Solvable Local and Stochastic Volatility Models: Supersymmetric Methods in Option Pricing
نویسنده
چکیده
In this paper we provide an extensive classification of one and two dimensional diffusion processes which admit an exact solution to the Kolmogorov (and hence Black-Scholes) equation (in terms of hypergeometric functions). By identifying the one-dimensional solvable processes with the class of integrable superpotentials introduced recently in supersymmetric quantum mechanics, we obtain new analytical solutions. In particular, by applying supersymmetric transformations on a known solvable diffusion process (such as the Natanzon process for which the solution is given by a hypergeometric function), we obtain a hierarchy of new solutions. These solutions are given by a sum of hypergeometric functions, generalizing the results obtained in the paper ”Black-Scholes Goes Hypergeometric” [2]. For two-dimensional processes, more precisely stochastic volatility models, the classification is achieved for a specific class called gauge-free models including the Heston model, the 3/2-model and the geometric Brownian model. We then present a new exact stochastic volatility model belonging to this class.
منابع مشابه
Option pricing under the double stochastic volatility with double jump model
In this paper, we deal with the pricing of power options when the dynamics of the risky underling asset follows the double stochastic volatility with double jump model. We prove efficiency of our considered model by fast Fourier transform method, Monte Carlo simulation and numerical results using power call options i.e. Monte Carlo simulation and numerical results show that the fast Fourier tra...
متن کاملNumerical Solution of Pricing of European Put Option with Stochastic Volatility
In this paper, European option pricing with stochastic volatility forecasted by well known GARCH model is discussed in context of Indian financial market. The data of Reliance Ltd. stockprice from 3/01/2000 to 30/03/2009 is used and resulting partial differential equation is solved byCrank-Nicolson finite difference method for various interest rates and maturity in time. Thesensitivity measures...
متن کاملMartingale Property and Pricing for Time-homogeneous Diffusion Models in Finance by
The thesis studies the martingale properties, probabilistic methods and efficient unbiased Monte Carlo simulation methods for various time-homogeneous diffusion models commonly used in mathematical finance. Some of the popular stochastic volatility models such as the Heston model, the Hull-White model and the 3/2 model are special cases. The thesis consists of the following three parts: Part I ...
متن کاملImplied and Local Volatilities under Stochastic Volatility
For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some näıve pricing and hedg...
متن کاملRegime Switching Stochastic Volatility with Perturbation Based Option Pricing
Volatility modelling has become a significant area of research within Financial Mathematics. Wiener process driven stochastic volatility models have become popular due their consistency with theoretical arguments and empirical observations. However such models lack the ability to take into account long term and fundamental economic factors e.g. credit crunch. Regime switching models with mean r...
متن کامل