Moment Methods for Exotic Volatility Derivatives

Albanese, Claudio; Osseiran, Adel

doi:10.2139/ssrn.1021401

Cited by 5 publications

(6 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The moment of order k is m k (ξ 3 ) = (3k)! and the density to be used next is f (x) = 1 3 x −2/3 e −x 1/3 , x > 0. Take this f as a density of X 0 , calculate v, write the SDE and get a stationary diffusion Markov process X with all required properties.…”

Section: None Is Exponentially Integrabilitymentioning

confidence: 99%

See 1 more Smart Citation

Moment Properties of Probability Distributions Used in Stochastic Financial Models

Stoyanov¹

2016

Recent Advances in Financial Engineering 2014

View full text Add to dashboard Cite

Our goal in this paper is to look at stochastic financial models and especially on those properties of the involved distributions which are expressed in terms of the moments. The questions discussed and the results presented reveal the role which the moments play in the analysis of distributions. Interesting conclusions can be derived in both cases when available are finitely many moments and when we know all moments.Among the results included in the paper are sharp lower and/or upper bounds for option prices in terms of a finite number of moments. However the main attention is paid to the determinacy of distributions by their moments. While any light tailed distribution is uniquely determined by its moments, the uniqueness may fail for heavy tailed distributions. And, here is the point. Heavy tailed distributions are essentially involved in stock market modelling, and most of them are nonunique in terms of the moments. This is why the phenomenon non-uniqueness of distributions deserves a special attention.We treat distributions on the positive half-line used to describe, e.g., stock prices and option prices, and distributions on the whole real line describing logreturns. For reader's convenience we give a brief and unified general picture of existing results about uniqueness and non-uniqueness of distribution in terms of their moments. Some of the results are classical and well-known. In several cases we provide here new arguments along with presenting new recent results on the moment determinacy of random variables and their non-linear transformations and also of stochastic processes which are solutions to SDEs.

show abstract

Section: None Is Exponentially Integrabilitymentioning

confidence: 99%

“…It turns out that the moments of distributions can be involved in option pricing when following semi-definite optimization methods. Pricing of options based on moments is treated in [1,24,32,38,60].…”

Section: Final Comments 91 Briefly On Relevant Topics Not Discussed Herementioning

confidence: 99%

Moment Properties of Probability Distributions Used in Stochastic Financial Models

Stoyanov¹

2016

Recent Advances in Financial Engineering 2014

View full text Add to dashboard Cite

show abstract

“…This section is based on work in collaboration with Adel Osseiran, see (Albanese and Osseiran 2007).…”

Section: Univariate Moment Expansions On Bridgesmentioning

confidence: 99%

“…In Finance, operator methods have been developed along two independent and non-overlapping streams of research, one by Ait-Sahalia, Hansen and Scheinkman who focused on econometric estimations in a series of papers reviewed in (Ait-Sahalia et al 2005), see also (Ait-Sahalia 1996), (Hansen et al 1998), (Hansen and Scheinkman 1995). The second stream of research is by the author and collaborators who instead worked on derivative pricing for path dependent and correlation derivatives, see (Albanese et al -2006b, (Albanese and Chen 2004a), (Albanese and Chen 2004b), (Albanese and Kusnetsov 2005), (Albanese and Lawi 2004), (Albanese et al 2006a), (Albanese and Trovato 2006), (Albanese and Trovato 2005), (Albanese and Vidler 2007), (Albanese and Jones 2007) and (Albanese and Osseiran 2007). In this paper, we attempt to systematize the mathematical framework of pricing theory in the operator formalism from our own viewpoint, reserving to future work the task of pursuing overlaps with the econometric literature.…”

mentioning

confidence: 99%

Operator Methods, Abelian Processes and Dynamic Conditioning

Albanese¹

2007

SSRN Journal

Self Cite

View full text Add to dashboard Cite

A mathematical framework for Continuous Time Finance based on operator algebraic methods offers a new direct and entirely constructive perspective on the field. It also leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are uniquely suited to execute large matrix manipulations. This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can effectively use block-factorization algorithms. Finally, we discuss the method of dynamic conditioning that allows one to dynamically correlate over possibly even hundreds of processes in a numerically noiseless framework while preserving marginal distributions.

show abstract

“…Several instances of these processes found engineering applications already, see (Albanese and Vidler 2007), (Albanese and Trovato 2005), (Albanese 2007a), (Albanese et al 2006), (Albanese and Osseiran 2007). As an example, we discuss here the case of the sup processes of a one-dimensional diffusion, i.e.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Integrals and Abelian Processes

Albanese¹

2008

SSRN Journal

Self Cite

View full text Add to dashboard Cite

Abstract. We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coefficients and an adapted, non-resonant Abelian process. The prototypical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coefficients. The range of applicability includes also a broader class of processes of practical relevance, such as the sup process and certain discrete time summations we discuss.We discretize the space coordinate in uniform steps and assume that time is either continuous or finely discretized as in a fully explicit Euler method and the Courant condition is satisfied. We show that the Fourier transform of the joint kernel of a diffusion and a stochastic integral converges in a uniform graph norm associated to the Markov generator. Convergence also implies smoothness properties for the Fourier transform of the joint kernel. Stochastic integrals are straightforward to define for finite triangulations and the convergence result gives a new and entirely constructive way of defining stochastic integrals in the continuum. The method relies on a reinterpretation and extension of the classic theorems by Feynman-Kac, Girsanov, Ito and Cameron-Martin, which are also reobtained.We make use of a path-wise analysis without relying on a probabilistic interpretation. The Fourier representation is needed to regularize the hypo-elliptic character of the joint process of a diffusion and an adapted stochastic integral. The argument extends as long as the Fourier analysis framework can be generalized. This condition leads to the notion of non-resonant Abelian process.

show abstract

Moment Methods for Exotic Volatility Derivatives

Cited by 5 publications

References 6 publications

Moment Properties of Probability Distributions Used in Stochastic Financial Models

Moment Properties of Probability Distributions Used in Stochastic Financial Models

Operator Methods, Abelian Processes and Dynamic Conditioning

Stochastic Integrals and Abelian Processes

Contact Info

Product

Resources

About