GARCH in Option Trading

Econometrics

GARCH models in option trading For many aspects of financial analysis uncertainty is of major importance. The conditional variance of financial time series is used for calculating measures of risk of holding an asset, the construction of hedge portfolios and option pricing. For some financial models, such as the CAPM (Capital Asset Pricing Model) and Black-Scholes model, returns are assumed to be independent (normally) distributed and variances in returns are assumed to be constant. This assumption about returns is typically not satisfied in financial timeseries. One of the most important characteristics of returns of financial assets or indices is that their mean appears to be constant while their variance changes over time. Furthermore, returns exhibit volatility clustering: large changes in returns are likely to be followed by further large changes. These characteristics can be seen from Figure 1, that shows the daily log return of the Euro Stoxx 50 Index, a weighted index of 50 European stocks.

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Karlijn Juttmann

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studied Econometrics at the University of Amsterdam. This year she finished her Master in Financial Econometrics. She did an internship at the Asset Management department of Insinger de Beaufort where she wrote her thesis ‘GARCH Models in Option Trading’. Last year she worked as an intern at Bain & Company in Chicago. Currently Karlijn is working at AlpInvest Partners as a Portfolio and Risk Analyst.

4 2 0 -2 -4 -6 -8 1992

1994

1996

1998

2000

2002

2004

2006

R

Figure 1: Daily log returns of Euro Stoxx, rt=100*ln(St/St-1)

One of the first economists to model the above mentioned characteristics of returns was Robert Engle when he introduced the Autoregressive Conditional Heteroskedasticity (ARCH) model in 1982. Bollerslev (1986) extended this model to the Generalised ARCH model (GARCH). Since their introduction a great number of papers appeared applying these models and extensions to it. The biggest strength of the GARCH models is their ability to describe the characteristics above in the variance of financial return data. There is extensive evidence that GARCH models are highly significant in-sample, but there is less evidence they also will provide good forecasts of future return volatility. The volatility forecast ability of the GARCH models is not straightforward to test, because actual volatility is not observed. In this paper I will test the forecast ability of the GARCH model by using the volatility forecasts in option trading. I will construct option trading strategies based on a comparison of volatility forecasts given by the GARCH model and the

volatilities that are implied by the Black-Scholes model and the market option prices. This implied volatility can be regarded as the market’s prediction of future volatility. If the GARCH model is capable of giving a better volatility forecast, this forecast can be used to construct a profitable strategy of buying and selling options. For the estimation of the GARCH model I will use daily return data of the Euro Stoxx 50 Index between 1992 and 2006. The options used for trading are European put options with the Euro Stoxx index as the underlying asset. The remainder of this article is organized as follows. The next section describes the GARCH model in some more detail. After that I will discuss the implied volatilities from the BlackScholes model and in the last sections I will explain how the option trading strategies are constructed and what the results from these strategies are. GARCH Models As noted before, GARCH models are well suited to describe most characteristics of return data. This section describes the GARCH model. Define the log return on a financial asset as

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rt=ln(St/St-1), where S is the price of the asset, in this case the index. We can model this log return as:

rt

E ¬ªrt | :t 1 ¼º  H t

(1)

ztV t , zt  IID(0,1)

(2)

E ª¬H t2 | :t 1 º¼

(3)

with

Ht and

V t2

Var ª¬rt | :t 1 º¼

where :t 1 is the information available up to time t-1. Equation 2 implies that εt is distributed with mean zero and a time varying vari2 ance V t . In the standard GARCH(1,1) model the variance at time t is modelled as a linear function of the square of the past shock and the conditional variance at time t-1.

V t2

Z  DH t21  EV t21, Z ! 0, D , E t 0.

(4)

It is possible to include more lagged squared shocks or lagged variances in the model but applied work frequently shows that the GARCH(1,1) model is suitable for representing the majority of financial time series. Under certain restrictions this model exhibits mean reversion to an unconditional variance. In most studies there is a high persistence in volatility, which means the reversion to the unconditional variance goes very slowly.

1 so that mt approaches ω very slowly. I estimate these models with the Euro Stoxx return data.1 All of the models above are able to describe the characteristics of the return data in-sample; they can remove the heteroskedasticity from the data. The asymmetric or TARCH model is preferred to the standard GARCH model, based on the value of its log likelihood and the significance of the parameters in the model. Based on the parameter estimates of these models I make volatility forecasts of the future volatility during the lifetime of the options used for trading. Implied Volatility Another way to construct a forecast of future volatility during the lifetime of an option is by means of the Black-Scholes model and option market prices. The Black-Scholes model is the most famous model to price European options. One of the determinants of option prices is volatility, which is assumed to be a constant. By putting all other determinants and market option prices in the model we can calculate the volatility implied by the model. This can be seen as a volatility forecast over the lifetime of the option which I will compare with the volatility forecasts of the previous section.

Several extensions to this model exist, which I include in my investigation. One of them is the TARCH (Threshold GARCH) model, in which a negative shock has a larger impact on future volatility than a positive shock of the same size.

V t2

Z  DH t21  JH t21I ª¬H t21  0º¼  EV t21

(5)

A second extension I consider is the Component TARCH or CTARCH model. This model allows mean reversion to a level mt that changes over time:

V t2

mt  D (H t21  mt 1 ) 

J (H t21  mt 1)I[H t 1  0]  E (V t21  mt 1 ) (6) mt

Z  U (mt 1  Z )  M(H t21  V t21 )

In this model V t converges to the time varying long run volatility mt , while mt converges to ω with powers of ρ. ρ is usually between 0.99 and 2

Figure 2: Implied volatility and GARCH volatility forecasts over the remaining lifetime of an option

Figure 2 shows the implied volatility together with the volatility forecasts given by the different GARCH models. On every trading day between 2001 and 2006 I take an at-the-money option with the lowest time to maturity. For this option I calculate both volatility forecasts. The figure shows that the three different GARCH forecasts are quite similar. They also move close together with the implied volatility, but the latter is on average slightly higher than the GARCH forecasts. This is especially the case during the years 2003 and 2004. Based on the comparison of these two volatilities I will con-

All models where estimated with the assumption of student-t distributed error terms. This led to better results than under the assumption of normally distributed error terms.

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Figure 3: The change in a delta neutral portfolio given a change in de underlying asset for a) a long position in the portfolio and b) a short position in the portfolio

struct an option trading strategy for every day in the sample. Trading Strategies For each trading day between 2001 and 2006 implied volatilities and GARCH forecasts are constructed in the previous sections. Furthermore, I construct a delta neutral portfolio on each day in the sample. This portfolio consists of one option and an amount of delta2 invested in the index. Figure 3 shows how the value of a delta neutral portfolio changes with a change in the price of the index. Based on the comparison of the two volatility forecasts I create buy and sell signals for the portfolios on every trading day. If the GARCH forecast exceeds the implied volatility of the option a long position in the portfolio is taken, which is held for one day. This means we buy the portfolio and sell it one day later. If the GARCH forecast is correct and the change over this day in the index is larger than expected by the market, this position will be worth more at the end of the holding period. Figure 3a shows this. Selling the position one day later will then lead to a positive return. If the GARCH forecast will be lower than the implied volatility a sell signal is given and a short position in the portfolio is taken. If the forecast is correct and the change in the index will be relatively small, this also will lead to a positive return as can be seen in Figure 3b. Each trading day the above trading strategy will generate a profit/loss of the difference between the portfolio values (Π) at times t and t+1. I calculate the daily returns (Rt) as the profit as a percentage of the underlying index price (S).

Rt

(3 t 1  3 t ) for a long position and St

Rt

(3 t  3 t 1 ) for a short position St

(7)

In this way I construct return series of strategies based on the GARCH, TARCH and CTARCH models. It is sometimes argued that options are consistently overpriced which means that implied volatility will be always higher than realised volatility (Engle and Rosenberg, 1994). In this case a strategy of always selling options will lead to significant profits. As can be seen from Figure 2 implied volatility exceeds the GARCH forecasts on most days and a sell signal is given. If the GARCH strategies generate positive returns this could be the result of overpriced options. Therefore I also consider the returns of a strategy of selling the option every day to investigate if the GARCH model strategy has some additional value over an always sell strategy. I also construct a second, filtered strategy where a buy or sell signal is given only if the difference between the GARCH forecast and the implied volatility exceeds a certain value. Results Table 1 shows the returns from trading put options according to the different strategies. These returns are only relative because of the way they are calculated. If a trade in two options was made instead of one, returns would be twice as large. However, this way of calculating returns allows us to compare the results of the different trading strategies considered here. Returns from all strategies turn out to be significantly positive for the full sample, 2001-2006.

The delta of an option is defined as the rate of change of the option price with respect to the change in the price of the underlying asset and calculated as the first derivative of the Black-Scholes pricing formula to the asset price.

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GARCH

TARCH

CTARCH

allways sell

Full sample

0.013 (0.199)

0.009 (0.199)

0.013 (0.199)

0.012 (0.199)

Sample 20012002

-0.003 (0.265)

-0.003 (0.265)

0.003 (0.265)

-0.004 (0.265)

Sample 20032004

0.030 (0.193)

0.026 (0.193)

0.025 (0.194)

0.030 (0.193)

Sample 20052006

0.009 (0.108)

0.003 (0.108)

0.009 (0.108)

0.006 (0.108)

Table 1: Average daily returns from trading delta hedged put options (st.dev.)

Considering the three different sub samples, we see that positive returns are only generated in the sample containing the years 2003 and 2004. This result implies a superior forecast ability of the GARCH models over the volatility forecast implied by the option prices in the period 2003-2004. However, Figure 2 shows that in this sub sample implied volatility exceeds the GARCH forecast on most of the days. In 90% of the days a sell signal is given. Moreover, the GARCH strategy returns do not exceed the returns of the always sell strategy. This suggests that the positive returns in 2003-3004 are the result of overpriced options in this sub sample. Another way to show this is to consider the returns after a buy and sell signal separately for the GARCH strategies. The average returns generated after selling the portfolio are significantly positive. The returns after a buy signal are not. This means that the GARCH models are unable to detect the days in the sample where implied volatility is too low and options are underpriced.

for example the risk free asset, on days no buy or sell signal is given by the model. Although the filtered strategies outperform the unfiltered and always sell strategies, this model is still unable to detect the days on which implied volatility is too low. Conclusions In the literature about GARCH models, there is extensive evidence that the model performs well in describing in-sample return data. There is less evidence that the model is also capable of making accurate forecasts about future volatility. This paper showed again that the GARCH models performed well in-sample. I tested their forecast ability by using the forecasts to create profitable option trading strategies. The models only performed well in the years 2003-2004 where a transition took place from a period of high volatility to a period of low volatility. The options in this sample were overpriced and the GARCH models were able to detect this. The filtered strategy performed better and was capable of generating positive returns in the period 2001-2002 as well. However, none of the models were able to detect the days on which implied volatility was too low and options were underpriced.

The models perform well in the 2003-2004 sub sample. Figures 1 and 2 show that in this period a transition takes place from a high to a low volatility period. The option prices in the market are not able to adjust fast enough to this change in volatility. The GARCH models on the other hand make a faster adjustment to a lower volatility and therefore sell signals are given on most of the days in this sample, leading to positive returns. Higher returns are generated by the filtered strategies, where we only buy or sell options if the difference between the GARCH forecast and implied volatility is sufficiently large. The filtered TARCH model gives the best results and outperforms the always sell model in the period from 2001-2004. In the last sub sample, 20052006, hardly any trades are made and most of the returns in this period are consequently zero. Therefore, a good way to use the filtered strategy is to make an investment in another asset,

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