Abstract:
We study hedging and pricing of unattainable contingent claims in a non-Markovian regime-switching financial model. Our financial market consists of a bank account and a risky asset whose dynamics are driven by a Brownian motion and a multivariate counting process with stochastic intensities. The interest rate, drift, volatility and intensities fluctuate over time and, in particular, they depend on the state (regime) of the economy which is modelled by the multivariate counting process. Hence, we can allow for stressed market conditions. We assume that the trajectory of the risky asset is continuous between the transition times for the states of the economy and that the value of the risky asset jumps at the time of the transition. We find the hedging strategy which minimizes the instantaneous mean-variance risk of the hedger's surplus and we set the price so that the instantaneous Sharpe ratio of the hedger's surplus equals a predefined target. We use Backward Stochastic Differential Equations. Interestingly, the instantaneous mean-variance hedging and instantaneous Sharpe ratio pricing can be related to no-good-deal pricing and robust pricing and hedging under model ambiguity. We discuss key properties of the optimal price and the optimal hedging strategy. We also use our results to price and hedge mortality-contingent claims with financial components (equity-linked insurance claims) in a combined insurance and regime-switching financial model.

We consider general regime switching stochastic volatility models where both the asset and the volatility dynamics depend on the values of a Markov jump process. Due to the stochastic volatility and the Markov regime switching, this financial market is thus incomplete and perfect pricing and hedging of options are not possible. Thus, we are interested in finding formulae to solve the problem of pricing and hedging options in this framework. For this, we use the local risk minimization approach to obtain pricing and hedging formulae based on solving a system of partial differential equations. Then we get also formulae to price volatility and variance swap options on these general regime switching stochastic volatility models.

Abstract:
We propose optimal mean-variance dynamic hedging
strategies in discrete time under a multivariate Gaussian regime-switching
model. The methodology, which also performs pricing, is robust to time-varying
and clustering risk observed in financial time series. As such, it overcomes
the main theoretical drawbacks of the Black-Scholes model. To support our
approach, we provide goodness-of-fit tests to validate the model and for
choosing the appropriate number of regimes, and we illustrate the methodology
using monthly S & P 500 vanilla options prices. Then, we present the
associated out-of-sample hedging results in the context of harvesting the
implied versus realized volatility premium. Using the proposed methodology, the
Sharpe ratio derived from the strategy doubles over the Black-Scholes
delta-hedging methodology.

Abstract:
We present a method of hedging Conditional Value at Risk of a position in stock using put options. The result leads to a linear programming problem that can be solved to optimise risk hedging.

Abstract:
At first, we solve a problem of finding a risk-minimizing hedging strategy on a general market with ratings. Next, we find a solution to this problem on Markovian market with ratings on which prices are influenced by additional factors and rating, and behavior of this system is described by SDE driven by Wiener process and compensated Poisson random measure and claims depend on rating. To find a tool to calculate hedging strategy we prove a Feynman-Kac type theorem. This result is of independent interest and has many applications, since it enables to calculate some conditional expectations using related PIDE's. We illustrate our theory on two examples of market. The first is a general exponential L\'{e}vy model with stochastic volatility, and the second is a generalization of exponential L\'{e}vy model with regime-switching.

Abstract:
We solve the problem of mean-variance hedging for general semimartingale models via stochastic control methods. After proving that the value process of the associated stochastic control problem has a quadratic structure, we characterize its three coefficient processes as solutions of semimartingale backward stochastic differential equations and show how they can be used to describe the optimal trading strategy for each conditional mean-variance hedging problem. For comparison with the existing literature, we provide alternative equivalent versions of the BSDEs and present a number of simple examples.

Abstract:
We consider the mean-variance hedging problem under partial Information. The underlying asset price process follows a continuous semimartingale and strategies have to be constructed when only part of the information in the market is available. We show that the initial mean variance hedging problem is equivalent to a new mean variance hedging problem with an additional correction term, which is formulated in terms of observable processes. We prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differential equations and the filtered wealth process of the optimal hedging strategy is characterized as a solution of a linear forward equation.

Abstract:
We investigate a continuous-time version of the mean-variance portfolio selection model with jumps under regime switching. The portfolio selection is proposed and analyzed for a market consisting of one bank account and multiple stocks. The random regime switching is assumed to be independent of the underlying Brownian motion and jump processes. A Markov chain modulated diffusion formulation is employed to model the problem. 1. Introduction The jump diffusion process has come to play an important role in many branches of science and industry. In their book [1], ？ksendal and Sulem have studied the optimal control, optimal stopping, and impulse control for jump diffusion processes. In mathematical finance theory, many researchers have developed option pricing theory, for example, Merton [2] was the first to use the jump processes to describe the stock dynamics, and Bardhan and Chao [3] were amongst the first authors to consider market completeness in a discontinuous model. The jump diffusion models have been discussed by Chan [4], F？llmer and Schweizer [5], El Karoui and Quenez [6], Henderson and Hobson [7], and Merculio and Runggaldier [8], to name a few. On the other hand, regime-switching models have been widely used for price processes of risky assets. For example, in [9] the optimal stopping problem for the perpetual American put has been considered, and the finite expiry American put and barrier options have been priced. The asset allocation has been discussed in [10], and Elliott et al. [11] have investigated volatility problems. Regime-switching models with a Markov-modulated asset have already been applied to option pricing in [12–14] and references therein. Moreover, Markowitz's mean-variance portfolio selection with regime switching has been studied by Yin and Zhou [15], Zhou and Yin [16], and Zhou and Li [17]. Portfolio selection is an important topic in finance; multiperiod mean-variance portfolio selection has been studied by, for example, Samuelson [18], Hakansson [19], and Pliska [20] among others. Continuous-time mean-variance hedging problems were attacked by Duffie and Richardson [21] and Schweizer [22] where optimal dynamic strategies were derived, based on the projection theorem, to hedge contingent claims in incomplete markets. In this paper, we will extend the results of Yin and Zhou [15] to SDEs with jumps under regime switching. After dealing with the difficulty from the jump processes, we obtain similar results to those of Yin and Zhou [15]. 2. SDEs under Regime Switching with Jumps Throughout this paper, let be a fixed complete

Abstract:
This article discusses an adjusted regime switching model in the context of portfoliooptimization and compares the attained portfolio weights and the performance to aclassical mean-variance set-up as introduced by Markowitz (1952). The model postulatesdifferent asset price dynamics under different regimes, and jumps between regimes are drivenby a Markov process. For examples, ’bear’ and ’bull’ markets could be such regimes. Givena particular regime, portfolio weights are set based on the conditional means and variancecovariancestructure of the asset dynamics. The model is evaluated in an out-of-sampleperiod of the last three years with a moving window and a forecast of only one period. It isfound that with the adjusted regime switching portfolio selection algorithm as applied here,the performance of the optimal portfolio is highly improved even where portfolio weights areconstrained to realistic values.

Abstract:
We study the pricing and the hedging of claim {\psi} which depends on the default times of two firms A and B. In fact, we assume that, in the market, we can not buy or sell any defaultable bond of the firm B but we can only trade defaultable bond of the firm A. Our aim is then to find the best price and hedging of {\psi} using only bond of the firm A. Hence, we solve this problem in two cases: firstly in a Markov framework using indifference price and solving a system of Hamilton-Jacobi-Bellman equations, secondly, in a more general framework, using the mean variance hedging approach and solving backward stochastic differential equations (BSDE).