Related papers: From characteristic functions to implied volatilit…

Our derivation of the distribution function for future returns is based on the risk neutral approach which gives a functional dependence for the European call (put) option price, C(K), given the strike price, K, and the distribution…

Let $Sf$ be a discrete martingale square function. Then, for any set $V$ of positive probability, we have $\mathbb{E} S(\mathbf{1}_V)^2 \geq \eta \mathbb{P}(V)$ for an absolute constant $\eta >0$. We extend this to wavelet square functions,…

Consider additive functionals of a Markov chain $W_k$, with stationary (marginal) distribution and transition function denoted by $\pi$ and $Q$, say $S_n=g(W_1)+...+g(W_n)$, where $g$ is square integrable and has mean 0 with respect to…

We investigate whether it is possible to formulate option pricing and hedging models without using probability. We present a model that is consistent with two notions of volatility: a historical volatility consistent with statistical…

Let $\xi=(\xi_t, t\ge 0)$ be a real-valued L\'evy process and define its associated exponential functional as follows \[ I_t(\xi):=\int_0^t \exp\{-\xi_s\}{\rm d} s, \qquad t\ge 0. \] Motivated by important applications to stochastic…

We study multi-default model which satisfies the quasi-left-continuity, the martingale representation property, the drift multiplier assumption and the full viability. We use $\natural$-model to construct one such model.

We present two explicit rational formulae for Bachelier, or normal, implied volatility. The formulae take the option price, forward, strike, and expiry as inputs and return the implied normal volatility without iteration. They follow the…

A dynamical model based on a continuous addition of colored shot noises is presented. The resulting process is colored and non-Gaussian. A general expression for the characteristic function of the process is obtained, which, after a scaling…

We consider the exponential functional $A_{\infty}=\int_0^{\infty} e^{\xi_s} ds$ associated to a Levy process $(\xi_t)_{t \geq 0}$. We find the asymptotic behavior of the tail of this random variable, under some assumptions on the process…

This paper concerns the estimation of sums of functions of observable and unobservable variables. Lower bounds for the asymptotic variance and a convolution theorem are derived in general finite- and infinite-dimensional models. An explicit…

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same light-tailed subexponential distribution. The examples of a Poisson and geometric number of summands serve as an…

This investigation establishes a formal equivalence between the generalized Black-Scholes equation under a Quadratic Normal Volatility (QNV) specification and the stationary Schr\"odinger equation for a hyperbolic P\"oschl-Teller potential.…

An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional…

This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of…

Asymptotic expansions are derived for associated Legendre functions of degree $\nu$ and order $\mu$, where one or the other of the parameters is large. The expansions are uniformly valid for unbounded real and complex values of the argument…

Let $\mathbb{Q}$ and $\mathbb{P}$ be equivalent probability measures and let $\psi$ be a $J$-dimensional vector of random variables such that $\frac{d\mathbb{Q}}{d\mathbb{P}}$ and $\psi$ are defined in terms of a weak solution $X$ to a…

We derive a backward and forward nonlinear PDEs that govern the implied volatility of a contingent claim whenever the latter is well-defined. This would include at least any contingent claim written on a positive stock price whose payoff at…

In this paper we aim to present two general results regarding, on one hand, the openness stability of set-valued maps and, on the other hand, the metric regularity behavior of the implicit multifunction related to a generalized variational…

Let $X=(X_t)_{t\geq 0}$ be a one-dimensional L\'evy process such that each $X_t$ has a $C^1_b$-density w.r.t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions…

This paper expands traditional stochastic volatility models by allowing for time-varying skewness without imposing it. While dynamic asymmetry may capture the likely direction of future asset returns, it comes at the risk of leading to…