Delft University of Technology Faculty of Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics Application of State Space Hidden Markov Models to the approximation of (embedded) option prices A thesis submitted to the Delft Institute of Applied Mathematics in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in APPLIED MATHEMATICS by Josephine Alberts Delft, the Netherlands November 2016 Copyright © 2016 by Josephine Alberts. All rights reserved. MSc THESIS APPLIED MATHEMATICS \Application of State Space Hidden Markov Models to the approximation of (embedded) option prices" Josephine Alberts Delft University of Technology Daily Supervisors Responsible Professor Prof. Dr. Ir. C.W. Oosterlee Prof. Dr. Ir. C.W. Oosterlee Dr. Ir. L.A. Grzelak Other thesis committee members Ir. S.N. Singor Dr. P. Cirillo November 2016 Delft, the Netherlands Acknowledgments This thesis has been submitted for the degree Master of Science in Applied Mathematics at Delft University of Technology. The responsible professor is Kees Oosterlee, professor at the Numerical Analysis group of Delft Institute of Applied Mathematics. Research for this project was carried out at Ortec Finance, under the supervision of Stefan Singor. Ortec Finance is a company aiming to improve investment decision-making by providing consistent solutions for risk and return management through a combination of market knowledge, mathematical models and information technology. First of all, I would like to thank Kees Oosterlee, Stefan Singor and Lech Grzelak for their close involvement in this project and their valuable advice. I would also like to thank Pasquale Cirillo for being part of the examination committee. Furthermore I would like to thank my colleagues at Ortec Finance for providing a pleasant and inspiring working environment. Lastly, I would like to thank my family and friends for all their encouragement and moral support over the whole duration of my studies. 1 Abstract This thesis discusses dimension reduction of the risk drivers that determine embedded option values by using the class of State Space Hidden Markov Models. As embedded options are typically valued by nested Monte Carlo simulations, this dimension reduction leads to a major reduction in computing time. This is especially important for insurance companies that are dealing with many embedded option valuations in order to determine the market value of their liabilities. To achieve the dimension reduction of the risk driver process, this thesis proposes a specific Hidden Markov Model approach. An overview on current methods for state and parameter inference within this class of models is presented. For the state-of-the-art CPF- SAEM method insights are obtained by investigating an example of the dimension reduction model. Furthermore, the satisfactory behavior of this HMM approach is investigated in more detail for multiple (market) cases. Lastly, the dimension reduction model is applied to calibration of the Heston model parameters to market data. It is shown that this approach avoids overfitting issues and results in a more stable model than direct calibration of the parameters. 2 Contents 1 Introduction 5 1.1 General setting . 5 1.2 Research objectives . 8 1.3 Organization of the report . 9 2 Overview of State Space Hidden Markov Models 10 2.1 Introduction to Hidden Markov Models . 10 2.2 State inference . 12 2.3 Combined state and parameter inference . 20 3 Dimension reduction in option valuation models by a HMM approach 26 3.1 Model description . 26 3.2 Black-Scholes example . 27 3.3 Benchmark: Kalman Filter within the EM framework . 28 3.4 Solving the BS example with the CPF-SAEM method . 31 3.5 Influence of the underlying HMM . 42 3.6 Conclusions . 45 4 Test cases for the HMM approach 46 4.1 Non-linear example . 46 4.2 Extensive example: Heston model . 47 4.3 Market example: basket of S&P-500 index options . 49 4.4 Conclusions . 52 5 Application to reduction of overfitting in the Heston model 53 5.1 Calibration of the Heston model . 53 5.2 Overfitting . 56 5.3 Hidden Markov Model approach . 58 5.4 Out-of-sample testing . 64 5.5 Conclusions . 65 6 Conclusions 67 6.1 Summary and conclusions . 67 6.2 Future research . 69 References 70 A Conditioning on the particle with highest weight 75 3 B The Unscented Kalman Filter 76 C Correlation matrices for the risk drivers in Section 4.3 79 D Market Data for the Heston Calibration in Chapter 5 80 E Alternative conditionings for the second out-of-sample test in Section 5.4 81 4 CHAPTER 1 Introduction 1.1 General setting Asset and Liability Management Asset and liability management (ALM) plays an important role in the strategic decision making of liability driven companies, such as insurers, pension funds, housing corporations and banks. It refers to the practice of managing the risks faced by a company that arise due to a mis- match between assets and liabilities [68]. Within ALM the maximum allowable risk with respect to the objectives and constraints of the stakeholders is determined by analyzing the balance sheet. Thereafter, it helps specifying policies which provide optimal returns given that maxi- mum risk. ALM models are used as guidance to determine for example contribution, premium, indexation and investment policies. Besides this, the models also need to provide insight and transparency for regulating authorities and for other stakeholders, especially after the financial crisis of 2008. In most ALM problems a lot of different stakeholders are involved. In a pension plan the stakeholders are for example the sponsor, employees and beneficiaries (retired and non-active members of the plan). An insurance company has to consider for example policy holders and shareholders. Both also need to take into consideration indirect stakeholders such as regulators, government and accountants. This wide variety of stakeholders can have conflicting interests and requirements. For example, for shareholders it is important to have stable and high returns on their invested equity. However, investing in risky assets which provide higher expected returns implies more solvency risks and this is not allowed by the regulator [69]. In practice ALM problems are approached with scenario analysis, in which external uncertainties are modeled by a set of possible plausible future developments, called scenarios. The external uncertainties concern both future development of economic variables such as interest rates, risk premiums of equity and inflation, and the development of non-economical variables such as coverage ratio and the size and composition of the group of policyholders. The scenarios are constructed to capture as many stylized facts of the market as possible, based on historical data and assumptions (market models and expert views). These scenarios form the input for an ALM- model which determines scores on the required ALM-criteria with respect to the objectives and 5 constraints that the management of the company has set. In Figure 1.1 a visualization of the scenario approach for ALM problems is given. We refer to recent Ortec Finance papers for a complete overview [59] and the relevance [60] of this scenario approach. In the comprehensive handbook [68, 69] more information about general ALM techniques can be found. Figure 1.1: ALM approach by scenario analysis, adapted from [69] Valuation of embedded options Following the financial crisis of 2008 new regulatory frameworks and accounting standards (e.g. Solvency II) were introduced. Insurance companies and pension funds are now obliged to value their liabilities at market value instead of at book value (which means that future cash flows were simply discounted with a fixed interest rate) [62]. Especially for insurance companies, asset and liability management has become much more complicated because they need to determine the amount of capital they have to hold against unforseen losses. A difficult aspect of this calculation is the market valuation of so-called embedded options. An embedded option is build into the structure of a financial security and it gives one of the parties the right, but not the obligation, to exercise some action by a certain date on terms that are established in advance. These options typically have a long term contract duration and are very sensitive to interest rates. An example is a policy conversion option that gives the insurance policyholder the right to convert from the current policy to another at pre-specified conditions [55]. Ortec Finance has developed an advanced simulation framework in which these complicated insurance liabilities can be modeled and many other questions concerning investment decisions can be answered. The value of an embedded option calculated by Ortec Finance is denoted by V (r1; : : : ; rn): In their valuation model, the price thus depends on n economical and non-economical variables, the so-called risk drivers [51]. To value an embedded option on time step t > 0, real world scenarios are generated based on assumed distributions for all of these risk drivers under the real 6 world measure P. These distributions correspond to appropriate time-series models for specific variables, for example a Hull-White model for modeling interest rates. As mentioned before, real world scenarios are thus instances of all of the risk drivers (^r1;:::; r^n)t at each time step t > 0. Since the valuation function V (^r1;:::; r^n) is typically not known in closed form, multiple Monte Carlo simulations have to be generated under the risk neutral measure Q in order to determine the option value. Note that these so-called risk neutral scenarios have to be calculated for every real world scenario at each time step. This leads to time-consuming nested Monte Carlo simulations (see Figure 1.2). To reduce computing times for the clients of Ortec Finance, we would like to reduce the dimension of the risk driver process in the option valuation model.