Pricing commodity futures options in the Schwartz multi factor model with stochastic volatility: An asymptotic method

In this paper we investigate the applicability of the asymptotic approach developed in Fouque et al. (2000) for pricing commodity futures options in a Schwartz (1997) multi factor model, featuring both stochastic convenience yield and stochastic volatility. We show that the zero order term in the expansion coincides with the Schwartz (1997) two factor term, with expected long-term volatility replacing the constant volatility term, and provide an explicit expression for the first order correction term. Using empirical data from the natural gas futures market, we demonstrate that a significantly better calibration can be achieved by involving the correction term as compared to the standard Schwartz (1997) two factor expression. This improvement comes at virtually no extra effort

Pricing derivatives with stochastic volatility

This Ph.D. thesis contains 4 essays in mathematical finance with a focus on pricing Asian option (Chapter 4), pricing futures and futures option (Chapter 5 and Chapter 6) and time dependent volatility in futures option (Chapter 7). In Chapter 4, the applicability of the Albrecher et al.(2005)'s comonotonicity approach was investigated in the context of various benchmark models for equities and com- modities. Instead of classical Levy models as in Albrecher et al.(2005), the focus is the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and the Schwartz (1997) two-factor model. It is shown that the method delivers rather tight upper bounds for the prices of Asian Options in these models and as a by-product delivers super-hedging strategies which can be easily implemented. In Chapter 5, two types of three-factor models were studied to give the value of com- modities futures contracts, which allow volatility to be stochastic. Both these two models have closed-form solutions for futures contracts price. However, it is shown that Model 2 is better than Model 1 theoretically and also performs very well empiri- cally. Moreover, Model 2 can easily be implemented in practice. In comparison to the Schwartz (1997) two-factor model, it is shown that Model 2 has its unique advantages; hence, it is also a good choice to price the value of commodity futures contracts. Fur- thermore, if these two models are used at the same time, a more accurate price for commodity futures contracts can be obtained in most situations. In Chapter 6, the applicability of the asymptotic approach developed in Fouque et al.(2000b) was investigated for pricing commodity futures options in a Schwartz (1997) multi-factor model, featuring both stochastic convenience yield and stochastic volatility. It is shown that the zero-order term in the expansion coincides with the Schwartz (1997) two-factor term, with averaged volatility, and an explicit expression for the first-order correction term is provided. With empirical data from the natural gas futures market, it is also demonstrated that a significantly better calibration can be achieved by using the correction term as compared to the standard Schwartz (1997) two-factor expression, at virtually no extra effort. In Chapter 7, a new pricing formula is derived for futures options in the Schwartz (1997) two-factor model with time dependent spot volatility. The pricing formula can also be used to find the result of the time dependent spot volatility with futures options prices in the market. Furthermore, the limitations of the method that is used to find the time dependent spot volatility will be explained, and it is also shown how to make sure of its accuracy

Simulation of Fragmentation Characteristics of Projectile Jacket Made of Tungsten Alloy after Penetrating Metal Target Plate using SPH Method

A smooth particle hydrodynamics (SPH) model was used to simulate the fragmentation process of the jacket during penetrator with lateral efficiency (PELE) penetrating the metal target plate to study the fragmentation characteristics of PELE jacket made of tungsten alloy. The validity of the SPH model was verified by experimental results. Then the SPH model was used to simulate the jacket fragmentation under different impact velocity and thickness of target plate. The influence of impact velocity and thickness of target plate on the jacket fragmentation was obtained by analysing the mass distribution and quantity distribution of the fragments formed by the jacket. The results show that the dynamic fragmentation of tungsten alloy can be simulated effectively using the SPH model, Johnson-Cook strength model, maximum tensile stress failure criterion and stochastic failure model. When the thickness of target plate is fixed, the greater the impact velocity, the greater the pressure produced by the projectile impacting the target plate; with the increase of impact velocity, the mass of residual projectile decreases, the number of fragments formed by fragmentation of jacket increases linearly, and the average mass of fragments decreases exponentially. When the impact velocity is constant, the greater the thickness of the target plate, the longer the pressure duration by the projectile impacting the target plate; with the increase of the thickness of target plate, the mass of residual projectile decreases, the number of fragments formed by fragmentation of jacket increases linearly, and the average mass of fragments decreases exponentially. The numerical calculation model and research method adopted in this paper can be used to study the impact fragmentation of solid materials effectively

Spatio-Temporal Patterns and Impacts of Sediment Variations in Downstream of the Three Gorges Dam on the Yangtze River, China

Spanning the Yangtze River of China, the Three Gorges Dam (TGD) has received considerable concern worldwide with its potential impacts on the downstream side of the dam. This work investigated the spatio-temporal variations of suspended sediment concentration (SSC) at the downstream section of Yichang-to-Chenglingji from 2002 to 2015. A random forest model was developed to estimate SSC using MODIS ground reflectance products, and the spatio-temporal distributions of SSC were retrieved with this model to investigate the characteristics of water-silt variation. Our results revealed that, relatively, SSC before 2003 was evenly distributed in the downstream Yangtze River, while this spatial distribution pattern changed ce 2003 when the dam started storing water. Temporally, the SSC demonstrated a W-shaped curve of seasonal variation as one peak occurred in September and two troughs in March and November, and showed a significantly decreasing trend after three-stage impoundment. After official operation of the TGD in 2009, the SSC was reduced by over 40% than before 2003. Spatially, the most significant changes occurred in the upper Jingjiang section, where the SSC dropped by 45%. During all stages of impoundment, the water impoundment to 135 m in 2003 had the most significant impact on suspended sediment. The decreased SSC has led to emerging risks of bank failure, aggravated erosion of water front and aggressive down-cutting erosion along the downstream of the dam, as well as other ecological and environmental issues that require urgent attention by the government

Fragmentation Behaviour of Radial Layered PELE Impacting Thin Metal Target Plates

The fragmentation mechanism of the penetrator with lateral effect (PELE) after perforating a thin target plate has been summarised and analysed firstly. Then the fragmentation of radial layered PELE was analysed qualitatively and verified by experiment. In the experiment, the target plates were made of 45# steel and 2A12 aluminium respectively. Qualitative analysis and experimental results show that: for normal PELE without layered, after perforating the thin metal target plate, from the bottom to the head of the projectile, the number of fragments formed by the jacket gradually increases, and the mass of the fragment decreases correspondingly. Compared with the normal PELE without layered, the radial layered PELE is less likely to break into fragments, when impacting the thin metal target plate with the same material and thickness under the same impact velocity. However, from the mechanism of the PELE, when the resistance of the target plate is large enough, and the duration of pressure is long enough, the radial layered PELE also can break into fragments with transverse velocity component. The resistance of the target plate plays an important role in the fragmentation of radial layered PELE. The radial layered PELE produced massive fragments with transverse velocity component when impacting the 45# steel plate with5 mm thickness under the impact velocity of 657.2 m/s