In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

Thesis (MSc)--Stellenbosch University, 2015 / AFRIKAANSE OPSOMMING : Die groeiende belangstelling in kalibrering en modelrisiko is ’n redelik resente ontwikkeling in finansiële wiskunde. Hierdie proefskrif fokusseer op hierdie sake, veral in verband met die prysbepaling van vanielje-en eksotiese opsies, en vergelyk die prestasie van verskeie Lévy modelle. ’n Nuwe metode om modelrisiko te meet word ook voorgestel (hoofstuk 6). Ons kalibreer eers verskeie Lévy modelle aan die log-opbrengs van die S&P500 indeks. Statistiese toetse en grafieke voorstellings toon albei aan dat suiwer sprongmodelle (VG, NIG en CGMY) die verdeling van die opbrengs beter beskryf as die Black-Scholes model. Daarna kalibreer ons hierdie vier modelle aan S&P500 indeks opsie data en ook aan "CGMY-wˆ ereld" data (’n gesimuleerde wÃłreld wat beskryf word deur die CGMY-model) met behulp van die wortel van gemiddelde kwadraat fout. Die CGMY model vaar beter as die VG, NIG en Black-Scholes modelle. Ons waarneem ook ’n effense verskil tussen die nuwe parameters van CGMY model en sy wisselende parameters, ten spyte van die feit dat CGMY model gekalibreer is aan die "CGMYwêreld" data. Versperrings-en terugblik opsies word daarna geprys, deur gebruik te maak van die gekalibreerde parameters vir ons modelle. Hierdie pryse word dan vergelyk met die "ware" pryse (bereken met die ware parameters van die "CGMY-wêreld), en ’n beduidende verskil tussen die modelpryse en die "ware" pryse word waargeneem. Ons eindig met ’n poging om hierdie modelrisiko te kwantiseer / ENGLISH ABSTRACT : The growing interest in calibration and model risk is a fairly recent development in financial mathematics. This thesis focussing on these issues, particularly in relation to the pricing of vanilla and exotic options, and compare the performance of various Lévy models. A new method to measure model risk is also proposed (Chapter 6). We calibrate only several Lévy models to the log-return of S&P500 index data. Statistical tests and graphs representations both show that pure jump models (VG, NIG and CGMY) the distribution of the proceeds better described as the Black-Scholes model. Then we calibrate these four models to the S&P500 index option data and also to "CGMY-world" data (a simulated world described by the CGMY model) using the root mean square error. Which CGMY model outperform VG, NIG and Black-Scholes models. We observe also a slight difference between the new parameters of CGMY model and its varying parameters, despite the fact that CGMY model is calibrated to the "CGMY-world" data. Barriers and lookback options are then priced, making use of the calibrated parameters for our models. These prices are then compared with the "real" prices (calculated with the true parameters of the "CGMY world), and a significant difference between the model prices and the "real" rates are observed. We end with an attempt to quantization this model risk.

<p>We discuss the Normal inverse Gaussian (NIG) distribution in modeling volatility in the financial markets. Refining the work of Barndorff-Nielsen (1997) and Andersson (2001), we introduce a new parameterization of the NIG distribution to build the GARCH(p,q)-NIG model. This new parameterization allows the model to be a strong GARCH in the sense of Drost and Nijman (1993). It also allows us to standardized the observed returns to be i.i.d., so that we can use standard inference methods when we evaluate the fit of the model.</p><p>We use the realized volatility (RV), calculated from intraday data, to standardize the returns of the ECU/USD foreign exchange rate. We show that normality cannot be rejected for the RV-standardized returns, i.e., the Mixture-of-Distributions Hypothesis (MDH) of Clark (1973) holds. {We build a link between the conditional RV and the conditional variance. This link allows us to use the conditional RV as a proxy for the conditional variance. We give an empirical justification of the GARCH-NIG model using this approximation.</p><p>In addition, we introduce a new General GARCH(p,q)-NIG model. This model has as special cases the Threshold-GARCH(p,q)-NIG model to model the leverage effect, the Absolute Value GARCH(p,q)-NIG model, to model conditional standard deviation, and the Threshold Absolute Value GARCH(p,q)-NIG model to model asymmetry in the conditional standard deviation. The properties of the maximum likelihood estimates of the parameters of the models are investigated in a simulation study.</p>

We discuss the Normal inverse Gaussian (NIG) distribution in modeling volatility in the financial markets. Refining the work of Barndorff-Nielsen (1997) and Andersson (2001), we introduce a new parameterization of the NIG distribution to build the GARCH(p,q)-NIG model. This new parameterization allows the model to be a strong GARCH in the sense of Drost and Nijman (1993). It also allows us to standardized the observed returns to be i.i.d., so that we can use standard inference methods when we evaluate the fit of the model. We use the realized volatility (RV), calculated from intraday data, to standardize the returns of the ECU/USD foreign exchange rate. We show that normality cannot be rejected for the RV-standardized returns, i.e., the Mixture-of-Distributions Hypothesis (MDH) of Clark (1973) holds. {We build a link between the conditional RV and the conditional variance. This link allows us to use the conditional RV as a proxy for the conditional variance. We give an empirical justification of the GARCH-NIG model using this approximation. In addition, we introduce a new General GARCH(p,q)-NIG model. This model has as special cases the Threshold-GARCH(p,q)-NIG model to model the leverage effect, the Absolute Value GARCH(p,q)-NIG model, to model conditional standard deviation, and the Threshold Absolute Value GARCH(p,q)-NIG model to model asymmetry in the conditional standard deviation. The properties of the maximum likelihood estimates of the parameters of the models are investigated in a simulation study.

This thesis focuses on the parameter stability of additive normal tempered stable processes when calibrating a volatility surface. The studied processes arise as a generalization of Lévy normal tempered stable processes, and their main characteristic are their time-dependent parameters. The theoretical background of the subject is presented, where its construction is discussed taking as a starting point the definition of Lévy processes. The implementation of an option valuation model using Fourier techniques and the calibration process of the model are described. The thesis analyzes the parameter stability of the model when it calibrates the volatility surface of a market index (EURO STOXX 50) during three time spans. The time spans consist of the periods from Dec 2016 to Dec 2017 (after the Brexit and the US presidential elections), from Nov 2019 to Nov 2020 (during the pandemic caused by COVID-19) and a more recent time period, April 2023. The findings contribute to the understanding of the model itself and the behavior of the parameters under particular economic conditions.

This essay investigates how realized variance affects the GARCH-models (GARCH, EGARCH, GJRGARCH) when added as an external regressor. The GARCH models are estimated with three different distributions; Normal-, Student’s t- and Normal inverse gaussian distribution. The results are ambiguous - the models with realized variance improves the model fit, but when applied to forecasting, the models with realized variance are performing similar Value at Risk predictions compared to the models without realized variance.

因為股票市場常具有厚尾、偏態和峰態的特性且在國際的股票市場之間，股票報酬長存在有尾端相依的情況，所以我們的資產模型不能選用Gaussian分配。 近幾年來，常用GH 分配建構單維度的股票報酬。這篇文章將利用多元仿射JD、多元仿射VG 和多元仿射NIG分配去建構風險性資產的報酬並請應用到資產配置。 建構風險性資產的報酬後，我們提供兩種不同形式的投資組合並且可以導出投資組合的期望值、變異數、偏態和峰態。我們嘗試以投資組合的期望值、變異數、偏態和峰態當成我們的目標函數，然後得出未來最佳的投資組合的權重。為了讓我們的資產配置更加動態和有效率，我們重新估計模型的參數、選擇最佳的投資組合權重，然後重新評估最佳的資產配置在每個決策日期。實證結果發現當股票市場的表現好的時候，我們建議資產配置應使用偏態當成我們的目標函數，但是當股票市場的表現太好的時候，我們建議資產配置應使用變異數當成我們的目標函數。 / Since the stock markets always have the characteristics of heavy-tailness, skewness and kurtosis and there exists tail dependence among the international stock markets, we can’t use the Gaussian distribution as our model. Recently, the generalized hyperbolic (GH) distribution has been suggested to fit the single stock returns. This article will use the multivariate affine JD (MAJD), multivariate affine variance gamma (MAVG) and multivariate affine normal inverse Gaussian (MANIG) distributions to construct the risky asset returns, and apply them to asset allocation. After constructing the risky asset returns, we provide two different forms of portfolio and obtain the mean, variance, skewness, kurtosis of portfolio. We can try to select the optimal weights of portfolio by using the mean, variance, skewness, kurtosis of portfolios as our objective functions. To make our asset allocation more dynamic and efficient, we re-estimate all parameters for our models, select the optimal weights of portfolio, and re-assess the optimal asset allocation at each decision date. Empirically, when the performances of stock markets are good, we suggest that our asset allocation uses the skewness as the objective function. When the performances of stock markets are not good, we suggest that our asset allocation uses the variance as the objective function.