Browsing by Author "Badescu, Alexandru"
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- ItemOpen AccessAffine GARCH option pricing models, stochastic interest rates, and diffusion limits(2022-09) Gu, Zhouzhou; Badescu, Alexandru; Qiu, Jinniao; Swishchuk, AnatoliyThis article proposes a derivative pricing framework when the asset returns and the short term rate process are modelled with affine GARCH models driven by correlated Gaussian innovations. The risk neutral dynamics are derived based on a co-variance dependent pricing kernel and semi-closed form solutions are derived for European style options and bond prices. We further derive the weak diffusion limits of the underlying processes under both physical and risk-neutral measure and we investigate the consistency between the proposed pricing kernel with the well-known Girsanov principle in continuoustime. A variety of numerical exercises are provided to analyze the validity of our pricing formulae, the sensitivity of the option prices relative to the pricing kernel parameters, and the convergence of option prices to those based on the limiting diffusions. Finally, we illustrate an empirical analysis based on a joint estimation using historical asset returns and short-term rates, and cross sections of options and bond prices.
- ItemOpen AccessBayesian Option Pricing with Asymmetric Variance Processes(2016-01-06) Zhu, Heng; Badescu, Alexandru; Scollnik, David; Lee, Ryan; Qiu, ChaoEmpirical studies have shown that financial time series exhibit negative skewness and excess kurtosis. GARCH models can be successfully used to model security returns. This thesis utilizes NGARCH and Normal Mixture NGARCH models to price options. The Gibbs sampling method is explained and implemented for parametric inference, and Bayesian inference results are compared with those obtained with Maximum Likelihood Estimates. Pricing option contracts requires the derivation of risk neutral return dynamics of the underlying asset. There is an infinite number of risk neutral measures under the incomplete market GARCH framework. In this thesis, we study the conditional Esscher transform and the Extended Girsanov Principle as the martingale measure candidates. We use the Radon Nikodym derivatives from both risk neutral measures to derive and compare the option prices for GARCH models based on Gaussian and Mixture of Gaussian innovations.
- ItemOpen AccessCaptial requirements and optimal investment for insurance companies(2012) Wang, Bo; Badescu, AlexandruAsset allocation is one of the central issues in banking, finance and insurance industries. Using ruin probability and expected loss at ruin as measures of risk, we aimed to investigate the optimal investment problem of an investor in finance and insurance in a static one-period setting. Mean - Variance, Value at Risk (VaR) and Conditional Value at Risk (CVaR) modeling were three approaches investigated in this thesis. However, mathematically VaR had some serious limitations, such as lack of sub-additivity. In the case of a finite number of scenarios, VaR is a non-smooth, non-convex function, making it difficult to control and optimize. This fact stimulate our development of new optimization algorithms presented here, by introducing ruin probability and expected loss at ruin. Optimization problems with a certain level of expected return in finance field were first analyzed and then with premium and claim loss added into consideration, the optimization methodologies were applied to the insurance field. After introducing ruin probability and expected loss at ruin as risk measures, the optimization results became more accurate and feasible. Using our proposed new method, we investigated the capital required and asset allocation problem.
- ItemOpen AccessCausal Inference With Non-probability Sample and Misclassified Covariate(2022-09) Sevinc, Emir; Shen, Hua; Lu, Xuewen; Deardon, Robert; Shen, Hua; Badescu, AlexandruCausal inference refers to the study of analyzing data that is explicitly defined on a question of causality. The problems motivating many, if not most studies in social and biological sciences, tend to be causative and not associative. A well defined and systematically representative sample tends to be the base in such studies. However, sometimes a sample may result from a non-probability process. This often provides a unique challenge in estimating the probability of an individual being in the sample, and generalizing the causality conclusions made off of the non-probability samples to the target population. Additionally, due to issues such as difficulty of precise measurements and human error, certain variables may be classified incorrectly. In this thesis, we address both challenges by implementing causal inferential methods in a case where we have a main non-probability sample with response available, and a probability sample with auxiliary information only. We deal with the presence of incorrectly classified confounder in the non-probability sample only, or both samples. We examine the consequences of naively ignoring misclassification, and develop a latent-variable based method via an Expectation-Maximization algorithm to correct for the misclassified confounder. We incorporate this method with a double-robust mean estimator requiring only the correct specification of either the regression model or the non-probability sample selection model to estimate the average treatment effect. We demonstrate the effectiveness of our methodology via simulation studies, and implement it on smoking data from the Centre of Disease Control and Prevention (CDC).
- ItemOpen AccessCredit Risk Pricing via Epstein-Zin Pricing Kernel(2017) Ogunsolu, Mobolaji; Sezer, Deniz; Frei, Christoph; Badescu, Alexandru; Ware, Antony; David, Alexander; Liao, WenyuanWe present an equilibrium framework for pricing corporate bonds with information delay in an Epstein-Zin setting. As in structural models of credit risk, the default time is modeled as the first hitting time of a default boundary by the unobservable process; the firm's asset value. The observable state variables; log consumption and volatility are affine processes which drive the unobservable firm's value process. The stochastic pricing kernel is expressed in terms of the state variables. The price of a zero-coupon bond is expressed as the solution of a multidimensional partial differential equation which is solved numerically. Our equilibrium price model is also calibrated to fit available corporate bond and consumption data. Finally, we analyze the implications of investor’s preferences and information delay on the credit yield spreads.
- ItemOpen AccessDeep Learning-based Numerical Methods for Stochastic Partial Differential Equations and Applications(2021-03-14) Yao, Yao; Qiu, Jinniao; Ware, Antony; Badescu, AlexandruIn this thesis, we are concerned with approximating solutions of stochastic partial differential equations (SPDEs) and their applications. Inspired by Huré, Pham, and Warin [15], we propose and study the deep learning-based methods for both the forward and backward SPDEs. In particular, the forward SPDEs may allow for Neumann boundary conditions. We also prove the convergence analysis of the proposed algorithms. The numerical results indicate that the performance of the algorithm is quite effective for solving the SPDEs, even in high-dimensional cases. The applications include various pricing problems under exchange rate target zone models as well as under rough volatility models.
- ItemOpen AccessDerivatives Pricing with Fractional Discrete-time Models(2022-07-07) Jayaraman, Sarath Kumar; Badescu, Alexandru; Godin, Frederic; Qiu, Jinniao; Swishchuk, Anatoliy; Ware, AntonyThis thesis studies the pricing of European style derivatives with various affine models. Most of this thesis focuses on the impact of long memory on asset return modelling and option pricing. We propose a general discrete-time pricing framework based on affine multi-component volatility models that admit ARCH(∞) representations. It not only nests a large variety of option pricing models from the literature, but also allows for the introduction of novel fractionally integrated processes for option valuation purposes. Using an infinite sum characterization of the log-asset price’s cumulant generating function, we derive semi-explicit expressions for European option prices under a variance-dependent stochastic discount factor. We carry out an extensive empirical analysis which includes estimations based on different combinations of returns and options of the S&P 500 index for a variety of short- and long-memory models. Our results indicate that the inclusion of long memory into return modelling substantially improves the option pricing performance. Using a set of out-of-sample option pricing errors, we show that long-memory models outperform richer parametrized one- and two-component models with short-memory dynamics. The last part of the thesis studies the pricing of volatility derivatives with affine models. We propose semi-closed form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options under affine GARCH models based on Gaussian and Inverse Gaussian distributions. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.
- ItemOpen AccessGarch option valuation(2011) Zhong, Wanyong; Badescu, Alexandru
- ItemOpen AccessHydrological Time Series Modelling and Applications(2016) Asadzadeh, Ilnaz; Ware, Antony; Badescu, Alexandru; Swishchuk, Anatoliy; Zinchenko, YuriyWe consider the problem of measuring reliability of a hydro reservoir over a finite horizon with a stochastic optimal control technique. To apply this technique, we need to model the underlying stochastic process which is the inflow of water to the reservoir. Typical time series models for such problems only capture linear dependency (simple correlation) in the data. Alternative approaches include artificial neural network methods but these lack a theoretical foundation and a systematic procedure for the construction of the model. To overcome both of these limitations, we propose a new framework based on the application of copulas to univariate time series modelling. Our model shows that some important statistical characteristics of hydrological time series, such as upper and lower tail dependencies, persistence, etc., can be described with the aid of copulas. In turn, this provides insight regarding the qualitative properties of the underlying time series. Our main contribution is a new method of estimation based on a semi-parametric technique. By semi-parametric we mean using empirical autocopula (copula of a time series with itself with different lags), and parametric marginal distributions. Goodness of fit analysis is carried out and numerical results are illustrated with variety of concrete examples and sample data sets. We then benchmark and compare our scheme to alternative methods such as parametric models and various other time series modelling techniques. The final part of the dissertation proposes an application of stochastic optimal control to measure the reliability function (the probability that a system will perform the required function for a specified period of time under stated conditions) of the reservoir. For this section, we work with both uncorrelated and correlated inflow time series. For the first case, we generate independent inflow series using some probability distribution and for the second assumption, correlated inflow series, we employ the values of inflow generated using the autocopula method.
- ItemOpen AccessMeasure change and filtering(2011) Deng, Jia; Elliott, Robert J.; Badescu, Alexandru
- ItemOpen AccessMerton Investment Problem for the Hawkes-based Risk Model(2022-09-21) Nova, Mushfika Hossain; Qiu, Jinniao; Swishchuk, Anatoliy; Badescu, Alexandru; Jiang, WenjunWe study the Merton investment problem in insurance where the risk process is based on the general compound Hawkes process. That means the arrival of claims modeled with a Hawkes process and the modeled claim sizes follow a finite number of fixed jump sizes governed by a Markov chain evolution. The Merton investment problem in insurance is an optimal control problem and we use the dynamic programming method to derive the stochastic Hamilton-Jacobi-Bellman (SHJB) equation satisfied by the value function. The stochastic HJB equation yields a means to obtain the optimal control and thus the optimally controlled stochastic differential equation. Finally, using the claim size from the empirical data set, we simulate the optimal investment portfolio and risk process.
- ItemOpen AccessMultivariate General Compound Hawkes and Point Processes with Financial Applications(2022-10-20) Guo, Qi; Swishchuk, Anatoliy; Qiu, Jinniao; Badescu, Alexandru; Ware, Antony; Hyndman, Cody; Swishchuk, AnatoliyThe Hawkes process (HP) significantly affected the financial modeling area in the past 15 years, especially high-frequency trading. This thesis focuses on various new Hawkes processes and considers their applications in the limit order book (LOB). Preexisting studies of the HP in the LOB showed that the arrivals of orders could be modeled by univariate or multivariate HP because of its long memory property and clustering effect. Therefore, we propose the multivariate general compound Hawkes process (MGCHP), a stochastic model for the mid-price in the LOB. For the MGCHP, we prove the Law of Large Numbers (LLN) and two Functional Central Limit Theorems (FCLT); the latter provides insights into the link between price volatilities and order flows in limit order books with several assets. The parameter estimation for the high-dimensional Hawkes process is always time-consuming. This motivates us to consider a generalization of the MGCHP. We replace the multivariate HP with a more general point process, and we call it the multivariate general compound point process (MGCPP). We also prove limit theorems for the MGCPP and compared numerical simulations for the MGCPP with the MGCHP. The MGCHP model provides us with a perfect framework for the stock price dynamics in the LOB. It’s natural to apply it to other financial applications. We extend the MGCHP to the exponential MGCHP (EMGCHP) and consider the corresponding asset-liability management problem. Risky assets are molded by the EMGCHP while the liability follows a Brownian motion with drift. We derive the Hamilton–Jacobi–Bellman equation and transformed it into a system of PDEs. With the FCLT, we can approximate the EMGCHP to a geometric Brownian motion in the LOB and apply Xie et al.’s results. Numerical simulations for the Hawkes-based model and comparisons with the Poisson-based model are also provided. In the last part of the thesis, we give an option pricing formula under the EMGCHP framework. We believe our study can provide a strong tool for not only researchers but also traders in the high-frequency market.
- ItemOpen AccessOn pricing and hedging options in regime-switching models with feedback effect(Elsevier, 2011) Elliott, Robert; Siu, Tak Kuen; Badescu, AlexandruWe study the pricing and hedging of European-style derivative securities in a Markov, regime-switching, model with a feedback e ect depending on the economic condition. We adopt a pricing kernel which prices both nancial and economic risks explicitly in a dynamically incomplete market and we provide an equilibrium analysis. A martingale representation for a European-style index option's price is established based on the price kernel. The martingale representation is then used to construct the local risk-minimizing strategy explicitly and to characterize the corresponding pricing measure.
- ItemOpen AccessOn Sharpe-ratio-based Optimal Insurance Design(2024-02-08) Liu, Jianan; Jiang, Wenjun; Swishchuk, Anatoliy; Badescu, AlexandruAs an important risk-hedging tool, insurance can increase an individual’s expected utility or reduce her risk exposure. However, pursuing both goals is rarely considered in the literature of insurance contracting. This thesis delves into the optimal insurance design problem by striking a balance between the expected utility and the associated risk. To tackle this objective, we resort to the notion of the Sharpe ratio to identify the optimal contract, which is located on the efficient frontier. The focus of this thesis centers on utilizing Value at Risk (VaR) and Tail Value at Risk (TVaR) as risk measures. We derive parametric forms of the optimal indemnity function in scenarios where a decision maker (DM) seeks to maximize end-of-period expected utility subject to a pre-set acceptable risk level. Since the closed-form or analytical solution for such a contract is rather difficult to derive, we present numerous numerical examples to comprehensively explore various aspects of this methodology. As shown by the results, the Shapre-ratio-based contract is relatively robust except in the Pareto case under VaR preference, and increasing the probability level or risk loading factor adversely affects the ratio. Furthermore, we numerically analyze the popular industrial contract specifically the limited excess-of-loss contract, under the framework of VaR. Our findings reveal that the optimal policy is achieved when the upper limit coverage equals VaR minus the deductible amount. This finding bears a strong resemblance to the optimal contract in our proposed model. The results complement the study of Jiang and Ren (2021).
- ItemOpen AccessOptimal captial investment for an insurance company(2012) Asanga, Sujith; Badescu, AlexandruThe purpose of this study was to investigate a common problem of managing economic capital for insurance companies. In particular, the role of determining the optimal investment capital and investing the capital in a well diversified portfolio were examined. The optimization problem incorporated in our methodology, was constructed to solve the optimal investment capital under ruin probability constraint in such a way that it jointly optimizes the asset allocation among a diversified portfolio. Our approach has a semiparametric nature with appropriate transformations and approximations that enable numerical techniques such as Monte Carlo simulation. Asset returns were modelled using Multivariate Generalized Autoregressive Conditional Heteroskedastic (MV GARCH) framework. We implemented different correlation models targeting two risky assets. A rolling window technique was employed to ensure the robustness of our models. It was revealed that the characteristics of the risky and risk-less assets and their behaviour over the moving investment horizon, have great impact on investment plans. Our results indicated the optimal asset allocation in a well-diversified portfolio provided that investment capital is optimized subject to regulatory constraints.
- ItemOpen AccessOptimal Reinsurance with Vajda Condition and Range-Value-at-Risk(2022-09) Wang, Ye; Jiang, Wenjun; Ambagaspitiya, Rohana; Badescu, AlexandruIn this project we study an optimal reinsurance problem where the insurer’s risk-adjusted liability gets minimized. To better reflect the spirit of reinsurance, we impose exogenous Vajda condition on indemnity functions which requires the reinsurer to pay an increasing proportion of loss. To consider both robustness and tail risk, the insurer is assumed to apply Range-Value-at-Risk (RVaR) to evaluate its risk. Under the expected value premium principle, we derive the closed-form solution to our problem, which includes the results in Chi and Weng (2013) as special cases. Some comparative studies and sensitivity analysis are also carried out through numerical examples. Results from a simulation study indicate that the policy with Vajda condition is superior to that without Vajda condition if the insurer is very aversive to the tail risk.
- ItemOpen AccessOptimization Approaches for Intensity Modulated Proton Therapy Treatment Planning(2023-07) Mousazadeh, Bahar; Zinchenko, Yuriy; Greenberg, Matthew; Badescu, AlexandruRadiation therapy is a critical modality in the field of oncology. The primary goal of radiation therapy is to destroy or control the growth of cancerous cells while minimizing damage to healthy tissues. Intensity Modulated Proton Therapy (IMPT) is a type of radiation therapy that utilizes protons to irradiate the tumor. The unique physical properties of protons enable precise control over the radiation dose distribution within the tumor and more effective sparing of healthy tissues. Typically, radiation therapy treatment planning is posed as a multi-criteria optimization problem, whereby the challenge is finding the best possible treatment plan. In this study, we formulate and compare two optimization approaches for IMPT treatment planning. We first explore a linear programming (LP) approach, followed by a moment-based approach where we incorporate the dose-volume requirements into the fluence map optimization (FMO) problem. The evaluation of these models is conducted using anonymized patient data corresponding to a lung cancer case, with a focus on generating a good-quality initial plan that is amenable to further refinement. The moment-based approach has a drawback in terms of its high memory usage. To mitigate this limitation, we explore several sparsification strategies aimed at reducing memory requirements. Employing an aggressive sparsification method, we demonstrate that the moment-based approach outperforms the LP model in dosimetric outcomes and computational run-time. We highlight a trade-off between the quality of the treatment plan and computational run-time when utilizing different sparcification strategies for the moment-based approach. By adopting a less strict sparsification method, we anticipate achieving higher-quality treatment plans at the expense of increased computational run-time.
- ItemOpen AccessOption Pricing Under Rough Heston Model With Jumps(2022-09) Jin, Yazhao; Qiu, Jinnao; Badescu, Alexandru; Lu, Xuewen; Swishchuk, AnatoliyThe rough Heston model with jumps is proposed and studied in the thesis which is inspired by some well-known models, including the Heston model, rough Heston model, and Kou's jump model. Our new model considers adding a Poisson jump process to the rough Heston model, which not only keeps the roughness of the volatility process, and also adds an extra noise to improve the model performance. To have better comparisons across models and to understand well how the jumps and roughness improve the model performance, we involve another interesting model called the Heston model with jumps where we used the similar jump process as in Kou's model. The concerned topics are reviewed in the front part of the thesis including option pricing methods, Fourier-inversion techniques, Black-Scholes implied volatilities, and the introduction of related models such as model dynamics and their characteristic functions/moment-generating-functions. In the empirical analysis section, two option pricing methods were compared in the dimension of accuracy and efficiency by the results from Monte Carlo's simulation and their CPU computational time. The calibration based on implied volatilities were processed by four discussed models we mentioned above: the Heston model, rough Heston model, Heston model with jumps, and rough Heston model with jumps. We have the in- and out-of-sample tests to monitor the performances of the models by using the 2014-2019 S&P 500 options, where the former test focuses on the calibrated implied volatilities and the latter conducts a time-series forecast based on the in-sample test results. Significant improvements are shown in the rough Heston model with jumps in both tests which lead us to the conclusion that the combination of the volatility roughness and the add-on Poisson jump process can help the model to reach a better performance.
- ItemOpen AccessPricing tranches of Collateralize Debt Obligation (CDO) using the one factor Gaussian Copula model, structural model and conditional survival model(2017-12-21) Ofori, Elizabeth; Sezer, Deniz; Badescu, Alexandru; Brudnyi, AlexIn this thesis we focus on the pricing of tranches of a synthetic collateralized Debt Obligation (synthetic CDO) which is a vehicle for trading portfolio of credit risk. Our purpose is not to create any new concept but we explore three different models to price the tranches of a synthetic CDO. These three models include the one factor Gaussian copula model, structural model and the conditional survival model To this end, we provide a step by step description of the one factor Gaussian Copula model as proposed by Li, structural model as by Hull Predecu and White and conditional survival model by Peng and Kou. This thesis implement all the three models using the pricing procedure discussed in Peng and Kou paper\cite{cluster}. For practical purpose, we use MATLAB to calculate a synthetic CDO tranche price based on the computation of a non-homogeneous portfolio of three reference entities under the one factor Gaussian copula model, structural model and conditional survival model. We calibrate the structural model to three cooperate bonds data to generate marginal probability of default key to all the three models. The pricing result of the three models are very close for the risky tranches whiles that of the less risky are a little different which is attribute to the fact that the three models are affected by other parameters such as correlation parameter and loading factor. Comparisons are then made between the one factor Gaussian Copula and the structural model and the result tally with the observation Hull, Predescu and White made concerning Gaussian copula model and structural model.
- ItemOpen AccessRisk neutral measures and GARCH model calibration(2012-09-24) LI, SHENG; Badescu, AlexandruEmpirical studies have shown that GARCH models can be successfully used to describe option prices. Pricing such option contracts requires the risk neutral return dynamics of underlying asset. Since under the GARCH framework the market is incomplete, there is more than one risk neutral measure. In this thesis, we study the locally risk neutral valuation relationship, the mean correcting martingale measure, the conditional Esscher transform and the second order Esscher transform as martingale measure candidates. All these methods lead to the respective risk neutral return dynamics. We empirically examine in-sample and out-ofsample performance of Gaussian-TGARCH and Normal inverse Gaussian (NIG)-TGARCH models under these risk neutral measures.