We consider the exact path sampling of the squared Bessel process and some other continuous-time Markov processes, such as the CIR model, constant elasticity of variance diffusion model, and hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale transformation, and/or change of measure. All these diffusions are broadly used in mathematical finance for modelling asset prices, market indices, and interest rates. We show how the probability distributions of a squared Bessel bridge and a squared Bessel process with or without absorption at zero are reduced to randomized gamma distributions. Moreover, for absorbing stochastic processes, we develop a new bridge sampling technique based on conditioning on the first hitting time at zero. Such an approach allows us to simplify simulation schemes. New methods are illustrated with pricing path-dependent options.
This paper develops bridge sampling path integral algorithms for pricing path-dependent options under a new class of nonlinear state dependent volatility models. Path-dependent option pricing is considered within a new (dual) Bessel family of semimartingale diffusion models, as well as the constant elasticity of variance (CEV) diffusion model, arising as a particular case of these models. The transition p.d.f.s or pricing kernels are mapped onto an underlying simpler squared Bessel process and are expressed analytically in terms of modified Bessel functions. We establish precise links between pricing kernels of such models and the randomized gamma distributions, and thereby demonstrate how a squared Bessel bridge process can be used for exact sampling of the Bessel family of paths. A Bessel bridge algorithm is presented which is based on explicit conditional distributions for the Bessel family of volatility models and is similar in spirit to the Brownian bridge algorithm. A special rearrangement and splitting of the path integral variables allows us to combine the Bessel bridge sampling algorithm with either adaptive Monte Carlo algorithms, or quasi-Monte Carlo techniques for significant numerical efficiency improvement. The algorithms are illustrated by pricing Asian-style and lookback options under the Bessel family of volatility models as well as the CEV diffusion model.
Path integral algorithms are developed for evaluating European-style Asian options within three new families of multi-parameter local volatility models. The forward price process is a martingale under an assumed risk-neutral measure and the transition probability densities for such models are given in analytically closed form. Implied volatility surfaces for these models exhibit a wide range of pronounced smiles and skews of the type observed in the option markets. By concatenating the pricing kernels in time, we obtain an exact multidimensional path integral representation for any discrete-time path-dependent option on these models. Nonlinear variable transformations and numeraire changes are exploited in simplifying the path integrals. Efficient weighted Monte Carlo schemes for accurate evaluation of the integrals are then presented. The algorithms combine variance reduction techniques such as the control variate method with distributed computing to significantly increase efficiency. Our numerical results show that distributed Monte Carlo simulations scale almost linearly with the increase in independent processors on a high-performance computer cluster. As a competitive approach we develop a multinomial lattice method that is based on the backward propagation of the option value function using an updating rule. The paper includes simulation results for two kinds of Asian options: average price and average strike options.Path integral pricing, Asian options, State-dependent volatility models,
We present some further developments in the construction and classification of new solvable one‐dimensional diffusion models having transition densities, and other quantities that are fundamental to derivatives pricing, representable in analytically closed form. Our approach is based on so‐called diffusion canonical transformations that produce a large class of multiparameter nonlinear local volatility diffusion models that are mapped onto various simpler diffusions. Using an asymptotic analysis, we arrive at a rigorous boundary classification as well as a characterization with respect to probability conservation and the martingale property of the newly constructed diffusions. Specifically, we analyze and classify in detail four main families of driftless regular diffusion models that arise from the underlying squared Bessel process (the Bessel family), Cox–Ingersoll–Ross process (the confluent hypergeometric family), the Ornstein‐Uhlenbeck diffusion (the OU family), and the Jacobi diffusion (the hypergeometric family). We show that the Bessel family is a superset of the constant elasticity of variance model without drift. The Bessel family, in turn, is nested by the confluent hypergeometric family. For these two families we find further subfamilies of conservative strict supermartingales and nonconservative martingales with an exit boundary. For the new classes of nonconservative regular diffusions we also derive analytically exact first exit time densities that are given in terms of generalized inverse Gaussians and extensions. As for the two other new models, we show that the OU family of processes are conservative strict martingales, whereas the Jacobi family are nonconservative nonmartingales. Considered as asset price diffusion models, we also show that these models demonstrate a wide range of local volatility shapes and option implied volatility surfaces that include various pronounced skew and smile patterns.
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