Option pricing in fractional models

This thesis deals with application of the fractional Black-Scholes and mixed fractional Black-Scholes models to evaluate different type of options. These assessments are considered in four individual papers. In the first articles, the problem of geometric Asian and power options pricing is investigated when the stock price follows a time changed mixed fractional model. In this model, an inverse subordinator process in the mixed fractional Black-Scholes model replaces the physical time. The aim of the third paper is to evaluate the European currency option in a fractional Brownian motion environment by the time-changed strategy. Also, the impact of time step and long range dependence are obtained under transaction costs. Conditional mean hedging under fractional Black-Scholes model is the propose of the second article. The conditional mean hedge of the European vanilla type option with convex or concave positive payoff under transaction costs is obtained. In the fourth article, the mixed fractional Brownian motion with jump process are incorporated to analyze European options in discrete time case. By a mean delta hedging strategy, the pricing model is proposed for European option under transaction costs.Väitöskirja tarkastelee fraktionaalisen Black–Scholes -mallin ja sekoitetun fraktionallisen Black–Scholes -mallin käyttöä erityyppisten optioiden arvottamisessa. Tätä tutkitaan neljässä artikkelissa. Ensimmäisessä artikkelissa tarkastellaan geometrisia aasialaisia optioita ja potenssioptioita, kun osakehinta noudattaa aikamuunnettua sekoitettua fraktionaalista mallia. Tässä mallissa sekoitun fraktionaalisen Black–Scholes -mallin käänteinen subordinaattoriprosessi korvaa fysikaalisen ajan. Kolmannen artikkelin tarkoitus on hinnoitella eurooppalainen valuuttaoptio fraktionaalisen Brownin liikkeen mallissa aikamuunnetulla strategialla. Lisäksi aika-askeleen ja pitkän aikavälin riippuvuuden vaikutusta tutkitaan transaktiokulujen alaisuudessa. Ehdollinen keskiarvosuojaaminen fraktionaalisessa Black–Sholes -mallissa on toisen artikkelin aihe. Ehdollinen keskiarvosuojaus eurooppalaiselle vaniljaoptiolle, jolla on konveksi tai konkaavi positiivinen tuottofunktio transaktiokulujen vallitessa, on artikkelin päätulos. Neljännessä artikkelissa tutkitaan eurooppalaisia optioita diskreetissä ajassa mallissa, joka on hypyllinen sekoitettu fraktionaalinen Brownin liike. Käyttäen keskiarvoista deltasuojausstrategiaa artikkelissa johdetaan hinnoittelumalli eurooppalaisille optioille transaktiokulujen vallitessa.fi=vertaisarvioitu|en=peerReviewed

Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method

This work deals with the put option pricing problems based on the time-fractional Black-Scholes equation, where the fractional derivative is a so-called modified Riemann-Liouville fractional derivative. With the aid of symbolic calculation software, European and American put option pricing models that combine the time-fractional Black-Scholes equation with the conditions satisfied by the standard put options are numerically solved using the implicit scheme of the finite difference method

The random diffusivity approach for diffusion in heterogeneous systems.

164 p.The two hallmark features of Brownian motion are the linear growth of the meansquared displacement (MSD) with diffusion coefficient D in d spatial dimensions, andthe Gaussian distribution of displacements. With the increasing complexity of thestudied systems deviations from these two central properties have been unveiledover the years. Recently, a large variety of systems have been reported in which theMSD exhibits the linear growth in time of Brownian (Fickian) transport, however, thedistribution of displacements is pronouncedly non-Gaussian (Brownian yet non-Gaussian, BNG). A similar behaviour is also observed for viscoelastic-type motionwhere an anomalous trend of the MSD is combined with a priori unexpected non-Gaussian distributions (anomalous yet non-Gaussian, ANG). This kind of behaviourobserved in BNG and ANG diffusions has been related to the presence ofheterogeneities in the systems and a common approach has been established toaddress it, that is, the random diffusivity approach.This dissertation explores extensively the field of random diffusivity models. Startingfrom a chronological description of all the main approaches used as an attempt ofdescribing BNG and ANG diffusion, different mathematical methodologies aredefined for the resolution and study of these models.The processes that are reported in this work can be classified in threesubcategories, i) randomly-scaled Gaussian processes, ii) superstatistical modelsand iii) diffusing diffusivity models, all belonging to the more general class of randomdiffusivity models.Eventually, the study focuses more on BNG diffusion, which is by now wellestablishedand relatively well-understood. Nevertheless, many examples arediscussed for the description of ANG diffusion, in order to highlight the possiblescenarios which are known so far for the study of this class of processes.The second part of the dissertation deals with the statistical analysis of randomdiffusivity processes. A general description based on the concept of momentgeneratingfunction is initially provided to obtain standard statistical properties of themodels. Then, the discussion moves to the study of the power spectral analysis andthe first passage statistics for some particular random diffusivity models. Acomparison between the results coming from the random diffusivity approach andthe ones for standard Brownian motion is discussed. In this way, a deeper physicalunderstanding of the systems described by random diffusivity models is alsooutlined.To conclude, a discussion based on the possible origins of the heterogeneity issketched, with the main goal of inferring which kind of systems can actually bedescribed by the random diffusivity approach

The random diffusivity approach for diffusion in heterogeneous systems

The two hallmark features of Brownian motion are the linear growth $\langle x^2(t) \rangle = 2 D d t$ of the mean squared displacement (MSD) with diffusion coefficient $D$ in $d$ spatial dimensions, and the Gaussian distribution of displacements. With the increasing complexity of the studied systems deviations from these two central properties have been unveiled over the years. Recently, a large variety of systems have been reported in which the MSD exhibits the linear growth in time of Brownian (Fickian) transport, however, the distribution of displacements is pronouncedly non-Gaussian (Brownian yet non-Gaussian, BNG). A similar behaviour is also observed for viscoelastic-type motion where an anomalous trend of the MSD, i.e., $\langle x^2(t) \rangle \sim t^\alpha$, is combined with a priori unexpected non-Gaussian distributions (anomalous yet non-Gaussian, ANG). This kind of behaviour observed in BNG and ANG diffusions has been related to the presence of heterogeneities in the systems and a common approach has been established to address it, that is, the random diffusivity approach. This dissertation explores extensively the field of random diffusivity models. Starting from a chronological description of all the main approaches used as an attempt of describing BNG and ANG diffusion, different mathematical methodologies are defined for the resolution and study of these models. The processes that are reported in this work can be classified in three subcategories, i) randomly-scaled Gaussian processes, ii) superstatistical models and iii) diffusing diffusivity models, all belonging to the more general class of random diffusivity models. Eventually, the study focuses more on BNG diffusion, which is by now well-established and relatively well-understood. Nevertheless, many examples are discussed for the description of ANG diffusion, in order to highlight the possible scenarios which are known so far for the study of this class of processes. The second part of the dissertation deals with the statistical analysis of ran- dom diffusivity processes. A general description based on the concept of moment- generating function is initially provided to obtain standard statistical properties of the models. Then, the discussion moves to the study of the power spectral analysis and the first passage statistics for some particular random diffusivity models. A comparison between the results coming from the random diffusivity approach and the ones for standard Brownian motion is discussed. In this way, a deeper physical understanding of the systems described by random diffusivity models is also outlined. To conclude, a discussion based on the possible origins of the heterogeneity is sketched, with the main goal of inferring which kind of systems can actually be described by the random diffusivity approach.BERC.2018-202

Fitting a function to time-dependent ensemble averaged data

Time-dependent ensemble averages, i.e., trajectory-based averages of some observable, are of importance in many fields of science. A crucial objective when interpreting such data is to fit these averages (for instance, squared displacements) with a function and extract parameters (such as diffusion constants). A commonly overlooked challenge in such function fitting procedures is that fluctuations around mean values, by construction, exhibit temporal correlations. We show that the only available general purpose function fitting methods, correlated chi-square method and the weighted least squares method (which neglects correlation), fail at either robust parameter estimation or accurate error estimation. We remedy this by deriving a new closed-form error estimation formula for weighted least square fitting. The new formula uses the full covariance matrix, i.e., rigorously includes temporal correlations, but is free of the robustness issues, inherent to the correlated chi-square method. We demonstrate its accuracy in four examples of importance in many fields: Brownian motion, damped harmonic oscillation, fractional Brownian motion and continuous time random walks. We also successfully apply our method, weighted least squares including correlation in error estimation (WLS-ICE), to particle tracking data. The WLS-ICE method is applicable to arbitrary fit functions, and we provide a publically available WLS-ICE software.Comment: 47 pages (main text: 15 pages, supplementary: 32 pages