Lévy-Based Cross-Commodity Models and Derivative Valuation

From Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options Menghui Cao This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options. The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor of βσ √ ∆t. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one

Implementation of the modified Monte Carlo simulation for evaluate the barrier option prices

ABSTRACTStochastic volatility model with jumps fits almost perfectly the empirical implied volatility surface. Under this model, this paper considers continuously monitored barrier options pricing by Monte Carlo simulation. Based on quadratic exponential scheme, this paper develops an algorithm for pricing barrier options and provides convergence of the algorithm by moment-matching techniques. Variance reduction technique based on control variates further improves the efficiency of the algorithm. The algorithm is also extended to stochastic volatility model with contemporaneous jumps in variance and stock price. Simulations show that the proposed algorithm is efficient and easy to implement. Compared to contemporaneous jumps in variance and stock price, only jumps in stock price produce more profound impact on barrier options prices.

Asian barrier options are barrier options whose trigger is based on an average underlying price. They provide the advantages of both Asian options and barrier options. This paper introduces the first quadratic-time lattice algorithm to price European-style Asian barrier options. It is by far the most efficient lattice algorithm with convergence guarantees. The algorithm relies on the Lagrange multipliers to optimally distribute the number of states for each node of the multinomial lattice. We also show experiment results to demonstrate effectiveness and efficiency of our algorithm by comparing with Monte Carlo simulations.

This study develops a new methodology for combining density forecast accuracy tests and assessing the relevance of psychological indicators in predicting commodity returns. Density forecasts provide a complete description of the uncertainty associated with a prediction and are highly requested by policymakers, central bankers, and financial operators to define policy actions, manage financial risks, and assess portfolio selection. The proposed methodology combines different tests and derives the p-value of the resulting test statistic by Monte Carlo simulations. To assess the power of the proposed methodology, we implement a set of experiments for several data-generating processes. Based on an empirical forecasting exercise applied to agricultural, energy, and metal commodities, we find that sentiment variables and psychological factors improve the density forecasts of commodity futures returns, especially for agricultural commodities. Additionally, combinations of sentiment variables are more powerful in predicting returns than considering them separately.

The role that clustering in activity and/or severity plays in catastrophe modeling and derivative valuation is a key aspect that has been overlooked in the recent literature. Here, we propose two marked point processes to account for these features. The first approach assumes the points are driven by a stochastic hazard rate modulated by a Markov chain while marks are drawn from a regime-specific distribution. In the second approach, the points are driven by a self-exciting process while marks are drawn from an independent distribution. Within this context, we provide a unified approach to efficiently value catastrophe options—such as those embedded in catastrophe bonds—and show that our results are within the 95% confidence interval computed using Monte Carlo simulations. Our approach is based on deriving the valuation PIDE and utilizes transforms to provide semi-analytical closed-form solutions. This contrasts with most prior works where the valuation formulae require computing several infinite sums together with numerical integration.

Valuing three-asset barrier options and autocallable products via exit probabilities of Brownian bridge

Aims. Implement a matched filter (MF) cross-correlation algorith m in multipole space and compare it to the standard Angular Cross Power Spectrum (ACPS) method. Apply both methods on a Integrated Sachs Wolfe (ISW) - Large Scale Structure (LSS) cross correlation scenario and study how sky masks influence the multipo le range where the cross correlation signal arises and its co mparison to theoretical predictions. Methods. The MF requires the inversion of a multipole covariance matrix that, under non-full sky coverage ( fsky < 1), is generally non-diagonal and singular. We choose a Singular Value Decomposition (SVD) approach that enables the identificatio n of those modes carrying most of the information from those more likely to introduce numerical noise, (that are dropped from the analysis). We compare the MF to the ACPS in ISW-LSS Monte Carlo simulations, focusing on the effect that a limited sky coverage has on the cross-correlation results. Results. Within the data model s = t+αm where t is Gaussian noise and m is a known filter, we find that the MF performs comparatively better than the ACPS for smaller values of fsky and scale dependent (non-Poissonian) noise fields. In the co ntext of ISW studies both methods are comparable, although the MF performs slightly more sensitively under more restrictive masks (smaller values of fsky). A preliminary analytical study of the ISW‐LSS cross correlation signal to noise (S/N) ratio shows that most of it should be found in the very large scales (50% of the S/N at l< 10, 90% at l< 40− 50), and this is confirmed by Monte Carlo simulations. The statistical significance of our cross-correlation stat istics reaches its maximum when considering l∈ [2, lmax], with lmax∈ [5, 40] for all values of fsky observed, despite of the smoothing and power aliasing that aggressive masks introduce in Fourier space. This l-confinement of the ISW-LSS cross correlation should enable a safe distinction from other secondary effects arising at smaller (higher l-s) angular scales.

Thermal fluctuations are ubiquitous in mesoscopic and microscopic systems. Take nematic liquid crystals (LCs) as an example; their director fluctuations can strongly scatter light and give rise to random motions and rotations of topological defects and solid inclusions. These stochastic processes contain important information about the material properties of the LC and dictate the transport of the immersed colloidal particles. However, modeling thermal fluctuations of the nematic field remains challenging. Here, we introduce a new Monte Carlo simulation method, namely the Fourier-space Monte Carlo (FSMC) method, which is based on the Oseen–Frank elastic distortion energy model. This method accurately models the thermal fluctuations of a nematic LC’s director field. In contrast to the traditional real-space MC method, which perturbs the director locally, the FSMC method samples different eigenmodes of the director distortions in the Fourier space, aligning with the equipartition theorem. We apply FSMC to study defect fluctuations and trajectories in a two-dimensional nematic LC confined to various geometries. Our results show that FSMC can effectively sample degenerate defect configurations and reproduce long-range elastic interactions between defects. In addition, we conduct three-dimensional molecular dynamics simulations using a coarse-grained Gay–Berne potential, which corroborates the findings from FSMC. Taken together, we have developed a new Monte Carlo method to accurately model thermal fluctuations in nematic LCs, which can be useful for searching global free-energy minimum states in nematic, smectic, and other LC mesophases and can also be helpful in modeling the thermal motions of defects and inclusions in LCs.

In this paper, we present our research on pricing window barrier options under a hybrid stochastic-local volatility (SLV) model in the foreign exchange (FX) market. Due to the hybrid effect of the local volatility and stochastic volatility components of the model, the SLV model can reproduce the market implied volatility surface, and can improve the pricing accuracy for exotic options at the same time. In this paper, numerical techniques such as Monte Carlo and finite difference methods for standard exotic barrier options under the SLV model are extended to pricing window barrier options and numerical results produced by the SLV model are used to examine the performance and accuracy of the model for pricing window barrier options.

The energy commodities market is inherently volatile, driven by geopolitical events, supply-demand imbalances, regulatory changes, and macroeconomic factors. This volatility poses significant challenges for traders, investors, and policymakers, necessitating advanced methodologies to stabilize trading outcomes and improve profitability. This study explores cutting-edge techniques for mitigating market fluctuations in energy commodities trading, focusing on algorithmic trading strategies, predictive analytics, risk management frameworks, and portfolio optimization models. One of the primary techniques discussed is algorithmic trading, where high-frequency trading (HFT) and machine learning-driven models enhance trade execution by capitalizing on micro-price fluctuations. These automated systems utilize historical and real-time data to identify patterns and execute trades with minimal latency, ensuring improved risk-adjusted returns. Predictive analytics, leveraging artificial intelligence (AI) and big data, further refines market forecasting by analyzing large datasets, including geopolitical developments, supply chain disruptions, and weather patterns. By incorporating neural networks and regression models, predictive analytics reduces uncertainty and enables proactive trading strategies. To mitigate risk exposure, robust hedging strategies are employed, utilizing futures contracts, options, and swaps to offset adverse price movements. These derivative instruments help traders lock in prices, ensuring more predictable revenue streams. Additionally, value-at-risk (VaR) modeling, stress testing, and Monte Carlo simulations provide comprehensive risk assessments, allowing traders to adjust their positions dynamically. Portfolio diversification, incorporating a mix of energy commodities such as crude oil, natural gas, and renewable energy credits, further minimizes market susceptibility. The implementation of blockchain technology enhances market transparency and efficiency by providing immutable transaction records and reducing counterparty risk. Furthermore, smart contracts automate trade settlements, reducing operational inefficiencies and ensuring compliance with regulatory frameworks. Real-time market monitoring systems, equipped with sentiment analysis, track global news and social media trends, offering traders critical insights to anticipate price swings. By integrating these advanced systems and processes, energy commodities trading firms can enhance market stability, optimize decision-making, and safeguard against extreme price volatility. The study underscores the importance of technology-driven innovations in achieving sustainable profitability in energy trading.

A story in Aristotle's Politics is often cited as the first published discussion of a financial derivative, an instrument whose value is derived from other assets or events. In Greece of the fifth century B.C., the philosopher Thales, believing six months in advance that the spring weather would be exceptionally favorable to olive growth, negotiated the right to rent oil presses for the next harvest at a bargain rate from press owners who wanted to hedge their bets against the possibility of a poor harvest. In today's parlance, Thales purchased a call option. When spring brought a lush olive harvest, Thales rented the machines to others at a much higher rate, making a tidy profit to support his philosophical inquiries. Options and other financial derivatives have become extremely popular in recent years, for better or worse. Overinvestment in derivatives of questionable value has led to notorious financial disasters such as the bankruptcy of Orange County, California, and the collapse of Barings Bank. Warren Buffett famously described derivatives as "financial weapons of mass destruction" in the Berkshire Hathaway Annual Report of 2002. Nonetheless, derivatives are undoubtedly here to stay; in 2002, trade in derivatives worldwide was estimated at $100 trillion or more, equaling or exceeding the total value of "real" equities. Readers of SIAM Review can take a special interest in derivatives because their analysis is an arena in which mathematics has had a huge impact on finance. In particular, the 1970s papers of Black and Scholes (1973) and Merton (1976) began a flood of research on options pricing that continues to this day. This issue's Survey and Review paper focuses on the special case of a spread option, a derivative that is based on the difference in value of two assets. Suppose you have some money to invest and the choice between two stocks to buy. You suspect that one will do much better than the other over the next few years, but you don't know which one. An ideal situation might be if you could buy one of them and hold it for a certain length of time, and then have the option of trading it in for the other stock if it turns out that that one did better. Another way to achieve the same benefit would be if someone agreed to pay you the difference in value between the two stocks at some specified time if you discover with hindsight that you made the wrong decision. Of course no one will offer you such an option free of charge, and so the mathematical question is how to price this option. This question is even more challenging if the option has a strike price that must be paid in order to exercise it. While spread options aren't publicly traded on arbitrary pairs of stocks, a variety of complex spread options are available in many markets, and are particularly popular in energy markets. Energy prices are extremely volatile compared to other commodities, a fact that most consumers are well aware of and that greatly affects producers, distributors, and speculators. Spread options are an important hedging and speculation tool, and are available in many different forms. For example, options may be based on the difference between the prices of two different forms of oil (the crack spread), between the prices of oil and electricity (the spark spread), or between the prices of a single commodity at two different times (calendar spreads) or locations (locational spreads). In the following article by Carmona and Durrleman, section 2 is devoted to a detailed description of these and other spread options that are used in practice. The authors then summarize the classical Black--Scholes pricing paradigm for options based on a single commodity, and the multivariate extensions needed for the analysis of spread options. The underlying stochastic differential equations can be solved by various approaches, and several approximations are illustrated and compared. Particular difficulties associated with energy markets are discussed throughout and are a main focus of the latter sections. Financial mathematics is as exciting growth area for applied mathematics, and we hope this paper will introduce readers to some challenging problems of current interest in this field.

Motivated by the changing nature of the natural gas industry in the European Union, driven by the liberalisation process, we focus on the introduction and pricing of gas swing options. These options are embedded in typical gas sales agreements in the form of offtake flexibility concerning volume and time. The gas swing option is actually a set of several American puts on a spread between prices of two or more energy commodities. This fact, together with the fact that the energy markets are fundamentally different from traditional financial security markets, is important for our choice of valuation technique. Due to the specific features of the energy markets, the existing analytic approximations for spread option pricing are hardly applicable to our framework. That is why we employ Monte Carlo methods to model the spot price dynamics of the underlying commodities. The price of an arbitrarily chosen gas swing option is then computed in accordance with the concept of risk-neutral expectations. Finally, our result is compared with the real payoff from the option realised at the time of the option execution and the maximum ex-post payoff that the buyer could generate in case he knew the future, discounted to the original time of the option pricing.

In this work, we are concerned with valuing the option to invest in a project when the project value and the investment cost are both mean-reverting. Previous works on stochastic project and investment cost concentrate on geometric Brownian motions (GBMs) for driving the factors. However, when the project involved is linked to commodities, mean-reverting assumptions are more meaningful. Here, we introduce a model and prove that the optimal exercise strategy is not a function of the ratio of the project value to the investment V/I – contrary to the GBM case. We also demonstrate that the limiting trigger curve as maturity approaches traces out a nonlinear curve in (V, I) space and derive its explicit form. Finally, we numerically investigate the finite-horizon problem, using the Fourier space time-stepping algorithm of Jaimungal and Surkov [2009. Lev´y based cross-commodity models and derivative valuation. SIAM Journal of Financial Mathematics, to appear. http://www.ssrn.com/abstract=972837]. Numerically, the optimal exercise policies are found to be approximately linear in V/I; however, contrary to the GBM case they are not described by a curve of the form V*/I*=c(t). The option price behavior as well as the trigger curve behavior nicely generalize earlier one-factor model results.

Parallel magnetic resonance imaging (pMRI) is a technique of accelerating the acquisition of MRI data with high spatial and temporal resolutions. It aims to reconstruct the reduced Field-of-View (FOV) images from under-sampled data in the Fourier space (k-space) by using multiple receiver coils. Therefore, several reconstruction techniques of full FOV image have been proposed. Currently, the most used technique is SENSitivity Encoding (SENSE). However, reconstructed images by SENSE are tainted by artifacts, mainly caused by the noise and inaccurate sensitivity maps. The objective of this paper is to propose a new regularization technique to improve the reconstruction of pMRI images by taking into account the sensitivity maps errors. Our technique is developed in a Bayesian framework using a Markov Chain Monte Carlo (MCMC) sampling scheme and accounts for complex-valued signals. The proposed approach is validated on both simulated and real data. The obtained results show the performance of the proposed technique for the restoration of complex-valued images with sensitivity maps errors.