Leunglung Chan

Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching

A model is developed for pricing volatility derivatives, such as variance swaps and volatility swaps under a continuous‐time Markov‐modulated version of the stochastic volatility (SV) model developed by Heston. In particular, it is supposed that the parameters of this version of Hestons SV model depend on the states of a continuous‐time observable Markov chain process, which can be interpreted as the states of an observable macroeconomic factor. The market considered is incomplete in general, and hence, there is more than one equivalent martingale pricing measure. The regime switching Esscher transform used by Elliott et al. is adopted to determine a martingale pricing measure for the valuation of variance and volatility swaps in this incomplete market. Both probabilistic and partial differential equation (PDE) approaches are considered for the valuation of volatility derivatives.

Pricing options under a generalized Markov-Modulated jump-diffusion model

Abstract We consider the pricing of options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump-diffusion model. We suppose that the market interest rate, the drift and the volatility of the underlying risky asset switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The measure process is defined to be a generalized mixture of Poisson random measure and encompasses a general class of processes, for example, a generalized gamma process, which includes the weighted gamma process and the inverse Gaussian process. Another interesting feature of the measure process is that jump times and jump sizes can be correlated in general. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the underlying risky asset. We employ the generalized regime-switching Esscher transform to determine an equivalent martingale measure in the incomplete market setting. A system of coupled partial-differential-integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Simulation results of the model have been presented and discussed.

OPTION PRICING FOR GARCH MODELS WITH MARKOV SWITCHING

In this paper we develop a method for pricing derivatives under a Markov switching version of the Heston-Nandi GARCH (1, 1) model by using a well known tool from actuarial science, namely the Esscher transform. We suppose that the dynamics of the GARCH process switch over time according to one of the regimes described by the states of an observable Markov chain process. By augmenting the conditional Esscher transform with the observable Markov switching process, a Markov switching conditional Esscher transform (MSCET) is developed to identify a martingale measure for option valuation in the incomplete market described by our model. We provide an alternative approach for the derivation of an analytical option valuation formula under the Markov switching Heston-Nandi GARCH (1, 1) model. The use of the MSCET can be justified by considering a utility maximization problem with respect to a power utility function associated with the Markov switching risk-averse parameters.

On pricing barrier options with regime switching

We consider the valuation of both European-style and American-style barrier options in a Markovian, regime-switching, Black-Scholes-Merton economy, where the price process of an underlying risky asset is governed by a Markovian, regime-switching, geometric Brownian motion. Both the probabilistic and partial differential equation (PDE), approaches are used to price the barrier options. For the probabilistic approach to value a European-style barrier option, we employ the fundamental matrix solution and the Fourier transform space to derive a (semi)-analytical solution. The PDE approach is employed to value an American barrier option, where we obtain a system of free-boundary, coupled PDEs and an analytical quadratic approximation to the price by solving the free-boundary problem.

Perpetual American options with fractional Brownian motion

In this paper, we derive a closed from solution for the value of a perpetual American option when the logreturn of a stock is driven by a fractional Brownian motion, with Hurst parameter H ↦ (0,1). A special case of our model would be the model driven by standard Brownian motion

Pricing and hedging of long dated variance swaps under a 3/2 volatility model

This paper investigates the pricing and hedging of variance swaps under a 3/2 volatility model using explicit formulae. Pricing and hedging is performed under the benchmark approach, which only requires the existence of the numeraire portfolio. The growth optimal portfolio is used as numeraire together with the real world probability measure as pricing measure. This real world pricing concept provides minimal prices for variance swaps even when an equivalent risk neutral probability measure does not exist.

Option valuation under a regime-switching constant elasticity of variance process

We investigate the pricing of both European and American-style options when the price dynamics of the underlying risky assets are governed by a Markov-modulated constant elasticity of variance process. Both probabilistic and partial differential equation approaches are considered in deriving the value of a European-style option. For the case of an American-style option, we consider a probabilistic approach and derive an integral representation for the early exercise premium.

A DUPIRE EQUATION FOR A REGIME-SWITCHING MODEL

A forward equation, which is also called the Dupire formula, is obtained for European call options when the price dynamics of the underlying risky assets are assumed to follow a regime-switching local volatility model. Using a regime-switching version of the adjoint formula, a system of coupled forward equations is derived for the price of the European call over different states of the economy.

Pricing volatility derivatives under the modified constant elasticity of variance model

This paper studies volatility derivatives such as variance and volatility swaps, options on variance in the modified constant elasticity of variance model using the benchmark approach. The analytical expressions of pricing formulas for variance swaps are presented. In addition, the numerical solutions for variance swaps, volatility swaps and options on variance are demonstrated.

Pricing of long dated equity-linked life insurance contracts

Abstract This article adopts an approach to pricing of equity-linked life insurance contracts, which only requires the existence of the numéraire portfolio. An equity-linked life insurance contract is equivalent to a sum of the guaranteed amount and the value of an option on the equity index with some mortality risk attached. The numéraire portfolio equals the growth optimal portfolio and is used as numéraire or benchmark, where the real-world probability measure is taken as pricing measure. To obtain tractable solutions the short rate is modelled as a quadratic form of some Gaussian factor processes. Furthermore, the dynamics of the mortality rate is modelled as a threshold life table. The dynamics of the discounted equity market index or benchmark is modelled by a time transformed squared Bessel process. The equity-linked life insurance contracts are evaluated analytically.