arXiv subjects

Piergiacomo Sabino

Publications and source records attributed to Piergiacomo Sabino.

At least 19 recordsLinked to original sources

arXiv2024

Conditions for a quadratic quantum speedup in nonlinear transforms with applications to energy contract pricing

Computing nonlinear functions over multilinear forms is a general problem with applications in risk analysis. For instance in the domain of energy economics, accurate and timely risk management demands for efficient simulation of millions of scenarios, largely benefiting from computational speedups. We develop a novel hybrid quantum-classical algorithm based on polynomial approximation of nonlinear functions, computed through Quantum Hadamard Products, and we rigorously assess the conditions for its end-to-end speedup for different implementation variants against classical algorithms. In our setting, a quadratic quantum speedup, up to polylogarithmic factors, can be proven only when forms are bilinear and approximating polynomials have second degree, if efficient loading unitaries are available for the input data sets. We also enhance the bidirectional encoding, that allows tuning the balance between circuit depth and width, proposing an improved version that can be exploited for the calculation of inner products. Lastly, we exploit the dynamic circuit capabilities, recently introduced on IBM Quantum devices, to reduce the average depth of the Quantum Hadamard Product circuit. A proof of principle is implemented and validated on IBM Quantum systems.

quant-ph↗

arXiv2023

Energy risk analysis with Dynamic Amplitude Estimation and Piecewise Approximate Quantum Compiling

We generalize the Approximate Quantum Compiling algorithm into a new method for CNOT-depth reduction, which is apt to process wide target quantum circuits. Combining this method with state-of-the-art techniques for error mitigation and circuit compiling, we present a 10-qubit experimental demonstration of Iterative Amplitude Estimation on a quantum computer. The target application is the derivation of the Expected Value of contract portfolios in the energy industry. In parallel, we also introduce a new variant of the Quantum Amplitude Estimation algorithm which we call Dynamic Amplitude Estimation, as it is based on the dynamic circuit capability of quantum devices. The algorithm achieves a reduction in the circuit width in the order of the binary precision compared to the typical implementation of Quantum Amplitude Estimation, while simultaneously decreasing the number of quantum-classical iterations (again in the order of the binary precision) compared to the Iterative Amplitude Estimation. The calculation of the Expected Value, VaR and CVaR of contract portfolios on quantum hardware provides a proof of principle of the new algorithm.

quant-ph↗

arXiv2022

Exchange option pricing under variance gamma-like models

In this article we focus on the pricing of exchange options when the dynamic of logprices follows either the well-known variance gamma or the recent variance gamma++ process introduced in Gardini et al [19]. In particular, for the former model we can derive a Margrabe's type formula whereas, for the latter one we can write an "integral free" formula. Furthermore, we show how to construct a general multidimensional versions of the variance gamma++ processes preserving both the mathematical and numerical tractability. Finally we apply the derived models to German and French energy power markets: we calibrate their parameters using real market data and we accordingly evaluate exchange options with the derived closed formulas, Fourier based methods and Monte Carlo techniques.

q-fin.CP↗

arXiv2022

Efficient Simulation of $p$-Tempered $α$-Stable OU Processes

We develop efficient methods for simulating processes of Ornstein-Uhlenbeck type related to the class of $p$-tempered $α$-stable ($\ts$) distributions. Our results hold for both the univariate and multivariate cases and we consider both the case where the $\ts$ distribution is the stationary law and where it is the distribution of the background driving Lévy process (BDLP). In the latter case, we also derive an explicit representation for the transition law as this was previous known only in certain special cases and only for $p=1$ and $α\in[0,1)$. Simulation results suggest that our methods work well in practice.

math.PR↗

arXiv2021

Normal Tempered Stable Processes and the Pricing of Energy Derivatives

In this study we consider the pricing of energy derivatives when the evolution of spot prices is modeled with a normal tempered stable driven Ornstein-Uhlenbeck process. Such processes are the generalization of normal inverse Gaussian processes that are widely used in energy finance applications. We first specify their statistical properties calculating their characteristic function in closed form. This result is instrumental for the derivation of non-arbitrage conditions such that the spot dynamics is consistent with the forward curve without relying on numerical approximations or on numerical integration. Moreover, we conceive an efficient algorithm for the exact generation of the trajectories which gives the possibility to implement Monte Carlo simulations without approximations or bias. We illustrate the applicability of the theoretical findings and the simulation algorithms in the context of the pricing of different contracts, namely, strips of daily call options, Asian options with European style and swing options. Finally, we present an extension to future markets.

q-fin.CP↗

arXiv2021

Pricing Energy Derivatives in Markets Driven by Tempered Stable and CGMY Processes of Ornstein-Uhlenbeck Type

In this study we consider the pricing of energy derivatives when the evolution of spot prices follows a tempered stable or a CGMY driven Ornstein- Uhlenbeck process. To this end, we first calculate the characteristic function of the transition law of such processes in closed form. This result is instrumental for the derivation of non-arbitrage conditions such that the spot dynamics is consistent with the forward curve. Moreover, based on the results of Cufaro Petroni and Sabino (2020), we also conceive efficient algorithms for the exact simulation of the skeleton of such processes and propose a novel procedure when they coincide with compound Poisson processes of Ornstein-Uhlenbeck type. We illustrate the applicability of the theoretical findings and the simulation algorithms in the context of the pricing different contracts namely, strips of daily call options, Asian options with European style and swing options. Finally, we present an extension to future markets.

q-fin.CP↗

arXiv2020

Tempered stable distributions and finite variation Ornstein-Uhlenbeck processes

Constructing \Levy-driven Ornstein-Uhlenbeck processes is a task closely related to the notion of self-decomposability. In particular, their transition laws are linked to the properties of what will be hereafter called the \emph{a-reminder} of their self-decomposable stationary laws. In the present study we fully characterize the Lévy triplet of these a-reminder s and we provide a general framework to deduce the transition laws of the finite variation Ornstein-Uhlenbeck processes associated with tempered stable distributions. We focus finally on the subclass of the exponentially-modulated tempered stable laws and we derive the algorithms for an exact generation of the skeleton of Ornstein-Uhlenbeck processes related to such distributions, with the further advantage of adopting a procedure computationally more efficient than those already available in the existing literature.

math.PR↗

arXiv2020

A bivariate Normal Inverse Gaussian process with stochastic delay: efficient simulations and applications to energy markets

Using the concept of self-decomposable subordinators introduced in Gardini et al. [11], we build a new bivariate Normal Inverse Gaussian process that can capture stochastic delays. In addition, we also develop a novel path simulation scheme that relies on the mathematical connection between self-decomposable Inverse Gaussian laws and Lévy-driven Ornstein-Uhlenbeck processes with Inverse Gaussian stationary distribution. We show that our approach provides an improvement to the existing simulation scheme detailed in Zhang and Zhang [23] because it does not rely on an acceptance-rejection method. Eventually, these results are applied to the modelling of energy markets and to the pricing of spread options using the proposed Monte Carlo scheme and Fourier techniques

q-fin.CP↗

arXiv2020

Correlating Lévy processes with Self-Decomposability: Applications to Energy Markets

Based on the concept of self-decomposability, we extend some recent multivariate Lévy models built using multivariate subordination with the aim of capturing situations in which a sudden event in one market is propagated onto related markets after a certain stochastic time delay. Consequently, we study the properties of such processes, derive closed form expressions for the characteristic function and detail how a Monte Carlo scheme can be easily implemented. We illustrate the applicability of our approach in the context of gas and power Energy markets focusing on the calibration and on the pricing of spread options written on different underlying assets using simulations techniques.

q-fin.PR↗

arXiv2020

Exact Simulation of Variance Gamma related OU processes: Application to the Pricing of Energy Derivatives

In this study we define a three-step procedure to relate the self-decomposability of the stationary law of a generalized Ornstein-Uhlenbeck process to the law of the increments of such processes. Based on this procedure and the results of Qu et al. (2019), we derive the exact simulation, without numerical inversion, of the skeleton of a Variance Gamma, and of a symmetric Variance Gamma driven Ornstein-Uhlenbeck process. Extensive numerical experiments are reported to demonstrate the accuracy and efficiency of our algorithms. These results are instrumental to simulate the spot price dynamics in energy markets and to price Asian options and gas storages by Monte Carlo simulations in a framework similar to the one discussed in Cummins et al. (2017, 2018).

q-fin.CP↗

arXiv2020

Fast Pricing of Energy Derivatives with Mean-reverting Jump-diffusion Processes

Most energy and commodity markets exhibit mean-reversion and occasional distinctive price spikes, which results in demand for derivative products which protect the holder against high prices. To this end, in this paper we present exact and fast methodologies for the simulation of the spot price dynamics modeled as the exponential of the sum of an Ornstein-Uhlenbeck and an independent pure jump process, where the latter one is driven by a compound Poisson process with (bilateral) exponentially distributed jumps. These methodologies are finally applied to the pricing of Asian options, gas storages and swings under different combinations of jump-diffusion market models, and the apparent computational advantages of the proposed procedures are emphasized.

q-fin.CP↗

arXiv2020

Gamma Related Ornstein-Uhlenbeck Processes and their Simulation

We investigate the distributional properties of two generalized Ornstein-Uhlenbeck (OU) processes whose stationary distributions are the gamma law and the bilateral gamma law, respectively. The said distributions turn out to be related to the self-decomposable gamma and bilateral gamma laws, and their densities and characteristic functions are here given in closed-form. Algorithms for the exact generation of such processes are accordingly derived with the advantage of being significantly faster than those available in the literature and therefore suitable for real-time simulations.

math.PR↗

arXiv2017

Correlated Poisson processes and self-decomposable laws

We analyze a method to produce pairs of non independent Poisson processes $M(t),N(t)$ from positively correlated, self-decomposable, exponential renewals. In particular the present paper provides the family of copulas pairing the renewals, along with the closed form for the joint distribution $p_{m,n}(s,t)$ of the pair $\big(M(s),N(t)\big)$, an outcome which turns out to be instrumental to produce explicit algorithms for applications in finance and queuing theory. We finally discuss the cross-correlation properties of the two processes and the relative timing of their jumps

math.PR↗

arXiv2016

Cointegrating Jumps: an Application to Energy Facilities

Based on the concept of self-decomposable random variables we discuss the application of a model for a pair of dependent Poisson processes to energy facilities. Due to the resulting structure of the jump events we can see the self-decomposability as a form of cointegration among jumps. In the context of energy facilities, the application of our approach to model power or gas dynamics and to evaluate transportation assets seen as spread options is straightforward. We study the applicability of our methodology first assuming a Merton market model with two underlying assets; in a second step we consider price dynamics driven by an exponential mean-reverting Geometric Ornstein-Uhlenbeck plus compound Poisson that are commonly used in the energy field. In this specific case we propose a price spot dynamics for each underlying that has the advantage of being treatable to find non-arbitrage conditions. In particular we can find close-form formulas for vanilla options so that the price and the Greeks of spread options can be calculated in close form using the Margrabe formula (if the strike is zero) or some other well known approximation.

q-fin.PR↗

arXiv2011

Multidimensional Quasi-Monte Carlo Malliavin Greeks

We investigate the use of Malliavin calculus in order to calculate the Greeks of multidimensional complex path-dependent options by simulation. For this purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the multidimensional case. The multidimensional setting shows the convenience of the Malliavin Calculus approach over different techniques that have been previously proposed. Indeed, these techniques may be computationally expensive and do not provide flexibility for variance reduction. In contrast, the Malliavin approach exhibits a higher flexibility by providing a class of functions that return the same expected value (the Greek) with different accuracies. This versatility for variance reduction is not possible without the use of the generalized integral by part formula of Malliavin Calculus. In the multidimensional context, we find convenient formulas that permit to improve the localization technique, introduced in Fournié et al and reduce both the computational cost and the variance. Moreover, we show that the parameters employed for variance reduction can be obtained \textit{on the flight} in the simulation. We illustrate the efficiency of the proposed procedures, coupled with the enhanced version of Quasi-Monte Carlo simulations as discussed in Sabino, for the numerical estimation of the Deltas of call, digital Asian-style and Exotic basket options with a fixed and a floating strike price in a multidimensional Black-Scholes market.

q-fin.CP↗

arXiv2010

Convenient Multiple Directions of Stratification

This paper investigates the use of multiple directions of stratification as a variance reduction technique for Monte Carlo simulations of path-dependent options driven by Gaussian vectors. The precision of the method depends on the choice of the directions of stratification and the allocation rule within each strata. Several choices have been proposed but, even if they provide variance reduction, their implementation is computationally intensive and not applicable to realistic payoffs, in particular not to Asian options with barrier. Moreover, all these previously published methods employ orthogonal directions for multiple stratification. In this work we investigate the use of algorithms producing convenient directions, generally non-orthogonal, combining a lower computational cost with a comparable variance reduction. In addition, we study the accuracy of optimal allocation in terms of variance reduction compared to the Latin Hypercube Sampling. We consider the directions obtained by the Linear Transformation and the Principal Component Analysis. We introduce a new procedure based on the Linear Approximation of the explained variance of the payoff using the law of total variance. In addition, we exhibit a novel algorithm that permits to correctly generate normal vectors stratified along non-orthogonal directions. Finally, we illustrate the efficiency of these algorithms in the computation of the price of different path-dependent options with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross markets.

q-fin.CP↗

arXiv2009

Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulation. We assume a Black-Scholes market with time-dependent volatilities and show how to compute the deltas by the aid of the Malliavin Calculus, extending the procedure employed by Montero and Kohatsu-Higa (2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. We present a new and fast Cholesky algorithm for block matrices that makes the Linear Transformation even more convenient. Moreover, we propose a new-path generation technique based on a Kronecker Product Approximation. This construction returns the same accuracy of the Linear Transformation used for the computation of the deltas and the prices in the case of correlated asset returns while requiring a lower computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004).

q-fin.PR↗

arXiv2007

Implementing Quasi-Monte Carlo Simulations with Linear Transformations

Pricing exotic multi-asset path-dependent options requires extensive Monte Carlo simulations. In the recent years the interest to the Quasi-monte Carlo technique has been renewed and several results have been proposed in order to improve its efficiency with the notion of effective dimension. To this aim, Imai and Tan introduced a general variance reduction technique in order to minimize the nominal dimension of the Monte Carlo method. Taking into account these advantages, we investigate this approach in detail in order to make it faster from the computational point of view. Indeed, we realize the linear transformation decomposition relying on a fast ad hoc QR decomposition that considerably reduces the computational burden. This setting makes the linear transformation method even more convenient from the computational point of view. We implement a high-dimensional (2500) Quasi-Monte Carlo simulation combined with the linear transformation in order to price Asian basket options with same set of parameters published by Imai and Tan. For the simulation of the high-dimensional random sample, we use a 50-dimensional scrambled Sobol sequence for the first 50 components, determined by the linear transformation method, and pad the remaining ones out by the Latin Hypercube Sampling. The aim of this numerical setting is to investigate the accuracy of the estimation by giving a higher convergence rate only to those components selected by the linear transformation technique. We launch our simulation experiment also using the standard Cholesky and the principal component decomposition methods with pseudo-random and Latin Hypercube sampling generators. Finally, we compare our results and computational times, with those presented in Imai and Tan.

math.PR↗