Fast binomial procedures for pricing Parisian/ParAsian options

The discrete procedures for pricing Parisian/ParAsian options depend, in general, by three dimensions: time, space, time spent over the barrier. Here we present some combinatorial and lattice procedures which reduce the computational complexity to second order. In the European case the reduction was already given by Lyuu-Wu \cite{WU} and Li-Zhao \cite{LZ}, in this paper we present a more efficient procedure in the Parisian case and a different approach (again of order 2) in the ParAsian case. In the American case we present new procedures which decrease the complexity of the pricing problem for the Parisian/ParAsian knock-in options. The reduction of complexity for Parisian/ParAsian knock-out options is still an open problem.Les méthodes à temps discret pour le pricing des options parisienne et parAsian dépendent généralement de trois paramètres : Le temps, l'espace et le temps écoulé proche de la barrière. Dans ce travail, nous présentons des procédures combinatoires et de treillis qui permettent de réduire d'ordre 2 la complexité du calcul. Ces simplifications ont déjà été utilisées par Lyuu-Wu et Li-Zhao dans le cas des options européennes. Dans cet article, une technique plus efficace est employée pour les options parisienne et parAsian. Nous introduisons aussi de nouvelles méthodes rapides pour les options américaines applicables aux parisiennes et parAsians knock-in. La généralisation de ce type de procédures aux options parisienne/parAsian knock-out reste un problème ouvert

Double Auctions in Markets for Multiple Kinds of Goods

Motivated by applications such as stock exchanges and spectrum auctions, there is a growing interest in mechanisms for arranging trade in two-sided markets. Existing mechanisms are either not truthful, or do not guarantee an asymptotically-optimal gain-from-trade, or rely on a prior on the traders' valuations, or operate in limited settings such as a single kind of good. We extend the random market-halving technique used in earlier works to markets with multiple kinds of goods, where traders have gross-substitute valuations. We present MIDA: a Multi Item-kind Double-Auction mechanism. It is prior-free, truthful, strongly-budget-balanced, and guarantees near-optimal gain from trade when market sizes of all goods grow to $\infty$ at a similar rate.Comment: Full version of IJCAI-18 paper, with 2 figures. Previous names: "MIDA: A Multi Item-type Double-Auction Mechanism", "A Random-Sampling Double-Auction Mechanism". 10 page

The wiener-hopf technique and discretely monitored path-dependent option pricing

Fusai, Abrahams, and Sgarra (2006) employed the Wiener-Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black-Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path-dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first-touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener-Hopf technique yields a novel formula for double-barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Padé approximation. A Gaussian Black-Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Lévy processes generally

A Domain Specific Approach to High Performance Heterogeneous Computing

Users of heterogeneous computing systems face two problems: firstly, in understanding the trade-off relationships between the observable characteristics of their applications, such as latency and quality of the result, and secondly, how to exploit knowledge of these characteristics to allocate work to distributed computing platforms efficiently. A domain specific approach addresses both of these problems. By considering a subset of operations or functions, models of the observable characteristics or domain metrics may be formulated in advance, and populated at run-time for task instances. These metric models can then be used to express the allocation of work as a constrained integer program, which can be solved using heuristics, machine learning or Mixed Integer Linear Programming (MILP) frameworks. These claims are illustrated using the example domain of derivatives pricing in computational finance, with the domain metrics of workload latency or makespan and pricing accuracy. For a large, varied workload of 128 Black-Scholes and Heston model-based option pricing tasks, running upon a diverse array of 16 Multicore CPUs, GPUs and FPGAs platforms, predictions made by models of both the makespan and accuracy are generally within 10% of the run-time performance. When these models are used as inputs to machine learning and MILP-based workload allocation approaches, a latency improvement of up to 24 and 270 times over the heuristic approach is seen.Comment: 14 pages, preprint draft, minor revisio

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Fluctuation identities with continuous monitoring and their application to the pricing of barrier options

We present a numerical scheme to calculate fluctuation identities for exponential Lévy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-Laplace domain and require numerical inverse transforms. Thus we cover a gap in the literature that has mainly studied the discrete monitoring case; indeed, there are no existing numerical methods that deal with the continuous case. As a motivating application we price continuously monitored barrier options with the underlying asset modelled by an exponential Lévy process. We perform a detailed error analysis of the method and develop error bounds to show how the performance is limited by the truncation error of the sinc-based fast Hilbert transform used for the Wiener–Hopf factorisation. By comparing the results for our new technique with those for the discretely monitored case (which is in the Fourier-z domain) as the monitoring time step approaches zero, we show that the error convergence with continuous monitoring represents a limit for the discretely monitored scheme

Optimization in Quasi-Monte Carlo Methods for Derivative Valuation

Computational complexity in financial theory and practice has seen an immense rise recently. Monte Carlo simulation has proved to be a robust and adaptable approach, well suited for supplying numerical solutions to a large class of complex problems. Although Monte Carlo simulation has been widely applied in the pricing of financial derivatives, it has been argued that the need to sample the relevant region as uniformly as possible is very important. This led to the development of quasi-Monte Carlo methods that use deterministic points to minimize the integration error. A major disadvantage of low-discrepancy number generators is that they tend to lose their ability of homogeneous coverage as the dimensionality increases. This thesis develops a novel approach to quasi-Monte Carlo methods to evaluate complex financial derivatives more accurately by optimizing the sample coordinates in such a way so as to minimize the discrepancies that appear when using lowdiscrepancy sequences. The main focus is to develop new methods to, optimize the sample coordinate vector, and to test their performance against existing quasi-Monte Carlo methods in pricing complicated multidimensional derivatives. Three new methods are developed, the Gear, the Simulated Annealing and the Stochastic Tunneling methods. These methods are used to evaluate complex multi-asset financial derivatives (geometric average and rainbow options) for dimensions up to 2000. It is shown that the two stochastic methods, Simulated Annealing and Stochastic Tunneling, perform better than existing quasi-Monte Carlo methods, Faure' and Sobol'. This difference in performance is more evident in higher dimensions, particularly when a low number of points is used in the Monte Carlo simulations. Overall, the Stochastic Tunneling method yields the smallest percentage root mean square relative error and requires less computational time to converge to a global solution, proving to be the most promising method in pricing complex derivativesImperial Users onl

Hilbert transform, spectral filters and option pricing

We show how spectral filters can improve the convergence of numerical schemes which use discrete Hilbert transforms based on a sinc function expansion, and thus ultimately on the fast Fourier transform. This is relevant, for example, for the computation of fluctuation identities, which give the distribution of the maximum or the minimum of a random path, or the joint distribution at maturity with the extrema staying below or above barriers. We use as examples the methods by Feng and Linetsky (Math Finance 18(3):337–384, 2008) and Fusai et al. (Eur J Oper Res 251(4):124–134, 2016) to price discretely monitored barrier options where the underlying asset price is modelled by an exponential Lévy process. Both methods show exponential convergence with respect to the number of grid points in most cases, but are limited to polynomial convergence under certain conditions. We relate these rates of convergence to the Gibbs phenomenon for Fourier transforms and achieve improved results with spectral filtering