Computing Tails of Compound Distributions Using Direct Numerical Integration

An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation errors significantly with little extra effort. High precision results of the 0.999 quantile and CVaR were obtained for compound losses with heavy tails and a very wide range of loss frequencies using the DNI, Fast Fourier Transform (FFT) and Monte Carlo (MC) methods. These results, particularly relevant to operational risk modelling, can serve as benchmarks for comparing different numerical methods. We found that the adaptive DNI can achieve high accuracy with relatively coarse grids. It is much faster than MC and competitive with FFT in computing high quantiles and CVaR of compound distributions in the case of moderate to high frequencies and heavy tails

Calculation of aggregate loss distributions

Estimation of the operational risk capital under the Loss Distribution Approach requires evaluation of aggregate (compound) loss distributions which is one of the classic problems in risk theory. Closed-form solutions are not available for the distributions typically used in operational risk. However with modern computer processing power, these distributions can be calculated virtually exactly using numerical methods. This paper reviews numerical algorithms that can be successfully used to calculate the aggregate loss distributions. In particular Monte Carlo, Panjer recursion and Fourier transformation methods are presented and compared. Also, several closed-form approximations based on moment matching and asymptotic result for heavy-tailed distributions are reviewed

A general methodology to price and hedge derivatives in incomplete markets

We introduce and discuss a general criterion for the derivative pricing in the general situation of incomplete markets, we refer to it as the No Almost Sure Arbitrage Principle. This approach is based on the theory of optimal strategy in repeated multiplicative games originally introduced by Kelly. As particular cases we obtain the Cox-Ross-Rubinstein and Black-Scholes in the complete markets case and the Schweizer and Bouchaud-Sornette as a quadratic approximation of our prescription. Technical and numerical aspects for the practical option pricing, as large deviation theory approximation and Monte Carlo computation are discussed in detail.Comment: 24 pages, LaTeX, epsfig.sty, 5 eps figures, changes in the presentation of the method, submitted to International J. of Theoretical and Applied Financ

Non-Gaussian gravitational clustering field statistics

In this work we investigate the multivariate statistical description of the matter distribution in the nonlinear regime. We introduce the multivariate Edgeworth expansion of the lognormal distribution to model the cosmological matter field. Such a technique could be useful to generate and reconstruct three-dimensional nonlinear cosmological density fields with the information of higher order correlation functions. We explicitly calculate the expansion up to third order in perturbation theory making use of the multivariate Hermite polynomials up to sixth order. The probability distribution function for the matter field includes at this level the two-point, the three-point and the four-point correlation functions. We use the hierarchical model to formulate the higher order correlation functions based on combinations of the two-point correlation function. This permits us to find compact expressions for the skewness and kurtosis terms of the expanded lognormal field which can be efficiently computed. The method is, however, flexible to incorporate arbitrary higher order correlation functions which have analytical expressions. The applications of such a technique can be especially useful to perform weak-lensing or neutral hydrogen 21 cm line tomography, as well as to directly use the galaxy distribution or the Lyman-alpha forest to study structure formation.Comment: 20 pages, 2 figures; accepted in MNRAS 2011 August 22, in original form 2010 December 14 published, Publication Date: 03/201

Formation and Evolution of Primordial Black Hole Binaries in the Early Universe

The abundance of primordial black holes (PBHs) in the mass range $0.1 - 10^3 M_\odot$ can potentially be tested by gravitational wave observations due to the large merger rate of PBH binaries formed in the early universe. To put the estimates of the latter on a firmer footing, we first derive analytical PBH merger rate for general PBH mass functions while imposing a minimal initial comoving distance between the binary and the PBH nearest to it, in order to pick only initial configurations where the binary would not get disrupted. We then study the formation and evolution of PBH binaries before recombination by performing N-body simulations. We find that the analytical estimate based on the tidally perturbed 2-body system strongly overestimates the present merger rate when PBHs comprise all dark matter, as most initial binaries are disrupted by the surrounding PBHs. This is mostly due to the formation of compact N-body systems at matter-radiation equality. However, if PBHs make up a small fraction of the dark matter, $f_{\rm PBH} \lesssim 10\%$, these estimates become more reliable. In that case, the merger rate observed by LIGO imposes the strongest constraint on the PBH abundance in the mass range $2 - 160 M_\odot$. Finally, we argue that, even if most initial PBH binaries are perturbed, the present BH-BH merger rate of binaries formed in the early universe is larger than $\mathcal{O}(10)\,{\rm Gpc}^{-3} {\rm yr}^{-1}\, f_{\rm PBH}^3$Comment: 32pages, 12 figures, typos corrected, references added, figures updated, matches version published in JCA

General closed-form basket option pricing bounds

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms; hence, they do not suffer from the curse of dimensionality and can be applied also to high-dimensional problems where most existing methods fail. In particular, we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate

Modelling FX smile : from stochastic volatility to skewness

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Applications of Laplace transform for evaluating occupation time options and other derivatives

The present thesis provides an analysis of possible applications of the Laplace Transform (LT) technique to several pricing problems. In Finance this technique has received very little attention and for this reason, in the first chapter we illustrate with several examples why the use of the LT can considerably simplify the pricing problem. Observed that the analytical inversion is very often difficult or requires the computation of very complicated expressions, we illustrate also how the numerical inversion is remarkably easy to understand and perform and can be done with high accuracy and at very low computational cost. In the second and third chapters we investigate the problem of pricing corridor derivatives, i.e. exotic contracts for which the payoff at maturity depends on the time of permanence of an index inside a band (corridor) or below a given level (hurdle). The index is usually an exchange or interest rate. This kind of bond has evidenced a good popularity in recent years as alternative instruments to common bonds for short term investment and as opportunity for investors believing in stable markets (corridor bonds) or in non appreciating markets (hurdle bonds). In the second chapter, assuming a Geometric Brownian dynamics for the underlying asset and solving the relevant Feynman-Kac equation, we obtain an expression for the Laplace transform of the characteristic function of the occupation time. We then show how to use a multidimensional numerical inversion for obtaining the density function. In the third chapter, we investigate the effect of discrete monitoring on the price of corridor derivatives and, as already observed in the literature for barrier options and for lookback options, we observe substantial differences between discrete and continuous monitoring. The pricing problem with discrete monitoring is based on an appropriate numerical scheme of the system of PDE's. In the fourth chapter we propose a new approximation for pricing Asian options based on the logarithmic moments of the price average