Application of Operator Splitting Methods in Finance

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps

Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments

Environmental valuation, ecosystem services and aquatic species

The thesis consists of an introduction and four articles that can be read independently of each other. The common topic is environmental valuation and cost-benefit analysis. The applications relates to the growing concern of invasive species, and to waterpower externalities. In broad terms, all of the articles relates to water management. Article 1: "A Cost-Benefit analysis of introducing a non-native species: the case of signal crayfish in Sweden", assesses the economic impact of introducing the signal crayfish into a Swedish lake. Two scenarios are set up and compared. The first one assumes that there is no introduction of signal crayfish, so that the noble crayfish is preserved. In the second scenario, the signal crayfish is introduced, which immediately wipes out the entire stock of noble crayfish. The values of noble- and signal crayfish populations are measured as present values of their net future revenues. The values are than compared and net benefit of an introduction is calculated. The result indicate that net benefit of an introduction is positive if the intrinsic growth rate or the carrying capacity of the noble crayfish is below 40 % that of the signal crayfish. Article 2: "Assessing management options for weed control with demanders and non-demanders in a choice experiment", estimates the benefits of having a weed management program for a lake in Sweden, and then compares them with corresponding costs. The policy recommendation from a simple cost-benefit rule is to control the weed at some specific sites of the lake. This paper also suggest how to distinguish those that have a positive WTP for at least one of the attributes (demanders) from those that have zero WTP for all attributes (non-demanders). The advantage of the suggested approach is that it facilitates to more clearly distinguish between conditional and unconditional willingness to pay. The suggested approach could also overcome some of the problems in the literature with negative welfare measures. Article 3: "Assessing transfer errors in the benefit transfer method: An application of invasive weed management using choice experiment", tests the accuracy of transferring benefits of a weed management program from one lake to another using choice experiment. The transfer errors are assessed and the convergent validity hypothesis is tested. Estimating the accuracy of benefit transfer for weed management is policy relevant as there are a number of lakes in Sweden infested with the water weed. The convergent validity was rejected for three out of five welfare estimates with a ten per cent significance level. Article 4: "Willingness to pay for environmental improvements in hydropower regulated rivers", assesses the benefits of environmental improvements along hydropower regulated rivers using choice experiments. Remedial measures that improve the conditions for fish, benthic invertebrates and river-margin vegetation were found to have a significant welfare increasing impact. The results can be of value for the implementation of the Water Framework Directives in Sweden, which aims to reform the use of all surface water and ground water in the member states

Measuring welfare gains from better quality infrastructure

Projects and reforms targeting infrastructure services can affect consumer welfare through changes in the price, coverage, or quality of the services provided. The benefits of improved service quality-while significant-are often overlooked because they are difficult to quantify. This paper reviews methods of evaluating the welfare implications of changes in the quality of infrastructure services within the broader theoretical perspective of welfare measurement. The study outlines the theoretical assumptions and data requirements involved, illustrating each method with examples that highlight common methodological features and differences. The paper also presents the theoretical underpinnings and potential applications of a new approach to analyzing the effects of interruptions in the supply of infrastructure services on household welfare.Energy Production and Transportation,Town Water Supply and Sanitation,Markets and Market Access,Economic Theory&Research,Water and Industry

(WP 2011-01) It Takes Two: The Incidence and Effectiveness of co-CEOs

This study examines the phenomenon of co-CEOs within publicly traded firms. Although shared executive leadership is not widespread, it occurs within some very prominent firms. We find that co-CEOs generally complement each other in terms of educational background or executive responsibilities. Our results show that firms most likely to appoint co-CEOs have lower leverage, a more limited firm focus, less independent board structure, fewer advising directors, lower institutional ownership and greater levels of merger activity. The governance structure of co-CEO firms suggest that co-CEOships can serve as an alternative governance mechanism, with co-CEO mutual monitoring substituting for board or external monitoring and co-CEO complementary skills substituting for board advising. An event study indicates that the market reacts positively to appointments of co-CEOs while a propensity score analysis shows that the presence of co-CEOs increases firm valuation

Regime switching in stochastic models of commodity prices: An application to an optimal tree harvesting problem

This paper investigates a regime switching model of stochastic lumber prices in the context of an optimal tree harvesting problem. Using lumber derivatives prices, two lumber price models are calibrated: a regime switching model and a single regime model. In the regime switching model, the lumber price can be in one of two regimes in which different mean reverting price processes prevail. An optimal tree harvesting problem is specified in terms of a linear complementarity problem which is solved using a fully implicit finite difference, fully-coupled, numerical approach. The land value and critical harvesting prices are found to be significantly different depending on which price model is used. The regime switching model shows promise as a parsimonious model of timber prices that can be incorporated into forestry investment problems.optimal tree harvesting, regime switching, calibration, lumber derivatives prices, fully implicit finite difference approach

Valuation of options for hedging against exchange rate exposure

>Magister Scientiae - MScThe risk associated with currency exposure is one of the main sources of risk in terms of internationally diversi ed portfolios. Controlling the risk is important for improving the performance of international investments. One approach to hedging against exchange rate exposure is by employing financial derivatives, particularly, foreign currency options. Currency options provide insurance against unfavorable exchange rate fluctuation, but also make provision to lock in a pro t when the exchange rate fluctuation are favorable. However, these instruments cannot be traded or managed without the relevant valuation techniques. In this dissertation we discuss one of the approaches to cover the risk associated with currency exposure. In particular, we focus on the partial differential equation (PDE) valuation of currency options by employing various finite difference schemes. We commence by introducing the mathematical tools required for the valuation of financial derivatives. Thereafter we study the valuation of European options. This involves deriving the famous Black-Scholes PDE for pricing options on stocks that do not yield dividends. Using the Black-Scholes PDE we derive the Black-Scholes formula for pricing European options. This derivation involves transforming the Black-Scholes PDE into the heat equation and by solving the heat equation we obtain the Black-Scholes formula. After completing the pricing of European options we now move to the pricing of American options. The early exercise facility associated with American options, leads to a free boundary problem which makes the pricing process of American options a challenging task. As in the case of the European options, we first derive the Black-Scholes inequality for American options and then transform this inequality for application to the heat equation to value American options. In the absence of an explicit formula for pricing American options we use numerical methods. Thus, we discuss the finite difference methods quite extensively with a focus on the implicit and Crank-Nicholson finite difference methods

The Effect of Non-Smooth Payoffs on the Penalty Approximation of American Options

This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. While convergence of the penalised solution for sufficiently smooth obstacles is well established in the literature, sharp rates of convergence and particularly the effect of gradient discontinuities (i.e., the omni-present `kinks' in option payoffs) on this rate have not been fully analysed so far. This effect becomes important not least when using penalisation as a numerical technique. We use matched asymptotic expansions to characterise the boundary layers between exercise and hold regions, and to compute first order corrections for representative payoffs on a single asset following a diffusion or jump-diffusion model. Furthermore, we demonstrate how the viscosity theory framework in [Jakobsen, 2006] can be applied to this setting to derive upper and lower bounds on the value. In a small extension to [Bensoussan & Lions, 1982], we derive weak convergence rates also for option sensitivities for convex payoffs under jump-diffusion models. Finally, we outline applications of the results, including accuracy improvements by extrapolation.Comment: 34 Pages, 10 Figure