Asset Pricing Under The Quadratic Class

We identify and characterize a class of term structure models where bond yields are quadratic functions of the state vector. We label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application. We consider asset pricing in general and derivative pricing in particular under the quadratic class. We provide two general transform methods in pricing a wide variety of fixed income derivatives in closed or semi­closed form. We further illustrate how the quadratic model and the transform methods can be applied to more general settings.quadratic class; interest rates; term structure models; state price density; Markov process.

Topics in computational finance: : The Barndorff-Nielsen & Shepard Stochastic volatility model

Hur uppför sig aktiepriser? den frågan ställde sig forskarna Ole Barndorff-Nielsen och Neil Shephard i början av detta årtusende. Resultatet av deras funderingar blev den matematiska modell för aktiepriser som nu bär deras namn. Aktiepriser antas av tradition kunna beskrivas av en normalfördelning, vilket grundliga studier emellertid har visat är en bristfällig antagelse. Risken för stora prisförändringar blir klart undervärderad, dvs man kan inte förklara de svarta dagarna på marknaden då börsen rasar. Dessutom visar sig förändringarna ofta inte vara jämnt fördelade. Vi ser små stegvisa ökningar och stora ras. Stora förändringar leder också ofta till stor aktivitet och aktiviteten på marknaden tenderar att vara stor i perioder för att sedan lugna ner sig. Alla dessa saker kan inte förklaras med den klassiska teorin för aktiepriser men får med hjälp av Barndorff-Nielsen och Shephards model en teoretisk förklaring. Denna avhandling studerar vilka konsekvenser det får för optionshandeln om vi ersätter den klassiska teorin med en modell som bättre beskriver verkligheten. En bättre beskrivelse innebär färre förenklingar vilket medför en mer komplicerad modell. Den ökade komplexiteten medför att vi enbart kan lösa problemen genom att använda avancerade datorsimuleringar. Tyngdpunkten i arbetet ligger på att utarbeta och använda metoder för att beräkna priser på optioner och andra finansiella kontrakt om vi antar att aktiepriser förklaras av BNS-modellen. Resultaten visar att vi får priser som ligger närmare de verkliga marknadspriserna och vi försöker utifrån resultaten dra slutsatser om investerarnas preferenser. Arbetet är utfört vid Centre of Mathematics for Applications under handledning av Fred Espen Benth, Professor i Finansmatematikk

Valuation of default sensitive claims under imperfect information.

We propose an evaluation method for financial assets subject to default risk, when investors face imperfect information about the state variable triggering the default. The model we propose generalizes the one by Duffie and Lando (2001) in the following way:(i)it incorporates informational noise in continuous time, (ii) it respects the (H) hypothesis, (iii) it precludes arbitrage from insiders. The model is sufficiently general to encompass a large class of structural models. In this setting we show that the default time is totally inaccessible in the market’s filtration and derive the martingale hazard process. Finally, we provide pricing formulas for default-sensitive claims and illustrate with particular examples the shapes of the credit spreads and the conditional default probabilities. An important feature of the conditional default probabilities is they are non Markovian. This might shed some light on observed phenomena such as the ”rating momentum”.hybrid models; default sensitive claims;

Keynesian Resurgence: Financial Stimulus And Contingent Claims Modelling

Since the commencement of the Global Financial Crisis, a worldwide resurgence in applying Keynesian modelling has occurred, and has been cited as a major factor in averting a worldwide economic depression. A key aspect of Keynesian modelling is that governments gain contingent claims on firms in exchange for financial stimulus. However, there exist few mathematical finance models examining Keynesian modelling, stimulus modelling and the valuation of such government contingent claims. In this paper we provide a new mathematical finance framework for modelling firms and financial stimulus under a Keynesian framework; we apply a stochastic differential equation model, rather than the standard time series models. Our model incorporates fundamental concepts of Keynesian modelling and Keynesian stimulus, which is a new characteristic to current financial models. We model the government's contingent claim on the firm as a real call option, and derive a closed form solution for the value of this option which takes into account firm stimulus. We also derive a solution for the minimum firm value required to exercise the option. We conduct numerical experiments for different firm equilibrium values, firm values, economic cycles and analyse the impact on option and stimulus values

Pricing swing options and other electricity derivatives

The deregulation of regional electricity markets has led to more competitive prices but also higher uncertainty in the future electricity price development. Most markets exhibit high volatilities and occasional distinctive price spikes, which results in demand for derivative products which protect the holder against high prices. A good understanding of the stochastic price dynamics is required for the purposes of risk management and pricing derivatives. In this thesis we examine a simple spot price model which is the exponential of the sum of an Ornstein-Uhlenbeck and an independent pure jump process. We derive the moment generating function as well as various approximations to the probability density function of the logarithm of this spot price process at maturity T. With some restrictions on the set of possible martingale measures we show that the risk neutral dynamics remains within the class of considered models and hence we are able to calibrate the model to the observed forward curve and present semi-analytic formulas for premia of path-independent options as well as approximations to call and put options on forward contracts with and without a delivery period. In order to price path-dependent options with multiple exercise rights like swing contracts a grid method is utilised which in turn uses approximations to the conditional density of the spot process. Further contributions of this thesis include a short discussion of interpolation methods to generate a continuous forward curve based on the forward contracts with delivery periods observed in the market, and an investigation into optimal martingale measures in incomplete markets. In particular we present known results of q-optimal martingale measures in the setting of a stochastic volatility model and give a first indication of how to determine the q-optimal measure for q=0 in an exponential Ornstein-Uhlenbeck model consistent with a given forward curve

The Heston stochastic volatility model in Hilbert space

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein-Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein-Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modelling of forward curves in energy markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics

Asset Pricing under the Quadratic Class

We identify and characterize a class of term structure models where bond yields are quadratic functions of the state vector. We label this class the quadratic class and aim to lay a solid theoretical foundation for its future empirical application. We consider asset pricing in general and derivative pricing in particular under the quadratic class. We provide two general transform methods in pricing a wide variety of fixed income derivatives in closed or semi-closed form. We further illustrate how the quadratic model and the transform methods can be applied to more general setting

Which Method for Pricing Weather Derivatives ?

Since the introduction of the first weather derivative in the United-States in 1997, a significant number of work was directed towards the pricing of this product and the modelling of the daily average temperature which characterizes most of the traded weather instruments. The weather derivatives were created to enable companies to hedge against climate risks. They respond more to a need to cover seasonal variations which may cause loss of profits for companies than to a coverage need in property damage. Despite the abundance of work on the topic, no consensus has emerged so far about the methodology for evaluating weather derivatives. The major problems of these instruments are on one hand, they are based on an meteorological index that is not traded on financial market which does not allow the use of traditional pricing methods and on the other hand, it is difficult to get round this obstacle by susbtituting the underlying for a linked exchanged security since the weather index is weakly correlated with prices of other financial assets. To further the question of evaluation, we propose in this paper to, firstly, shed light on the difficulties of implementing the three major pricing approaches suggested in the literature for the weather derivatives (actuarial, arbitrage-free and consumption-based methods) and, secondly, to compute the prices of a weather contract by the three methodologies for comparison.weather derivatives; arbitrage-free pricing method; actuarial pricing approach; consumption-based pricing model; risk-neutral distribution; market price of risk; finite difference method; Monte-Carlo simulations.

Lie Analysis for Partial Differential Equations in Finance

Weather derivatives are financial tools used to manage the risks related to changes in the weather and are priced considering weather variables such as rainfall, temperature, humidity and wind as the underlying asset. Some recent researches suggest to model the amount of rainfall by considering the mean reverting processes. As an example, the Ornstein Uhlenbeck process was proposed by Allen [3] to model yearly rainfall and by Unami et al. [52] to model the irregularity of rainfall intensity as well as duration of dry spells. By using the Feynman-Kac theorem and the rainfall indexes we derive the partial differential equations (PDEs) that governs the price of an European option. We apply the Lie analysis theory to solve the PDEs, we provide the group classification and use it to find the invariant analytical solutions, particularly the ones compatible with the terminal conditions