Local Volatility

Akademisk avhandling som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras tisdagen den 29 november 2022, 13.15 i Kappa, Mälardalens universitet, Västerås.

Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volati...

Akademisk avhandling som för avläggande av filosofie doktorsexamen i matematik/tillämpad matematik vid Akademin för utbildning, kultur och kommunikation kommer att offentligen försvaras tisdagen den 29 november 2022, 13.15 i Kappa, Mälardalens universitet, Västerås.

We present an efficient and accurate computational algorithm for reconstructing a local volatility surface from given market option prices. The local volatility surface is dependent on the values of both the time and underlying asset. We use the generalized Black-Scholes (BS) equation and finite difference method (FDM) to numerically solve the generalized BS equation. We reconstruct the local volatility function, which provides the best fit between the theoretical and market option prices by minimizing a cost function that is a quadratic representation of the difference between the two option prices. This is an inverse problem in which we want to calculate a local volatility function consistent with the observed market prices. To achieve robust computation, we place the sample points of the unknown volatility function in the middle of the expiration dates. We perform various numerical experiments to confirm the simplicity, robustness, and accuracy of the proposed method in reconstructing the local volatility function.

The original Ait-Sahalia model of the spot interest rate proposed by Ait-Sahalia assumes constant volatility. As supported by several empirical studies, volatility is never constant in most financial markets. From application viewpoint, it is important we generalise the Ait-Sahalia model to incorporate volatility as a function of delay in the spot rate. In this paper, we study analytical properties for the true solution of this model and construct several new techniques of the truncated Euler-Maruyama (EM) method to study properties of the numerical solutions under the local Lipschitz condition plus Khasminskii-type condition. Finally, we justify that the truncated EM approximate solution can be used within a Monte Carlo scheme for numerical valuations of some financial instruments such as options and bonds.

This paper presents an original probabilistic method for the numerical computations of Greeks (i.e. price sensitivities) in finance. Our approach is based on the integration-by-parts formula, which lies at the core of the theory of variational stochastic calculus, as developed in the Malliavin calculus. The Greeks formulae, both with respect to initial conditions and for smooth perturbations of the local volatility, are provided for general discontinuous path-dependent payoff functionals of multidimensional diffusion processes. We illustrate the results by applying the formula to exotic European options in the framework of the Black and Scholes model. Our method is compared to the Monte Carlo finite difference approach and turns out to be very efficient in the case of discontinuous payoff functionals.

We provide a thorough explanation of stochastic (random) processes, specifically Brownian motion, before rigorously introducing Itô Calculus. This allows us to evaluate integrals with respect to Brownian motion and solve stochastic differential equations (SDE). We then develop the SDE and measure-theory machinery to derive the famous Black-Scholes model for pricing options. A detailed overview of the Girsanov Theorem, the Feynman-Kac Formula, and the concept of arbitrage is included to tie together intuition, theory, and application.

Models for pricing interest rate claims, developed under the Heath-Jarrow-Morton paradigm, differ according to the volatility structure imposed on forward rates. For most general HJM structures the resultant path dependence creates implementation problems. Ritchken and Sankarasubramanian have recently identified necessary and sufficient conditions on the class of volatility structures of forward rates that enable the term structure dynamics to be captured by a finite set of state variables. The class is quite rich. The instantaneous spot rate volatility may be quite general, but the model curtails the structure of forward rate volatilities relative to this spot rate volatility. This paper provides empirical tests for this class of volatility structures. Unlike other studies, the volatility structure is examined over the a broad section of maturities in the yield curve. Using Treasury data over the period 1982-1994, we find support for this class. Furthermore, unlike other studies, n...

We consider a regime switching stochastic volatility model where the stock volatility dynamics is a semi-Markov modulated square root mean reverting process. Under this model assumption, we find the locally risk minimizing price of European type vanilla options. The price function is shown to satisfy a non-local degenerate parabolic PDE which can be viewed as a generalization of the Heston PDE. The related Cauchy problem involving the PDE is shown to be equivalent to an integral equation(IE). The existence and uniqueness of solution to the PDE is carried out by studying the IE and using the semigroup theory.

Recently, the Johannesburg Stock Exchange (JSE) launched a new range of exotic products, namely the Can-Do. The main idea behind the Can-Do products is to help financial institutions list some of their exotic options on the exchange. Cando’s are market to model. These products are all modelled with the celebrated Black-Scholes (Black & Scholes, 1973) partial differential equation (PDE) but with exotic features. Due to the inherent complexity of these features in the modeling equations, one can rarely find closed-form analytical solutions to these options. Therefore one has to resort to numerical methods. In the literature, four main families of methods where developed and are used extensively to solve option pricing problems. These include lattice methods (Breen, 1991), (Cox, et al., 1979), Monte Carlo simulations (Barraquand & Martineau, 1995), (Broadie & Glasserman, 1997), finite difference (FD) methods (Brennan & Schwartz, 1977), (Brennan & Schwartz, 1978) and analytical approximations (Carr & Faguet, 1994), (Geske & Johnson, 1984).

The use of finite difference methods for solving PDEs on a computer goes back almost to the 1950s since its invention. In finance, these methods were introduced in the 1970’s after the derivation of the Black-Scholes model. The sophistication of these methods in finance has become a field of extensive research. These include alternating finite difference (ADI) methods (Hout & Foulon, 2010), adaptive grids (Persson & Sydow, 2010), grids stretching (Oosterlee et al., 2005) and compact finite differences (Tangman et al., 2008), to name but a few. The interested reader is referred to two important textbooks (Daniel J, 2006), (Tavella & Randall, 2000) in the literature that cover financial derivatives pricing with finite difference methods exclusively.

This document provides a base for the valuation methods used in the Can-Do product space. Thus, we will not re-document already well-documented models and methodologies. This document will rather focus on the application of these methods to the JSE Can-Do products.

By Gyongy's theorem, a local and stochastic volatility (LSV) model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented in , provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, we prove existence to the associated Fokker-Planck equation. Thanks to [12], we then deduce existence of a new class of fake Brownian motions. We then extend these results to the special case of the LSV model called regime switching local volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, we prove existence to the associated Fokker-Planck PDE. Finally, we deduce existence of the calibrated model by extending the results in .

By Gyongy's theorem, a local and stochastic volatility (LSV) model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented in , provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, we prove existence to the associated Fokker-Planck equation. Thanks to [12], we then deduce existence of a new class of fake Brownian motions. We then extend these results to the special case of the LSV model called regime switching local volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, we prove existence to the associated Fokker-Planck PDE. Finally, we deduce existence of the calibrated model by extending the results in .

We propose a simple and robust numerical algorithm to estimate a time-dependent volatility function from a set of market observations, using the Black–Scholes (BS) model. We employ a fully implicit finite difference method to solve the BS equation numerically. To define the time-dependent volatility function, we define a cost function that is the sum of the squared errors between the market values and the theoretical values obtained by the BS model using the time-dependent volatility function. To minimize the cost function, we employ the steepest descent method. However, in general, volatility functions for minimizing the cost function are nonunique. To resolve this problem, we propose a predictor-corrector technique. As the first step, we construct the volatility function as a constant. Then, in the next step, our algorithm follows the prediction step and correction step at half-backward time level. The constructed volatility function is continuous and piecewise linear with respect...

We present some recent developments in the construction and classification of new analytically solvable one-dimensional diffusion models for which the transition densities and other quantities that are fundamental to derivatives pricing are represented in closed form. Our approach is based on so-called diffusion canonical transformations that allow us to uncover new multiparameter processes that are mapped onto various simpler diffusions. Using an asymptotic analysis of the boundary behavior of the processes, we arrive at a rigorous characterization or classification of the newly constructed diffusions with respect to probability conservation and the martingale property. The approach is applicable to a fairly general class of nonlinear volatility diffusions. Specifically, we analyze and classify in detail four main families of diffusion models that arise from the underlying squared Bessel process (the Bessel family), CIR process (the confluent hypergeometric family), the Ornstein-Uhlenbeck diffusion (the OU family) and the Jacobi diffusion (hypergeometric family). We show that the Bessel family is a superset of the constant elasticity of variance (CEV) models. The Bessel family, in turn, is generalized by the confluent hypergeometric family. For these two families we find further subfamilies of conservative strict supermartingales and absorbed martingales. For the new classes of absorbed diffusions we also derive analytically exact first-passage time densities for paths hitting an absorbing endpoint boundary, with densities given in terms of generalized inverse gaussians and extensions that involve the modified Bessel functions and confluent hypergeometric (Kummer) functions. As for the two other models, we show that the OU family of processes are strictly conservative martingales, whereas the Jacobi family is formed by absorbed non-martingales. Considered as asset price diffusion models, we also show that these models demonstrate a large range of local volatility shapes including pronounced skew and smile patterns. A discussion of applications to option pricing concludes the paper.

In this paper we use the 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 the volatility in the sense of Malliavin calculus.

Local volatility models are popular because they can be simply calibrated to the market of European options. We extend the results of [4], [3] for such models, i.e. we propose a modified Leland method which allows us to approximately replicate a European contingent claim when the market is under proportional transaction costs.

Recent evolutions in the business of exotic products have rendered the use of stochastic volatility models necessary. Calibration of single stochastic volatility models has already been discussed in several articles. The purpose of this paper is to build a parametrization of the correlation matrix of a multidimensional model with stochastic volatility, given that: • the correlation between each asset and its volatility is specified; • the correlations between different assets are specified. In the following, we propose a parametrization of the set satisfying the two constraints. This parametrization can be used to calibrate the correlations between single underlying models, and provides a range of attainable prices for standard exotic products.

This article presents a generic hybrid numerical method to price a wide range of
options on one or several assets, as well as assets with stochastic drift or volatility. In particular
for equity and interest rate hybrid with local volatility.

Following a trend of sustained and accelerated growth, the VIX futures and options market has become a closely followed, active and liquid market. The standard stochastic volatility models-which focus on the modeling of instantaneous variance-are unable to fit the entire term structure of VIX futures as well as the entire VIX options surface. In contrast, we propose to model directly the VIX index, in a mean-reverting local volatility-of-volatility model, which will provide a global fit to the VIX market. We then show how to construct the local volatility-of-volatility surface by adapting the ideas in Carr (Local variance gamma. Bloomberg Quant Research, New York, 2008) and Andreasen and Huge (Risk Mag 76-79, 2011) to a mean-reverting process.

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, even simple nonlinear financial derivatives as vanilla European options are typically priced by means of Monte–Carlo (MC) simulations which are more computationally demanding than similar MC schemes for standard stochastic volatility models. In this paper, we provide a proof of the prediction law for general Gaussian Volterra processes. The prediction law is then utilized to obtain an adapted projection of the future squared volatility — a cornerstone of the proposed pricing approximation. Firstly, a decomposition formula for European option prices under general Volterra volatility models is introduced. Then we focus on particular models wit...

A non-uniform generation of the points for discretization of the spatial variables in pricing stochastic volatility jump models, such as the model of Bates, is given. The distribution attempts to concentrate on the hotzone at which the price of the option should be given with high precision. The approach is quadratically convergent in space and for the jump term, as well as with quartic rate of convergence in time. We show that the numerical procedure has conditional stability and is fast in pricing both American and European vanilla options under such models. In fact, enough criteria for global convergence of the discrete equations are discussed. Computational results for various options show the superiority and efficiency of our approach.

A non-uniform generation of the points for discretization of the spatial variables in pricing stochastic volatility jump models, such as the model of Bates, is given. The distribution attempts to concentrate on the hotzone at which the price of the option should be given with high precision. The approach is quadratically convergent in space and for the jump term, as well as with quartic rate of convergence in time. We show that the numerical procedure has conditional stability and is fast in pricing both American and European vanilla options under such models. In fact, enough criteria for global convergence of the discrete equations are discussed. Computational results for various options show the superiority and efficiency of our approach.

We derive generalizations of Dupire formula to the cases of general stochastic drift and/or stochastic local volatility. First, we handle a case in which the drift is given as difference of two stochastic short rates. Such a setting is natural in foreign exchange context where the short rates correspond to the short rates of the two currencies, equity single-currency context with stochastic dividend yield, or commodity context with stochastic convenience yield. We present the formula both in a call surface formulation as well as total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes and present the limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion including one or more stochastic local volatility terms. In the general setting, our derivation allows the computation and calibration of the leverage function for stochastic local volatility models.

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods. Contents

We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for β = 1, namely ∂τ u = σ 2 1 2 (∂ 2 x u − ∂xu) + νρ∂x∂σu + 1 2 ν 2 ∂ 2 σ u + κ(θ − σ)∂σ, by choosing ν as small parameter. This yields u = u 0 + νu 1 + ν 2 u 2 +. . ., with u j independent of ν. The terms u j are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of u that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" τ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of ν, similar to Hagan's formula, but including also the mean reverting term. We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility. Contents 2 O. GRISHCHENKO, X. HAN, AND V. NISTOR 2. Applications: approximations of the derivatives and of the implied volatility 18 2.1. Approximation of derivatives 18 2.2. Implied Volatility 19 3. Model calibration and market tests 20 3.1. Description of the data 20 3.2. General description of the method 20 3.3. The first type of market data tests: implied volatility 21 3.4. The second type of market data tests: actual prices 23 3.5. The third type of market data tests: log-prices 24 4. Numerical tests 25 4.1. The residual of the approximations: substituting in the PDE 25 4.2. Comparison of our implied volatility formula with Hagan's formula for maket data 26 4.3. A Monte Carlo simulation 27 4.4. Comparison with the Finite Difference approximate solution 27 5. Extensions of the method 31 5.1. The log-normal model with mean-reverting volatility 32 5.2. The SABR model (with general β) 34 References 35 mean-reverting stochastic volatility model.

We present an explicit hedging strategy, which enables to prove arbitrageness of market incorporating at least two assets depending on the same random factor. The implied Black-Scholes volatility, computed taking into account the form of the graph of the option price, related to our strategy, demonstrates the "skewness" inherent to the observational data.

Local volatility models are popular because they can be simply calibrated to the market of European options. We extend the results of [4], [3] for such models, i.e. we propose a modified Leland method which allows us to approximately replicate a European contingent claim when the market is under proportional transaction costs.

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, even simple nonlinear financial derivatives as vanilla European options are typically priced by means of Monte–Carlo (MC) simulations which are more computationally demanding than similar MC schemes for standard stochastic volatility models. In this paper, we provide a proof of the prediction law for general Gaussian Volterra processes. The prediction law is then utilized to obtain an adapted projection of the future squared volatility — a cornerstone of the proposed pricing approximation. Firstly, a decomposition formula for European option prices under general Volterra volatility models is introduced. Then we focus on particular models wit...

We present a framework for calibrating a pricing model to a prescribed set of option prices quoted in the market. Our algorithm yields an arbitrage-free di usion process that minimizes the relative entropy distance to a prior di usion. We solve a constrained (minimax) optimal control problem using a nite-di erence scheme for a Bellman parabolic equation combined with a gradient-based optimization routine. The number of unknowns in the optimization step is equal to the number of option prices that need to be matched, and is independent of the mesh-size used for the scheme. This results in an e cient, non-parametric calibration method that can match an arbitrary number of option prices to any desired degree of accuracy. The algorithm can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics. The stability and qualitative properties of the computed volatility surface are discussed, including the e ect of the Bayesian prior on the shape of the surface and on the implied volatility smile/skew. The method is illustrated by calibrating to market prices of Dollar-Deutschemark over-thecounter options and computing interpolated implied-volatility curves.

Implied volatility skew and smile are ubiquitous phenomena in the financial derivative market especially after the Black Monday 1987 crash. Various stochastic volatility models have been proposed to capture volatility skew and smile in derivative pricing and hedging. Almost 30 years after the advent of the first type of stochastic volatility model calibrating them to the market volatility surface still remains challenging, especially when stochastic interest rate has to be also taken into account for long-dated options. Many techniques have been applied to tackle this problem, including Fast Fourier Transform, singular perturbation expansion, heat kernel expansion, Markovian projection, to name a few. Although they have achieved some success in deriving either a close-form solution for a specific type of model or asymptotic solution in more general, none of them can really solve the calibration problem satisfying our need in term of both efficiency and accuracy. Monte Carlo method is a flexible numerical pricing method but has not been considering for calibration because of its slow convergence. However, with the great advance in computational power, in particular, parallel computation and the invention of other variance reduction techniques fast and accurate calibration using Monte Carlo becomes possible. This paper presents a Monte Carlo calibration method for stochastic volatility models with stochastic interest rate, which reduces simulation dimension by conditional expectation and further improves speed by vectorization. Numerical experiments show that both the calibration speed and accuracy of this generic method are satisfactory for almost all applications.

We derive generalizations of Dupire formula to the cases of general stochastic drift and/or stochastic local volatility. First, we handle a case in which the drift is given as difference of two stochastic short rates. Such a setting is natural in foreign exchange context where the short rates correspond to the short rates of the two currencies, equity single-currency context with stochastic dividend yield, or commodity context with stochastic convenience yield. We present the formula both in a call surface formulation as well as total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes and present the limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion including one or more stochastic local volatility terms. In the general setting, our derivation allows the computation and calibration of the leverage function for stochastic local volatility models. Despite being implicit, the generalized Dupire formulae can be used numerically in a fixed-point iterative scheme.

ABSTRACTObjective: This article aims to solve the non-linear Black Scholes (BS) equation for European call options using Radial Basis Function (RBF) Multi-Quadratic (MQ) Method.Methodology / Approach: This work uses the MQ RBF method applied to the solution of two complex models of nonlinear BS equation for prices of European call options with modified volatility. Linear BS models are also solved to visualize the effects of modified volatility.  Additionally, an adaptive scheme is implemented in time based on the Runge-Kutta-Fehlberg (RKF) method.

We tackle the calibration of the so-called Stochastic-Local Volatility (SLV) model. This is the class of financial models that combines the local and stochastic volatility features and has been subject of the attention by many researchers recently. More precisely, given a local volatility surface and a choice of stochastic volatility parameters, we calibrate the corresponding leverage function. Our approach makes use of regularization techniques from the inverse-problem theory, respecting the integrity of the data and thus avoiding data interpolation. The result is a stable and robust algorithm which is resilient to instabilities in the regions of low probability density of the spot price and of the instantaneous variance. We substantiate our claims with numerical experiments using simulated as well as real data.

Usually, in the Black-Scholes pricing theory the volatility is a positive real parameter. Here we explore what happens if it is allowed to be a complex number. The function for pricing a European option with a complex volatility has essential singularities at zero and infinity. The singularity at zero reflects the put-call parity. Solving for the implied volatility that reproduces a given market price yields not only a real root, but also infinitely many complex roots in a neighbourhood of the origin. The Newton-Raphson calculation of the complex implied volatility has a chaotic nature described by fractals.

Usually, in the Black-Scholes pricing theory the volatility is a positive real parameter. Here we explore what happens if it is allowed to be a complex number. The function for pricing a European option with a complex volatility has essential singularities at zero and infinity. The singularity at zero reflects the put-call parity. Solving for the implied volatility that reproduces a given market price yields not only a real root, but also infinitely many complex roots in a neighbourhood of the origin. The Newton-Raphson calculation of the complex implied volatility has a chaotic nature described by fractals.

Usually, in the Black-Scholes pricing theory the volatility is a positive real parameter. Here we explore what happens if it is allowed to be a complex number. The function for pricing a European option with a complex volatility has essential singularities at zero and infinity. The singularity at zero reflects the put-call parity. Solving for the implied volatility that reproduces a given market price yields not only a real root, but also infinitely many complex roots in a neighbourhood of the origin. The Newton-Raphson calculation of the complex implied volatility has a chaotic nature described by fractals.

I would like to express my sincere thanks to Prof. Johan Tysk for his helpful comments, suggestions and for the continuous support in achieving this work. I thank as well the Mathematics Department of Eduardo Mondlane University for the given opportunity to take my masters studies in one of the best Universities in the world. I am also very grateful to International Science Program team for their hospitality. I would also like to express my sincere thanks to my family, friends and loved ones for their continuous love and encouragement.

I would like to express my sincere thanks to Prof. Johan Tysk for his helpful comments, suggestions and for the continuous support in achieving this work. I thank as well the Mathematics Department of Eduardo Mondlane University for the given opportunity to take my masters studies in one of the best Universities in the world. I am also very grateful to International Science Program team for their hospitality. I would also like to express my sincere thanks to my family, friends and loved ones for their continuous love and encouragement.

In this paper we use the 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 the volatility in the sense of Malliavin calculus.

This paper introduces a time-inhomogeneous parameterization of the forward LIBOR volatilities and analyzes its implications for the valuation of Bermudan swaptions. The model approximates the actual term structure of volatilities with a curve from a given set defined by the parametric volatility specification and the structure of a continuous time Markov chain that modulates the volatility function. The first stochastic volatility specification generates jump discontinuities in volatility and shape-preserving evolution of the volatility term structure in the future. The second specification allows, in addition, for changes in the shape of the volatility curve. Simulated values of Bermudan swaptions in a LIBOR market model with these volatility structures were obtained and compared to the prices from standard deterministic volatility specifications. The model allows for the assessment of the forward volatility risk of Bermudan swaptions.

Further praise for Exotic Options and Hybrids "This book brings a practitioner's prospective into an area that has seen little treatment to date. The challenge of writing a logical, rigorous, accessible and readable account of a vast and diverse field that is structuring of exotic options and hybrids is enthusiastically taken up by the authors, and they succeed brilliantly in covering an impressive range of products." Vladimir Piterbarg, Head of Quantitative Research, Barclays "What is interesting about this excellent work is that the reader can measure clearly that the authors are sharing a concrete experience. Their writing approach and style bring a clear added value to those who want to understand the structuring practices, Exotics pricing as well as the theory behind these." Younes Guemouri, Chief Operating Officer, Sophis "The book provides an excellent and compressive review of exotic options. The purpose of using these derivatives is well exposed, and by opposition to many derivatives' books, the authors focus on practical applications. It is recommended to every practitioner as well as advanced students looking forward to work in the field of derivatives." Dr Amine Jalal, Vice President, Equity Derivatives Trading, Goldman Sachs International "Exotic Options and Hybrids is an exceptionally well written book, distilling essential ingredients of a successful structured products business. Adel and Mohamed have summarized an excellent guide to developing intuition for a trader and structurer in the world of exotic equity derivatives." Anand Batepati, Structured Products Development Manager, HSBC, Hong Kong "A very precise, up-to-date and intuitive handbook for every derivatives user in the market." Amine Chkili, Equity Derivatives Trader, HSBC Bank PLC, London "Exotic Options and Hybrids is an excellent book for anyone interested in structured products. It can be read cover to cover or used as a reference. It is a comprehensive guide and would be useful to both beginners and experts. I have read a number of books on the subject and would definitely rate this in the top three." Ahmed Seghrouchni, Volatility Trader, Dresdner Kleinwort, London "A clear and complete book with a practical approach to structured pricing and hedging techniques used by professionals. Exotic Options and Hybrids introduces technical concepts in an elegant manner and gives good insights into the building blocks behind structured products." Idriss Amor, Rates and FX Structuring, Bank of America, London "Exotic Options and Hybrids is an accessible and thorough introduction to derivatives pricing, covering all essential topics. The reader of the book will certainly appreciate the alternation between technical explanations and real world examples." Khaled Ben-Said, Quantitative Analyst, JP Morgan Chase, London "A great reference handbook with comprehensive coverage on derivatives, explaining both theory and applications involved in day-today practices. The authors' limpid style of writing makes it a must-read for beginners as well as existing practitioners involved in day-today structuring, pricing and trading." Anouar Cedrati, Structured Products Sales, HSBC, Dubai "A good reference and an excellent guide to both academics and experts for its comprehensive coverage on derivatives through real world illustrations and theory concepts." Abdessamad Issami, Director of Market Activities, CDG Capital "Exotic Options and Hybrids offers a hands-on approach to the world of options, giving good insight into both the theoretical and practical side of the business. A good reference for both academics and market professionals as it highlights the relationship between theory and practice."

We study the problem of option replication in general stochastic volatility markets with transaction costs using a new form for enlarged volatility in Leland's algorithm [25]. The asymptotic results recover the existing works in the Leland spirit and enable us to fix the under-hedging property pointed out by Kabanov and Safarian in [20]. We analyze possible relationships between the present setting and high frequency markets with proportional transaction costs. Possibilities to improve the convergence rate and reduce the option price inclusive transaction costs are also discussed.

Prices of European call options in a regime-switching local volatility model can be computed by solving a parabolic system which generalises the classical Black and Scholes equation, giving these prices as functionals of the local volatilities. We prove Lipschitz stability for the inverse problem of determining the local volatilities from quoted call option prices for a range of strikes, if the calls are indexed by the different states of the continuous Markov chain which governs the regime switches.

In this paper we propose an extension of the Libor market model with a high-dimensional specially structured system of square root volatility processes, and give a road map for its calibration. As such the model is well suited for Monte Carlo simulation of derivative interest rate instruments. As a key issue, we require that the local covariance structure of the market model is preserved in the stochastic volatility extension. In a case study we demonstrate that the extended Libor model allows for stable calibration to the cap-strike matrix. The calibration algorithm is FFT based, so fast and easy to implement.

While the main conceptual issue related to deposit insurances is the moral hazard risk, the main technical issue is inaccurate calibration of the implied volatility. This issue can raise the risk of generating an arbitrage. In this paper, first, we discuss that by imposing the no-moral-hazard risk, the removal of arbitrage is equivalent to removing the static arbitrage. Then, we propose a simple quadratic model to parameterize implied volatility and remove the static arbitrage. The process of removing the static risk is as follows: Using a machine learning approach with a regularized cost function, we update the parameters in such a way that butterfly arbitrage is ruled out and also implementing a calibration method, we make some conditions on the parameters of each time slice to rule out calendar spread arbitrage. Therefore, eliminating the effects of both butterfly and calendar spread arbitrage make the implied volatility surface free of static arbitrage.

We propose to discuss a new technique to derive an good approximated solution for the price of a European Vanilla options, in a market model with stochastic volatility. In particular, the models that we have considered are the Heston and SABR(for β = 1). These models allow arbitrary correlation between volatility and spot asset returns. We are able to write the price of European call and put, in the same form in which one can see in the Black-Scholes model. The solution technique is based upon coordinate transformations that reduce the initial PDE in a straightforward one-dimensional heat equation.