Forecasting Stock Market Volatility And The Application Of Volatility Trading Models

“Forecasting Stock Market Volatility and the Application of Volatility Trading Models” by Jason Laws* (Liverpool Business School and CIBEF **) and Andrew Gidman*** (George Petch International, Brussels and CIBEF**)

November 2000

Abstract This paper examines the ability of GARCH(1,1) and GARCH(1,1) + Implied Volatility models to forecast stock market volatility on the FTSE100 index. Comparing the volatility forecasts with the implied volatility of the corresponding at-the-money index option contract, it is investigated whether successful volatility trading models can be developed. An at-the-money index call was bought/sold if the volatility forecast was above/below the implied volatility by a certain threshold. Eight different trading strategies were developed combining the different methods of forecasting, different activation thresholds and different weightings. The strategies were analysed and performance assessed in terms of the profit / loss generated. It was found that forecasting techniques that include both market based information and times series information produce better forecasts. The combined models also produced more profitable signals. On the evidence of the research presented in this paper, the conclusion is that options markets appear to be inefficient and/or the option pricing formulae used are incorrect. This is a direct inference from the fact that volatility forecasts have been used to identify mispriced options and profitable trading rules have been established based on the implied volatility of the option and the forecasted volatility of the corresponding index.

* Jason Laws is a Lecturer in International Finance at Liverpool Business School and a Senior Researcher with CIBEF (E-mail: [email protected]). ** CIBEF – Centre for International Banking, Economics and Finance, JMU, John Foster Building, 98 Mount Pleasant, Liverpool L3 5UZ. *** Andrew Gidman is a Fund Analysts at George Petch International, 23 Avenue des Sorbiers, 1950 Kraainem, Belgium.

1. Introduction Volatility and the measure of it are very important to equity traders and derivatives traders alike. Traders are interested in which direction the market is moving and the future direction of the market but they are also interested in the velocity of such movements. Volatility is a key and, perhaps, the most important factor when calculating the price of an option using the traditional methods. If the underlying stock has a fairly low volatility then the option contract on it will have a relatively low value because it is considered that the price of the underlying is less likely to hit the exercise price and subsequently go above for a call or below for a put. This is in contrast to high volatility markets where there can be extreme price movements, meaning that there is more of a possibility of the strike price being hit and, hence, the option is more expensive. Since volatility is such a vital component of option pricing in particular and of measuring financial risks in general, any successful method of forecasting it is a benefit to users. Advances in time series modelling such as ARCH/GARCH models and stochastic volatility models have made it possible to do this. GARCH models will be used in this paper to forecast stock market volatility. When the volatility of the FTSE100 index is forecasted in this paper, GARCH(1,1) and GARCH(1,1) + Implied Volatility models will be used as in Kroner et al (1995) and Dunis et al (2000). There are many different variations of Generalised Autoregressive Conditional Heteroskedastic (GARCH) models such as stationary GARCH, unconstrained GARCH, integrated GARCH and exponential GARCH. These can all be used to forecast stock market volatility with varying degrees of success as discussed in Chong, Ahmad and Abdullah (1999). Since it is not the main aim of this report to compare the performance of different GARCH models, we will concentrate on the GARCH(1,1) model and the GARCH(1,1) + Implied Volatility model. Justification for the use of the GARCH(1,1) model will be provided based on the empirical evidence of Walsh and Tsou (1998), Akgiray (1989), Corhay and Rad (1994) and Vasilellis and Meade (1996) in particular. Regarding the GARCH(1,1) + Implied Volatility model, particular references will be made to Kroner, Kneafsey and Claessens (1995) and Dunis, Laws and Chauvin (2000). The paper is organised as follows: Section two presents the motivation for the study; Section three provides a review of the literature and discusses how previous authors have forecast volatility in this type of environment and, also, how these forecasts have been incorporated into a trading strategy; Section four provides details of the source of the data and discusses how the data was manipulated into a form suitable for the forecasting process. Details of how the forecasts were constructed are also provided in this section; Section five describes in detail how and why the various strategies were formulated. Section six includes the results of the out of sample trading simulation. Finally, we conclude and provide some suggestions for refinements to the trading strategies.

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2. Motivation The Black-Scholes Option Pricing formula is one of the most popular methods of pricing an option. To calculate the price, five inputs are required i.e. spot price of the underlying, exercise price of the option, risk-free rate of return, time to expiry and the volatility of asset returns. Of these five inputs, four are measurable with near certainty. The exception to this is the volatility of the underlying asset over the period of the option’s life. For example, one could calculate the price of an option using the best possible guess about volatility over the period. However, when the result is produced, the price of the option may well be different to the price of the option in the market. One possible explanation for this could be that the market is using a different pricing model. If the assumption is made that the pricing models are the same, however, then it must be the case that the inputs into the model are different. Since the spot price, exercise price, interest rate and time to expiry are observable directly from the market and, hence, unlikely to be incorrect, it must mean that a different volatility has been perceived by the market. The volatility perceived by the market can be found by holding all the other four inputs constant and solving the model with respect to volatility. This volatility figure which is used by the market is known as the ‘Implied Volatility’ of the option and is the volatility which must be used in the option pricing formula to yield an option value identical to that in the marketplace. The implied volatility of an option is constantly changing due to the fact that option prices, time to expiry, spot price etc. are always evolving. This paper aims to exploit the differences between the market’s estimate of future volatility and the estimates produced by various forecasting models. In theory, if the future volatility could be predicted accurately, a trader would be able to look at differences between an option’s theoretical value and its value in the market. In turn, they could sell any options which were overpriced in relation to the theoretical value and buy any which were under-priced. 3. Literature Review It is clear from the proposal above that the success or failure of these trading models depend crucially on our ability to generate accurate volatility forecasts. There is quite a strong body of literature advocating the use of the GARCH family of models to forecast volatility. See for example Chong et al. (1999), Walsh and Tsou (1998), Akgiray (1989), Corhay and Rad (1993) and Vasilellis and Meade (1996). Given that, in this paper, we will be required to generate a forecast daily and on a continuous basis we are not in a position to select each day the most appropriate GARCH configuration, we, therefore, propose to adopt a “one size fits all” and will select one GARCH configuration to make all of the forecasts. The work of Corhay and Rad is particularly useful in helping us select this model. Corhay and Rad (1993) investigated whether autoregressive conditional heteroskedastic models could adequately describe stock price behaviour in European capital markets. The reason they chose to do this was due to the fact that so much work had been applied to American markets as in Akgiray (1989) and they wanted to investigate whether the models were suitable in markets which are ‘generally much smaller and thinner’ than the American ones. The markets they

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chose to look at were France, Germany, Italy, the Netherlands and the UK. Estimating ARCH and GARCH models of various orders they found that with the exception of Italy, the GARCH (1,1) model generally outperformed other (G)ARCH models. To confirm the findings of the literature review that GARCH (1,1) models are the best at forecasting stock market volatility, different types of GARCH models were compared using the data to be used in the forecasting process. Firstly, the models were compared according to their goodness-of-fit statistics, namely Akaike’s Information Criterion (AIC) and Schwarz’s Bayesian Information Criterion (SBC). The models compared were a GARCH (1,1), GARCH (2,1) and a GARCH (2,2) model. The GARCH (1,1) was indeed the best performer, showing the smallest values for both criterion1. Next, a series of one-step-ahead forecasts were carried out for each of the models. The best model was defined as being the one with the lowest forecast error over the forecast period. The forecast errors were compared using two loss functions; Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE). Again, the GARCH (1,1) method was found to be the best model2. A major criticism of these models made by Kroner et al. (1995) is that they ignore the market’s expectations of the future volatility and rely only on past information. Kroner et al. acknowledge that GARCH forecasts seem to provide the best forecasts of volatility (based on evidence from literature) but also that Implied volatility based forecasts can still be used to explain some of the forecasting error from the GARCH models. For this reason, they combined the two forms of forecasting techniques to produce their ‘COMBt,T’ forecasting model. The model was as follows:ln St – ln St-1 = µ + εt εt I ϑt-1 ~ N(0,ht) ht = ω + α ε2t-1 + βht-1+ δσ2t-1

[1]

In total, Kroner et al. were able to forecast volatility in six different ways (3 x impied standard deviations (ISD), 2 x Time-Series and a combined model (COMB)). The data used were futures add options on futures for cocoa, corn, cotton, gold, silver, sugar and wheat. The time period was roughly January 1987 to November 1990. With the exception of silver, GARCH and COMB models were much more successful at forecasting the volatility forecasts. GARCH had the smallest mean squared forecast error (MSFE) for four out of seven commodities and was second best on two other occasions. Kroner et al. say that their results confirm those of Lamoureux and Lastrapes (1993), Day and Lewis (1992) and others who found that GARCH based forecasts outperform ISD forecasts. However, when the combined model was analysed, it was said to be ‘very promising’ and that the forecasts were better than those which existed in the literature at the time. These views are reinforced in Dunis et al. (2000) who find that, at the 21 and 63 day horizon the combined GARCH and implied volatility “perform best most of the time (compared to a large number of time series and market based measures)” in forecasting the volatility of currency returns. Kroner at al. conclude by noting the implications of the success of the combined model, saying ‘the history of the time-series contains information about future volatility which is not captured by market expectations…This suggests that options markets are inefficient and/or the option-pricing formula we used incorrect’. They go 1 2

In order to conserve space these results are not shown. Again, in order to conserve space these results are not shown.

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on to say ‘This implies it is possible that our volatility forecast can be used to identify mispriced options, and a profitable trading rule could be established based on the difference between the ISD and the COMB volatility forecast’. The conclusions drawn by Kroner et al. are key to this paper. If their assumptions are correct, then it should be possible for a volatility trading model to be successfully implemented based on a combined model. In fact Dunis and Gavridris (1997) recognised this potential inefficiency in the options market but used a GARCH(1,1) model, rather than the combined model of Kroner et al to generate the volatility forecasts. The trading models they developed were applied to six exchange rates: USD/DEM, USD/CHF, USD/JPY, GBP/DEM, GBP/USD and DEM/JPY over the period of 2 January 1991 to 30 August 1996. One-month volatility forecasts were generated using a GARCH(1,1) model. A threshold was set whereby if a forecasted volatility was different from the implied one-month volatility by more than that amount, a volatility position was initiated. The volatility positions were initiated by buying or selling one-month at-the-money forward straddles. The volatility forecasts were split up into five categories: large up, small up, no change, large down and small down as in Dunis (1996). To avoid taking positions too often, the change threshold defining the boundary between small and large movements was determined as a confirmation filter. In the paper, Dunis and Gavridis limited themselves to buying straddles where the one-month volatility forecast was above the one-month implied volatility by more that a threshold. The straddles were assumed to be held until expiry (one month). They acknowledged, however, that this was not the optimal strategy due to the drop in time value during the life of the option and because of this, they also evaluated what happens to the equity curve if the holding period was five and ten trading days. To avoid the accumulation of a number of positions at the same time, Dunis and Gavridis evaluated two alternative strategies. The first strategy allowed just one position per month and the second allowed one position per week to be triggered. In-house (Global Market Research, BNP, London) daily databank figures were used for implied volatilities and exchange rates. The profitability of the positions was determined by comparing the level of implied volatility at the inception of the position to the prevailing one-month historical volatility at maturity. This was then weighted by the amount of the position taken. The authors note, therefore, that ‘profitability is defined as a volatility profit, but realised returns could only be estimated by comparing the actual profit of the straddle at expiry against the premium paid at inception’. However, with the practise for OTC Forex options to be quoted in volatility terms this was not possible. Dunis and Gavridis found that most stable and profitable models were found to be those for USD/CHF and USD/JPY. The volatility trading models adopted in this paper follow those of Dunis and Gavridis. 4. Data and Methodology Daily closing prices for the FTSE100 index was obtained from ‘Datastream’. The FTSE 100 Index option data was extracted from the ‘LIFFEdata Euro-Out Products – Tick Data’ CD-Rom. The CD contained information on index options from 1992 to 1998.After public holidays had been removed, there were 1,503 observations. The

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first 749 observations (2 January 1992 to 16 December 1994) were used to generate the initial forecasts. The final 754 observations (19 December 1994 to 18 December 1997) were used as the trading period i.e. forecasts were generated for this period. The LIFFE CD contained data for every single tick during the period. Since the trading model introduced in this paper is based on the use of at-the-money options, the option which was closest to the money on each day of the period was filtered out at the end of the day from the data3. The LIFFE CD is time stamped and, in addition to the option premium it also includes the corresponding spot price. Given that the time to expiry and exercise price are known by default, it was possible to back out the implied volatility using the Black and Scholes Option Pricing Model4. The periods used to generate the volatility forecasts were based on a one-day ‘sliding’ system. For example, to generate the first one-day forecast, the data period was split up as follows: The period used to generate the forecast was 5 January 1992 to 19 December 1994. A one-month volatility forecast was then generated to coincide with the expiry of the January 1995 FTSE100 Index option contract, i.e. the option contract had one month until expiry and the forecast generated was for one month. The next forecast was generated by a data period which had been moved forward by one day. The forecast generated was then for (1 month – 1 day) to coincide again with the time left until expiry of the option. Care was taken to ensure that periods did not overlap i.e. although it would have been possible to buy or sell an August option contract, for example, many months before, in this paper, the first possible date of purchase would be the first trading day following the expiry of the July contract. This prevented multiple positions being taken on the same day on options with different expiry dates. This ‘sliding’ process was then repeated until all of the data had been exhausted, i.e. to the final one-day forecast for 18 December 1997. The result of the forecasts was a matrix of volatility forecasts for each day of the forecast period. For each day, there were volatility forecasts for 25, 24, 23,…0 days. As an example, if on 1 August, an option had 20 days until expiry, from the matrix, the forecasted volatility figure to be used would be the one which corresponded to ‘1 August, 20 days’. The volatility forecasting techniques used in this paper are the GARCH(1,1) method and the GARCH(1,1) + Implied Volatility method. The GARCH(1,1) process is given by: ht = ω + α ε2t-1 + β ht-1

[2]

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The reason for this was because of the ‘volatility smile’ effect. It has been shown by Chance (1998) that the volatility of an option tends to increase the more that the money is out-of-the money. However, the implied volatility should only be based on the volatility of the stock. This indicates therefore that the implied volatility of an option is also influenced by how close to the money the option is. At-the-money options were also used based on the evidence of previous authors, most notably Beckers (1981) who found that the at-the-money options produce more accurate forecasts of future volatility than deep in or out of the money options. 4 To calculate the implied volatility of each option, ‘FinancialCAD’ was used in Excel. The function taken from the program was ‘aaBL_iv’ which calculated the implied volatility based on the Black (1976) model. The data for the risk-free rate were the LIBOR one-month values. Although it is recognised that it would have been better to match the maturity of the option to the interest rate duration it must be noted, however, that the Rho of an option is usually very low.

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The one-step-ahead volatility forecast for this process is immediately given by using recursive substitution. Baillie and Bollerslev (1992) give the n-step-ahead forecast for the GARCH (1,1) process as: ht+n = ω [ 1 + ( α + β ) +…+ ( α + β )n-2 ] + ω + α ε2t +β ht

[3]

The one-step ahead volatility forecast for the GARCH(1,1) + Implied Volatility model is given by: ht = ω + α ε2t-1 + β ht-1 + γ IMPt-1

[4]

By following the recursive procedure above and taking into account the fact that the last information on implied volatility available at time t is IMPt, the GARCH(1,1) + Implied Volatility n-step ahead forecast becomes: ht+n = ω [ 1 + ( α + β ) +…+ ( α + β )n-2 ] + ω + α ε2t +β ht + γ IMPt

[5]

5. Strategies There were two classifications of trading strategies used in the paper and within these two ‘classes’ were a further two ‘types’ of strategy. The two classes of strategy were the ‘Monthly’ and ‘Weekly’ strategies. The ‘Monthly’ strategy allowed no more than one position to be initiated per month. Once initiated, the contract was held until expiry. This was the case even if a position was triggered on the very first forecast day for that option contract, no more positions were initiated for the remainder of the month. The ‘Weekly’ strategy allowed no more than one position to be generated per week. This meant that a maximum of four positions were initiated per one-month period. Again, once a trade was triggered, the option was held until expiry. Both classes of trading model were evaluated by combining them with the two ‘types’ of strategy. These were the ‘Weighted’ and the ‘Naïve’ strategies. This meant that eight different trading strategies were evaluated in the paper: ‘Monthly Weighted’, ‘Monthly Naïve’, ‘Weekly Weighted’ and ‘Weekly Naïve’. These four strategies were evaluated based on both the GARCH (1,1) and GARCH (1,1) + Implied Volatility models. The ‘Naïve’ strategy was so-called because it failed to distinguish between and account for the size of the difference between the implied volatility in the marketplace and the forecasted volatility when initiating a strategy. The threshold was fixed at 2%. In other words, if the difference between the implied volatility and the forecasted volatility was >= 2%, one call option would be bought / sold depending on which volatility was higher. If the implied volatility is higher than the forecasted volatility, then according to the forecasts, it would imply that the option was over-priced in the market and the call would be sold. On the other hand, if the forecasted volatility was higher than the implied volatility, it would imply that the option was under-priced in the market and the call would be bought. On the other hand, the ‘Weighted’ strategy varied the number of calls bought / sold based on the size of the difference in the implied volatility and the forecasted

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volatility. A table of the weighting is shown below where d = difference between the volatilities. Weightings of ‘Weighted’ Strategy 2% <= d <= 3.99% 4% <= d <= 5.99% 6% <= d <= 7.99% 8% <= d

One Option Two Options Three Options Four Options

So, if the difference between the volatilities was 4.5%, two calls were bought / sold. This strategy therefore placed a greater emphasis on those options where there was a greater difference between the implied and forecasted volatilities. Earlier in the paper, it was noted that Dunis and Gavridis (1997) measured the success / failure of their trading models on the ‘volatility profit’ they realised. In this paper, however, since we have the option premiums at the inception and end of the strategy, we are able to compute the sterling value of the profit. In an attempt to give an indication of the amount of each position was at risk, the average drawdown was calculated for each of the strategies. The drawdown was defined as being the loss that would be incurred should the position(s) be closed before expiry i.e. enforced closure of the position. For example, had a call initially been written, it would have to be bought to close the position. If the premium for the call had gone up, this would result in a loss when the position was closed. This loss was then quoted as a percentage of the initial profit from the position. Obviously, a loss would not be realised on every day in the period due to the fluctuations of the option premium. The drawdown was therefore only calculated for the days where there would be a loss. These figures were then averaged to produce an average drawdown for each day that produced a loss. 6 Results 6.1 Strategy 1 – ‘GARCH (1,1) Naï ve Monthly’ This strategy produced trading signals in each of the 36 months of the trading period. As stated above for a position to be activated in the naïve strategy, the difference between the implied volatility and the forecasted volatility had to be greater than or equal to 2%. Thirty-five of the 36 positions triggered were found to be profitable with the overall profit of the period being 1821 pence5. The 35 profitable positions were write signals and the only ‘buy’ signal resulted in a loss. The reason behind so many write signals was the difference between the implied volatility and the forecasted volatility. The forecasts were lower than the implied volatility on an almost daily basis. Since there was such a difference between the two volatilities throughout the whole period, positions were triggered early in each period (month). As the implied volatilities were greater than the forecasted volatilities, it indicates that the options were overpriced in the market and, hence, the calls were sold (written). 5

Figure is for one option

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The result of this was a positive cash inflow. Often, the inflow would be large because the option was at-the-money. When the positions were closed, i.e. the calls bought back, the premiums had often fallen and big profits were obtained. The fall in premium throughout the period was due to the fall in time value of the option. The results of the strategy would indicate that it would be profitable to sell an at-themoney FTSE100 index option approximately one month before expiry and then hold until expiry. Since the calls were written, it would imply that large profits could have been generated from a limited cash outflow i.e. transaction costs and margin payments. The success rate of the trades was 97.22% which produced an average profit per trade of 50.58 pence. The average drawdown of the strategy was 15.23%. This indicated that on average, 15.23% of the profits were at risk if the positions had to be closed before expiry and the option premiums had moved unfavourably. 6.2 Strategy 2 – ‘GARCH (1,1) Naï ve Weekly’ Trading signals were generated in almost every week of the trading period. The result was that 154 positions were triggered. Over the period, there were 158 weeks and, hence, there were four weeks when positions were not triggered. Again, as with the monthly strategy, only one signal was to buy the option. The rest of the positions triggered were short positions i.e. the calls were written. All of the 154 positions triggered were profitable with the overall profit being 5780 pence. This was 3.17 times greater than the profit for the monthly strategy and yet there were 4.28 times more positions triggered. This indicates, that although the strategy appears to be successful, in comparison to the monthly strategy, it is not as efficient due to time decay. The average profit per position was 37.53 pence as opposed to 50.58 pence for the monthly strategy. Since there are more positions being taken, the transaction costs would obviously be greater. In the real world, this would further reduce the profitability of the ‘naïve-weekly’ strategy. The average drawdown for the strategy was 13.70%. 6.3 Strategy 3 – ‘GARCH (1,1) Weighted Monthly’

As with the ‘Naïve Monthly’ strategy, positions were triggered in each of the 36 months in the trading period. All but one of the 36 positions were found to be profitable and resulted in an overall profit of 4684 pence. This equates to an average profit per signal of 130.11 pence. Obviously, this profit is much greater than the profit generated by the ‘naïve’ strategy. The reason for this is that the number of positions taken was dependant on the size of the difference between the implied volatility and the forecasted volatility. Since the same implied volatility and forecasted volatility figures were used, as in the ‘naïve’ model and the differences were quite large, the result was that a large number of positions were taken for each signal. Since most of the positions were profitable in the ‘naïve’ model, an increase in the number of positions as in the ‘weighted’ model naturally resulted in greater profits.

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The average number of positions taken for each signal in the ‘weighted monthly’ strategy was 2.28. This was calculated by dividing the 82 positions taken by the 36 signals generated. The average profit per trade (position taken) was 57.12 pence. The success rate of the signals was 97.22% as with the ‘naïve’ model. All signals generated were to write the calls. The average drawdown of the strategy was 15.23%. 6.4 Strategy 4 – ‘GARCH (1,1) Weighted Weekly’ A total of 154 trading signals were generated throughout the period. Since each signal was weighted, the actual number of positions taken was 322. The profit generated by the strategy over the trading period was 14,161 pence. This equates to an average profit per signal of 91.98 pence. Since there was an average weighting per signal of 2.09 trades, the average profit per individual trade was 43.98 pence. The average drawdown of the strategy was 13.70%. 6.5 Strategy 5 – ‘COMB Naï ve Monthly’ All 36 signals were profitable in this model with the overall profit being 1,949 pence. There were no ‘buy’ signals with all the signals being to ‘write’. The COMB model’s forecasts of the volatility were, again, lower than the implied volatility figures, however, not by as much as the GARCH (1,1) forecasts, since they had taken into account the implied volatility. The strategy yielded a 100% success rate for the signals generated. The average profit per trade was 54.14 pence. The average drawdown was 14.54%. 6.6 Strategy 6 – ‘COMB Naï ve Weekly’ There were 155 signals generated by this model, all of which were profitable resulting in an overall profit of 5,749 pence. Again, there was just one signal to buy an option, the remaining 154 were ‘write’ signals. The average profit per trade was 37.09 pence. The average drawdown of the strategy was 16.33%. 6.7 Strategy 7 – ‘COMB Weighted Monthly’ The ‘COMB Weighed Monthly’ strategy produced 36 trading signals. All of the 36 signals were profitable and the overall profit totalled 4,919 pence. This meant an average profit per signal of 136.64 pence. The average number of positions taken per signal was 2.14 with 77 positions being taken. This was slightly less (6%) than the corresponding GARCH (1,1) strategy. It was expected that this would be the case since the difference between the implied volatility and the volatility forecasts were slightly less for the COMB strategies hence resulting in some of the weightings being less. The average profit per trade was 63.88 pence. 6.8 Strategy 8 – ‘COMB Weighted Weekly’ Over the three year trading period, 154 trading signals were generated by this model. All of the signals were to write the call options. Only one month in the period produced a loss. The overall profit obtained by using this strategy was 13,757 pence. This resulted in an average profit per signal of 89.33 pence. The average weighting per signal was 2.01 resulting in a profit per trade of 44.52 pence. The average drawdown was 16.33%.

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6.9 Strategy Conclusions Overall, greater average profits per trade were obtained for the ‘COMB’ as opposed to the GARCH (1,1) strategies. Regarding the weighted strategies, the ‘COMB Monthly’ model was ranked first with the GARCH (1,1) Monthly strategy second. The ‘COMB Weekly’ model strategy was ranked third whilst the was ranked fourth. This result for the weighted strategies was not expected since the COMB model actually predicted the volatility better. Since the forecasts of the COMB model were more accurate and, hence, closer to the actual implied volatility of the options, obviously the difference between the two values was less. When the weighted strategies were applied, fewer positions were taken due to this smaller difference. Since all of the positions were so profitable, it was thought that the increased weighting of the GARCH (1,1) strategies would, effectively, increase their overall profit. This was the case for the weekly trading strategy. However, when the monthly strategies were analysed, it was found that although the GARCH (1,1) model had a greater average weighting, the overall profit generated by it was surprisingly less that that of the ‘COMB’ model. Model Rankings (average profit per signal) 1 2 3 4 5 6 7 8

COMB Weighted Monthly GARCH (1,1) Weighted Monthly COMB Naïve Monthly GARCH (1,1) Naïve Monthly COMB Weighted Weekly GARCH (1,1) Weighted Weekly GARCH (1,1) Naïve Weekly COMB Naïve Weekly

63.88 pence 57.12 pence 54.14 pence 50.58 pence 44.52 pence 43.98 pence 37.53 pence 37.09 pence

The results clearly show that the monthly strategies performed better over the trading period. On the whole, it can be said that the ‘COMB’ strategies were the best performers, being ranked first, third, fifth and eighth.

Strategy GARCH (1,1) Naïve GARCH (1,1) Weighted COMB Naïve COMB Weighted

Monthly Trading Strategy Results Ave. Signals Profitable Signals Weighting 36 35 N/A 36 35 2.28 36 36

36 36

N/A 2.14

Profit (p) 1,821 4,684

Ave. Profit per Signal 50.58 57.12

1,949 4,919

54.14 63.88

Not only does the ‘COMB’ model produce the best average profit per trade, but the model is more successful. Profitable signals were generated 100% of the time.

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Weekly Trading Strategy Results Strategy Signals Profitable Ave. Signals Weighting GARCH (1,1) Naïve 154 153 N/A GARCH (1,1) 154 153 2.09 Weighted COMB Naïve 155 154 N/A COMB Weighted 155 154 2.01

Profit (p) 5,780 14,161

Ave. Profit per Signal 37.53 43.98

5,749 13,757

37.09 44.52

The most surprising aspect of the results was the number of trading signals. This stemmed from the large difference between the volatility forecasts and the implied volatility. Had the forecasts been closer to the implied volatility, then fewer positions may have been triggered. At the beginning of the study, it was expected that the forecasted volatility figures would be close to the implied volatility figures. As can be seen from the results, this was not the case. Although it is widely documented that the implied volatility of an option often over-estimates the actual realised volatility, the size of the difference observed in this study was unexpected. In their 1997 paper, Laws and Thompson conclude that ‘It is evident from the analysis that implied volatility consistently overestimates realised volatility’. In fact, the implied volatility of the nearest to-the-money $/DM call option over-predicted realised volatility on twelve out of sixteen occasions in their analysis. The differences between the forecasted volatilities and the implied volatilities over the trading period are shown in the chart below.

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Comparison of GARCH (1,1) and COMB Forecasts with the Implied Volatility of the Options (19 December 1994 – 18 December 1997)

70

60

Volatility %

50

40

30

20

10

10/12/97

27/10/97

9/9/97

24/07/97

6/6/97

21/04/97

3/3/97

15/01/97

10/10/96

27/11/1996

8/7/96

Date

22/08/1996

2/4/96

22/05/1996

2/1/96

15/02/1996

13/11/1995

28/09/1995

11/8/95

28/06/1995

10/5/95

22/03/1995

3/2/95

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Implied Volatility (%) COMB Forecasted Volatility GARCH(1,1) Forecasted Volatility

The chart clearly shows that, overall, the ‘COMB’ forecasts are closer to the implied volatility than the GARCH (1,1) forecasts. However, since the ‘COMB’ forecasts take into account information about the implied volatility, when there are ‘spikes’ in the implied volatility figures as observed towards the end of the period, there are also spikes in the forecasts. This may have been the reason behind the increased number of trades since the distance between the volatility forecast and the implied volatility was maintained above the threshold level of 2%. 6.10 Risky Naked Call Write Trades The vast majority of the trades signalled in the strategies were ‘write’ signals. The reason for these signals was that the implied volatility was greater than the volatility forecasts. This indicated that the options were overpriced in the market and hence should be sold. In this scenario, the volatility is effectively being sold. Short volatility trades are very risky. By writing a call option, the writer is giving the buyer the option to choose between calling the stock from them at the specified exercise price or not. The writer has no say in the decision but will have an idea as to what the holder will do. If the stock price is above the exercise price at expiry, the writer will have to deliver the stock at the exercise price. If the stock is not already owned, it will have to be acquired by the writer. The price paid for it in the market would be greater than the amount received and, hence, a loss would be incurred. Often, it is actually easier to just buy back the call and incur the same loss. Since

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there is no upper limit to share prices or the level of a stock index, it can be very risky to write call options. There is a limited profit potential and an unlimited loss potential. If the stock price falls below the exercise price then the holder would not exercise the option and it would expire worthless. This would result in the call writer essentially buying it back at zero. This is the best outcome for naked call writers. This was often the case in this study where big profits were made because the premiums were often very low at expiry. All other things being equal, the passing of time results in the fall of the option prices. This is beneficial if the option has been written. Connolly (2000) comments that ‘near-the-money options suffer the most time decay’ (i.e. have the largest theta) and hence, shorting at-the-money options provides the most profit. This provides another explanation as to why the strategies were so profitable in this study since most of the calls were written and at-the-money options were used. Simply writing naked calls is very risky and is equivalent to establishing a short position in the underlying shares resulting in big losses should the underlying price rise. To offset this risk, a hedge can be generated whereby a short call option position is perfectly hedged with a long stock position. The ‘greek’ which determines the relationship between the change in stock price (in this case the FTSE 100 Index level) and the option is called the delta6. From the Black-Scholes option pricing model, this is given as N(d1) and ranges from zero to 1. In a Black-Scholes world, it is possible to create a risk-free hedge called a delta hedge by buying stock and selling calls. To maintain the risk-free position, the number of options per share must be adjusted continuously according to the hedge ratio. The result of the delta hedge is a delta neutral position. In the study, suppose there had been a ‘write’ signal and the delta of the index option was 0.512. A delta hedge could have been formed by buying 512 ‘units’ of the FTSE 100 Index and writing 1,000 calls. If the FTSE 100 fell by a small amount (say 10 points), 5,120 would have been lost on the stock. On the other hand, the option price would have also fallen by approximately 5.12. Multiplying by 1,000 gives a fall of 5,120. Since the options had been written, the loss in value of the options would result in a gain when they were bought back and the position closed. This gain would offset the loss on the stock. A similar result would be obtained if the underlying value went up. Delta hedging in this way works for small stock price movements. Things become complicated when there are significant price movements however. If the stock price rises significantly the option component always loses more than is made by the stock component. If the stock price falls significantly, the losses on the stock component always exceed the profits on the option component. The bigger the move, the bigger the losses. The reason for the losses is due to the curvature of the option price.

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The Delta of an option is the rate of change of an option’s price with respect to a change in the underlying. The delta is a measure of the sensitivity of the option price to changes in the price of the underlying. Also known as the hedge ratio.

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7. Conclusions On the face of it, one could argue that the empirical evidence provided in this paper suggests that it is possible to produce a volatility trading model that can take advantage of the mispricing of options based on the differences between the implied and forecasted volatility. It implies, therefore, that the option market is inefficient and / or the option pricing formula used is incorrect. This is a direct inference from the fact that volatility forecasts have been used to identify mispriced options and profitable trading rules have been established. The only problem would appear to be the riskiness of the strategies since the majority of the trading signals are to write calls. The difference between the volatility forecasts and the implied volatilities could be attributed to the well documented fact that the implied volatility is often overestimated. It was not expected that the difference would be quite so large however. If anything, the differences actually helped to make the strategies so profitable. Since the forecasted volatility was often lower than the implied volatility, the trading signals were to write at-the-money calls. These at-the-money calls offered a large amount of time decay which resulted in big profits when the positions were closed. Since the majority of the trades made were risky naked call writes a suggestion for further work would be to develop a model which hedges the risk of such trades. The problem with this, however, is that it would be very difficult to maintain the perfect hedge. Positions would have to be constantly changed due to the continually changing underlying price. The model would be more realistic, however, since in real life, traders would be unlikely to want to pursue such a risky strategy as proposed in this paper.

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DUNIS, C., LAWS, J. and CHAUVIN, S. (2000) ‘FX Volatility Forecasts and the Informational Content of Market Data for Volatility’, CIBEF working paper, May 2000. DUNIS, C.L. and GAVRIDIS, M. (1997) ‘Volatility Trading Models: an Application to Daily Exchange Rates’ Global Markets Research – Working Papers in Financial Economics, Issue 1, pp 1-3 DUNIS, C.L. (1996) ‘The Economic Value of Neural Network Systems for Exchange Rate Forecasting’ Neural Network World, Vol.1/96, pp 43-45. KRONER, K.F., KNEAFSEY, K.P. and CLAESSENS, S. (1995) ‘Forecasting Volatility in Commodity Markets’ Journal of Forecasting, 14, pp 77-95 LAMOUREUX, C.G. and LASTRAPES, W.D. (1993) ‘Forecasting Stock Return Variance: Toward an Understanding of Stochastic Implied Volatilities’ Review of Financial Studies, 6(2), pp 293 – 326. LAWS, J. and THOMPSONM, J.L. (1997) ‘The Informational Content of Weighted Implied Volatilities derived from $/DM Foreign Currency Options’ Version 1.2b. Presented at International Symposium on Forecasting, 1997. VASILELLIS, G. and MEADE, N. (1996) ‘Forecasting Volatility for Portfolio Selection’ Journal of Business Finance and Accounting, 23, pp 125-143 WALSH, D.M. , TSOU, G.Y.-G. (1998) ‘Forecasting Index Volatility: Sampling interval and non-trading effects’ Applied Financial Economics, 8, pp477-485

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