A general framework is proposed for pricing both continuously and discretely monitored Asian options under onedimensional Markov processes. For each type (continuously monitored or discretely monitored), we derive the double transform of the Asian option price in terms of the unique bounded solution to a related functional equation. In the special case of continuous-time Markov chain (CTMC), the functional equation reduces to a linear system that can be solved analytically via matrix inversion. Thus the Asian option prices under a one-dimensional Markov process can be obtained by first constructing a CTMC to approximate the targeted Markov process model, and then computing the Asian option prices under the approximate CTMC by numerically inverting the double transforms. Numerical experiments indicate that our pricing method is accurate and fast under popular Markov process models, including the CIR model, the CEV model, Merton's jump diffusion model, the double-exponential jump diffusion model, the variance gamma model, and the CGMY model.

The stochastic alpha-beta-rho (SABR) model becomes popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves observed in the marketplace. However, no analytical solution to the SABR model exists that can be simulated directly. This paper explores the possibility of exact simulation for the SABR model. Our contribution is threefold. (i) We propose an exact simulation method for the forward price and its volatility in two special but practically interesting cases, i.e., when the elasticity β 1, or when β < 1 and the price and volatility processes are instantaneously uncorrelated. Primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the Hartman-Watson and the noncentral chi-squared distribution functions, respectively. Two novel simulation schemes are proposed to achieve numerical accuracy, efficiency, and stability. One stems from numerical Laplace inversion and Asian option literature, and the other is based on recent developments in evaluating the noncentral chi-squared distribution functions in a robust way. Numerical examples demonstrate that our method is fast and accurate under various market environments. (ii) When β < 1 but the price and volatility processes are correlated, our simulation method becomes a semi-exact one. Numerical results suggest that it is still quite accurate when the time horizon is not long, e.g., no greater than one year. For long time horizons, a piecewise semi-exact simulation scheme is developed that reduces the biases substantially. (iii) For European option pricing under the SABR model, we propose a conditional simulation method, which reduces the variance of the plain simulation significantly, e.g., by more than 99%.

BackgroundThe relationship between oxidative balance score (OBS) and diabetes remains poorly understood and may be gender-specific. We conducted a cross-sectional study to investigate the complex association between OBS and diabetes among US adults.MethodsOverall, 5,233 participants were included in this cross-sectional study. The exposure variable was OBS, composed of scores for 20 dietary and lifestyle factors. Multivariable logistic regression, subgroup analysis, and restricted cubic spline (RCS) regression were applied to examine the relationship between OBS and diabetes.ResultsCompared to the lowest OBS quartile group (Q1), the multivariable-adjusted odds ratio (OR) (95% confidence interval (CI) for the highest OBS quartile group (Q4) was 0.602 (0.372–0.974) (p for trend = 0.007), and for the highest lifestyle, the OBS quartile group was 0.386 (0.223–0.667) (p for trend &lt; 0.001). Moreover, gender effects were found between OBS and diabetes (p for interaction = 0.044). RCS showed an inverted-U relationship between OBS and diabetes in women (p for non-linear = 6e−04) and a linear relationship between OBS and diabetes in men.ConclusionsIn summary, high OBS was negatively associated with diabetes risk in a gender-dependent manner.

No abstract

The prices of Asian options, which are among the most important options in financial engineering, can often be written in terms of Laplace transforms. However, computable error bounds of the Laplace inversions are rarely available to guarantee their accuracy. We conduct a thorough analysis of the inversion of the Laplace transforms for continuously and discretely monitored Asian option prices under general continuous-time Markov chains (CTMCs), which can be used to approximate any one-dimensional Markov process. More precisely, we derive computable bounds for the discretization and truncation errors involved in the inversion of Laplace transforms. Numerical results indicate that the algorithm is fast and easy to implement, and the computable error bounds are especially suitable to provide benchmark prices under CTMCs. The online supplement is available at https://doi.org/10.1287/ijoc.2017.0805 .

This thesis covers two topics related to the computational methods in financial engineering. The stochastic alpha-beta-rho (SABR) model is popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves. However, no analytical solution to the SABR model exists that can be simulated directly. In the first topic, we explore the possibility of exact simulation for the SABR model. Our contribution is threefold. (i) We propose an exact simulation method in two special but practically interesting cases. Primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the Hartman-Watson and the non-central chi-squared distribution functions, respectively. Two novel simulation schemes are proposed to achieve numerical accuracy, efficiency, and stability. (ii) In general cases, we propose a semi-exact simulation scheme, which turns out to be accurate when the time horizon is not long, e.g., no longer than 1 year. When the time horizon is long, a piecewise semi-exact simulation scheme is developed that reduces the biases substantially. (iii) For European option pricing, we propose a conditional simulation method, which reduces the variance of plain simulation significantly by, e.g., 99%. The time-inhomogeneous diffusion models, such as the local volatility models and term structure models for interest rates, are widely used in financial applications. However, analytical pricing formulas are usually not available even for vanilla options. In the second topic, we derive a series representation for the European-type option price with a general payoff under time-inhomogeneous diffusions. Its convergence is proved rigorously under some regularity conditions. Our series representation provides a unified framework for analytically approximating European-type option prices with general payoffs through different parametrization expansion schemes. Besides, a systematic method based on Wiener-Itô Chaos expansion is xiii developed to derive explicit expressions for the related analytical approximations. Numerical results demonstrate that our series representation method is accurate and efficient. xiv

No abstract

The constant elasticity of variance (CEV) model is widely used in modeling commodity futures prices, but it may not perform well in calibrating corresponding futures options. We consider two variations of the CEV model, that is, CEV with jumps and CEV with regime switching, and compare their performance in calibrating the Chinese futures options market. In particular, we propose a unified framework for pricing American futures options by combining the continuous‐time Markov chain approximation and the dynamic programming method. Results show that the inverse leverage effect in the soybean meal options market can be better described by the CEV regime‐switching model.