Alexander Szimayer

Elliptical copulas: applicability and limitations

We study copulas generated by elliptical distributions. We show that their tail dependence can be simply computed with default routines on Students t-distribution given Kendalls [tau] and the tail index. The copula family generated by the sub-Gaussian [alpha]-stable distribution is unable to cover the size of tail dependence observed in financial data.

A Multinomial Approximation for American Option Prices in Levy Process Models

This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Levy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Levy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Levy process has infinite activity.

Ornstein-Uhlenbeck processes and extensions

This paper surveys a class of Generalised Ornstein-Uhlenbeck (GOU) processes associated with Levy processes, which has been recently much analysed in view of its applications in the financial modelling area, among others. We motivate the Levy GOU by reviewing the framework already well understood for the “ordinary” (Gaussian) Ornstein-Uhlenbeck process, driven by Brownian motion; thus, defining it in terms of a stochastic differential equation (SDE), as the solution of this SDE, or as a time changed Brownian motion. Each of these approaches has an analogue for the GOU. Only the second approach, where the process is defined in terms of a stochastic integral, has been at all closely studied, and we take this as our definition of the GOU (see Eq. (12) below).

The Uncertain Mortality Intensity Framework: Pricing and Hedging Unit-Linked Life Insurance Contracts

We study the valuation and hedging of unit-linked life insurance contracts in a setting where mortality intensity is governed by a stochastic process. We focus on model risk arising from different specifications for the mortality intensity. To do so we assume that the mortality intensity is almost surely bounded under the statistical measure. Further, we restrict the equivalent martingale measures and apply the same bounds to the mortality intensity under these measures. For this setting we derive upper and lower price bounds for unit-linked life insurance contracts using stochastic control techniques. We also show that the induced hedging strategies indeed produce a dynamic superhedge and subhedge under the statistical measure in the limit when the number of contracts increases. This justifies the bounds for the mortality intensity under the pricing measures. We provide numerical examples investigating fixed-term, endowment insurance contracts and their combinations including various guarantee features. The pricing partial differential equation for the upper and lower price bounds is solved by finite difference methods. For our contracts and choice of parameters the pricing and hedging is fairly robust with respect to misspecification of the mortality intensity. The model risk resulting from the uncertain mortality intensity is of minor importance.

A Parsimonious Multi-Asset Heston Model: Calibration and Derivative Pricing

We propose a parsimonious multi-asset Heston model and provide an easy-to-implement calibration algorithm. The model is customized to pricing multi-asset options in markets with liquidly traded single-asset options but no liquidly traded cross-asset options. In this situation, single-asset model parameters can be calibrated from option price data, however, cross-asset parameters cannot. We formulate a parsimonious model specification such that all single-asset models are Heston models, which are affine allowing for efficient calibration of the respective parameters. The single-asset models are correlated using cross-asset correlations only. Cross-asset correlations are observable, in contrast to correlations of latent variables such as volatilities, and serve as basis for calibration. A hybrid calibration approach for identifying the model parameters consistent with option price data and asset price data is outlined and illustrated by a case study. In banking practice the approach is referred to as correlation adjustment.

Valuing Executive Stock Options: Performance Hurdles, Early Exercise and Stochastic Volatility

Accounting standards require companies to assess the fair value of any stock options granted to executives and employees. We develop a model for accurately valuing executive and employee stock options, focusing on performance hurdles, early exercise and uncertain volatility. We apply the model in two case studies and show that properly computed fair values can be significantly lower than traditional Black-Scholes values. We then explore the implications for pay-for-performance sensitivity and the design of effective share-based incentive schemes. We find that performance hurdles can require a much greater fraction of total compensation to be a fixed salary, if pre-existing incentive levels are to be maintained. Copyright (c) 2008 The Authors. Journal compilation (c) 2008 AFAANZ.

The effect of policyholders’ rationality on unit-linked life insurance contracts with surrender guarantees

We study the valuation of unit-linked life insurance contracts with surrender guarantees. Instead of solving an optimal stopping problem, we propose a more realistic approach accounting for policyholders’ rationality in exercising their surrender option. The valuation is conducted at the portfolio level by assuming the surrender intensity to be bounded from below and from above. The lower bound corresponds to purely exogenous surrender and the upper bound represents the limited rationality of the policyholders. The valuation problem is formulated by a valuation PDE and solved with the finite difference method. We show that the rationality of the policyholders has a significant effect on average contract value and hence on the fair contract design. We also present the separating boundary between purely exogenous surrender and endogenous surrender. This provides implications on the predicted surrender activity of the policyholders.

Valuation of American options in the presence of event risk

Abstract.This paper studies the valuation of American options in the presence of external/non-hedgeable event risk. When the event occurs, the American option is terminated and a rebate is paid instead of the promised pay-off profile. Consequently, the presence of event risk influences the exercise strategy of the option holder. For the financial market in a diffusion setting, the probabilistic structure in terms of equivalent martingale measures is briefly analysed. Then, for a given equivalent martingale measure the optimal stopping problem of the American option is solved. As a main result, no-arbitrage bounds for American option values in the presence of event risk are derived, as well as hedging strategies corresponding to the no-arbitrage bounds.

Testing for Mean Reversion in Processes of Ornstein-Uhlenbeck Type

In this paper, we analyse processes of Ornstein-Uhlenbeck (OU) type, driven by Lévy processes. This class is designed to capture mean reverting behaviour if it exists; but the data may in fact be adequately described by a pure Lévy process with no OU (autoregressive) effect. For an appropriate discretised version of the model, we utilise likelihood methods to test for such a reduction of the OU process to Lévy motion, deriving the distribution of the relevant pseudo-log-likelihood ratio statistics, asymptotically, both for a refining sequence of partitions on a fixed time interval with mesh size tending to zero, and as the length of the observation window grows large. These analyses are non-standard in that the mean reversion parameter vanishes under the null of a pure Lévy process for the data. Despite this we are able to give a very general analysis with no technical restrictions on the underlying processes or parameter sets, other than a finite variance assumption for the Lévy process. As a special case, for Brownian motion as driving process, we deduce the limiting distribution in a quite explicit way, finding results which generalise the well-known Dickey-Fuller (‘unit-root’) theory.

A reduced form model for ESO valuation

Abstract.In this paper we extend a reduced form model for the valuation of employee share options (ESOs) to incorporate employee departure, and company takeover. We also allow for performance linked vesting and other exotic features specific to ESOs. We clarify the assumptions underlying the reduced form model, and discuss their implications. We analyze the probabilistic structure of the model which includes an explicit characterization of the set of equivalent martingale measures, as well as the computation of the variance optimal martingale measure and the minimal martingale measure. Moreover, we deduce an additive decomposition of the relative entropy. Particular ESO specifications are studied emphasizing different aspects of the proposed framework. In this context, we also provide strict no-arbitrage bounds for ESO prices by applying optimal stopping. Furthermore, possible limitations of the proposed model are explored by examining departures from the crucial assumptions of no-arbitrage, i.e. by considering the effects of the employee having inside information.