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# Valuation of financial derivatives under regime switching models

thesis

posted on 2022-03-29, 00:38 authored by Kun FanFollowing the global financial crisis of 2008, the impacts of changes in (macro)-economic conditions and business cycles have attracted increasing interests in the new millennium. Regime-switching models have been considered as a natural tool of pricing financial derivatives by both academic researchers and industrial practitioners since Hamilton (1989) introduced this class of models into financial econometrics. Regime switching models typically use the states of the modulating Markov chain to represent the states of an economy, depicted by some (macro)-economic indicators. By adopting this methodology, regime-switching models can incorporate the impacts of structural changes in (macro)-economic conditions. Consequently, it is practical to consider the valuation of financial derivatives under regime-switching models.
In this thesis, the Markov chain we adopt is a continuous-time and finite-state Markov chain, either observable or unobservable, under regime-switching models. Our modelling setup includes regime-switching diffusion models, regime-switching jump diffusion models, regime-switching stochastic interest rate models, double regime-switching models and hidden Markov models. Under regime-switching models, the additional uncertainty leads to an incomplete financial market. The selection of a pricing kernel in an incomplete market has long been discussed as more than one equivalent martingale measures may exist in an incomplete market. In this thesis, both the regime-switching Esscher transform and the minimal martingale measure approach are considered in selecting a pricing kernel under regime-switching models. To obtain analytical pricing formulae for the financial derivatives, we mainly focus on the applications of the fast Fourier transform under regime-switching models.
In Chapter 1, we first briefly introduce the pricing models of financial derivatives. We further provide a literature review on topics including option valuation under different kinds of regime-switching models, hidden Markov models, stochastic interest rate models, approaches to determine an equivalent martingale measure in incomplete markets, and fast Fourier transform. Then the mathematical tools to be used in this thesis are introduced. More specifically, we give the mathematical representation of the modulating Markov chain and describe how to obtain analytical pricing formulae via Fourier transform and discretize the pricing formulas via the fast Fourier transform 20method. Finally, we give an overview of the papers to be included in this thesis.
In Chapters 2-4, we consider the valuations of various options under different regime-switching models where model parameters are modulated by a continuous-time, finite-state and observable Markov chain. Analytical pricing formulae for these options are obtained via the inverse Fourier transform and calculated via the fast Fourier transform, providing an easier and neater way to calculate the option prices.
In Chapter 2, we consider the valuation of foreign equity options, settling in one currency while the underlying assets are denominated in a different currency, under a Markovian regime-switching mean-reversion lognormal model. Intuitively, the valuation of the so-called foreign equity options has to deal with the joint dynamics of both the foreign equity and the exchange rate. Chapter 2 considers two kinds of foreign equity options, one with a strike price in the foreign currency (FEOF ) and the other with a strike price in the domestic currency (FEOD), under the assumption that the foreign exchange rate follows a mean-reversion lognormal model with regime-switching. To utilize the fast Fourier transform method, the characteristic function of the logarithmic entity price is needed. For FEOF , a measure change technique has to be applied first to take the expectation of the foreign exchange rate as the numeraire. For FEOD, a simple summation of the characteristic functions of the logarithmic foreign equity price and the logarithmic foreign exchange rate are calculated. Then we derive analytical pricing formulae for both FEOF options and FEOD options.
The valuation of power options are discussed in Chapter 3 by considering a Markovian regime-switching jump-diffusion model, where a Poisson random measure is adopted to depict the jump component. Power options provide investors with a choice of financial products with nonlinear payoff functions. This feature is attractive to many investors, especially in a financial market that involves many types of nonlinear risks. A unique equivalent martingale measure is selected by adopting a version of regimeswitching Esscher transform. Then, under the risk-neutral probability measure, a standard application of the inverse Fourier transform is applied to obtain analytical pricing formulae for the power option. In the numerical analysis, particular parametric forms of the compensator measure, including the Markov-modulated inverse Gaussian process and the Markov-modulated Merton jump-diffusion model, are considered.
Chapter 4 discusses the option valuation under a regime-switching stochastic interest rate model, which may increase the long-term effectiveness of the model. We start with a risk-neutral probability measure. To take the zero-coupon bond value as the numeraire, a measure change technique is applied to change the risk-neutral probability measure into a forward measure. By deriving the formulae for the characteristic function of the logarithmic return of the underlying asset, the fast Fourier transform can then be applied.
Chapters 5-6 consider the valuation of insurance products with embedded-option features under regime-switching models. In Chapter 5, we investigate the valuation of equity-linked annuities with mortality risk under a double regime-switching model,where the modulating Markov chain is a continuous-time, finite-state, observable chain. In addition to the assumption that model parameters will change when the states of the Markov chain switch, we also assume that a jump of the price level of the underlying investment fund will be triggered when a state transition of the chain occurs. One of the main features of the double regime-switching models is to provide an endogenous way to determine the regime-switching risk. To specify a unique pricing kernel, we employ both the generalized version of regime-switching Esscher transform and the minimal martingale method to determine a unique equivalent martingale measure, respectively. Note that the transition matrix of the Markov chain will also change. There have been many works adopting the option valuation techniques to price insurance policies with embedded-option features. In Chapter 5, we write the payoff of the equity-linked annuities as a combination of the payoffs of several European-style options. Consequently,we can utilize option valuation techniques to price the equity-linked annuities. Here, the technique of the fast Fourier transform applied to option valuation can be used to price such equity-linked annuities. Numerical analysis and sensitivity analysis provide us with intuitive understandings of the valuation of equity-linked annuities.
Chapter 6 discusses the valuation of dynamic fund protection under a hidden Markov model, where the states of the modulating Markov chain are unobservable. To protect investors from downside risk, a dynamic fund protection plan is an effective and convenient tool. The payoff function of a dynamic fund protection plan can bewritten as a product of the payoff of a fixed strike lookback call and the exchange rate. Consequently, option valuation techniques can also be utilized to price dynamic fund protection plans. In this chapter, the approach of partial differential equations is adopted to price dynamic fund protection plans by a three-stage estimation method. It consists of the Baum-Welch algorithm to estimate the model parameters of the hidden Markov chain, the Viterbi algorithm to select the most-probable path for the chain, and the maximum likelihood method to estimate the model parameters. The three-stage estimation method is intuitively appealing and easy to implement in practice.