Josep Vives

University of Barcelona | UB · Department of Probability, Logic and Statistics

In this paper, we present a stochastic optimal control model to optimize an insurance firm problem in the case where its cash-balance process is assumed to be described by a stochastic differential equation driven by Teugels martingales. Noticing that the insurance firm is able to control its cash-balance dynamics by regulating the underlying premi...

Topological data analysis provides a new perspective on many problems in the domain of complex systems. Here, we establish the dependency of mean values of functional p-norms of ’persistence landscapes’ on a uniform scaling of the underlying multivariate distribution. Furthermore, we demonstrate that average values of p-norms are decreasing, when t...

In this paper we investigate, since both, the theoretical and the empirical point of view, the pricing of European call options under a hybrid CEV-Heston model. CEV-Heston model captures two typical behaviors of financial assets: (i) the leverage effect and (ii) the stochastic volatility. We prove theoretically that the CEV-Heston model covers the...

The research presented in this paper provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenges alongside. Most noticeably, ev...

We show that the integration by parts formula based on Malliavin-Skorohod calculus techniques for additive processes helps us to compute quantities like E(LT h(LT)), or moregenerally E(H(LT)), for different suitable functions h or H and different models for the cumulative loss process L.These quantities are important in Insurance and Finance. For e...

In the present paper we investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that give rise to time-inconsistency of the decision maker. We consider equilibrium policies within the class of open-loop controls, that are characterized, in our context, by means of a variational method which leads...

In this paper we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alos (2012) for Heston (1993) SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular clas...

The research presented in this article provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenges alongside. Most noticeably,...

In the present paper, a decomposition formula for the call price due to Al\`{o}s is transformed into a Taylor type formula containing an infinite series with stochastic terms. The new decomposition may be considered as an alternative to the decomposition of the call price found in a recent paper of Al\`{o}s, Gatheral and Radoi\v{c}i\'{c}. We use th...

In this paper we obtain a Dyson type formula for integrable functionals of a pure jump Lévy process. We represent the conditional expectation of a F T -measurable random variable F at a time t≤T as an exponential formula involving Malliavin derivatives evaluated along a frozen path. The series representation of this exponential formula turns out to...

In this paper, we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs [(2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation, Finance...

We obtain a Hull and White type option price decomposition for a general local volatility model. We apply the obtained formula to CEV model. As an application we give an approximated closed formula for the call option price under a CEV model and an approximated short term implied volatility surface. These approximated formulas are used to estimate...

The paper [12] examines a concept of equilibrium policies instead of optimal controls in stochastic optimization to analyze a mean-variance portfolio selection problem. We follow the same approach in order to investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that give rise to time-inconsist...

A filtered process Xk is defined as an integral of a deterministic kernel k with respect to a stochastic process X. One of the main problems to deal with such processes is to define a stochastic integral with respect to them. When X is a Brownian motion one can use the Gaussian properties of Xk to define an integral intrinsically. When X is a jump...

We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model, extending a previous decomposition formula for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used for this purpose are non-anticipative. In particular, we also s...

The goal of this survey article is to present in detail a method that, for a financial derivative under a certain stochastic volatility model, allows to obtain a decomposition of its pricing formula that distinguishes clearly the impact of correlation and jumps. This decomposed pricing formula, usually called Hull and White type formula, can be pot...

In this paper, we present a new, simple and efficient calibration procedure that uses both the short and long-term behavior of the Heston model in a coherent fashion. Using a suitable Hull and White-type formula, we develop a methodology to obtain an approximation to the implied volatility. Using this approximation, we calibrate the full set of par...

We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model, extending a previous decomposition formula for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used for this purpose are nonanticipative. In particular, we also se...

We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model extending the decomposition obtained by E. Al\`os in [2] for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used are non anticipative. In particular, we see also t...

In this paper we develop a Malliavin-Skorohod type calculus for additive processes in the $L^0$ and $L^1$ settings, extending the probabilistic interpretation of the Malliavin-Skorohod operators to this context. We prove calculus rules and obtain a generalization of the Clark-Hausmann-Ocone formula for random variables in $L^1$. Our theory is then...

In this paper, we obtain sharp asymptotic formulas with error estimates for the Mellin convolution of functions, and use these formulas to characterize the asymptotic behavior of marginal distribution densities of stock price processes in mixed stochastic models. Special examples of mixed models are jump-diffusion models and stochastic volatility m...

Since Itô (1956) it is known that Lévy processes enjoy the chaotic representation property in a certain generalized form. In other words, the space of square integrable functionals of a certain independent random measure associated to a Lévy process has Fock space structure. The Fock space structure gives the possibility to develop a formal calculu...

In this paper a Malliavin calculus for L\'evy processes based on a family of true derivative operators is developed. The starting point is an extension to L\'evy processes of the pioneering paper by Carlen and Pardoux [8] for the Poisson process, and our approach includes also the classical Malliavin derivative for Gaussian processes. We obtain a s...

We study the existence of a unique solution for linear stochastic differential equations driven by a Lévy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying filtration. Towards this end, we extend a method based on Girsanov transformation on Wiener space and developed by R. Buckdahn [P...

The present paper is devoted to applications of mathematical analysis to the study of distribution densities arising in stochastic stock price models. We consider uncorrelated Stein-Stein, Heston, and Hull-White models and their perturbations by compound Poisson processes with jump amplitudes distributed according to a double exponential law. Simil...

We consider uncorrelated Stein-Stein, Heston, and Hull-White models and their perturbations by compound Poisson processes with jump amplitudes distributed according to a double exponential law. Similar perturbations of the Black-Scholes model were studied by S. Kou. For perturbed stochastic volatility models, we obtain two-sided estimates for the s...

In this paper, generalizing results in Alòs, León and Vives (2007b), we see that the dependence of jumps in the volatility under a jump-diffusion stochastic volatility model, has no effect on the short-time behaviour of the at-the-money implied volatility skew, although the corresponding Hull and White formula depends on the jumps. Towards this end...

In this paper, we use the Malliavin calculus techniques to obtain an anticipative version of the change of variable formula for Lévy processes. Here the coefficients are in the domain of the anihilation (gradient) operator in the "future sense", which includes the family of all adapted and square-integrable processes. This domain was introduced on...

We obtain a Hull and White type formula for a general jump-diffusion stochastic volatility model, where the involved stochastic volatility process is correlated not only with the Brownian motion driving the asset price but also with the asset price jumps. Towards this end, we establish an anticipative Itô's formula, using Malliavin calculus techni...

We study the existence and uniqueness of pathwise solutions to backward and forward stochastic differential equations on the Poisson space. We obtain the structure of these pathwise solutions to give the relationship between them. Also, in the bilinear case, we calculate the explicit form of their chaos decompositions.

A suitable canonical Lévy process is constructed in order to study a Malliavin calculus based on a chaotic representation property of Lévy processes proved by Itô using multiple two-parameter integrals. In this setup, the two-parameter derivative Dt,x is studied, depending on whether x=0 or x[not equal to]0; in the first case, we prove a chain rule...

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process as the examples in section 7 show. This expression depends on the derivative of...

We study the asymptotic behavior in Sobolev norm of the local time of the $d$-dimensional fractional Brownian motion with $N$-parameters when the space variable tends to zero, both for the fixed time case and when simultaneously time tends to infinity and space variable to zero.

Without Abstract

According to many recent studies, Lévy processes with stochastic volatility seem to be the best candidates for replacing geometric Brownian motion (GBM) as a price process model. This means that the GBM model has to be generalised by introducing the possibility of jumps and allowing the volatility to be a stochastic process. In this paper, we prese...

We find a Stroock formula in the setting of generalized chaos expansion introduced by Nualart and Schoutens for a certain class of Lévy processes, using a Malliavin-type deriva-tive based on the chaotic approach. As applications, we get the chaotic decomposition of the local time of a simple Lévy process as well as the chaotic expansion of the pric...

We give the Wiener-Ito chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters and study its smoothness in the Sobolev-Watanabe spaces.

This paper deals with some additive functionals of the fractional Brownian motion that arise as limits in law of some occupation times of this process. In concrete, this functionals are obtained via the Cauchy principal value and the Hadamard finite part. We derive some regularity properties of theses functionals in Sobolev-Watanabe sense.

Recent work by Nualart and Schoutens (2000), where a kind of chaotic property for Lévy processes has been proved, has enabled us to develop a Malliavin calculus for Lévy processes. For simple Lévy processes some useful formulas for computing Malliavin derivatives are deduced. Applications for option hedging in a jump-diffusion model are given.

In this paper we establish the conditions on a L2-process u for the existence of its anticipating Stratonovich integral with respect to a normal martingale belonging to a certain class. This class includes the Azema's martingales and the compensated Poisson processes.

This paper aims to study the possibility to define anticipating Integrals with respect to an indefinite Skorohod integral with respect to the Poisson process. Different Stratonovich and Skorohod type integrals are introduced and their relationships are studied. Our approach extends the results of the adapted and non-adapted stochastic calcutus with...

We derive the chaotic expansion of the product of nth- and first-order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulae for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits pr...

We study the relationship between the translation operator, its dual and the pathwise integral on the Poisson space with weak conditions on the processes.

We derive the chaotic expansion of the product of n-th and first order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulas for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits pr...

In the mathematical treatment of financial derivatives, and specially that of options, the defining stochastic differential equation coupled with the arbitrage-free pricing condition leads to a deterministic partial differential equation. The solution of this equation under appropriate boundary conditions is interpreted as the price of the asset. A...

Double intersection local times α(x,.) of Brownian motion which measure the size of the set of time pairs (s, t), s ≠ t, for which Wt and Ws + x coincide can be developed into series of multiple Wiener-Ito integrals. These series representations reveal on the one hand the degree of smoothness of α(x,.) in terms of eventually negative order Sobolev...

We establish a duality formula for the chaotic derivative operator on the canonical Poisson space. The adjoint of this operator is proved to coincide with the stochastic integration on predictable processes.

In this paper we show that the local time of the Brownian motion belongs to the Sobolev space \mathbbDa\text,p\mathbb{D}^{\alpha {\text{,}}p} for any p \mathbbD - a\text,p\mathbb{D}^{ - \alpha {\text{,}}p} , for any >0 and p>1 such that +1/p>1.

We study the relationship between the translation operator, its dual and the pathwise integral on the Poisson space with weak conditions on the processes.