A Path Integral Approach to Derivative Security Pricing: I. Formalism and Analytical Results

The Constant Elasticity of Variance (CEV) model significantly outperforms the Black-Scholes (BS) model in forecasting both prices and options. Furthermore, the CEV model has a marked advantage in capturing basic empirical regularities such as: heteroscedasticity, the leverage effect, and the volatility smile. In fact, the performance of the CEV model is comparable to most stochastic volatility models, but it is considerable easier to implement and calibrate. Nevertheless, the...

We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs numerically requires the approximation of nested conditional expectations, i.e., iterated integrals of previous approximations. Our approach allows us to compute and iteratively improve upper and lower bounds on the true solution starting from...

In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a series expansion of the deviation of its logarithm from that of a Gaussian distribution. Through this procedure, dubbed {\em exponent expansion}, the transition probability is obtained as a power series in $\Delta t$. This becomes asymptotic...

The paper discusses a path-wise approach to stock price modelling.

Recently, it was found that a new set of simple techniques allow one to conveniently express ordinary integrals through differentiation. These techniques add to the general toolbox for integration and integral transforms such as the Fourier and Laplace transforms. The new methods also yield new perturbative expansions when the integrals cannot be solved analytically. Here, we add new results, for example, on expressing the Laplace transform and its inverse in terms of derivat...

In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action functional whose potential term is an exponential function. This path integral is evaluated by use of the Feynman-Kac theorem. After working out some auxiliary integrations involving Bessel and Whittaker functions, we arrive at the spectra...

We discuss suitable classes of diffusion processes, for which functionals relevant to finance can be computed via Monte Carlo methods. In particular, we construct exact simulation schemes for processes from this class. However, should the finance problem under consideration require e.g. continuous monitoring of the processes, the simulation algorithm can easily be embedded in a multilevel Monte Carlo scheme. We choose to introduce the finance problems under the benchmark appr...

In the framework of risk management, for the study of the sensitivity of pricing and hedging in stochastic financial models to changes of parameters and to perturbations of the stock prices, we propose an error calculus which is an extension of the Malliavin calculus based on Dirichlet forms. Although useful also in physics, this error calculus is well adapted to stochastic analysis and seems to be the best practicable in finance. This tool is explained here intuitively and w...

We regard options on VIX and Realised Variance as solutions to path-dependent partial differential equations (PDEs) in a continuous stochastic volatility model. The modeling assumption specifies that the instantaneous variance is a $C^3$ function of a multidimensional Gaussian Volterra process; this includes a large class of models suggested for the purpose of VIX option pricing, either rough, or not, or mixed. We unveil the path-dependence of those volatility derivatives and...

This paper studies how to price and hedge options under stock models given as a path-dependent SDE solution. When the path-dependent SDE coefficients have Fr\'{e}chet derivatives, an option price is differentiable with respect to time and the path, and is given as a solution to the path-dependent PDE. This can be regarded as a path-dependent version of the Feynman-Kac formula. As a byproduct, we obtain the differentiability of path-dependent SDE solutions and the SDE represen...