Bilateral Trade, Openness and Asset Holdings

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.unit-linked life insurance contracts, surrender guarantee, limited rationality, fair contract analysis

A Limit Theorem for Copulas

We characterize convergence of a sequence of d-dimensional random vectors by convergence of the one-dimensional margins and of the copula. The result is applied to the approximation of portfolios modelled by t-copulas with large degrees of freedom, and to the convergence of certain dependence measures of bivariate distributions

Nonlinear Term Structure Dependence: Copula Functions, Empirics, and Risk Implications

This paper documents nonlinear cross-sectional dependence in the term structure of U.S. Treasury yields and points out risk management implications. The analysis is based on a Kalman filter estimation of a two-factor affine model which specifies the yield curve dynamics. We then apply a broad class of copula functions for modeling dependence in factors spanning the yield curve. Our sample of monthly yields in the 1982 to 2001 period provides evidence of upper tail dependence in yield innovations; i.e., large positive interest rate shocks tend to occur under increased dependence. In contrast, the best fitting copula model coincides with zero lower tail dependence. This asymmetry has substantial risk management implications. We give an example in estimating bond portfolio loss quantiles and report the biases which result from an application of the normal dependence model.affine term structure models, nonlinear dependence, copula functions, tail dependence, value-at-risk

Some asymptotic results on non-standard likelihood ratio tests, and Cox process modeling in finance

This dissertation consists of two parts. In the first part, the subject of hypothesis testing is addressed. Here, non-standard formulations of the null hypothesis are discussed, e.g., non-stationarity under the null, and boundary hypotheses. In the second part, stochastic models for financial markets are developed and studied. Particular emphasis is placed on the application of Cox processes. Part one begins with a survey of time-series models which allow for conditional heteroscedasticity and autoregression, AR-GARCH models. These models reduce to a white noise model, when some of the conditional heteroscedasticity parameters take their boundary value at zero, and the autoregressive component is in fact not present. The asymptotic distribution of the pseudo-log-likelihood ratio statistics for testing the presence of conditional heteroscedasticity and the autoregression term is reproduced. For financial market data, the model parameters are estimated and tests for the reduction to white noise are performed. The impact of these results on risk measurement is discussed by comparing several Value-at-Risk calculations assuming the alternative model specifications. Furthermore, the power function of these tests is examined by a simulation study of the ARCH(1) and the AR(1)-ARCH(1) models. First, the simulations are carried out assuming Gaussian innovations and then, the Gaussian distribution is replaced by the heavy tailed t-distribution. This reveals that a substantial loss of power is associated with the use of heavy tailed innovations. A related testing problem arises in the analysis of the Ornstein-Uhlenbeck (OU) model, driven by Levy processes. This model is designed to capture mean reverting behaviour if it exists; but the data may in fact be adequately described by a pure Levy process with no OU (autoregressive) effect. For an appropriate discretized version of the model, likelihood methods are utilized to test for such a reduction of the OU process to Levy 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 Levy process for the data. Despite this a very general analysis is conducted with no technical restrictions on the underlying processes or parameter sets, other than a finite variance assumption for the Levy process. As a special case, for Brownian Motion as driving process, the limiting distribution is deduced in a quite explicit way, finding results which generalise the well-known Dickey-Fuller ("unit-root") theory. Part two of this dissertation considers the application of Cox processes in mathematical finance. Here, a framework is discussed for the valuation of employee share options (ESO), and credit risk modeling. One popular approach for ESO valuation involves a modification of standard option pricing models, augmenting them by the possibility of departure of the executive at an exogenously given random time. Such models are called reduced form models, in contrast to structural models that require measures of the employee's utility function and other unobservable quantities. Here, an extension of the reduced form model for the valuation of ESOs is developed. This model incorporates and emphasises employee departure, company takeover, performance vesting and other exotic provisions specific to ESOs. The assumptions underlying the reduced form model are clearified, and discussed for their implications. Further, the probabilistic structure of the model is analysed which includes an explicit characterization of the set of equivalent martingale measures, as well as the computation of prominent martingale measures like, e.g., the variance optimal martingale measure and the minimal martingale measure. Particular ESO specifications are studied emphasizing different aspects of the proposed framework. In this context, also strict no-arbitrage bounds for ESO prices are provided 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. In a continuous time market model, credit risk modeling and pricing of credit derivatives is discussed. In the approach it is adopted that credit risk is described by the interest rate spread between a corporate bond and a government bond. This spread is modeled in terms of explaining variables. For this purpose, a specific market model consisting of four assets is considered where the default process of the company is incorporated in a risky money market by a Cox process. It is shown that this market model has a unique equivalent martingale measure and is complete. As a consequence, contingent claim valuation can be executed in the usual way. This is illustrated with the valuation of a convertible bond which fits naturally in the given setting

Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing

We unify and extend a number of approaches related to constructing multivariate Madan–Seneta Variance-Gamma models for option pricing. Complementing Grigelionis’ (2007) class, an overarching model is derived by subordinating multivariate Brownian motion to a subordinator from Thorin’s (1977) [58, 59] class of generalised Gamma convolutions. Multivariate classes developed by Pérez-Abreu and Stelzer (2014), Semeraro (2008) and Guillaume (2013) are submodels. The classes are shown to be invariant under Esscher transforms, and quite explicit expressions for canonical measures are obtained, which permit applications such as option pricing using PIDEs or tree based methodologies. We illustrate with best-of and worst-of European and American options on two assets.This research was partially supported by ARC grants DP1092502, DP0988483 and DP16010403

The Effects of Largest Claim and Excess of Loss Reinsurance on a Company's Ruin Time and Valuation

We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer’s point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company’s ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this

Own-company stockholding and work effort preferences of an unconstrained executive

We develop a framework for analyzing an executive's own-company stockholding and work effort preferences. The executive, characterized by risk aversion and work effectiveness parameters, invests his personal wealth without constraint in the financial market, including the stock of his own company whose value he can directly influence with work effort. The executive's utility-maximizing personal investment and work effort strategy is derived in closed form, and a utility indifference rationale is applied to determine his required compensation. Being unconstrained byperformance contracting, the executive's work effort strategy establishes a base case for theoretical or empirical assessment of the benefits or otherwise of constraining executives with performance contracting

Finite approximation schemes for Levy processes, and their application to optimal stopping problems

This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0, T], and having a finite number of states, for a pure jump Lévy process Lt. The sequences of discrete processes converge to the original process