162 research outputs found
Dirichlet forms methods, an application to the propagation of the error due to the Euler scheme
We present recent advances on Dirichlet forms methods either to extend
financial models beyond the usual stochastic calculus or to study stochastic
models with less classical tools. In this spirit, we interpret the asymptotic
error on the solution of an sde due to the Euler scheme in terms of a Dirichlet
form on the Wiener space, what allows to propagate this error thanks to
functional calculus.Comment: 15
An Application of Malliavin Calculus to Hedging Exotic Barrier Options
The thesis uses Malliavinâs Stochastic Calculus of Variations to identify the hedging strategies
for Barrier style derived securities. The thesis gives an elementary treatment of this
calculus which should be accessible to the non-specialist. The thesis deals also with extensions
of the calculus to the composition of a Generalized Function and a Stochastic
Variable which makes it applicable to the discontinuous payoffs encountered with Barrier
Structures. The thesis makes a mathematical contribution by providing an elementary
calculus for the composition of a Generalized function with a Stochastic Variable in the
presence of a conditional expectation
Topics in computational finance: : The Barndorff-Nielsen & Shepard Stochastic volatility model
Hur uppför sig aktiepriser? den frÄgan stÀllde sig forskarna Ole Barndorff-Nielsen och Neil Shephard i början av detta Ärtusende. Resultatet av deras funderingar blev den matematiska modell för aktiepriser som nu bÀr deras namn. Aktiepriser antas av tradition kunna beskrivas av en normalfördelning, vilket grundliga studier emellertid har visat Àr en bristfÀllig antagelse. Risken för stora prisförÀndringar blir klart undervÀrderad, dvs man kan inte förklara de svarta dagarna pÄ marknaden dÄ börsen rasar. Dessutom visar sig förÀndringarna ofta inte vara jÀmnt fördelade. Vi ser smÄ stegvisa ökningar och stora ras. Stora förÀndringar leder ocksÄ ofta till stor aktivitet och aktiviteten pÄ marknaden tenderar att vara stor i perioder för att sedan lugna ner sig.
Alla dessa saker kan inte förklaras med den klassiska teorin för aktiepriser men fÄr med hjÀlp av Barndorff-Nielsen och Shephards model en teoretisk förklaring. Denna avhandling studerar vilka konsekvenser det fÄr för optionshandeln om vi ersÀtter den klassiska teorin med en modell som bÀttre beskriver verkligheten. En bÀttre beskrivelse innebÀr fÀrre förenklingar vilket medför en mer komplicerad modell. Den ökade komplexiteten medför att vi enbart kan lösa problemen genom att anvÀnda avancerade datorsimuleringar. Tyngdpunkten i arbetet ligger pÄ att utarbeta och anvÀnda metoder för att berÀkna priser pÄ optioner och andra finansiella kontrakt om vi antar att aktiepriser förklaras av BNS-modellen. Resultaten visar att vi fÄr priser som ligger nÀrmare de verkliga marknadspriserna och vi försöker utifrÄn resultaten dra slutsatser om investerarnas preferenser. Arbetet Àr utfört vid Centre of Mathematics for Applications under handledning av Fred Espen Benth, Professor i Finansmatematikk
Quantum computing for finance
Quantum computers are expected to surpass the computational capabilities of
classical computers and have a transformative impact on numerous industry
sectors. We present a comprehensive summary of the state of the art of quantum
computing for financial applications, with particular emphasis on stochastic
modeling, optimization, and machine learning. This Review is aimed at
physicists, so it outlines the classical techniques used by the financial
industry and discusses the potential advantages and limitations of quantum
techniques. Finally, we look at the challenges that physicists could help
tackle
Application of Malliavin Calculus and Wiener chaos to option pricing theory.
This dissertation provides a contribution to the option pricing literature by means of some recent developments in probability theory, namely the Malliavin Calculus and the Wiener chaos theory. It concentrates on the issue of faster convergence of Monte Carlo and Quasi-Monte Carlo simulations for the Greeks, on the topic of the Asian option as well as on the approximation for convexity adjustment for fixed income derivatives. The first part presents a new method to speed up the convergence of Monte- Carlo and Quasi-Monte Carlo simulations of the Greeks by means of Malliavin weighted schemes. We extend the pioneering works of Fournie et al. (1999), (2000) by deriving necessary and sufficient conditions for a function to serve as a weight function and by providing the weight function with minimum variance. To do so, we introduce its generator defined as its Skorohod integrand. On a numerical example, we find evidence of spectacular efficiency of this method for corridor options, especially for the gamma calculation. The second part brings new insights on the Asian option. We first show how to price discrete Asian options consistent with different types of underlying densities, especially non-normal returns, by means of the Fast Fourier Transform algorithm. We then extends Malliavin weighted schemes to continuous time Asian options. In the last part, we first prove that the Black Scholes convexity adjustment (Brotherton-Ratcliffe and Iben (1993)) can be consistently derived in a martingale framework. As an application, we examine the convexity bias between CMS and forward swap rates. However, for more complicated term structures assumptions, this approach does not hold any more. We offer a solution to this, thanks to an approximation formula, in the case of multi-factor lognormal zero coupon models, using Wiener chaos theory
Pathwise functional calculus and applications to continuous-time finance
This thesis develops a mathematical framework for the analysis of continuous-
time trading strategies which, in contrast to the classical setting of
continuous-time finance, does not rely on stochastic integrals or other probabilistic
notions.
Using the recently developed `non-anticipative functional calculus', we
first develop a pathwise definition of the gain process for a large class of
continuous-time trading strategies which include the important class of delta-hedging
strategies, as well as a pathwise definition of the self-financing condition.
Using these concepts, we propose a framework for analyzing the performance
and robustness of delta-hedging strategies for path-dependent derivatives
across a given set of scenarios. Our setting allows for general path-dependent
payoffs and does not require any probabilistic assumption on the
dynamics of the underlying asset, thereby extending previous results on robustness
of hedging strategies in the setting of diffusion models. We obtain a
pathwise formula for the hedging error for a general path-dependent derivative
and provide sufficient conditions ensuring the robustness of the delta
hedge. We show in particular that robust hedges may be obtained in a large
class of continuous exponential martingale models under a vertical convexity
condition on the payoffs functional. Under the same conditions, we show that
discontinuities in the underlying asset always deteriorate the hedging performance.
These results are applied to the case of Asian options and barrier
options.
The last chapter, independent of the rest of the thesis, proposes a novel
method, jointly developed with Andrea Pascucci and Stefano Pagliarani, for
analytical approximations in local volatility models with L\ue9vy jumps. The
main result is an expansion of the characteristic function in a local L\ue9vy
model, which is worked out in the Fourier space by considering the adjoint
formulation of the pricing problem. Combined with standard Fourier methods,
our result provides effcient and accurate pricing formulae. In the case
of Gaussian jumps, we also derive an explicit approximation of the transition
density of the underlying process by a heat kernel expansion; the approximation
is obtained in two ways: using PIDE techniques and working in the
Fourier space. Numerical tests confirm the effectiveness of the method
Topics in volatility models
In this thesis I will present my PhD research work, focusing mainly on financial
modelling of assetâs volatility and the pricing of contingent claims (financial derivatives),
which consists of four topics:
1. Several changing volatility models are introduced and the pricing of European
options is derived under these models;
2. A general local stochastic volatility model with stochastic interest rates (IR)
is studied in the modelling of foreign exchange (FX) rates. The pricing of FX
options under this model is examined through the use of an asymptotic expansion
method, based on Watanabe-Yoshida theory. The perfect/partial hedging issues
of FX options in the presence of local stochastic volatility and stochastic IRs are
also considered. Finally, the impact of stochastic volatility on the pricing of FX-IR
structured products (PRDCs) is examined;
3. A new method of non-biased Monte Carlo simulation for a stochastic volatility
model (Heston Model) is proposed;
4. The LIBOR/swap market model with stochastic volatility and jump processes
is studied, as well as the pricing of interest rate options under that model.
In conclusion, some future research topics are suggested.
Key words: Changing Volatility Models, Stochastic Volatility Models, Local
Stochastic Volatility Models, Hedging Greeks, Jump Diffusion Models, Implied
Volatility, Fourier Transform, Asymptotic Expansion, LIBOR Market Model, Monte
Carlo Simulation, Saddle Point Approximation
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