Vigilant measures of risk and the demand for contingent claims

We examine a class of utility maximization problems with a non-necessarily law-invariant utility, and with a non-necessarily law-invariant risk measure constraint. Under a consistency requirement on the risk measure that we call Vigilance, we show the existence of optimal contingent claims, and we show that such optimal contingent claims exhibit a desired monotonicity property. Vigilance is satisfied by a large class of risk measures, including all distortion risk measures and some classes of robust risk measures. As an illustration, we consider a problem of optimal insurance design where the premium principle satisfies the vigilance property, hence covering a large collection of commonly used premium principles, including premium principles that are not law-invariant. We show the existence of optimal indemnity schedules, and we show that optimal indemnity schedules are nondecreasing functions of the insurable loss.     

Local risk minimization for vulnerable European contingent claims on nontradable assets under regime switching models

This article focuses on an optimal hedging problem of the vulnerable European contingent claims. The underlying asset of the vulnerable European contingent claims is assumed to be nontradable. The interest rate, the appreciation rate and the volatility of risky assets are modulated by a finite-state continuous-time Markov chain. By using the local risk minimization method, we obtain an explicit closed-form solution for the optimal hedging strategies of the vulnerable European contingent claims. Further, we consider a problem of hedging for a vulnerable European call option. Optimal hedging strategies are obtained. Finally, a numerical example for the optimal hedging strategies of the vulnerable European call option in a two-regime case is provided to illustrate the sensitivities of the hedging strategies.     

Pricing and hedging of American contingent claims in incomplete markets

0.IntroductionandSummaryThecelebratedpapersof[2]and[3],pavedthewayforpricingoptionsonstocks,onthebasisofthefollowingprinciple:inacompletemarket(suchastheoneinSection1.5),everycontingentclaimcanbeattainedexactlybyinvestinginthemarketandstartingwithala...     

Valuation of contingent claims with mortality and interest rate risks

We consider the pricing of life insurance contracts under stochastic mortality and interest rates assumed not independent of each other. Employing the method of change of measure together with the Bayes’ rule for conditional expectations, solution expressions for the value of common contracts are obtained. A demonstration of how to apply our proposed stochastic modelling approach to value survival and death benefits is provided. Using the Human Mortality Database and UK interest rates, we illustrate that the dependence between interest rate and mortality dynamics has considerable impact in the value of even a simple survival benefit.     

Valuation and hedging of contingent claims in the HJM model with deterministic volatilities

The aim of the present paper is mostly expository, namely, we intend to provide a concise presentation of arbitrage pricing and hedging of European contingent claims within the Heath, Jarrow and Morton frame-work introduced in Heath et al. (1992) under deterministic volatilities. Such a special case of the HJM model, frequently referred to as the Gaussian HJM model, was studied among others in Amin and Jarrow (1992), Jamshidian (1993), Brace and Musiela (1994a, 1994b). Here, we focus mainly on the partial differential equations approach to the valuation and hedging of derivative securities in the HJM framework. For the sake of completeness, the risk neutral methodology (more specifically, the forward measure technique) is also exposed.     

Duality and martingales: a stochastic programming perspective on contingent claims

The hedging of contingent claims in the discrete time, discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program. Application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics, namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale. The model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims. However, we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options. An extension of the model to incorporate pre-existing liabilities and endowments reveals the reasons why buyers and sellers trade in options. The model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets. Received: June 5, 2000 / Accepted: July 12, 2001?Published online December 6, 2001     

Full Bayesian analysis of claims reserving uncertainty

We revisit the gamma–gamma Bayesian chain-ladder (BCL) model for claims reserving in non-life insurance. This claims reserving model is usually used in an empirical Bayesian way using plug-in estimates for the variance parameters. The advantage of this empirical Bayesian framework is that allows us for closed form solutions. The main purpose of this paper is to develop the full Bayesian case also considering prior distributions for the variance parameters and to study the resulting sensitivities.     

On the pricing of American contingent claims under transaction costs and multiple risky assets

This paper addresses the hedging problem of American Contingents Claims (ACCs) in the framework of continuous-time Itô models for financial market. The special feature of this paper is that in the financial market the investor has to face fixed and proportional transaction costs when trading multiple risky assets. By using the auxiliary martingale approach and extending the results of Cvitanic and Karatzas [Cvitanic J, Karatzas I. Hedging and portfolio optimization under transaction costs: a martingale approach. Math Finance 1996;6:135–65] on pricing European contingent with transaction costs in the single-stock market, an arbitrage-free interval [h_low, h_up] is identified, and the end points are characterized by auxiliary martingales and stopping times in terms of auxiliary stochastic control problems. Here h_up and h_low are so-called the upper hedging price and the lower hedging price.     

Return smoothing mechanisms in life and pension insurance: Path-dependent contingent claims

Traditional with-profits pension saving schemes have been criticized for their opacity, plagued by embedded options and guarantees, and have recently created enormous problems for the solvency of the life insurance and pension industry. This has fueled creativity in the industry’s product development departments, and this paper analyzes a representative member of a family of new pension schemes that have been introduced in the new millennium to alleviate these problems. The complete transparency of the new scheme’s smoothing mechanism means that it can be analyzed using contingent claims pricing theory. We explore the properties of this pension scheme in detail and find that in terms of market value, smoothing is an illusion, but also that the return smoothing mechanism implies a dynamic asset allocation strategy which corresponds with traditional pension saving advice and the recommendations of state-of-the-art dynamic portfolio choice models.     

Asymptotic analysis of utility-based hedging strategies for small number of contingent claims

We study the linear approximation of utility-based hedging strategies for small number of contingent claims. We show that this approximation is actually a mean-variance hedging strategy under an appropriate choice of a numéraire and a risk-neutral probability. In contrast to previous studies, we work in the general framework of a semimartingale financial model and a utility function defined on the positive real line.     

On the representations of generating trading strategy for contingent claims in multidimensional diffusion model

In this paper, we find generating trading strategy for discounted security model in multidimensional diffusion model. Also, we observe generating trading strategy in its original model using discounted security model formula.     

Universal contingent claims in a general market environment and multiplicative measures: Examples and applications

We present and further develop the concept of a universal contingent claim introduced by the author in 1995. This concept provides a unified framework for the analysis of a wide class of financial derivatives.A universal contingent claim describes the time evolution of a contingent payoff. In the simplest case of a European contingent claim, this time evolution is given by a family of nonnegative linear operators, the valuation operators. For more complex contingent claims, the time evolution that is given by the valuation operators can be interrupted by discrete or continuous activation of external influences that are described by, generally speaking, nonlinear operators, the activation operators. For example, Bermudan and American contingent claims represent discretely and continuously activated universal contingent claims with the activation operators being the nonlinear maximum operators.We show that the value of a universal contingent claim is given by a multiplicative measure introduced by the author in 1995. Roughly speaking, a multiplicative measure is an operator-valued (in general, an abstract measure with values in a partial monoid) function on a semiring of sets which is multiplicative on the union of disjoint sets. We also show that the value of a universal contingent claim is determined by a, generally speaking, impulsive semilinear evolution equation.     

The non-Markovian approach to the valuation and hedging of European contingent claims on power with scaling spikes

We present and further develop a new approach to modeling power prices with spikes proposed earlier by the author. In contrast to other approaches, we model power prices with spikes as a non-Markovian stochastic process that allows for modeling spikes directly as self-reversing jumps. We show how this approach can be used to value European contingent claims on power with spikes as well as to value and dynamically hedge European contingent claims on forwards on power for power with spikes in a practically important special case of the scaling probability distribution for the magnitude of spikes.     

Mixed-integer second-order cone programming for lower hedging of American contingent claims in incomplete markets

We describe a challenging class of large mixed-integer second-order cone programming models which arise in computing the maximum price that a buyer is willing to disburse to acquire an American contingent claim in an incomplete financial market with no arbitrage opportunity. Taking the viewpoint of an investor who is willing to allow a controlled amount of risk by replacing the classical no-arbitrage assumption with a “no good-deal assumption” defined using an arbitrage-adjusted Sharpe ratio criterion we formulate the problem of computing the pricing and hedging of an American option in a financial market described by a multi-period, discrete-time, finite-state scenario tree as a large-scale mixed-integer conic optimization problem. We report computational results with off-the-shelf mixed-integer conic optimization software.     

A convex duality approach for pricing contingent claims under partial information and short selling constraints

We consider the pricing problem facing a seller of a contingent claim. We assume that this seller has some general level of partial information, and that he is not allowed to sell short in certain assets. This pricing problem, which is our primal problem, is a constrained stochastic optimization problem. We derive a dual to this problem by using the conjugate duality theory introduced by Rockafellar. Furthermore, we give conditions for strong duality to hold. This gives a characterization of the price of the claim involving martingale- and super-martingale conditions on the optional projection of the price processes.     

Composite multilinearity,epistemic uncertainty and risk achievement worth

Risk achievement worth is one of the most widely utilized importance measures. RAW is defined as the ratio of the risk metric value attained when a component has failed over the base case value of the risk metric. Traditionally, both the numerator and denominator are point estimates. Relevant literature has shown that inclusion of epistemic uncertainty (i) induces notable variability in the point estimate ranking and (ii) causes the expected value of the risk metric to differ from its nominal value. We investigate the conditions under which the equality of the nominal and expected values of a reliability risk metric holds. We then study how the presence of epistemic uncertainty affects RAW and the associated ranking. We propose an extension of RAW (called ERAW) which allows one to obtain a ranking robust to epistemic uncertainty. We discuss the properties of ERAW and the conditions under which it coincides with RAW. We apply our findings to a probabilistic risk assessment model developed for the safety analysis of NASA lunar space missions.     

Optimal investment under operational flexibility, risk aversion, and uncertainty

Traditional real options analysis addresses the problem of investment under uncertainty assuming a risk-neutral decision maker and complete markets. In reality, however, decision makers are often risk averse and markets are incomplete. We confirm that risk aversion lowers the probability of investment and demonstrate how this effect can be mitigated by incorporating operational flexibility in the form of embedded suspension and resumption options. Although such options facilitate investment, we find that the likelihood of investing is still lower compared to the risk-neutral case. Risk aversion also increases the likelihood that the project will be abandoned, although this effect is less pronounced. Finally, we illustrate the impact of risk aversion on the optimal suspension and resumption thresholds and the interaction among risk aversion, volatility, and optimal decision thresholds under complete operational flexibility.     

The Application of backward stochastic differential equation with stopping time in hedging American contingent claims

We consider a more general wealth process with a drift coefficient which is Lipschitz continuous and the portfolio process with convex constraint. We convert the problem of hedging American contingent claims into the problem of minimal solution of backward stochastic differential equation with stopping time. We adopt the penalization method for constructing the minimal solution of stochastic differential equations and obtain the upper hedging price of American contingent claims.