Related papers: On the probability density function of baskets
This work generalizes the subdiffusive Black-Scholes model by introducing the variable exponent in order to provide adequate descriptions for the option pricing, where the variable exponent may account for the variation of the memory…
We study the Heston model for pricing European options on stocks with stochastic volatility. This is a Black\--Scholes\--type equation whose spatial domain for the logarithmic stock price $x\in \RR$ and the variance $v\in (0,\infty)$ is the…
We consider a stochastic volatility model with L\'evy jumps for a log-return process $Z=(Z_{t})_{t\geq 0}$ of the form $Z=U+X$, where $U=(U_{t})_{t\geq 0}$ is a classical stochastic volatility process and $X=(X_{t})_{t\geq 0}$ is an…
Assuming that price of the underlying stock is moving in range bound, the Black-Scholes formula for options pricing supports a separation of variables. The resulting time-independent equation is solved employing different behavior of the…
The application of the Cauchy distribution has often been discussed as a potential model of the financial markets. In particular the way in which single extreme, or "Black Swan", events can impact long term historical moments, is often…
We show that our generalization of the Black-Scholes partial differential equation (pde) for nontrivial diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, this was proven for the…
We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with…
This study deals with continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves with various kinds of coding representations. These systems are essentially of a…
We study general properties such as the solution representation of a moving boundary value problem of the Black-Scholes equation, its min-max estimation, lower and upper gradient estimates, and strict monotonicity with respect to the…
We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black--Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market…
This paper develops power series expansions of a general class of moment functions, including transition densities and option prices, of continuous-time Markov processes, including jump--diffusions. The proposed expansions extend the ones…
Differential equations can be used to construct predictive models of a diverse set of real-world phenomena like heat transfer, predator-prey interactions, and missile tracking. In our work, we explore one particular application of…
In this paper we analyze a nonlinear Black--Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price $V$ is assumed to be a function of the underlying…
Explicit representations of densities for linear parabolic partial differential equations are useful in order to design computation schemes of high accuracy for a considerable class of diffusion models. Approximations of lower order based…
Volatility measures the amplitude of price fluctuations. Despite it is one of the most important quantities in finance, volatility is not directly observable. Here we apply a maximum likelihood method which assumes that price and volatility…
We analyze the Standard & Poor's 500 stock market index from the last 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong…
Black-Scholes equation, after a certain coordinate transformation, is equivalent to the heat equation. On the other hand the relativistic extension of the latter, the telegraphers equation, can be derived from the Euclidean version of the…
Constant and symmetric price impact functions, most commonly used in agent-based market modelling, are shown to give rise to paradoxical and inconsistent outcomes in the simplest case of arbitrage exploitation when open-hold-close actions…
We fit the volatility fluctuations of the S&P 500 index well by a Chi distribution, and the distribution of log-returns by a corresponding superposition of Gaussian distributions. The Fourier transform of this is, remarkably, of the Tsallis…
The generalized 5D Black-Scholes differential equation with stochastic volatility is derived. The projections of the stochastic evolutions associated with the random variables from an enlarged space or superspace onto an ordinary space can…