Fractional diffusion models of option prices in markets with jumps.

Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Lévy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Lévy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derivedFractional-Black–Scholes; Lévy-stable processes; FMLS; KoBoL; CGMY; Fractional calculus; Riemann–Liouville fractional derivative; Barrier options; Down-and-out; Up-and-out; Double knock-out;

Fractional diffusion models of option prices in markets with jumps.

Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Lévy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Lévy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derivedFractional-Black-Scholes; Lévy-Stable processes; FMLS; KoBoL; CGMY; Fractional calculus; Riemann-Liouville fractional derivative; Barrier options; Down-and-out; Up-and-out; Double knock-out;

Crossover between Levy and Gaussian regimes in first passage processes

We propose a new approach to the problem of the first passage time. Our method is applicable not only to the Wiener process but also to the non--Gaussian L$\acute{\rm e}$vy flights or to more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we particularly focus on the first passage time problems in the truncated L$\acute{\rm e}$vy flights (the so-called KoBoL processes), in which the arbitrarily large tail of the L$\acute{\rm e}$vy distribution is cut off. We find that the asymptotic scaling law of the first passage time $t$ distribution changes from $t^{-(\alpha +1)/\alpha}$-law (non-Gaussian L$\acute{\rm e}$vy regime) to $t^{-3/2}$-law (Gaussian regime) at the crossover point. This result means that an ultra-slow convergence from the non-Gaussian L$\acute{\rm e}$vy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated L$\acute{\rm e}$vy flight but also in the first passage time distribution of the flight. The nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cut-off length of the L$\acute{\rm e}$vy distribution are discussed.Comment: 18pages, 7figures, using revtex4, to appear in Phys.Rev.

Valuation Of Continuously Monitored Double Barrier Options And Related Securities

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/92059/1/j.1467-9965.2010.00469.x.pd

Fractional diffusion models and option pricing in jump models

Mestrado em Mathematical FinanceO problema de valorização de derivados tem sido o foco da investigação em Matemática Financeira desde a sua conceção. Mais recentemente, a literatura tem-se focado por exemplo em modelos que assumem que as dinâmicas do preço do ativo subjacente são governadas por um processo de Lévy (por vezes chamado um processo com saltos). Este tipo de modelo admite a possibilidade de eventos extremos (saltos), que não são devidamente capturados por modelos clássicos do tipo Black-Scholes, alicerçados no movimento Browniano. Foi também demonstrado ao longo da última década que se as dinâmicas do preço do ativo subjacente seguem certos processos de Lévy, tais como o CGMY , o FMLS e o KoBoL, os preços das opções satisfazem uma equação diferencial parcial fracionária. Nesta dissertação, iremos mostrar que se as dinâmicas do ativo subjacente seguem o denominado Processo Estável Temperado Generalizado, que admite como caso particular os suprareferidos processos CGMY e KoBoL, então os preços das opções satisfazem igualmente uma equação diferencial parcial fracionária. Além disso, iremos implementar um método simples de diferenças finitas para resolver numericamente a equação deduzida, e valorizar opções do tipo europeu.The problem of pricing financial derivatives has been the focal point of research within the field of Mathematical Finance since its conception. In recent years, one of the main areas of focus within the literature has been on models which assume that the dynamics of the price of the underlying asset are governed by a Lévy process (sometimes referred to as a jump process). This type of model admits the possibility of extreme events (jumps), which are not captured by classical Black-Scholes type models based on the Brownian motion. Over the last decades, the literature has further shown that if the dynamics of the price of the underlying is governed by certain Lévy processes, such as the CGMY , the FMLS and the KoBoL, the price processes of European-style options satisfy a variety of fractional partial differential equations (FPDEs). In this dissertation, we will show that if the underlying price dynamic follows a Generalized Tempered Stable process, which admits as particular cases the aforementioned CGMY and KoBoL processes, prices of options satisfy an FPDE of the same type. Further, we will implement a simple finite difference scheme to solve the FPDE numerically to price European-type options.info:eu-repo/semantics/publishedVersio

A comparison of numerical methods to solve fractional partial differential equations

A comparison of two numerical methods - ¯nite di®erence and Adomian decomposition method (ADM) - to solve a variety of fractional partial dif- ferential equations that occur in ¯nance are investigated. These fractional partial di®erential equations fall into the class of L¶evy models. They are known as the Finite Moment Log Stable (FMLS), CGMY and the extended Koponen (KoBol) models. Convergence criteria for these models under the numerical methods are studied. ADM fails to accurately price a claim writ- ten on these models. However, the ¯nite di®erence scheme works well for the FMLS and KoBol models