Pricing approximations and error estimates for local Lévy-type models with default

3siWe find approximate solutions of partial integro-differential equations, which arise in financial models when defaultable assets are described by general scalar Lévy-type stochastic processes. We derive rigorous error bounds for the approximate solutions. We also provide numerical examples illustrating the usefulness and versatility of our methods in a variety of financial settings.partially_openembargoed_20170402Lorig, Matthew; Pagliarani, Stefano; Pascucci, AndreaLorig, Matthew; Pagliarani, Stefano; Pascucci, Andre

Managing uncertainty:financial, actuarial and statistical modelling.

present value; Value; Actuarial;

The History of the Quantitative Methods in Finance Conference Series. 1992-2007

This report charts the history of the Quantitative Methods in Finance (QMF) conference from its beginning in 1993 to the 15th conference in 2007. It lists alphabetically the 1037 speakers who presented at all 15 conferences and the titles of their papers.

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

Essays in Quantitative Finance

This thesis contributes to the quantitative finance literature and consists of four research papers.Paper 1. This paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function that allows the model to simultaneously fit the implied volatility of commodity and interest rate options. Because liquid market prices are only available for options on commodity futures (not forwards), a convexity correction formula is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given cross-correlation structure between forward interest rates and commodity prices. When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), whereas in the case of futures, a (rapidly converging) iterative fitting procedure is presented. The cross-correlation fitting is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the “orthonormal Procrustes” problem. The calibration approach is demonstrated on market data for oil futures.Paper 2. This paper describes an efficient American Monte Carlo approach for pricing Bermudan swaptions in the LIBOR market model using the Stochastic Grid Bundling Method (SGBM) which is a regression-based Monte Carlo method in which the continuation value is projected onto a space in which the distribution is known. We demonstrate an algorithm to obtain accurate and tight lower–upper bound values without the need for the nested Monte Carlo simulations that are generally required for regression-based methods.Paper 3. The credit valuation adjustment (CVA) for over-the-counter derivatives are computed using the portfolio’s exposure over its lifetime. Usually, future exposure is approximated by Monte Carlo simulations. For derivatives that lack an analytical approximation for their mark-to-market (MtM) value, such as Bermudan swaptions, the standard practice is to use the regression functions from the least squares Monte Carlo method to approximate their simulated MtMs. However, such approximations have significant bias and noise, resulting in an inaccurate CVA charge. This paper extend the SGBM to efficiently compute expected exposure, potential future exposure, and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates how different measures, such as spot and terminal measures, can simultaneously be employed in the SGBM framework to significantly reduce the variance and bias.Paper 4. This paper presents an algorithm for simulation of options on Lévy driven assets. The simulation is performed on the inverse transition matrix of a discretised partial differential equation. We demonstrate how one can obtain accurate option prices and deltas on the variance gamma (VG) and CGMY model through finite element-based Monte Carlo simulations

Mod-Poisson approximation schemes: Applications to credit risk

We introduce a new numerical approximation method for functionals of factor credit portfolio models based on the theory of mod-$\phi$ convergence and mod-$\phi$ approximation schemes. The method can be understood as providing correction terms to the classic Poisson approximation, where higher order corrections lead to asymptotically better approximations as the number of obligors increases. We test the model empirically on two tasks: the estimation of risk measures ($\mathrm{VaR}$ and $\mathrm{ES}$) and the computation of CDO tranche prices. We compare it to other commonly used methods -- such as the recursive method, the large deviations approximation, the Chen--Stein method and the Monte Carlo simulation technique (with and without importance sampling) -- and we show that it leads to more accurate estimates while requiring less computational time.Comment: 42 pages, 7 figure