+T and ut+1ut where T is a threshold (5% in present case) then -1

If lt+1lt and ut+1ut then 1

If lt+1>lt, ut+1>ut and if lt+1-lt>ut+1­-ut then -1else 1

The above rules use the absolute value of the predictions. The threshold is incorporated to avoid changing position on a minor increase in the lower quantile.

As the direction indicates whether the price of a stock will increase (1) or decrease (-1), the trading strategy also follows the same direction for keeping a short or long position in the stock or index in this case. I.E. if the first prediction is positive, the investors adopt a long position (buy) and keep it till the price doesn’t goes down, (negative prediction). This strategy results in the cash flow at point of time when stock price moves down and the investor shorts their position in the market, (ignoring all the transaction costs). If at time t the direction is positive, the investor buys one dollar of the index and doesn’t change his/her position as long as the prediction is not negative. Once a negative prediction occurs (-1) indicating a decline in price at time t+n, the return for investor would be. A generalization of this strategy gives the profit at time T as

(12)

Where is the direction of price change at time t.

4. Results

Figure-1 and figure-2 give a confidence interval plot of first fifty [5%-95%] interval estimates for FTSE-100 volatility index returns and S&P-500 volatility index returns obtained from linear quantile regression (QR), kernel quantile regression (KQR) and quantile regression forests (QuantregForests). The dots in the graphs represent the actual return and the bar the predicted interval estimate, which shows that most of the actual returns lie within the estimated interval.

Figure 1: [5%-95%] Interval Estimates and Actual Returns for FTSE-100 Volatility Index (Year 2008)

Figure 2: [5%-95%] Interval Estimates and Actual Returns for S&P-500 Volatility Index (Year 2008)

Table-1 and table-2 gives the final returns from FTSE-100 price index and S&P-500 price index, after applying the trading strategy (as discussed in Section-) on the interval estimates. They also give the hold out return on these indices if the position is closed at the end of the testing (estimation) period for comparison. We can see that as the year 2008 suffers from GFC none of the models tested gives a positive return, but the KQR method improves on the negative hold out return for both of the markets. The results for year 2008 shows that the KQR [1%-99%] interval estimates outperforms the other estimates, which can be accounted for by the increased volatility in the markets during this period and hence the returns are lying more in the extremes than the other relatively normal market periods.

The results for the other periods (years 2009 and 2010) show that the KQR method again outperforms the other two estimation techniques, in particular in the [5%-95%] interval estimate it performs better in relatively normal market conditions (except for the S&P-500 in year 2010). The results clearly show that the KQR method performs consistently better than the final hold out return in normal market conditions for both the interval predictions. For linear quantile regression (QR) and quantile regression forests (QRF) we get inconsistent results.

Table 1: Returns observed with trading strategy applied on FTSE-100 price index

Trading Strategy Returns

Year

2008

2009

2010

[1%-99%]

[5%-95%]

[1%-99%]

[5%-95%]

[1%-99%]

[5%-95%]

QR

-54.2255

-23.8469

39.53271

5.229167

18.29091

11.51107

KQR

-24.9358

-46.2419

29.4114

30.91785

18.4969

23.37544

QRF

-46.9848

-36.1911

20.14434

-5.99551

-0.14102

-1.26365

Actual hold out return

-36.6482427

23.53985451

7.022388286

Table 2:Returns observed with trading strategy applied on S&P-500 price index.

Trading Strategy Returns

Year

2008

2009

2010

[1%-99%]

[5%-95%]

[1%-99%]

[5%-95%]

[1%-99%]

[5%-95%]

QR

-62.7601

-61.9122

13.98154

20.32935

-11.59

6.456039

KQR

-41.5446

-56.4594

15.53714

30.62247

20.88832

18.41588

QRF

-44.5796

-51.7369

8.040667

16.19486

-4.92525

14.55047

Actual hold out return

-48.58952303

24.98511848

6.66617096

5. Conclusion

We have used several new techniques based on the use of forward looking implied volatilities on two indices; the FTSE100 and the S&P500 to generate short and long trading strategies in the two indices. These strategies employ variants of quantile regression based techniques, including linear quantile regression, Kernel based quantile regression and quantile regression random forests to predict quantile intervals and employ changes in these to generate a trading strategy. Kernel based quantile regression methods appear to generate the greatest returns in our hold out sample periods and dominate buy and hold returns. We ignore transactions costs in this exercise but it is clear from the results in Tables 1 and 2 that the deduction of realistic transaction costs would not change the order of the results.

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