T his paper aims to extend the analytical tractability of the Black-Scholes model to alternative models with arbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixed-exponential distribution, which is a weighted average of exponential distributions but with possibly negative weights. The new model extends existing models, such as hyperexponential and double-exponential jump diffusion models, because the mixed-exponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavy-tailed distributions. The mixed-exponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for path-dependent options such as lookback and barrier options. The Laplace transforms can be inverted via the Euler inversion algorithm. Numerical experiments indicate that the formulae are easy to implement and accurate. The analytical solutions are made possible mainly because we solve a high-order integro-differential equation explicitly. A calibration example for SPY options shows that the model can provide a reasonable fit even for options with very short maturity, such as one day.

We obtain a closed-form solution for the double-Laplace transform of Asian options under the hyper-exponential jump diffusion model. Similar results were available previously only in the special case of the Black-Scholes model (BSM). Even in the case of the BSM, our approach is simpler as we essentially use only Itô's formula and do not need more advanced results such as those of Bessel processes and Lamperti's representation. As a by-product we also show that a well-known recursion relating to Asian options has a unique solution in a probabilistic sense. The double-Laplace transform can be inverted numerically via a two-sided Euler inversion algorithm. Numerical results indicate that our pricing method is fast, stable, and accurate; and it performs well even in the case of low volatilities.

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.

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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%.

We investigated the prevalence of β-lactamase genes and plasmid-mediated quinolone resistance (PMQR) determinants in 51 carbapenem-resistant Enterobacteriaceae (CRE) from five teaching hospitals in central China. The prevalence of carbapenem resistance in Enterobacteriaceae was 1·0% (51/5012). Of 51 CRE, 31 (60·8%) isolates were positive for one tested carbapenemase gene, while 10 (19·6%) were simultaneously positive for two tested carbapenemase genes. The positive rates of bla KPC-2, bla NDM-1, bla IMP-4, bla IMP-26 and bla IMP-8 were 54·9%, 17·6%, 11·8%, 11·8% and 3·9%, respectively. Of 10 CRE with two carbapenemase genes, three, five, one and one were positive for bla KPC-2 and bla IMP-4, bla KPC-2 and bla IMP-26, bla KPC-2 and bla IMP-8, and bla KPC-2 and bla NDM-1, respectively. Eight of nine bla NDM-1-positive isolates lacked carbapenemases by the modified Hodge test, while 27/28 isolates harbouring bla KPC-2 were positive for carbapenemases determined by this test; 41·2% of the CRE-positive isolates also harboured ESBL genes in various combinations (three and two positive for bla KPC-2 also carried bla DHA-1 and bla CMY-2). The positive rates of qnrS1, qnrA1, qnrB and aac-(6/)-Ib-cr in CRE were 25·5%, 9·8%, 23·5% and 15·7%, respectively. In particular, 7/9 isolates harbouring bla NDM-1 were positive for these quinolone resistance genes, of which five carried qnrS1 and two carried qnrS1 and qnrB4. All but two of 29 Klebsiella pneumoniae isolates were grouped into 20 clonal clusters by PFGE, with the predominant cluster accounting for four bla KPC-2-positive isolates distributed in the same hospital. We conclude that there is a high prevalence of bla NDM-1 and PMQR determinants in CRE isolates in central China. Multiple resistance determinants in various combinations co-exist in these strains and we report for the first time the co-existence of bla KPC-2 and bla IMP-26 in a strain of Klebsiella oxytoca.

In this paper we propose a closed-form asymptotic expansion approach to pricing discretely monitored Asian options in general one-dimensional diffusion models. Our expansion is a small-time expansion because the expansion parameter is selected to be the square root of the length of monitoring interval. This expansion method is distinguished from many other pricing-oriented expansion algorithms in the literature due to two appealing features. First, we illustrate that it is possible to explicitly calculate not only the first several expansion terms but also any general expansion term in a systematic way. Second, the convergence of the expansion is proved rigorously under some regularity conditions. Numerical experiments suggest that the closed-form expansion formula with only a few terms (e.g., four terms up to the third order) is accurate, fast, and easy to implement for a broad range of diffusion models, even including those violating the regularity conditions.

In this paper, we provide Laplace transform-based analytical solutions to pricing problems of various occupation-time-related derivatives such as step options, corridor options, and quantile options under Kou's double exponential jump diffusion model. These transforms can be inverted numerically via the Euler Laplace inversion algorithm, and the numerical results illustrate that our pricing methods are accurate and efficient. The analytical solutions can be obtained primarily because we derive the closed-form Laplace transform of the joint distribution of the occupation time and the terminal value of the double exponential jump diffusion process. Beyond financial applications, the mathematical results about occupation times of a jump diffusion process are of more general interest in applied probability.