Pricing and Static Hedging of Foreign Exchange Barrier Options: An Empirical Analysis in a Period of High Volatility

Transcription

1 HEC MONTRÉAL Pricing and Static Hedging of Foreign Exchange Barrier Options: An Empirical Analysis in a Period of High Volatility By Kacper Jurga Business Administration Finance This thesis is presented in partial fulfillment of the requirements for the degree of Master of Science (M.Sc.) (Finance Option) January 2012 c Kacper Jurga, 2012

2 i Résumé Dans ce mémoire nous évaluons l effi cacité de cinq différentes stratégies de duplication continue en répliquant des options barrière écrites sur taux de change dans trois modèles paramétriques différents durant une période de volatilité générale prononcée. Spécifiquement, les trois modèles paramétriques que nous considérons sont le modèle "de praticien" de Garman et Kohlhagen, le modèle de volatilité stochastique de Heston, ainsi que le modèle à sauts de Merton. Les cinq stratégies de duplication continue évaluées sont la stratégie de Carr et Chou avec trois options (CC3), la stratégie de Carr et Chou avec cinq options (CC5), la stratégie de Carr et Chou ajustée de façon uniforme (UNIF), la stratégie de Carr et Chou ajustée au smile (SMILE) ainsi que la stratégie symétrique put-call (PCS). Le but de notre mémoire est de déterminer la sensibilité de nos stratégies de duplication continue au risque de modèle, ainsi que d évaluer laquelle des cinq stratégies considérées fournit les meilleurs résultats de duplication pour chacun des trois modèles paramétriques. Ultimement, nous n avons trouvé aucun lien explicatif entre le coût d une stratégie de duplication et sa performance en termes de duplication terminale, malgré la présence d une corrélation. De plus, nous avons trouvé que la qualité de duplication terminale de nos stratégies diminue lorsque le niveau de la barrière de l option exotique augmente, c est-à-dire lorsque l option devient plus "exotique", peu importe le modèle paramétrique considéré. Nous trouvons aussi que la qualité de duplication terminale de nos stratégies s améliore à mesure qu une option est davantage dans la monnaie. Les stratégies de duplication continue mises en place dans le modèle de Garman et Kohlhagen obtiennent les meilleurs résultats. Pour les options plus "exotiques", les stratégies de duplication mises en place dans le modèle de Merton performent mieux que celles mises en oeuvre dans le modèle de Heston. Pour les options moins "exotiques", nous constatons le résultat opposé. En ce qui concerne la réplication du prix initial, la meilleure stratégie de duplication est la stratégie UNIF, étant donné qu elle reproduit parfaitement le prix de chaque option barrière respective pour les trois modèles. En terme de la précision terminale des stratégies de duplication, il n y a aucune stratégie qui surclasse inconditionnellement les autres pour les trois modèles, et ce pour toutes les options barrière. Cependant, nous trouvons que les stratégies UNIF et CC5 ont tendance à offrir les meilleurs résultats. La stratégie SMILE, elle, a tendance à offrir les pires résultats pour l ensemble des trois modèles. Mots clés: analyse empirique, options barrières, tarification, duplication continue, duplication strike-spread, taux de change, modèle de Garman et Kohlhagen, modèle de volatilité stochastique de Heston, modèle à sauts de Merton

3 ii Abstract In this thesis we evaluate the effi ciency of five different strike-spread static hedges by hedging foreign exchange down-and-out call options in three different parameterizations during a period of pronounced general market volatility. The aim of our empirical analysis is to determine these hedging strategies sensitivity to model risk, and to assess which hedging strategy provided the best results for each parameterization. The three models used were the Practitioner s Garman-Kohlhagen model, Heston s stochastic volatility model, and Merton s jump diffusion model. The five static hedges implemented were the 3-option Carr-Chou hedge (CC3), the 5-option Carr-Chou hedge (CC5), the 3-option Uniformly-Scaled hedge (UNIF), the 3-option Smile-Scaled hedge (SMILE), and the put-call symmetry hedge (PCS). We found no causality between a hedge s initial cost and its terminal hedging performance for all three parameterizations, despite the presence of correlation. We also found that as the exotic options barrier levels increased, the effi ciency of our hedging strategies decreased, irrespective of the parameterization in which they were implemented. Similarly, as the exotic options moneyness increased, each of our hedges quality decreased in all three of our models. Additionally, we found that static hedges implemented in the Practitioner s Garman-Kohlhagen parameterization generally produced the overall lowest initial pricing and terminal hedging errors. We thus argue that model risk is smallest for this parameterization. For more "exotic" options, static hedges implemented in the Merton jump diffusion parameterization tended to outperform those implemented in the Heston stochastic volatility parameterization. For less "exotic" options, we found the opposite to be true. With regards to initial price replication, the UNIF static hedge was best, given that it perfectly reproduced each respective barrier option s price in all three considered parameterizations. With regards to terminal hedge precision, there was no absolute best hedge which always unconditionally outperformed all others for all three parameterizations. However, we would argue that the UNIF and CC5 hedges tended to perform best. Alternatively, the SMILE hedge tended to broadly perform worst across all three parameterizations. Keywords: empirical analysis, barrier options, pricing, static hedging, strike-spread hedging, foreign exchange, Practitioner s Garman-Kohlhagen model, Heston stochastic volatility model, Merton jump diff usion model

4 iii Acknowledgments First and foremost, I wish to thank my thesis supervisor, Dr. Pascal François, for his guidance, knowledge and countless brilliant suggestions. Without his ideas and support, this thesis could not have been completed. I also wish to thank the faculty at HEC Montréal, especially the finance professors I had a chance to learn from, and rub shoulders with, on a daily basis during my time here. Their knowledge and understanding of finance was an endless source of information and inspiration. I wish to thank HEC Montréal for awarding me the Le Blanc-Dalphond bursary during the course of my studies. As well, I wish to thank Mr. Guy Le Blanc and Mr. Claude Dalphond for their generous donation and support in establishing the Le Blanc-Dalphond award for graduate students in finance. Last, although certainly not least, I wish to thank my family and friends who, with their support and encouragements, kept my morale elevated in moments of waning. I thank my parents for their understanding, and for instilling in me my intellectual curiosity. I thank my brother and sister for their continuous motivation. I thank my friends for their support, backing and encouragements. Finally, I thank my fiancée for her countless revisions, unwavering support and understanding, and her everlasting enthusiasm, patience and affection. This thesis is dedicated to her.

5 iv Contents Résumé i Abstract ii Acknowledgments iii List of Figures vi List of Tables vii 1 Introduction 1 2 Literature Review 4 3 Barrier Options Barrier Option Specifications Greeks Model Choice and the Volatility Smile Pricing Models Practitioner s Garman-Kohlhagen Vanilla Options Barrier Options Heston Stochastic Volatility Vanilla Options Barrier Options Merton Jump Diffusion Vanilla Options Barrier Options

6 v 5 Hedging Methods Calendar-Spread Static Hedging Static Super-Replication Hedging Strike-Spread Static Hedging Data and Methodology Data Option Prices Exchange Rate Risk-Free Interest Rates Methodology Volatility Smoothing Calibration Broad Methodology and Performance Evaluation Results and Analysis Initial Pricing Sensitivity Terminal Pricing Sensitivity Analysis and Discussion General Results Practitioner s Garman-Kohlhagen Results Heston Stochastic Volatility Results Merton Jump Diffusion Results Conclusion 91 Bibliography 93

7 vi List of Figures 1 Practitioner s Garman-Kohlhagen Model s Implied Volatility Surface Heston Stochastic Volatility Model s Implied Volatility Surface Merton Jump Diffusion Model s Implied Volatility Surface Adjusted Payoffs for a DOC and Two Strike-Spread Hedge Portfolios Daily VIX Index Closing Levels Daily EUR/USD Spot Exchange Rate Closing and Intra-Day Low Levels Cubic Spline Smoothing Function Applied to Consistent Data Cubic Spline Smoothing Function Applied to Inconsistent Data Terminal Relative Hedging Errors of Chosen Options Initial Pricing Errors Versus Terminal Hedging Errors

8 vii List of Tables 1 Literature Review Key Highlights Sample Summary Data Sample Option Data for Calibration Initial Parameters for Calibration of the Heston Model Initial Parameters for Calibration of the Merton Model Heston Model Parameter Statistics Merton Model Parameter Statistics Barrier Knock-Out Statistics Results of Calibrated Models Plain Vanilla Option Pricing Errors Initial Pricing Errors of the PCS Hedge for the PGK Model Initial Pricing Errors of the CC3 Hedge for the PGK Model Initial Pricing Errors of the CC5 Hedge for the PGK Model Initial Pricing Errors of the SMILE Hedge for the PGK Model Initial Pricing Errors of the PCS Hedge for the Heston Model Initial Pricing Errors of the CC3 Hedge for the Heston Model Initial Pricing Errors of the CC5 Hedge for the Heston Model Initial Pricing Errors of the SMILE Hedge for the Heston Model Initial Pricing Errors of the PCS Hedge for the Merton Model Initial Pricing Errors of the CC3 Hedge for the Merton Model Initial Pricing Errors of the CC5 Hedge for the Merton Model Initial Pricing Errors of the SMILE Hedge for the Merton Model Terminal Pricing Errors of the PCS Hedge for the PGK Model Terminal Pricing Errors of the CC3 Hedge for the PGK Model Terminal Pricing Errors of the CC5 Hedge for the PGK Model

9 viii 25 Terminal Pricing Errors of the UNIF Hedge for the PGK Model Terminal Pricing Errors of the SMILE Hedge for the PGK Model Terminal Pricing Errors of the PCS Hedge for the Heston Model Terminal Pricing Errors of the CC3 Hedge for the Heston Model Terminal Pricing Errors of the CC5 Hedge for the Heston Model Terminal Pricing Errors of the UNIF Hedge for the Heston Model Terminal Pricing Errors of the SMILE Hedge for the Heston Model Terminal Pricing Errors of the PCS Hedge for the Merton Model Terminal Pricing Errors of the CC3 Hedge for the Merton Model Terminal Pricing Errors of the CC5 Hedge for the Merton Model Terminal Pricing Errors of the UNIF Hedge for the Merton Model Terminal Pricing Errors of the SMILE Hedge for the Merton Model

10 Chapter 1 Introduction Over the past decade, financial derivatives markets have grown substantially. According to numerous sources, a subsection of these products, specifically the foreign exchange financial derivatives market, is now considered to be one of the largest in the world. The Bank for International Settlements (BIS), an intergovernmental organization of central banks, reported in its Semiannual Over-the-Counter Derivatives Markets publication, BIS (2011), that the over-the-counter (OTC) foreign exchange market in the G10 countries and Switzerland accounted for a notional amount outstanding of about USD $64.70 trillion and a gross market value of about USD $2.34 trillion as of June Foreign exchange options accounted for USD $11.34 trillion and USD $332 billion of these amounts respectively. Exchange-traded foreign exchange contracts accounted for a notional amount outstanding of only USD $389 billion according to the same document. The rise to prominence of foreign exchange OTC markets has helped sprout the growth of exotic derivatives, with barrier options undoubtedly representing one of the most popular forms of second-generation derivatives. As argued in Dupont (2001), barrier options are especially prominent in the foreign exchange and fixed income markets, and have various uses. Dupont (2001) claimed that they "[...] help traders who place directional bets enhance their leverage and investors who accept to keep some residual risk on their books reduce their hedging costs." As explained in François (2005), a barrier option is a type of exotic option which potentially offers the same payoff as a plain vanilla option, depending on the evolution of the underlying security s price throughout the life of the contract. As such, Dupont (2001) claims that barrier options are known as path dependent financial contracts, since they are contingent on both the final price of the underlying security, as well as its path throughout the life of the option in order to be properly valued. Dupont (2001) argues that, to reduce their risks, financial institutions often hedge these products in order to match their payoffs at all possible times, both throughout the life of the contract and at expiration. In parallel, market participants often try to replicate these securities, a process whereby they attempt to match their terminal payoffs at the same cost as that of the barrier options in question. However,

11 Chapter 1 - Introduction 2 in order to effi ciently hedge and replicate these second generation options, their writers and buyers must first correctly price them. As shown in Hirsa et al. (2003), this is not a straightforward task; parametric models can often produce very similar plain vanilla option prices, yet offer quite different barrier option prices. These points essentially lead us to the scope of this dissertation. In this thesis we seek to empirically compare different option pricing models with regards to barrier option hedging. Thus, we aim to determine the sensitivity of hedging strategies to model risk, specifically by analyzing a data set covering a period of high volatility ranging from mid-2008 to mid Our goal is to determine which hedge would have performed best for each of our parametric models during the aforementioned analysis period. As such, we will price OTC foreign exchange European down-and-out call (DOC) standard barrier options in three different classes of parametric models, each calibrated to the implied volatilities of European plain vanilla foreign exchange options. The foreign exchange rate we will be using is the EUR/USD rate, governed chiefly by the fact that it represents the most liquid and most traded foreign exchange rate available in currency markets. The three models used will be the Practitioner s Garman-Kohlhagen model, from the Black-Scholes family, as described in Garman and Kohlhagen (1983), with the "Practitioner" element presented in Dumas et al. (1998), the Heston model, from the stochastic volatility family, as elaborated in Heston (1993), and the Merton model, from the jump diffusion family, covered in Merton (1976). As mentioned previously, since barrier options are traded over-the-counter, it will be impossible for us to measure the quality of the pricing performance of our models. Because we do not have access to market participants OTC derivative prices, we will be simulating our prices through the use of Monte Carlo simulations for the Heston and Merton models, and implementing an analytical formula for the Garman-Kohlhagen model. We will be assuming that these prices represent the "true" prices for each model. As mentioned in the previous paragraph, we will proceed by calibrating the parametric models directly to daily implied volatility surfaces obtained from market prices (in a Black-Scholes framework) in order to price our EUR/USD one-month European down-and-out call options of various barrier and moneyness levels. We will have a maximum of seven strike levels for our exotic options, varying between 91% and 107% of the underlying s initial level, depending on the chosen barrier level. Our moneyness levels will thus be "deep" in-the-money ( K X t 0.91), in-the-money ( K X t 0.93 and 0.96), atthe-money ( K X t 0.99 and 1.01), out-of-the-money ( K X t 1.04), and "deep" out-of-the-money ( K X t 1.07). Of note, although we classify them as being "deep" in- or out-of-the-money, these strike levels are clearly within 10% of par moneyness levels, a range chosen intentionally in order to guarantee the use of the most common and liquid levels. We will incorporate four barrier levels, which will be set at 90%, 92%, 95% and 98% of the underlying s initial price. It is assumed that most, if not all, of the work presented in this thesis for down-and-out calls can

12 Chapter 1 - Introduction 3 easily be applied to other barrier types, and even to other exotic options (as will be covered in Chapter 3). The static hedging strategies scrutinized will be variations of the methodology known as strikespread static hedging. This strategy involves implementing a hedge portfolio composed of plain vanilla options with varying strikes, but identical maturities. The portfolio components, and their respective weights, do not vary until the barrier option s maturity, or until the occurrence of a barrier event. This thesis is organized as follows. In Chapter 2 we present a brief review of the available literature. Chapter 3 describes barrier options in more detail. Chapter 4 introduces the three pricing models used and discusses the pricing methodologies employed for both plain vanilla and barrier options. In Chapter 5 we provide a broad overview of popular alternative static hedging strategies, and thoroughly present the strike-spread static hedging methods which will be implemented for barrier options. Chapter 6 deals with the raw data and the methodology used in this paper, including our data smoothing procedure and the calibration of our parametric models. Chapter 7 offers the results of our work and presents an analysis of the outcome. Finally, we present our conclusions in Chapter 8.

13 Chapter 2 Literature Review Our literature review focuses primarily on the combination of exotic option pricing, model risk and static hedging. Past work on other pertinent elements, such as those concerning barrier options, hedging methodologies and different model parameterizations themselves, will be covered in their respective Chapters in the remainder of this thesis. With regards to the combination of exotic option pricing, model risk and static hedging, there are unfortunately few papers that focus on all three elements at once, while also implementing the same model parameterizations as those incorporated in this thesis. One paper that did cover these elements however was Nalholm and Poulsen (2006), and it served as a very important reference for this thesis. In Nalholm and Poulsen (2006), the authors test the simulated hedge precision of various static hedging strategies implemented for barrier options in five different parametric models; the Black-Scholes model, the Constant Elasticity of Variance model presented in Cox and Ross (1976), the Heston stochastic volatility model, the Merton jump diffusion model and the Variance Gamma model introduced by Madan et al. (1998). The authors show that static hedging generally offers important advantages over, and better results than, standard dynamic hedging. In fact, Nalholm and Poulsen (2006) determines that "[...] static hedge accuracy [is] robust across models producing similar prices of plain vanilla options (but diff erent barrier option prices)." In the three models that are most relevant to this thesis (the Black-Scholes, Heston and Merton models), the authors prove that strike-spread and calendar-spread static hedging strategies always offer vast reductions in the standard deviation of a replication portfolio when hedging both down-and-out and up-and-out call (UOC) barrier options. With regards to the mean replicating error, their results indicate that the implemented static hedges produce lower absolute values in a majority of cases, although for some static hedges in more complex models the mean error, in absolute terms, was shown to be greater than that of dynamic hedges (such as for strike-spread static hedges in Heston s stochastic volatility model for example). Overall, the strike-spread static hedging strategy (or one of its suggested adjusted versions) offers the lowest hedge standard deviation in five of the six relevant cases (which we consider to be the results for both the UOC and the DOC in the three aforementioned models). Only for the down-and-out call in the Black-Scholes model does the calendar-spread

14 Chapter 2 - Literature Review 5 methodology slightly beat out the strike-spread hedge (with an accuracy of 1.4% versus 1.9%). In some cases, as for an up-and-out call in Merton s jump diffusion model for example, the improvements obtained by implementing the strike-spread hedges were as high as a reduction of about 78% of the standard error of a traditional dynamic hedge. Furthermore, in six of the eight cases in which scaled strike-spread static hedges, which Nalholm and Poulsen (2006) explains are essentially adjusted standard strike-spread static hedges, were implemented, they outperformed both dynamic hedges and standard strike-spread static hedges in terms of hedge accuracy. Nalholm and Poulsen (2006) concludes that static hedges outperform dynamic delta hedges, and are much more effi cient in various model parameterizations. Additionally, the authors simulation results also indicate that for the static hedges themselves, the strike-spread strategies tend to beat out the calendar-spread strategies in terms of precision. This paper, amongst others, led us to focus on the performance of strike-spread static hedges from an empirical perspective. In terms of plain vanilla and exotic option pricing methodologies, we relied a great deal on Jessen and Poulsen (2010), which served as our primary guide, along with Nalholm and Poulsen (2006). In their paper, Jessen and Poulsen test the empirical foreign exchange barrier option pricing accuracy of five different parametric models; the Black-Scholes model, the Heston stochastic volatility model, the Merton jump diffusion model, the Constant Elasticity of Variance model and the Variance Gamma model (which are the same five models as in Nalholm and Poulsen s paper). Using plain vanilla and barrier option data, and calibrating the parametric models directly to plain vanilla options implied volatilities, Jessen and Poulsen (2010) empirically evaluates each model s precision in pricing EUR/USD barrier options (the authors mention that they obtained empirical barrier option prices from the Danske Bank for comparison). Jessen and Poulsen (2010) chooses the most precise model implementation methodologies for each parametric setting, according to the available literature, and empirically validates the mispricing of foreign exchange barrier options with regards to actual market prices. In this thesis, we incorporated the same pricing implementations based on Jessen and Poulsen s suggestions (except for pricing barrier options in Merton s jump diffusion model, which will be covered in Chapter 4). Of note, although we will only be working with standard down-and-out call options, Jessen and Poulsen (2010) prices multiple types of barrier options, in both standard and reverse formats. The authors show that although all five of their models produce plain vanilla option prices close to market prices, the Heston stochastic volatility model produces the lowest absolute pricing errors, followed closely by the Merton jump diffusion model. The same two models perform best when the authors test the prediction errors out-of-sample for up to five days for all five models, except that Merton s model instead comes out ahead, followed closely by Heston s model. For standard barrier options, Jessen and Poulsen (2010) demonstrates that the Black-Scholes, Constant Elasticity of Variance and Heston models produce the lowest in-sample absolute pricing errors. Out-of-sample, again for a time horizon of five days, the same three models obtain much better results than both the Merton and Variance Gamma models, which are deemed to perform very poorly and are shown to produce unacceptably large absolute errors according to Jessen and Poulsen (2010).

15 Chapter 2 - Literature Review 6 Unfortunately, there currently does not exist a wide array of papers which attempt to determine the hedging risk associated with barrier options in an empirical context. Although Jessen and Poulsen (2010) was very useful, especially from a data-based perspective, it unfortunately did not offer any hedging results (although the authors were adamant that future work would be focused on this area). Additionally, few, if any, papers combine an empirical comparison of various model settings with different static hedging methodologies, or more precisely calendar-spread and strike-spread static hedges (which will both be covered in Chapter 5 of the present thesis, with a more detailed emphasis on strike-spread static hedging as it is the main strategy we will employ). However, two papers that are often mentioned when discussing empirical studies pertaining to hedging and barrier options are those of Engelmann et al. (2006) and An and Suo (2009). In Engelmann et al. (2006), the authors empirically compare various static and dynamic hedging methodologies applied to barrier options in a setting based on market data. Using the Black-Scholes model and the Local Volatility model introduced by Derman and Kani (1994), and refined by Dupire (1994), the authors implement 17 various hedging strategies to hedge one-year reverse up-and-out call and down-and-out put (DOP) barrier options, each with specific strike and barrier levels, written on the Deutscher Aktien IndeX (DAX), a German stock market index. Engelmann et al. (2006) reviews and implements hedging methodologies from four broad families; dynamic hedges, calendar-spread static hedges, strike-spread static hedges and unified hedges (which the authors explain combine both static hedging methodologies). In order to evaluate their hedging methodologies, the authors rank each according to their profit-and-loss outcomes at expiration. Engelmann et al. (2006) proves that, according to the mean absolute deviation of the median hedging error, dynamic hedging strategies are much less effi cient for both option types (reverse UOC and reverse DOP) than the unified hedging strategies. However, the results most relevant to this thesis pertain to the strike-spread hedging strategies. Engelmann et al. (2006) shows that, depending on the barrier option in question, the strategies either did very well, or very poorly. As the authors results indicate, in the case of reverse UOC options, they provided the best hedging results, combined with very low standard deviations and the smallest extreme losses. For reverse DOP options, the strike-spread hedging strategies were amongst the least effective strategies tested, and were broadly in line with the various calendar-spread hedging strategies implemented in terms of poor results. However, the authors conclude that based on their overall results, from the static hedging subset of methodologies, the strike-spread static hedges perform much better than the calendar-spread static hedges in their empirical analysis. Engelmann et al. (2006) also mentions that there is no consistent relationship between hedging costs and hedging errors, as a hedge s cost appears to be linked to its performance in the case of certain options, whereas it appears to have no influence for other options. In the second empirical paper of great importance, An and Suo (2009), the authors evaluate the model risk inherent in exotic option pricing and hedging using four parametric model settings; a unique Practitioner s Black-Scholes model, Merton s jump diffusion model, as well

16 Chapter 2 - Literature Review 7 as a stochastic volatility and jump diffusion model, found in Bakshi et al. (1997), and its nojump alternative. After calibrating their models to EUR/USD European plain vanilla foreign exchange option prices, the authors determine the hedging performances of different hedging methodologies for an array of exotic options, including up-and-out call barrier options of varying maturities (between one month and one year) and moneyness levels (which are considered to be Xt K in An and Suo s paper, and range between 0.94 and 1.06). An and Suo (2009) evaluates the hedging performance of the models according to the mean dollar-value hedging error and the mean absolute deviation error. Although the authors unfortunately do not compare and contrast any static hedging strategies, they incorporate dynamic strategies that are more advanced than basic Black-Scholes delta hedges. Specifically, An and Suo (2009) implements minimum variance and delta-vega neutral dynamic hedges. Focusing exclusively on the results most relevant to this thesis, more precisely those dealing with short-term barrier options, An and Suo show that, using both evaluation metrics mentioned previously, the stochastic volatility model seems to consistently produce the smallest hedging errors overall, followed by the Black-Scholes model, across all moneyness levels for both the minimum variance and delta-vega neutral dynamic hedges. The jump diffusion model lags behind the first two for all relevant cases, in terms of both the mean dollar-value hedging error and the mean absolute deviation error. For all relevant cases, An and Suo (2009) shows that hedge precision also broadly decreases as moneyness decreases. An and Suo (2009) explicitly proves that shortterm out-of-the-money barrier options have much worse hedging results (errors further away from zero) than short-term in-the-money barrier options. An and Suo s results also prove that the closer the barrier level to the underlying, the worse the hedging performance, irrespective of the moneyness level of the underlying option. This essentially highlights the primary finding we retain from An and Suo (2009), whereby "[...] the more "exotic" the option, the poorer the hedging eff ectiveness." In this case, an out barrier level far from the underlying has less chance of being breached, thereby rendering the barrier option less exotic (and easier to hedge), as highlighted by An and Suo (2009). Unlike for empirical studies, there are, auspiciously, quite a few simulation studies with regards to barrier options, hedging strategies and various model dynamics. The one that primarily retained our attention was Nalholm and Poulsen (2006), chiefly because it was an important reference for this thesis due to the implemented models (as explained, it covered all three of the models we incorporate in this thesis) and the static hedging strategies compared (it also covered most of the hedges we incorporate in this thesis). The other study which we considered to be quite informative was that of Tompkins (2002). We found Tompkins (2002) to be very pertinent to this thesis, as the author compared the performance of dynamic and static hedge performances by implementing a simulation study for a wide array of exotic options, including barriers, under various market imperfections. Although Tompkins limited his paper to the simplest form of call barrier options, and established a straightforward delta hedging strategy and a basic put-call symmetry static hedge, as proposed by Carr (1994), his simulation results are nonetheless quite revealing. Tompkins (2002)

17 Chapter 2 - Literature Review 8 shows that the implemented simple static hedge outperforms the discrete dynamic hedge for down barrier options in both a constant and non-constant volatility setting, as well as when including transaction costs in both situations. For all of Tompkins simulated cases, the static hedging strategy was shown to reduce both of the author s hedge precision evaluation metrics (the hedging cost difference and its standard deviation). For up barrier options, Tompkins (2002) showed that both the dynamic and static hedging strategies are ineffi cient, with these poor results amplified when transaction costs or non-constant volatility were imposed on the simulations. In fact, in the most complex case, when incorporating both transaction costs and non-constant volatility, the standard deviation of the hedging cost relative to the up barrier option s theoretical value was 385.3% for the dynamic strategy, and 443.5% for the static strategy; the dynamic hedge was actually shown to outperform the static hedge. Tompkins (2002) essentially shows that, for down barrier options in a simulated environment, static hedges are less sensitive to market imperfections (such as non-constant volatility and the addition of transaction costs) than dynamic hedges, thereby laying the groundwork for the findings of Nalholm and Poulsen (2006) for out barrier options, and those found in Engelmann et al. (2006) pertaining to reverse up-and-out call options. In this literature review we also retain Toft and Xuan (1998), as it was one of the first papers of great significance to analyze static hedges sensitivity to different model settings. Furthermore, we choose to mention Figlewski and Green (1999) in this review as well, since this paper empirically proved the benefits of a basic dynamic hedging strategy for option writers. In Toft and Xuan (1998), using the calendar-spread static hedging methodology, the authors publish an important paper regarding static hedging sensitivity. Working with a Heston stochastic volatility parameterization for the underlying asset, and estimating a deterministic local volatility function along the lines of the one suggested in Dumas et al. (1998), Toft and Xuan (1998) constructs static hedges for one-year up-and-out barrier options using trinomial trees to match the hedge portfolios boundary values with that of specific options values at chosen times. Simulating both the underlying asset price and the volatility process, Toft and Xuan (1998) shows that when the volatility of volatility (the σ in the Heston model, which will be covered in detail in Chapter 4) is small, static hedges perform fairly well. Thus, even though not originally intended for a stochastic volatility parameterization, the calendar-spread static hedge, which Nalholm and Poulsen (2006) and Engelmann et al. (2006) argue is inferior to strike-spread static hedges, nonetheless holds up in a simulation setting incorporating stochastic volatility, as long as σ remained relatively small and values could be well matched on the authors trinomial tree. The primary finding of Toft and Xuan (1998) however is that when the volatility of volatility is large, the authors registered lower values for the terminal results of the hedge portfolios, as well as higher standard deviations and larger losses. Thus, Toft and Xuan (1998) contends that as the volatility s variability increases, the performance of the static hedge decreases. As argued in Fink (2003) this is quite suggestive, as "[...] poor performance when volatility is stochastic is a significant indictment of the reliability of

18 Chapter 2 - Literature Review 9 a static hedge." It should however be noted that Toft and Xuan showed that their hedging errors remained of reasonable size for "less exotic" barrier options, a result confirmed in An and Suo (2009) which showed this to be true for options with their barriers much higher than their underlying assets for up-and-out options. We also chose to retain the importance of Figlewski and Green (1999) in our review due to the authors focus on model risk in European option writing strategies. Specifically, using an empirical simulation, Figlewski and Green (1999) measured the impact of various risk sources on standard option writing strategies. Using a basic Black-Scholes parameterization with estimated volatilities, and adapting it to different assumptions and underlying assets, the authors priced European options of five different maturities for four different underlying assets (the Standard & Poor s 500 Index, the Deutschmark-USD contract, the USD LIBOR contract and the 10-year U.S. Treasury bond yield), and ultimately found that model risk can greatly affect the return (and risk) of a basic option writing strategy, even when a straightforward delta-neutral dynamic hedge is implemented. Figlewski and Green (1999) claims that, across all four assets and five corresponding maturities, writing options without hedges causes very large standard deviations, even if the volatility is forecasted, and returns are for the most part negative and losses large. When the authors implement a simple delta hedging strategy, they lower the standard deviation for all but one of their cases, and many more option writing strategies reverse their previous negative results. Moves to different volatility-forecasting techniques and moneyness levels dampen the primary results of Figlewski and Green (1999), both in terms of mean returns and standard deviations, although overestimated standard deviations for the most part increase the authors option writing strategies mean returns, which we however interpret as detrimental nonetheless given that hedges should not produce positive returns. The main conclusion we retain from Figlewski and Green s work however is that they prove, empirically, that the implementation of a basic hedging methodology greatly reduces the standard deviation of an option writing strategy. Combined with previously mentioned papers, this leads us to believe that static hedges, which have been shown to be superior to dynamic hedges, should perform even better. We also retain two more pertinent papers; Hirsa et al. (2003) and Schoutens et al. (2004). Both papers deal with the diffi culties of pricing exotic options using various model parameterizations calibrated to underlying option data. These papers focus on the pricing differences of exotic options that occur for models that produce similar plain vanilla option prices. We retain Hirsa et al. (2003) because, as mentioned above, this paper discusses the intricacies of pricing exotic options. Specifically, the authors argue that the use of plain vanilla option prices as inputs in the calibration of exotic option pricing models can lead to alternate exotic pricing results, even for similar initial inputs. Focusing exclusively on up-and-out call options, Hirsa et al. (2003) performs two exercises. In the first, Hirsa et al. (2003) compares exotic option prices obtained from the Variance-Gamma (VG) model and the Constant Elasticity of Variance (CEV) model, both calibrated to the same Standard & Poor s 500 Index out-of-the-

19 Chapter 2 - Literature Review 10 money options. In the second, Hirsa et al. (2003) instead calibrates the Local Volatility (LV) model and the VG with stochastic arrivals (VGSA) model to an entire surface of options. The authors calibrate the VG and CEV models to a single maturity, and they calibrate the LV and VGSA models to the entire range of option maturities. Hirsa et al. (2003) determines that all models offer respectable (and quite similar) results when pricing plain vanilla options. The main results occur however when the authors turn to pricing UOC options, whereby their terminal outcomes are quite different. Hirsa et al. (2003) shows that the simpler models consistently produce lower prices than the more complex models (the jump models in this case). In some cases, Hirsa et al. (2003) proves that the price in one model was more than 500% higher than in another (although both were shown to produce quasi-identical plain vanilla option prices). Thus, Hirsa et al. (2003) argues that no matter how similar plain vanilla option prices can be in an array of calibrated pricing models, these same models can produce vastly different barrier option prices. Hirsa et al. (2003) concludes by arguing that the authors results invariably indicate that not only is the pricing of barrier options model dependent, but that any potential static hedging methodologies would depend on the parameterization in which they would be implemented as well. Thus, the discrepancies presented for barrier option pricing could also be quite significant for barrier option hedging. In Schoutens et al. (2004), the authors perform a similar study to the one proposed in Hirsa et al. (2003). Schoutens et al. (2004) shows that models that produce indiscernible European plain vanilla option prices for an equity index, can also produce quite different exotic option prices. Using option data from the Eurostoxx 50 Index with maturities ranging from less than a month to about five years, Schoutens et al. (2004) calibrates seven stochastic volatility parametric models; the Barndorff-Nielsen-Shephard model, two versions of the Heston model, and four versions of Lévy models combined with processes rendering the evolution of time stochastic (specifically the Normal Inverse Gaussian, the Variance Gamma, the continuous CIR and the Gamma-OU processes). Using the Fast Fourier Transform (FFT) introduced in Carr and Madan (1998) to calibrate their models, the authors show that for four measures, European option prices are quasi-identical for all seven models. Implementing Monte Carlo simulations with one million asset paths each, Schoutens et al. (2004) then prices digital barrier options, standard barrier options (specifically down-and-out calls, down-and-in calls, up-and-in calls and up-and-out calls), lookback options and cliquet options. For the options most relevant to this thesis, the down-and-out calls, the Heston model systematically produces the smallest option prices. The results of Schoutens et al. (2004) indicate that as the barrier level moves away from the spot price however, the differences between the models diminish and the prices appear to converge towards one price. This finding is broadly in line with that of An and Suo (2009); the less "exotic" the option, the closer the prices for all of the models. As the barrier moves closer to the spot price on the other hand, the spread between the models prices grows further. From the authors results we can see that the price of the most expensive barrier option is about 84% greater than that of the smallest price (this difference was less than 1% when the barrier was furthest from the spot), which is quite significant when we consider that all seven models used to price these standard barrier options produced

20 Chapter 2 - Literature Review 11 plain vanilla option prices which were indistinguishable between one another. Schoutens et al. (2004) concludes by warning that we need to be very careful when implementing hedges for barrier options due to their dependence on model parameters, which echoes the warning found in Hirsa et al. (2003), especially for more "exotic" options. Recent literature has primarily focused on the comparison of static hedging strategies with "super-static" (also known as static super-replication) hedging methodologies. One such paper is Fengler et al. (2011), in which the authors empirically compare and contrast the effi ciency of the robust super-static hedge presented in Maruhn and Sachs (2009), which they execute using ten plain vanilla options, against the hedging results obtained by implementing the strike-spread static hedge, which they execute using five plain vanilla options, and a deltavega neutral dynamic hedge. Fengler et al. (2011) analyzes the hedging error of the aforementioned specific hedges, as well as the relationship between each hedge s cost and the respective barrier option s cost, by hedging reverse barrier options (up-and-out calls and down-and-out puts) written on the DAX over a sample period starting in early 2000 and ending at the end of The authors hedging strategies are implemented assuming a Heston stochastic volatility parameterization (for the robust super-static hedge), a Black-Scholes parameterization (for the strike-spread static hedge) and a Local Volatility parameterization (for the delta-vega neutral dynamic hedge). In terms of the cost analysis, Fengler et al. (2011) finds that, for reverse UOC options, the minimal assumption super hedge (MA), inspired by Maruhn and Sachs (2009), turns out to be much more expensive than the other two strategies (with the strike-spread hedge being the cheapest), whereas for reverse DOP options, it ranks in the middle (with the strike-spread hedge being the most expensive in this case). In order to analyze the discounted hedging error on the other hand, Fengler et al. (2011) divides the outcomes into two possible realizations; in the first the authors perform their analysis assuming that all trades are executed at the barrier level, whilst in the second they instead assume that these same trades are executed at the underlying asset s price immediately following a barrier event (or as close to post-event as possible). For the reverse UOC options, if the portfolio is sold-off directly at the barrier level, Fengler et al. (2011) shows that the strike-spread static hedge provides better traditional metrics than the MA hedge (it has a mean closer to zero and a lower standard deviation), although the MA hedge has the lowest probability of loss of all the hedges, and registers a minimum loss closest to zero. For the reverse DOP options, the MA hedge is shown to dominate the strike-spread static hedge (it has a lower standard deviation and a lower mean absolute deviation), although both hedges have a 0% probability of loss and both register extreme losses of zero. If the portfolio is instead traded assuming an underlying asset level different from that of the barrier level (i.e.; right after a barrier event), Fengler et al. (2011) shows that, for the reverse UOC options, the MA hedge dominates the strike-spread hedge (it has a mean closer to zero, a lower standard deviation, a lower mean absolute deviation and the smallest extreme loss) and has by far the lowest probability of loss. For the reverse DOP options, the results of Fengler et al. (2011) appear to indicate that there is no clear

21 Chapter 2 - Literature Review 12 winner between the MA hedge and the strike-spread hedge; the former has a mean closer to zero and a lower standard deviation, while the latter has a minimum loss closer to zero and a much lower probability of loss. Thus, although the authors prove that their implemented MA super-hedge does indeed perform very well, the basic strike-spread static hedge nonetheless provides decent results in all scenarios, especially given its cost. Ultimately, combining both its cost and its hedging results, Fengler et al. (2011) however shows that the MA static hedge does provide the best broad results, slightly ahead of the static strike-spread hedge, and a great deal ahead of the delta-vega neutral dynamic hedge. The authors nonetheless praise the strike-spread static hedge. Fengler et al. (2011) highlights that the strike-spread static hedge always guarantees a null hedging error for a hedged barrier option, as long as no barrier event occurs during the option s life. Finally, in order to highlight certain key differences and similarities between the present thesis and three of the papers which were most pertinent to our work, specifically Nalholm and Poulsen (2006), Engelmann et al. (2006), and An and Suo (2009), we refer the reader to Table 1. As can be seen, key parts of our work were guided by the three aforementioned papers, although we believe our overall approach and methodology were different. Nalholm and Engelmann An and This Poulsen (2006) et al. (2006) Suo (2009) Thesis Type of Study Simulation Empirical Empirical Empirical Models BS, CEV, JD, VG, SV BS, LV BS, JD, SV, SVJ PGK, JD, SV Dynamic (2) Dynamic (3) Dynamic (2) Strike-Spread (5) Hedging Strike-Spread (3) Strike-Spread (4) Strategies Calendar-Spread (3) Calendar-Spread (6) Unified (4) Underlying Single Stock DAX Index EUR/USD Rate EUR/USD Rate Type of Exotics Standard DOC Reverse DOP Standard UOC Standard DOC Reverse UOC Reverse UOC Compound Options Variations Single Barrier Single Barrier Multiple Barriers Multiple Barriers Single Strike Single Strike Multiple Strikes Multiple Strikes Single Maturity Single Maturity Multiple Maturities Single Maturity Time Frame Table 1: Table containing highlights of three key papers, specifically Nalholm and Poulsen (2006), Engelmann et al. (2006), and An and Suo (2009), as well as those of the present thesis. For the "Models" row, "BS" represents the Black-Scholes model, "PGK" represents the Practitioner s Garman-Kohlhagen model, "CEV" represents the Constant Elasticity of Variance model, "LV" represents the Local Volatility model, "JD" represents the Jump Diffusion model, "VG" represents the Variance Gamma model, "SV" represents the Stochastic Volatility model, and "SVJ" represents the Stochastic Volatility with Jumps model.