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

Electricity Forward Curves with Thin Granularity: Theory and Empirical Evidence in the Hourly EPEX Spot Market

We propose a constructive definition of electricity forward price curve with cross-sectional timescales featuring hourly frequency on. The curve is jointly consistent with both risk-neutral market information represented by baseload and peakload futures quotes, and historical market information, as mirrored by periodical patterns exhibited by the time series of day-ahead prices. From a methodological standpoint, we combine nonparametric filtering with monotone convex interpolation such that the resulting forward curve is pathwise smooth and monotonic, cross-sectionally stable, and time local. From an empirical standpoint, we exhibit these features in the context of EPEX Spot and EEX Derivative markets. We perform a backtesting analysis to assess the relative quality of our forward curve estimate compared to the benchmark market model of Benth, Koekebakker, and Ollmar (2007)

Moment-matching approximations for stochastic sums in non-Gaussian Ornstein-Uhlenbeck models

In this paper, we recall actuarial and financial applications of sums of dependent random variables that follow a non-Gaussian mean-reverting process and contemplate distribution approximations. Our work complements previous related studies restricted to lognormal random variables; we revisit previous approximations and suggest new ones. We then derive moment-based distribution approximations for random sums attuned to Asian option pricing and computation of risk measures of random annuities. Various numerical experiments highlight the speed–accuracy benefits of the proposed methods

Technical Note. On Matrix Exponential Differentiation with Application to Weighted Sum Distributions

In this note, we revisit the innovative transform approach introduced by Cai, Song, and Kou [(2015) A general framework for pricing Asian options under Markov processes. Oper. Res. 63(3):540–554] for accurately approximating the probability distribution of a weighted stochastic sum or time integral under general one-dimensional Markov processes. Since then, Song, Cai, and Kou [(2018) Computable error bounds of Laplace inversion for pricing Asian options. INFORMS J. Comput. 30(4):625–786] and Cui, Lee, and Liu [(2018) Single-transform formulas for pricing Asian options in a general approximation framework under Markov processes. Eur. J. Oper. Res. 266(3):1134–1139] have achieved an efficient reduction of the original double to a single-transform approach. We move one step further by approaching the problem from a new angle and, by dealing with the main obstacle relating to the differentiation of the exponential of a matrix, we bypass the transform inversion. We highlight the benefit from the new result by means of some numerical examples

Estimation of Multivariate Asset Models with Jumps

We propose a consistent and computationally efficient 2-step methodology for the estimation of multidimensional non-Gaussian asset models built using L´evy processes. The proposed framework allows for dependence between assets and different tail-behaviors and jump structures for each asset. Our procedure can be applied to portfolios with a large number of assets as it is immune to estimation dimensionality problems. Simulations show good finite sample properties and significant efficiency gains. This method is especially relevant for risk management purposes such as, for example, the computation of portfolio Value at Risk and intra-horizon Value at Risk, as we show in detail in an empirical illustration

Pricing exotic derivatives exploiting structure

In this paper we introduce a new fast and accurate numerical method for pricing exotic derivatives when discrete monitoring occurs, and the underlying evolves according to a Markov one-dimensional stochastic processes. The approach exploits the structure of the matrix arising from the numerical quadrature of the pricing backward formulas to devise a convenient factorization that helps greatly in the speed-up of the recursion. The algorithm is general and is examined in detail with reference to the CEV (Constant Elasticity of Variance) process for pricing different exotic derivatives, such as Asian, barrier, Bermudan, lookback and step options for which up to date no efficient procedures are available. Extensive numerical experiments confirm the theoretical results. The MATLAB code used to perform the computation is available online at http://www1.mate.polimi.it/∼marazzina/BP.htm. © 2013 Elsevier B.V. All rights reserved

A general closed-form spread option pricing formula

We propose a new accurate method for pricing European spread options by extending the lower bound approximation of Bjerksund and Stensland (2011) beyond the classical Black–Scholes framework. This is possible via a procedure requiring a univariate Fourier inversion. In addition, we are also able to obtain a new tight upper bound. Our method provides also an exact closed form solution via Fourier inversion of the exchange option price, generalizing the Margrabe (1978) formula. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. We test the performance of these new pricing algorithms performing numerical experiments on different stochastic dynamic models

General optimized lower and upper bounds for discrete and continuous arithmetic Asian options

We propose an accurate method for pricing arithmetic Asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique. In addition, we derive an upper bound to the error from the lower bound price approximation. Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. Our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases

Quantitative assessment of common practice procedures in the fair evaluation of embedded options in insurance contracts

This work analyses the common industry practice used to evaluate financial options written on with-profit policies issued by European insurance companies. In the last years regulators introduced, with the Solvency II directive, a market consistent valuation framework for determining the fair value of asset and liabilities of insurance funds. A relevant aspect is how to deal with the estimation of sovereign credit and liquidity risk, that are important components in the valuation of the majority of insurance funds, which are usually heavily invested in treasury bonds. The common practice is the adoption of the certainty equivalent approach (CEQ) for the risk neutral evaluation of insurance liabilities, which results in a deterministic risk adjustment of the securities cash flows. In this paper, we propose an arbitrage free stochastic model for interest rate, credit and liquidity risks, that takes into account the dependences between different government bond issuers. We test the impact of the common practice against our proposed model, via Monte Carlo simulations. We conclude that in the estimation of options whose pay-off is determined by statutory accounting rules, which is often the case for European traditional with-profit insurance products, the deterministic adjustment for risk of the securities cash flows is not appropriate, and that a more complete model such as the one described in this article is a viable and sensible alternative in the context of market consistent evaluations