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

Valuation of Employee Stock Options (ESOs) by means of Mean-Variance Hedging

We consider the problem of ESO valuation in continuous time. In particular, we consider models that assume that an appropriate random time serves as a proxy for anything that causes the ESO's holder to exercise the option early, namely, reflects the ESO holder's job termination risk as well as early exercise behaviour. In this context, we study the problem of ESO valuation by means of mean-variance hedging. Our analysis is based on dynamic programming and uses PDE techniques. We also express the ESO's value that we derive as the expected discounted payoff that the ESO yields with respect to an equivalent martingale measure, which does not coincide with the minimal martingale measure or the variance-optimal measure. Furthermore, we present a numerical study that illustrates aspects or our theoretical results

Pricing executive stock options under employment shocks

We obtain explicit expressions for the subjective, objective and market value of perpetual executive stock options (ESOs) under exogenous employment shocks driven by an independent Poisson process. Within this setup,we obtain the executive's optimal exercise policy which allows us to analyze the determinants of both, the subjective valuation by executives and the objective valuation by firms. The perpetual ESO is compared with the more realistic finite maturity ESO finding that the approximation is reasonably good. We also use the objective valuation's results for accounting purposes. Further,we analyze the objective valuation distribution when there is uncertainty about the employment shock parameter. Finally, the role of ESOs in the design of executive's incentives is also discussed.ESO, Risk Aversion, Undiversification, Incentives, FAS 123R.

Executive stock option exercise with full and partial information on a drift change point

We analyse the optimal exercise of an executive stock option (ESO) written on a stock whose drift parameter falls to a lower value at a change point, an exponentially distributed random time independent of the Brownian motion driving the stock. Two agents, who do not trade the stock, have differing information on the change point, and seek to optimally exercise the option by maximising its discounted payoff under the physical measure. The first agent has full information, and observes the change point. The second agent has partial information and filters the change point from price observations. This scenario is designed to mimic the positions of two employees of varying seniority, a fully informed executive and a partially informed less senior employee, each of whom receives an ESO. The partial information scenario yields a model under the observation filtration $\widehat{\mathbb{F}}$ in which the stock drift becomes a diffusion driven by the innovations process, an $\widehat{\mathbb{F}}$-Brownian motion also driving the stock under $\widehat{\mathbb{F}}$, and the partial information optimal stopping value function has two spatial dimensions. We rigorously characterise the free boundary PDEs for both agents, establish shape and regularity properties of the associated optimal exercise boundaries, and prove the smooth pasting property in both information scenarios, exploiting some stochastic flow ideas to do so in the partial information case. We develop finite difference algorithms to numerically solve both agents' exercise and valuation problems and illustrate that the additional information of the fully informed agent can result in exercise patterns which exploit the information on the change point, lending credence to empirical studies which suggest that privileged information of bad news is a factor leading to early exercise of ESOs prior to poor stock price performance.Comment: 48 pages, final version, accepted for publication in SIAM Journal on Financial Mathematic

Valuation of American Options and Employee Stock Options

Options play an important role in the financial world and are actively traded with huge trading volume. Most of the options traded on exchanges are American options. Spanning over a few decades, the American option pricing problem continues to intrigue scholars and practitioners in finance. The employee stock options (ESOs), a variant of American options, has been increasingly popular for firms to compensate, motivate and retain employees. ESOs importantly do not trade in markets nevertheless fair value must be determined – often by accountants. Unique features of ESOs however complicate the valuation. Our research, consisting of three papers, focuses on the improved lattice techniques for valuing American options and ESOs. Research paper 1 (Chapter 2) introduces an intelligent lattice search algorithm to efficiently locate the optimal exercise boundary for American options. The computational runtime can be reduced from over 18 minutes down to less than 3 seconds to estimate a 15,000-step CRR binomial tree. Research paper 2 (Chapter 3) introduces a set of lattice techniques to the Leisen-Reimer and Tian binomial models for American options pricing. A level of accuracy and efficiency combined can be achieve that surpass analytical solution models prominent in the literature. Moreover, lattices importantly afford an explicit trade-off locus between accuracy and speed that can be navigated according to predetermined precision tolerance levels and option types. These should have practical relevance to trading platforms that require real-time estimates of implied volatility. Research paper 3 (Chapter 4) proposes adjustments to the Hull-White ESO pricing model, based on insights developed by Boyle-Lau and Tian specifications. The proposed Hull-White-Boyle-Lau and Hull White-Tian revamps expand the practicable menu choice available to stakeholders tasked with the valuation of these ESOs. Accountants, across many jurisdictions, are subjected to higher demands for disclosure and fair valuation. The streamlined valuation approaches developed here may prove ii useful in expanding the tool kit of practicable/workable models. This improved efficiency can be harnessed even at the level of a basic spreadsheet and this this should assist in testing, validating and benchmarking valuation in lattices and in evaluating the newer generation of closed-form solutions emerging in the literature

The valuation of employee stock options : how good is the standard?

This study contributes to the valuation of employee stock options (ESO) in two ways: First, a new pricing model is presented, admitting a major part of calculations to be solved in closed form. Designed with a focus on good replication of empirics, the model fits with publicly observable exercise characteristics better than earlier models. In particular, it is able to account for the correlation of the time of exercise and the stock price at exercise, suspected of being crucial for the option value. The impact of correlation is weak, however, whereas cancellations play a central role. The second contribution of this paper is an examination to what extent the ESO pricing method of SFAS 123 is subject to discretion of the accountant. Given my model were true, the SFAS price would be a good proxy. Yet, outside shareholders usually cannot observe one of the SFAS input parameters. On behalf of an example I show that there is wide latitude left to the accountant

Evaluating Incentive Options

We provide an analytical and flexible framework to evaluate incentive options. Our model not only considers vesting periods and trading and hedging restrictions on the holders, but also specifically includes provisions of reloading and resetting to capture the fact that firms tend to grant more options after existing options are either exercised or become deep out of the money. By treating the incentive option as a flow of barrier options, we are able to obtain a near-explicit formula for the option value. Our model allows us to discuss many issues related to incentive options such as their issuing cost, exercising strategies, and induced incentives. Especially, we highlight some significant interactions among different features of incentive optionsExecutive Stock Options, Incentives, Resetting and Reloading, Subjective Valuation

Auctions and Efficiency

efficiency, allocation of resources, privatization