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| Type | Bachelor's Thesis |
| Scope | Discipline-based scholarship |
| Title | Dynamic Hedging of Swiss Stock Options during the Financial Crisis using GARCH: An Empirical Investigation |
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| Institution | University of Zurich |
| Faculty | Faculty of Economics, Business Administration and Information Technology |
| Date | 2011 |
| Zusammenfassung | Problem The monumental losses on derivative nancial instruments su ered by investors during the recent Financial Crisis may have been avoided through the implementa- tion of viable strategies for dynamic hedging. Due to the empirical observations of volatility clustering and excess kurtosis in return distributions, a hedging strategy in association with a pricing model is called for that accounts for the presence of stochastic volatility. Such a model is given by a GARCH(1,1) process. In a simple hedging strategy, a basic portfolio may be created that counteracts any price movements in derivative instruments such as option contracts. The empirical investigation in this thesis simulates the situation of an investor that purchases a Call option on the Swiss Market Index (SMI) at the beginning of the Financial Crisis and wishes to be hedged against any unexpected movements in the option price at every point of the crisis. A parsimonious replication portfolio consisting of an Exchange Traded Fund (ETF) on the SMI and a presumably risk-free Swiss Government Bond (Eidgenosse) serves as a hedging device. The e ectiveness of the hedging strategy is tested in a simulation study, with the additional intention of assessing the accuracy of the option pricing model and the ability of the strategy to capture the observed volatility dynamics. Procedure The analysis to follow is concerned with the e ectiveness of a hedging strategy for a Call option based on a GARCH(1,1) model to forecast volatilities and a binomial asset pricing model to price the option in a Cox-Ross-Rubinstein framework. The purpose of the simulation study is to evaluate how a dynamically adjusted portfolio can serve as a hedge against an exposure in an option contract on the SMI for the evaluation period given by the lifetime of the investigated option. The analyzed option is a European at-the-money Call issued in August 2006 with maturity in December 2009. Therefore the considered period for the simulation study covers the run-up, the main part and the aftermath of the Financial Crisis. With the time frame of the empirical analysis being mostly during the Financial Crisis, the e cacy of the strategy is intentionally tested in a harsh market environment. The Call option is priced in a CRR framework where the value of the option is derived using a binomial tree with stochastic volatility. At the initial point of estimation (the 25th of August 2006, after close), a GARCH(1,1) model is estimated based on ten years of daily and weekly historical log returns to estimate GARCH(1,1) parameters and to forecast volatilities for every single node of the binomial tree until maturity. The future volatilities are then incorporated into the binomial tree model to determine the price of the option and the parameters for the hedging portfolios. Since Delta hedging might be insu cient in the presence of stochastic volatility, a 'standard' volatility swap is considered to make the hedging portfolio i Vega neutral as well. However, this method is not only applied once, but every day or week until maturity of the option contract. This means that, in the daily measurement case, the call is purchased on the 28th of August 2006 along with the number of units of the ETF, the Eidgenosse and, in the case of additional Vega hedging, the 'standard' volatility swap as calculated on the evening of the 25th. At the end of the 28th, the whole estimation procedure is repeated to account for the new market data, meaning that a new GARCH(1,1) model is estimated based on the daily log returns of the past 10 years, new volatilities are forecasted until maturity of the option and eventually, a new fair value of the option and new hedge portfolios are calculated. On the next trading day, the portfolio components are adjusted so they re ect the calculated optimal values. If this procedure is conducted every day (or every week, in the analysis of weekly data) until the option matures, the private investor should be protected against the movements in the option's price. The performance of the overall portfolio (called the Net Portfolio) consisting of the option and the appropriate hedging portfolio can then be evaluated after every time unit. The comparison between daily and weekly estimation delivers insights as to whether a more active hedging strategy would result in a greater hedging e ectiveness. Furthermore, it is of interest which hedge portfolio will yield better e cacy, the simple Delta hedging strategy or the one that additionally hedges Vega exposure. The pricing and hedging errors for both Delta and additional Vega hedging are calculated as the Mean Squared Pricing Error (MSPE) and the Mean Squared Hedging Error (MSHE). Comparing the realized volatilities with the forecasted volatilities by the GARCH(1,1) models may also deliver some crucial insights regarding the capability of the GARCH(1,1) process to model the risk associated with stock indices, especially during periods of crisis. The thesis is structured as follows: the second section discusses the existing literature on dynamic hedging and the mathematical modelling of stochastic volatility. The third and fourth sections provide the theoretical background for the empirical investigation to follow. Section three covers the topic of estimating volatilities and volatility processes, whereas section four elaborates on the risk-neutral option pricing and hedging under stochastic volatility. Section 5 then introduces the setup and data background of the empirical investigation, after which the results are summarized in section 6. Finally, section 7 discusses some of the issues associated with the data treatment, econometric procedure and estimation, such as the major discrepancy between the physical and the risk-neutral probability measures. Results The ndings suggest that the applied model performs well in protecting the investor from the massive losses observed during the main part and the aftermath of the Financial Crisis. This result holds true especially for the daily data treatment, whereas in the weekly evaluation some signi cant disturbances are encountered. Furthermore, relevant pricing errors are observed in the months preceding the main part of the Financial Crisis in both the daily and the weekly data setting. ii The examined daily and weekly log returns show clear signs of volatility clustering for the period 1996-2009 and particularly for 2006-2009, which suggests signi cant correlation in the volatility processes during the Financial Crisis. Furthermore, distributions of the return series are found to exhibit excess kurtosis (fat tails and narrow peaks) for all time periods and data frequencies. These observations serve as a justi cation for modelling the heteroskedastic nature of the volatility processes with a GARCH model during the Financial Crisis. The estimated GARCH(1,1) parameters are stable in the daily analysis, however, in the weekly analysis some signi cant disturbances are discovered for the estimated 1 and coe cients, resulting in unrealistically high long run equilibrium volatility levels. The applied option pricing model is then found to show some relevant pricing errors during the rst year of the evaluation period, the run-up of the Financial Crisis, for both daily and weekly data, but thereafter also exhibits sequences during the center piece of the Financial Crisis of high pricing accuracy. Regarding the hedging e ectiveness, the Net Portfolio is mostly found to be negative for simple Delta hedging during the evaluation period. The hedging errors show the almost identical patterns as the pricing errors for both daily and weekly data. The results further imply that incorporating a 'standard' volatility swap to hedge plain Vega exposure has almost no e ects on the hedging strategy in both daily and weekly evaluation. Overall, the MSPE and MSHE are found to be almost three times larger for the weekly analysis than for the daily analysis, which suggests that more frequent rebalancing of the weights in the hedging portfolio is worth the e ort. Finally, the results show that the estimated GARCH models are able to estimate the general process of the realized volatility levels. This is seen in a signi cant rise of the estimated volatilities during the Financial Crisis. The predicted future levels of volatility show some obvious deviations from the realized levels. Volatilities that are forecasted further than one year into the future have no meaningful interpretation due to the mean reversion inherent to the model speci cations. General Evaluation Aside from certain assumptions that are in disagreement with the real market environments but are at the same time necessary for maintaining viability of the pricing model, the issue pertaining to the physical and risk-neutral probability measures represents one of the main limitations of the presented framework that should be accounted for in future studies. As an extension, the presented transition from the physical to the risk-neutral measure could be applied to the underlying return process for the GARCH model in order to compare the resulting pricing accuracy and hedging e ectiveness to the ndings in this thesis. Furthermore, the pricing accuracy and hedging e ectiveness could be compared to the parameters resulting from the Black-Scholes-Merton (BSM) model in order to evaluate the goodness of the presented methodology. Even if the restriction of constant asset volatilities suggested by BSM is not assumed to hold, practitioners still use models depending on that assumption. More speci cally, it may be an important topic for future research to compare the option valuation performance for iii discrete time GARCH models with continuous time stochastic volatility models. On the other hand, comparing the hedging e ectiveness for di erent GARCH speci- cations and distributions of the error terms may also give some additional insights. EGARCH, NGARCH or TGARCH may, for example, serve as alternative types of GARCH processes, whereas the Student's t-distribution or the Hansen's Skew t- Distribution could, for instance, be applied as assumptions for the distribution of the error terms. |
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