arXiv subjects

Giuseppe Campolieti

Publications and source records attributed to Giuseppe Campolieti.

4 recordsLinked to original sources

arXiv2025

Last Hitting Time Distributions for Solvable Diffusions

By considering any one-dimensional time-homogeneous solvable diffusion process,this paper develops a complete analytical framework for computing the distribution of the last hitting time, to any level, and its joint distribution with the process value on any finite time horizon. Our formalism allows for regular diffusions with any type of endpoint boundaries. We exploit the inherent link between last and first hitting times. The simpler known formula for the marginal distribution of the last hitting time on an infinite time horizon is easily recovered as a special limit. Furthermore, we derive general formulae for each component of the joint distribution, i.e., the jointly continuous, the partly continuous (defective) and the jointly defective portions. By employing spectral expansions of the transition densities and the first hitting time distributions, our derivations culminate in novel general spectral expansions for both marginal and joint distributions of the last hitting time and the process value on any finite time horizon. An additional main contribution of this paper lies in the application of our general formulae, giving rise to newly closed-form analytical formulae for several solvable diffusions. In particular, we systematically derive analytical expressions for each portion of the marginal and joint distributions of the last hitting time and the process value on any finite time horizon, without and with imposed killing at one or two interior points, for Brownian motion, Brownian motion with drift (geometric Brownian motion), the squared Bessel , squared radial Ornstein-Uhlenbeck (CIR) and Ornstein-Uhlenbeck processes. Most of our formulae are given in terms of spectral series that are rapidly convergent and efficiently implemented. We demonstrate this by presenting some numerical calculations of marginal and joint distributions using accurately truncated series.

math.PR↗

arXiv2014

Dual Stochastic Transformations of Solvable Diffusions

We present new extensions to a method for constructing several families of solvable one-dimensional time-homogeneous diffusions whose transition densities are obtainable in analytically closed-form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines smooth monotonic mappings and measure changes via Doob-h transforms. This gives rise to new multi-parameter solvable diffusions that are generally divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The theory is applicable to diffusions with either singular and/or non-singular endpoints. As part of the results in this paper, we also present a complete boundary classification and martingale characterization of the newly developed diffusion families.

q-fin.PR↗

arXiv2013

Pricing Step Options under the CEV and other Solvable Diffusion Models

We consider a special family of occupation-time derivatives, namely proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96 (1999)]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.

q-fin.PR↗

arXiv2011

Bridge Copula Model for Option Pricing

In this paper we present a new multi-asset pricing model, which is built upon newly developed families of solvable multi-parameter single-asset diffusions with a nonlinear smile-shaped volatility and an affine drift. Our multi-asset pricing model arises by employing copula methods. In particular, all discounted single-asset price processes are modeled as martingale diffusions under a risk-neutral measure. The price processes are so-called UOU diffusions and they are each generated by combining a variable (Ito) transformation with a measure change performed on an underlying Ornstein-Uhlenbeck (Gaussian) process. Consequently, we exploit the use of a normal bridge copula for coupling the single-asset dynamics while reducing the distribution of the multi-asset price process to a multivariate normal distribution. Such an approach allows us to simulate multidimensional price paths in a precise and fast manner and hence to price path-dependent financial derivatives such as Asian-style and Bermudan options using the Monte Carlo method. We also demonstrate how to successfully calibrate our multi-asset pricing model by fitting respective equity option and asset market prices to the single-asset models and their return correlations (i.e. the copula function) using the least-square and maximum-likelihood estimation methods.

q-fin.PR↗