The Ornstein-Uhlenbeck-Type Model with a Hybrid Dividend Strategy

We consider the Ornstein-Uhlenbeck-type model. We first introduce the model and then find the ordinary differential equations and boundary conditions satisfied by the dividend functions; closed-form solutions for the dividend value functions are given. We also study the distribution of the time value of ruin. Furthermore, the moments and moment-generating functions of total discounted dividends until ruin are discussed

Efficient valuation of exotic derivatives with path-dependence and early exercise features

The main objective of this thesis is to provide effective means for the valuation of popular financial derivative contracts with path-dependence and/or early-exercisable provisions. Starting from the risk-neutral valuation formula, the approach we propose is to sequentially compute convolutions of the value function of the contract at a monitoring date with the transition density between two dates, to provide the value function at the previous monitoring date, until the present date. A rigorous computational algorithm for the convolutions is then developed based on transformations to the Fourier domain. In the first part of the thesis, we deal with arithmetic Asian options, which, due to the growing popularity they enjoy in the financial marketplace, have been researched signicantly over the last two decades. Although few remarkable approaches have been proposed so far, these are restricted to the market assumptions imposed by the standard Black-Scholes-Merton paradigm. Others, although in theory applicable to Lévy models, are shown to suffer a non-monotone convergence when implemented numerically. To solve the Asian option pricing problem, we initially propose a flexible framework for independently distributed log-returns on the underlying asset. This allows us to generalize firstly in calculating the price sensitivities. Secondly, we consider an extension to non-Lévy stochastic volatility models. We highlight the benefits of the new scheme and, where relevant, benchmark its performance against an analytical approximation, control variate Monte Carlo strategies and existing forward convolution algorithms for the recovery of the density of the underlying average price. In the second part of the thesis, we carry out an analysis on the rapidly growing market of convertible bonds (CBs). Despite the vast amount of research which has been undertaken yet. This is due to the need for proper modelling of the CBs composite payout structure and the multi factor modelling arising in the CB valuation. Given the dimensional capacity of the convolution algorithm, we are now able to introduce a new jump diffusion structural approach in the CB literature, towards more realistic modelling of the default risk, and further include correlated stochastic interest rates. This aims at fixing dimensionality and convergence limitations which previously have been restricting the range of applicability of popular grid- based, lattice and Monte Carlo methods. The convolution scheme further permits flexible handling of real-world CB specications; this allows us to properly model the call policy and investigate its impact on the computed CB prices. We illustrate the performance of the numerical scheme and highlight the effects originated by the inclusion of jumps

From Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options

This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options. The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one formula. In Chapter 3 we introduce new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal-dual simulation algorithm (Andersen and Broadie 2004), we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements

Stochastic optimal control problems under model uncertainty

The stochastic optimal decision-making problem concerns the process of dynamically deciding actions to optimize pre-specified criteria based on specific stochastic models. It is, however, common that a decision-maker is unable to obtain complete information to formulate fully reliable models and faces the issue of model uncertainty. Existing empirical studies have shown that ignoring model uncertainty leads to improper decisions and causes losses in the financial market. Thus, it is important to incorporate model uncertainty into decision-making. To our best knowledge, no existing works on dividend optimization have taken model uncertainty into consideration. This thesis is an early attempt to fill such a gap in the actuarial literature. This thesis studies three popular optimization problems in the framework of model uncertainty, which involve different models with multiple control variables and various assumptions. It consists of three projects. The first project investigates an optimal risk exposure-dividend control problem under a diffusion model with model uncertainty. Due to the concerns about model uncertainty, the ambiguity averse insurer aims at finding the robust strategies such that a penalized reward function is maximized in the worst-case scenario. The problem is formulated as a zero-sum stochastic differential game between the insurer and the market. Explicit expressions for the value functions are obtained and the optimal dividend strategies are identified as barrier strategies. The second project incorporates model uncertainty into a dividend optimization problem of a singular type under the classical risk model with general assumptions on the claim size distribution. Using the standard stochastic control techniques, we characterize the value function as the smallest viscosity supersolution to the existing Hamilton-Jacobi-Bellman equation and show that the optimal strategies are of band type. The third project extends the second project by incorporating fixed and proportional transaction costs on dividend payments. The problem is an impulse control problem and the optimal dividend strategies are shown to be n-level lump sum strategies. Numerical studies are provided for each project and the economic implications of model uncertainty on insurer’s decision-making are discussed. It is shown that the insurer who is more averse to ambiguity tends to be more conservative in the optimal strategies

From Spectral Methods to the Geometrical Approximation for PDEs in Finance

The Thesis investigates the benefits of Spectral Methods, which are found to be an appealing numerical technique when the solution in closed form doesn't exist, but unfortunately it cannot be used in every case. A remarkable case in which it is possible to use the Spectral Methods is for pricing the Double Barrier Options as we have seen in Chapter 3. The main achievement of this Thesis is the introduction of two methods, that we have called Geometrical Approximation and Perturbative Method respectively, by which is possible to evaluate the fair option prices in the Heston and SABR market model. Both proposed methods can be generalised to other market models and for pricing other derivatives contracts, although, in order to show the above methodologies, we have chosen to pricing Options of only two kinds: Vanilla Options and knock-out Barrier Options. On the first, we have that the G. A. method intends to be an alternative method, which can be particularly convenient for sensible values of the model parameters, which allows computation of closed-form expressions of approximated option prices. The option price is approximated since we can get closed-form solutions for the PDE at the cost of modifying the Cauchy's condition, rather than looking for a numerical solution to the PDE with the exact Cauchy's condition. The proposed method has the advantage to compute a solution in closed form, therefore, we do not have the problems which plague the numerical methods. For example, one can consider the inverse Fourier transform method, in which we have to compute an integral between zero and infinity. In this case in fact, there is always some problem in order to define (in practice) the correct domain of integration; or equivalently, considering also the finite difference method, in which we have to define a suitable grid, in other words we have some problems about the choice of the grid's meshes. In the present work we have used the Geometrical Approximation method in the Heston model and in the SABR model, comparing the Vanilla Option price obtained with these computed by inverse Fourier transform, Monte-Carlo simulation and Finite Difference method or again the Implied Volatility method. The Geometrical Approximation method is more reliable for low values of the correlation between price and variance shocks. In this case, our numerical experiments in a specific but sensible case show that the difference with the Fourier method is of the order of 1%. Markets in which the price/volatility correlation is low, and thus the G. A. method seems more promising, are the Electricity Markets. Besides it is possible, through the G. A. method, get the Vanilla Option price by a strategy, whose price at time zero is equal to the sum of the option price with modified payoff and a bond price, so that, this one is equal to the difference between |S_{0}e^{\varepsilon_{0}}-E| capitalised at rate r. This strategy gives us, for every correlation value, a price higher than the Heston price around some percent, but in this way the writer of the Options is fully hedged. This strategy can be very useful for Banks and Institutions that write derivatives contracts. On the second, the Perturbative Method, we have elaborated another approximating approach, illustrated in Chapter 5, in which we have discussed a particular choice of the volatility price of risk in the Heston model, namely such that the drift term of the risk-neutral stochastic volatility process is zero. This allowed us to illustrate an alternative methodology for solving the pricing PDE in an approximate way, in which we have neglected some terms of the PDE, recovering a pricing formula which in this particular case, turn out to be simple, for Vanilla Options and Barrier Options. The approximating formulas give an accurate price close to that obtained by Fourier transform for the Vanilla options, and Down-knock-out Call options. The Perturbative method can be used for pricing several derivatives contracts and we are sure that manifold applications will follow

Exact Monte Carlo sampling of jump diffusions

The main objective of this thesis is to explore the theoretical foundations of the exact method for sampling jump diffusions proposed in [20] by Kay Giesecke and Dmitry Smelov, and implement it in order to compare the performance of the algorithm for pricing purposes against more traditional finite element methods, which generate biased samples. The method applies to a large class of models defined by a one-dimensional jump diffusion process, allowing us to generate exact simulations of a skeleton, a hitting time and other functionals of it, used for purposes like path-dependent option or interest rate derivatives pricing.O principal objetivo desta tese é explorar os fundamentos teóricos relativos ao método proposto em [20] por Kay Giesecke e Dmitry Smelov e implementá-lo de modo a comparar a sua performance face a métodos mais tradicionais de elementos finitos, que geram amostras enviesadas. O método aplica-se a uma grande parte dos modelos definidos por um processo de difusão com saltos unidimensional, permitindo gerar simulações de Monte Carlo exatas de um esqueleto, tempos de paragem e outros funcionais do mesmo, com finalidades como a avaliação de path-dependent options, derivados de taxa de juro ou outros instrumentos financeiros

Local and Stochastic Volatility Models: An Investigation into the Pricing of Exotic Equity Options

Faculty of Science; School of Computational and Applied Maths; MSC ThesisThe assumption of constant volatility as an input parameter into the Black-Scholes option pricing formula is deemed primitive and highly erroneous when one considers the terminal distribution of the log-returns of the underlying process. To account for the `fat tails' of the distribution, we consider both local and stochastic volatility option pricing models. Each class of models, the former being a special case of the latter, gives rise to a parametrization of the skew, which may or may not re°ect the correct dynamics of the skew. We investigate a select few from each class and derive the results presented in the corresponding papers. We select one from each class, namely the implied trinomial tree (Derman, Kani & Chriss 1996) and the SABR model (Hagan, Kumar, Lesniewski & Woodward 2002), and calibrate to the implied skew for SAFEX futures. We also obtain prices for both vanilla and exotic equity index options and compare the two approaches

Credit Modelling: Generating Spread Dynamics with Intensities and Creating Dependence with Copulas

The thesis is an investigation into the pricing of credit risk under the intensity framework with a copula generating default dependence between obligors. The challenge of quantifying credit risk and the derivatives that are associated with the asset class has seen an explosion of mathematical research into the topic. As credit markets developed the modelling of credit risk on a portfolio level, under the intensity framework, was unsatisfactory in that either: 1. The state variables of the intensities were driven by diffusion processes and so could not generate the observed level of default correlation (see Schönbucher (2003a)) or, 2. When a jump component was added to the state variables, it solved the problem of low default correlation, but the model became intractable with a high number of parameters to calibrate to (see Chapovsky and Tevaras (2006)) or, 3. Use was made of the conditional independence framework (see Duffie and Garleanu (2001)). Here, conditional on a common factor, obligors’ intensities are independent. However the framework does not produce the observed level of default correlation, especially for portfolios with obligors that are dispersed in terms of credit quality. Practitioners seeking to have interpretable parameters, tractability and to reproduce observed default correlations shifted away from generating default dependence with intensities and applied copula technology to credit portfolio pricing. The one factor Gaussian copula and some natural extensions, all falling under the factor framework, became standard approaches. The factor framework is an efficient means of generating dependence between obligors. The problem with the factor framework is that it does not give a representation to the dynamics of credit risk, which arise because credit spreads evolve with time. A comprehensive framework which seeks to address these issues is developed in the thesis. The framework has four stages: 1. Choose an intensity model and calibrate the initial term structure. 2. Calibrate the variance parameter of the chosen state variable of the intensity model. 3. When extended to a portfolio of obligors choose a copula and calibrate to standard market portfolio products. 4. Combine the two modelling frameworks, copula and intensity, to produce a dynamic model that generates dependence amongst obligors. The thesis contributes to the literature in the following way: • It finds explicit analytical formula for the pricing of credit default swaptions with an intensity process that is driven by the extended Vasicek model. From this an efficient calibration routine is developed. Many works (Jamshidian (2002), Morini and Brigo (2007) and Schönbucher (2003b)) have focused on modelling credit swap spreads directly with modified versions of the Black and Scholes option formula. The drawback of using a modified Black and Scholes approach is that pricing of more exotic structures whose value depend on the term structure of credit spreads is not feasible. In addition, directly modelling credit spreads, which is required under these approaches, offers no explicit way of simulating default times. In contrast, with intensity models, there is a direct mechanism to simulate default times and a representation of the term structure of credit spreads is given. Brigo and Alfonsi (2005) and Bielecki et al. (2008) also consider intensity modelling for the purposes of pricing credit default swaptions. In their works the dynamics of the intensity process is driven by the Cox Ingersoll and Ross (CIR) model. Both works are constrained because the parameters of the CIR model they consider are constant. This means that when there is more than one tradeable credit default swaption exact calibration of the model is usually not possible. This restriction is not in place in our methodology. • The thesis develops a new method, called the loss algorithm, in order to construct the loss distribution of a portfolio of obligors. The current standard approach developed by Turc et al. (2004) requires differentiation of an interpolated curve (see Hagan and West (2006) for the difficulties of such an approach) and assumes the existence of a base correlation curve. The loss algorithm does not require the existence of a base correlation curve or differentiation of an interpolated curve to imply the portfolio loss distribution. • Schubert and Schönbucher (2001) show theoretically how to combine copula models and stochastic intensity models. In the thesis the Schubert and Schönbucher (2001)framework is implemented by combining the extended Vasicek model and the Gaussian copula model. An analysis of the impact of the parameters of the combined models and how they interact is given. This is as follows: – The analysis is performed by considering two products, securitised loans with embedded triggers and leverage credit linked notes with recourse. The two products both have dependence on two obligors, a counterparty and a reference obligor. – Default correlation is shown to impact significantly on pricing. – We establish that having large volatilities in the spread dynamics of the reference obligor or counterparty creates a de-correlating impact: the higher the volatility the lower the impact of default correlation. – The analysis is new because, classically, spread dynamics are not considered when modelling dependence between obligors. • The thesis introduces a notion called the stochastic liquidity threshold which illustrates a new way to induce intensity dynamics into the factor framework. • Finally the thesis shows that the valuation results for single obligor credit default swaptions can be extended to portfolio index swaptions after assuming losses on the portfolio occur on a discretised set and independently to the index spread level