Hedging Cost Analysis of Put Option with Applications to Variable Annuities by Panpan Wu A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Statistics University of Toronto c Copyright 2013 by Panpan Wu Abstract Hedging Cost Analysis of Put Option with Applications to Variable Annuities Panpan Wu Doctor of Philosophy Graduate Department of Statistics University of Toronto 2013 Variable annuities (VA) are equity-linked annuity contracts which provide the opportu- nity for policy-holders to benefit from financial markets appreciation and at the mean time provide protection from the downside risks of the markets. They have been over- shadowing traditional fixed annuities to become the leading form of protected investment worldwide. However, the embedded guarantees in VA can bring significant downside risks to the insurer and need to be hedged. Among different hedging strategies, move-based discrete hedging strategies are widely adopted in practice but the cost analysis for move- based discrete hedging strategies are mathematically complex. In this thesis, we first ex- amine various move-based hedging strategies and show that a two-sided underlier-based hedging strategy is desirable for the return of premium guarantee. Then we assume a GBM model for the sub-account to develop a semi-analytic framework for the hedging cost analysis of this strategy and thereby propose a modified \Percentile Premium Prin- ciple", which imposes a significant \loading" on top of the regular charge to cover the costs arising from the discrete re-balances under the two-sided underlier-based strategy. We apply the modified \Percentile Premium Principle" to the pricing of various VA de- signs, including GMMB, annual ratchet VA, structured product based VA with both buffered and contingent protection. Finally, we advance the algorithm towards a more general model-GBM with regime switching-to allow a better representation of the VA sub-account. ii Acknowledgements Several people have influenced this thesis in various ways and it is my pleasure to express my thanks to them here. I owe a particular debt to my supervisor Prof. X. Sheldon Lin. His guidance illuminated my way towards independent research. His encouragement led me go the distance. Professor Andrei L. Badescu, Professor Samuel Broverman, Professor Radu V. Craiu, Professor Andrey Feuerverger, Professor Tak Kuen (Ken) Siu, and Professor Fang Yao carefully reviewed the draft version of my thesis and provided numerous constructive comments. Their patience and insights are warmly appreciated. Furthermore, Professor Broverman generously offered me many study materials during my preparation for the SOA preliminary exams, which has greatly accelerated my progress. My special thank goes to Professor Yichun Chi, for his enthusiastic discussions with me on a wide range of academic topics and his share of personal experiences in conducting excellent research. I thank Professor Sebastian Jaimungal. His solid mastery of financial modeling benefited me through his courses on derivative pricing theory. I would also like to thank Professor Radu V. Craiu, Professor Radford Neal, Professor Jeremy Quastel, Professor Nancy Reid, Professor Jeffrey Rosenthal, Professor Balint Vi- rag and Professor Fang Yao, for their interesting courses on various aspects of probability and statistics that prepare me well for the challenges in research. The staff members in my department, especially Ms. Andrea Carter and Ms. Angela Fleury, have always been forward in lending me a hand. Their professionalism is highly regarded. At last, I am dearly and forever grateful to my parents and grandparents, for they fill my heart with hope and love. iii Contents 1 Introduction 1 1.1 Background . 1 1.2 Basic Features of Variable Annuities . 3 1.3 Hedging Variable Annuities . 5 2 Option Pricing Basics 13 2.1 Dynamic Hedging of Derivatives . 13 2.2 The Black-Scholes Option Pricing Formula . 16 2.3 Option Greeks . 18 2.4 The Martingale Approach for Derivatives Pricing . 19 2.5 Geometric Brownian Motion . 21 3 The Move-Based Hedging Strategies 24 3.1 The Move-Based Hedging Strategies . 25 3.2 A First Look at the Cost Distribution . 27 3.3 Three Density Functions for the Hitting Time Distribution of Geometric Brownian Motion . 29 3.4 An Asymptotic Analysis for the Time-Based Hedging . 37 4 A Semi-Analytic Algorithm for the Cost Analysis of Put Option 39 4.1 The Main Idea . 39 iv 4.2 The Expected Cost . 40 4.3 Higher Moments . 46 4.4 Numerical Examples . 49 4.5 Quantile Calculation . 58 4.5.1 GEV Distribution . 60 4.5.2 Edgeworth Expansion . 61 5 The Percentile Principle Premium for Variable Annuities 63 5.1 Guaranteed Minimum Maturity Benefit . 64 5.2 Annual Ratchet VA . 69 5.3 Structured Product Based VA . 77 5.3.1 Structured Capital Strategies . 79 5.3.2 Structured Notes with Contingent Protection . 83 6 Extension to Regime Switching GBM 94 6.1 Regime Switching Models . 94 6.2 Hitting Time Distribution for GBM with Two Regimes . 96 6.3 The Expected Cost . 106 6.4 Expectation of the First Cost . 110 6.5 The Management Fee . 117 6.6 Application to Structured Product based VA . 119 7 Future Work and Conclusion 122 7.1 Preliminary Results on the High-Water-Mark VA . 122 7.2 Conclusion . 129 Bibliography 131 v Chapter 1 Introduction 1.1 Background Over the last decade, variable annuities (VA) have been overshadowing traditional fixed annuities to become the leading form of protected investment worldwide. In the US, the total sales peaked in 2007, just before the financial crisis, at $182.2 billion dollars, and the net assets almost doubled in the past ten years from $885.8 billion in 2001 to $1,505.0 billion in 2010. See Insured Retirement Institute (2011) for more detail. In UK, where it has recently started to thrive, the sales figures are catching up, with 538.7 million pounds in 2007, 1,153.3 million in 2008, 1,045.4 million in 2009 and 444.7 million in the first half of 2010. In Japan, where growth has been even more spectacular, from almost no market in 2001 to more than $100 billion in 2007. Variable annuities offer a tax-deferred way of investing in the capital markets. For an additional fee, many VA come with guarantees minimum benefits paid out even if the underlying funds dries up. In this sense, VA provides its policyholders the opportunity to gain from financial markets appreciation as well as the protection from the downside risks of the markets. 1 According to a report by Oliver Wyman Limited (2007)1, the popularity of these con- tracts is driven by the demographic changes taking place in most parts of the world. The over-50 population is getting larger, richer and more diverse in styles of life. They de- mand access to market appreciation in order to keep abreast with the rising cost of living, but also expect protection for their assets and well-being given increased uncertainty in the volatility of asset returns. With the various investment options and multiple forms of guarantees variable annuities could offer, they would be able to choose the best fit for their desired risk/return target. Though assets reached an all-time high of $1.61 trillion in the first quarter of 2012, the variable annuity market has been sending adverse signals lately. Sales are down, several key players have exited the market (Indeed, companies such as Sun Life, The Hartford, Jackson National, ING and MetLife have dialed down their variable annuity exposure or pulled-out entirely. See AnnuityDigest (2012) for more details). Critics say variable annuities are too costly, the relatively high fees annuity investors pay can eat up a significant amount of money over the long term. Variable annuities typically charge 1.25 percent to 1.60 percent mortality and expense fees on top of the fund expense ratios. So instead of paying around 1% for all-in participant fees, the participant pays at least 2.25 to 2.6 percent for annuity products and often even more. Some insurers respond to these criticism by designing \Structured Product Based Variable Annuities", whose payouts are based on the price changes of a reference asset (an index, equity, interest rate, etc.) but subject to buffers against losses and caps on gains. More specifically, structured products underlying these new annuity contracts pass on losses below some threshold or “buffer” to investors and limit the gains of investors at some \cap" (See Deng et al. (2012) and references therein). These new features allow the provision of 1The report discusses the European retirement protection market but the reasons given in the report are applicable to other countries too. 2 cheaper products by the insurers to boost their sales. 1.2 Basic Features of Variable Annuities Tax-Deferred Returns Payments and returns of the variable annuities are tax-deferred. All dividends, interest and capital gains are automatically reinvested without incurring local, state or federal taxes. All earnings are taxed at ordinary income tax rates when withdrawn. Therefore, returns compound more quickly without the erosive effects of taxes, and the value of the annuity is allowed to grow at a faster pace. Separate Account and Sub-Accounts Variable annuities provide the choice of stock and bond portfolios called \sub-accounts" that are similar to mutual funds. By law, this account must be kept separate from the insurance company's general account, so that all dividends, interest, gains and losses are separate and apart from the finances of the insurance company. Sub-accounts allow the owner to tailor an asset allocation model for specific investment objectives. The number of sub-account choices and the specific funds will depend on the individual variable annuity contract that the investor chooses.