Hedging for the long run

Enlarged Filtrations and Indistinguishable Processes

A Visual Classification of Local Martingales

Market Imperfections, Equilibrium and Arbitrage

Asymptotic optimality of the generalized [formula omitted] rule under model uncertainty

Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform

We use the Cox process (or a doubly stochastic Poisson process) to model the claim arrival process for catastrophic events. The shot noise process is used for the claim intensity function within the Cox process. The Cox process with shot noise intensity is examined by piecewise deterministic Markov process theory. We apply the model to price stop-loss catastrophe reinsurance contract and catastrophe insurance derivatives. The asymptotic distribution of the claim intensity is used to derive pricing formulae for stop-loss reinsurance contract for catastrophic events and catastrophe insurance derivatives. We assume that there is an absence of arbitrage opportunities in the market to obtain the gross premium for stop-loss reinsurance contract and arbitrage-free prices for insurance derivatives. This can be achieved by using an equivalent martingale probability measure in the pricing models. The Esscher transform is used for this purpose.

Let \(\{W_{t}\}_{t\geq 0}\) be a Brownian motion with respect to a probability measure \(\mathbb{P}\). Take a constant \(\theta\), and consider \(X_{t} = W_{t} +\theta t\), \(0 \leq t <\infty\), which is called a Brownian motion with drift. Our goal is to find a probability measure \(\mathbb{Q}\) for which X t , 0 ≤ t ≤ T, is a Brownian motion for some fixed T. We require an additional condition that \(\mathbb{Q}\) is equivalent to \(\mathbb{P}\), i.e., \(\mathbb{P}(A) = 0\) if and only if \(\mathbb{Q}(A) = 0\). In other words, an event occurs with positive \(\mathbb{P}\)-probability if and only if it happens with positive \(\mathbb{Q}\)-probability. Such a condition is important in financial applications since we have to deal with the same set of asset price movements even when we switch to a new probability measure. Igor Girsanov proved the existence of such a measure \(\mathbb{Q}\). We will find first a necessary condition for the existence of an equivalent probability measure \(\mathbb{Q}\) for which a Brownian motion with drift is a Brownian motion. Such a necessary condition will turn out to be crucial in defining \(\mathbb{Q}\).

Delta and Gamma Hedging of Mortality and Interest Rate Risk

We study mean-field control (MFC) problems with common noise using the control randomisation framework, where we substitute the control process with an independent Poisson point process, controlling its intensity instead. To address the challenges posed by the mean-field interactions in this randomisation approach, we reformulate the admissible control as L0-valued processes adapted only to the common noise. We then construct the randomised control problem from this reformulated control process, and show its equivalence to the original MFC problem. Thanks to this equivalence, we can represent the value function as the minimal solution to a backward stochastic differential equation (BSDE) with constrained jumps. Finally, using this probabilistic representation, we derive a randomised dynamic programming principle (DPP) for the value function, expressed as a supremum over equivalent probability measures.

This paper presents a unifying theory for valuing contingent claims under a stochastic term of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processeE for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results. IN RELATION TO the term of interest rates, arbitrage pricing theory has two purposes. The first, is to price all zero coupon (default free) bonds of varying maturities from a finite number of economic fundamentals, called state variables. The second, is to price all interest rate sensitive contingent claims, taking as given the prices of the zero coupon bonds. This paper presents a general theory and a unifying framework for understanding arbitrage pricing theory in this context, of which all existing arbitrage pricing models are special cases (in particular, Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), Ball and Torous (1983), Ho and Lee (1986), Schaefer and Schwartz (1987), and Artzner and Delbaen (1988)). The primary contribution of this paper, however, is a new methodology for solving the second problem, i.e., the pricing of interest rate sensitive contingent claims given the prices of all zero coupon bonds. The methodology is new because (i) it imposes its stochastic directly on the evolution of the forward rate curve, (ii) it does not require an inversion of the term structure to eliminate the market prices of risk from contingent claim values, and (iii) it has a stochastic spot rate process with multiple stochastic factors influencing the term structure. The model can be used to consistently price (and hedge) all contingent claims (American or European) on the term structure, and it is derived from necessary and (more importantly) sufficient conditions for the absence of arbitrage. The arbitrage pricing models of Vasicek (1977), Brennan and Schwartz (1979), Langetieg (1980), and Artzner and Delbaen (1988) all require an IFormerly titled Bond Pricing and the Term Structure of Interest Rates: A New Methodology.

We consider perpetual American options assuming, that under a chosen equivalent martingale measure the shock returns follow a Levy process. For put and call options, their analogs for more general payoffs, and a wide class, of Levy processes, which contains Brownian motion, normal inverse Gaussian processes, hyperbolic processes, truncated Levy processes and their mixtures, we obtain formulas for the optimal exercise price and the fair price of the option in terms of the factors in the Wiener-Hopf factorization formula, i.e., in terms of the resolvents of the supremum and infimum processes, and derive explicit formulas for these factors. For calls, puts and some other options, the results are valid for any Levy process.

An approach to Malliavin calculus for Levy processes, discrete in time and smooth in chance, is presented. Each Levy triple can be satisfied by a Levy process living on a fixed sample space Ω, which is, in a certain sense, a finite dimensional Euclidean space. The probability measures on Ω characterize the Levy processes. We compare these measures with the associated Levy measures, and present several examples. Using chaos expansions for Levy functionals, even for those having no moments, we can represent all these functionals by polynomials in several variables. There exists an effective method to compute the kernels of the chaos decomposition. Finally, we point out several applications, which are postponed to a succession of papers.

Forward Measures in a Ho and Lee Jump Diffusion Model

This paper investigates necessary conditions for an equilibrium to exist on a reinsurance market with short sale constraints. It establishes that, at the equilibrium, there exists an equivalent probability measure under which the reinsurance premium is the compensator of the jump process describing the risk (even if, a priori, the form of the premium does not allow “à la Girsanov” changes of probability). Besides, the equivalent probability is locally represented by the marginal utility of some insurance companies.

This chapter introduces the use of a time-homogenous Markov chain for the valuation of options. In the computational finance literature, three categories of numerical techniques have been explored extensively for the valuation of contingent claims. The first category involves the use of a lattice structure to approximate the price movement of the underlying asset under the risk neutral probability measure and then computes the price of a contingent claim as a discounted expected payoff. The lattice approach essentially discretizes both time and state in a particular way. The second category is the finite difference/element approach. This technique numerically solves the partial differential equation that the value function of a contingent claim must obey under the no-arbitrage condition. The third category is the Monte Carlo method, which simulates the system under the risk-neutral probability measure so that the expectation of a contingent payoff can be approximated by the sample average.