|dc.description.abstract||This thesis addresses the pricing and hedging issues on the newly-developed financial and insurance products, including simplified hedges for path-dependent options, variable annuities tied with state-dependent fees, and defaultable reverse mortgage contracts.
In Chapter 1, we present a method to construct a simplified alternative derivative that resembles a given highly path-dependent derivative. Path-dependent derivatives are typically difficult to hedge. Traditional dynamic delta hedging does not perform well because of the difficulty to evaluate the Greeks and the high cost of constantly rebalancing. We propose to price and hedge path-dependent derivatives by constructing simplified alternatives that preserve certain distributional properties of their terminal payoffs, and that can be hedged by semi-static replication. The method is illustrated by a geometric Asian option and by a lookback option in the Black-Scholes setting, for which explicit forms of the simplified alternatives exist. An extension to a Heston stochastic volatility model is discussed as well.
In Chapter 2, we model and study the benefits of charging state-dependent fees in variable annuities tied to the market volatility. Variable annuities (VAs) and other long-term equity-linked insurance products are typically difficult to hedge in incomplete markets. A state-dependent fee structure tied to market volatility is proposed in these products to contribute to the risk sharing mechanism between policyholders and insurers and also to reduce the hedging difficulty. We provide criteria for the fair-fee determination in the context of reducing the risk related to writing the VA contract. A method of optimal static hedging as a benchmark compared to other strategies is proposed in the stochastic volatility setting. We formulate our problem with guaranteed minimum accumulation benefits (GMABs), but it is also applicable to other equity-linked insurance contracts.
In Chapter 3, we propose a pricing scheme based on default risk models for Home Equity Conversion Mortgages (HECM). HECM Reverse mortgages are designed to allow elder homeowners aged 62 or over to convert the equity in their homes to regular revenues or a line of credit and to retain full ownership of their property for the whole life of the loan. Unlike a traditional mortgage, reverse mortgage loans do not need to be paid off as long as the borrowers remain in their home and pay due obligations such as home insurance and property taxes. HECM are non-recourse reverse mortgage loans insured by the Federal Housing Administration (FHA). HECM reverse mortgages confront a rising default risk in the wake of 2008, jeopardising the financial soundness of FHA's Mutual Mortgage Insurance Fund. The fairness of the HECM insurance premium has therefore been challenged. In this chapter, we initiate to price the reverse mortgage contract according to borrowers' individual credit and default risk. The proposed method achieves a closed-form valuation with mortality risk, interest rate risk, housing price risk, and default risk. The impact on fair HECM insurance premiums of these risks is then investigated. Our work demonstrates that the proposed pricing solution and the corresponding newly-designed rating system will provide HECM lenders a better payment arrangement for the risk management and also support the effectiveness of recent policy changes in the HECM program.
The products described as above are designed in incomplete markets, which renders perfect hedging of these contracts impossible. The goal of Chapter 4 is to develop optimal static hedging in the context of minimizing the shortfall risk either for path-dependent options, hedging liabilities with insufficient budget, or hedging liabilities under the stochastic volatility environment. The shortfall risk is defined as the expectation of the potential loss from the imperfect hedging strategy, weighted by some loss function reflecting the hedger's risk preferences. In Chapter 4, we take examples on the Asian option and the GMAB contract in Chapter 2 and further develop the optimal static hedging for our products under the Heston-type stochastic volatility environment.||en