Alexander Melnikov

Risk Analysis in Finance and Insurance

Historically, financial and insurance risks were separate subjects most often analyzed using qualitative methods. The development of quantitative methods based on stochastic analysis is an important achievement of modern financial mathematics, one that can naturally be extended and applied in actuarial mathematics.Risk Analysis in Finance and Insurance offers the first comprehensive and accessible introduction to the ideas, methods, and probabilistic models that have transformed risk management into a quantitative science and led to unified methods for analyzing insurance and finance risks. The authors approach is based on a methodology for estimating the present value of future payments given current financial, insurance, and other information, which leads to proper, practical definitions of the price of a financial contract, the premium for an insurance policy, and the reserve of an insurance company.Self-contained and full of exercises and worked examples, Risk Analysis in Finance and Insurance serves equally well as a text for courses in financial and actuarial mathematics and as a valuable reference for financial analysts and actuaries. Ancillary electronic materials will be available for download from the publishers Web site.

Efficient Hedging and Pricing of Equity-Linked Life Insurance Contracts on Several Risky Assets

The paper uses the efficient hedging methodology in order to optimally price and hedge equity-linked life insurance contracts whose payoff depends on the performance of several risky assets. In particular, we consider a policy which pays the maximum of the values of n risky assets at some maturity date T, provided that the policyholder survives to T. Such contracts incorporate financial risk, which stems from the uncertainty about future prices of the underlying financial assets, and insurance risk, which arises from the policyholders mortality. We show how efficient hedging can be used to minimize expected losses from imperfect hedging under a particular risk preference of the hedger. We also prove a probabilistic result, which allows one to calculate analytic pricing formulas for equity-linked payoffs with n risky assets. To illustrate its use, explicit formulas are given for optimal prices and expected hedging losses for payoffs with two risky assets. Numerical examples highlighting the implications of efficient hedging for the management of financial and insurance risks of equity-linked life insurance policies are also provided.

On drift parameter estimation in models with fractional Brownian motion

We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein–Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behaviour of the fractional derivative of the fractional Brownian motion is established.

Stochastic equations and Krylov's estimates for semimartingales

In 1969 N. V. Krylov obtained the following estimate where X is an Ito process in is the exit time of X from a bounded region is Ld-norm of a measurable nonnegative function , and N is a constant. We generalize estimates of this type to semimartingales and give applications to the theory of stochastic equations with respect to semimartingales. The questions of the existence, uniqueness, convergence and comparison of solutions of these equations are also studied.

Quantile hedging for a jump-diffusion financial market model

The paper is devoted to the problem of hedging contingent claims in the framework of a jump-diffusion model. Based on the results of H. Follmer and P. Leukert [1]-[2] in a general semimartingale setting, we study the question how an investor can maximize the probability of a successful hedge under the constraint that he invests not more than a fixed amount of capital which is strictly less than the price of the option. We derive explicit formulas for this so-called quantile hedging strategy.

On financial markets based on telegraph processes

The paper develops a new class of financial market models. These models are based on generalised telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black–Scholes fundamental differential equation is derived, but, in contrast with the Black–Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging.

On option pricing in binomial market with transaction costs

Abstract.Option replication is studied in a discrete-time framework with proportional transaction costs. The model represents an extension of the Cox-Ross-Rubinstein binomial option-pricing model to cover the case of proportional transaction costs for one risky asset with different interest rates on bank credit and deposit. Contingent claims are supposed to be 2-dimensional random variables. Explicit formulas for self-financing strategies are obtained for this case.

Efficient Hedging and Pricing of Life Insurance Policies in a Jump-Diffusion Model

Abstract This paper is devoted to the problem of hedging contingent claims in the framework of a two factors jump-diffusion model under initial budget constraint. We give explicit formulas for the so called efficient hedging. These results are applied for the pricing of equity linked-life insurance contracts.

On properties of strong solutions of stochastic equations with respect to semimartingales

The paper is devoted to strong solutions of stochastic integral equations with respect to semimartingales. We study existence (for non-Lipschitz coefficients) and asymptotic behaviour of strong solutions and obtain also a number of results on weak and square mean convergence of these solutions.

On Comparison Theorem and its Applications to Finance

This paper studies a comparison theorem for solutions of stochastic differential equations and its generalization to the multi-dimensional case. We show, that even though the proof of the generalized theorem follows that of the one-dimensional comparison theorem, the multi-dimensional case requires a different condition on the drift coefficient, known in the theory of differential equations as Kamke-Wazewski condition. We also present several examples of possible applications to option price estimation in finance.