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This paper discusses the value-at-risk (VaR) concept and assesses the financial adequacy of the price probability determined by frequency of trades at price p. We take the price definition as the ratio of executed trade value to volume and show that it leads to price statistical moments, which differ from those, generated by frequency price probability. We derive the price n-th statistical moments as ratio of n-th statistical moments of the value and the volume of executed transactions. We state that the price probability determined by frequency of trades at price p doesn’t describe probability of executed trade prices and VaR based on frequency price probability may be origin of unexpected and excessive losses. We explain the need to replace frequency price probability by frequency probabilities of the value and the volume of executed transactions and derive price characteristic function. After 50 years of the VaR usage main problems of the VaR concept are still open. We believe that VaR commitment to forecast the price probability for the time horizon T seems to be one of the most tough and expensive puzzle of modern finance.

A Fast Fourier transform-based method for estimation of parameters of a stochastic process from a historical time series is proposed. The main advantage of this method is that allows to fit distributions that are defined in terms of characteristic function, instead of more commonly used density-based estimation. In turn more exotic models can be calibrated to the past data, thereby extending these exotic models to applications in risk management.

A new model of TU game is introduced and studied. Here apart from the characteristic function, two additional functions are used: relations between the players and the probability of coalition formation. We suggest and study the sufficient properties of the probability function to transform it into a probability measure. The generalized Shapley's value is defined for a new class of games by analogy with the original Shapley approach. For the games under study, we give the axiomatization of value functions. It unambiguously characterizes the (generalized) Shapley's value.

In this work we show how the techniques introduced by Hurd and Zhou can be used to derive a pricing framework for rainbow options by using the joint characteristic function of the logarithm of the underlying assets. Semi-analytic formulae will be achieved by splitting the symbol, applying different dampings to its constituent components, and finally calculating the Fourier transforms of said components allowing the pricing of call and put options on the maximum and minimum of multiple assets; we shall also revisit Margrabe’s classical formula. Our work extends a number of classical results that focus on Geometric Brownian motion dynamics, and provides an alternative pricing methodology to that proposed in some of the most recent literature.

A simple expression for the characteristic function of the lognormal distribution has eluded researchers. This is useful for computing the sum of lognormal variables, either with each other or with other statistical variables. In this paper, we provide a simple formula for the characteristic function.

In this paper, enlightened by the asymptotic expansion methodology developed by Li [(2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380] and Li and Chen [(2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 51–70], we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Lévy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein–Uhlenbeck model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method.

Multivariate return distributions consistent with bilateral gamma marginals are formulated and termed multivariate bilateral gamma (MBG). Tail probability distances and Wasserstein distances between return data, model simulations and their squares evaluate the model performance. A full Gaussian copula [Formula: see text] is taken as an alternate test model and the MBG delivers a comparatively better performance for equity pairs. The MBG is however inadequate for the S&P 500 index return when paired with the VIX returns. Applying MBG to the S&P 500 the index and regression residuals of VIX on the S&P 500 index return is successful. This model is termed MBGR. The residual taken as an independent bilateral gamma, delivers the model MBGIR. Characteristic function estimations are employed to develop asset-specific VIX levels and their joint returns with the asset return are studied. The CBOE SKEW index is generalized to be asset-specific and triples of returns for the asset, its VIX and its SKEW are studied using all four models and performance statistics. The model MBGR continues to deliver a good performance.

We estimate and test for multiple structural breaks in distribution with unknown break dates via a characteristic function approach. By minimizing the sum of squared generalized residuals, we can consistently estimate the break fractions. We propose a sup-F type test for structural breaks in distribution and suggest an information criterion and a sequential testing procedure to determine the number of breaks. We further construct a class of derivative tests to gauge the possible source of structural breaks, which is asymptotically more powerful than the nonparametric-based tests for structural breaks. Simulation studies show that our method performs well in determining the appropriate number of breaks and estimating the unknown breaks. Furthermore, the proposed tests have reasonable size and excellent power. In an application to exchange rate returns, our tests are able to detect structural breaks in distribution and locate the break dates. Our derivate tests indicate that the documented breaks appear to occur in variance and higher order moments, but not so often in mean.

VIX futures and options are the most popular contracts traded in the Chicago Board Options Exchange. The bid-ask spreads of traded VIX derivatives remain to be wide, possibly due to the lack of reliable pricing models. In this paper, we consider pricing VIX derivatives under the consistent model approach, which considers joint modeling of the dynamics of the S&P index and its instantaneous variance. Under the affine jump-diffusion formulation with stochastic volatility, analytic integral formulas can be derived to price VIX futures and options. However, these integral formulas invariably involve Fourier inversion integrals with cumbersome hyper-geometric functions, thus posing various challenges in numerical evaluation. We propose a unified numerical approach based on the willow tree algorithms to price VIX derivatives under various common types of joint process of the S&P index and its instantaneous variance. Given the analytic form of the characteristic function of the instantaneous variance of the S&P index process in the Fourier domain, we apply the fast Fourier transform algorithm to obtain the transition density function numerically in the real domain. We then construct the willow tree that approximates the dynamics of the instantaneous variance process up to the fourth order moment. Our comprehensive numerical tests performed on the willow tree algorithms demonstrate high level of numerical accuracy, runtime efficiency and reliability for pricing VIX futures and both European and American options under the affine model and 3/2-model. We also examine the implied volatility smirks and the term structures of the implied skewness of VIX options.

Affine jump diffusion models in general and affine stochastic volatility models in particular are important modeling tools in finance. Their popularity resides in their exibility coupled with their analytical tractability, especially with respect to characteristic functions and polynomial moments. Within a generic affine jump diffusion model and a generic stochastic volatility model, nested in the former as a special case, this paper collects explicit expressions for various characteristic functions and polynomial moments.

In this paper we propose a statistically derived measure as an alternative to the simple average PD to provide more accurate risk assessment at portfolio level. The theoretical analysis is followed by a numerical example in sections 3 and 4. We then assess the accuracy and representativeness between the simple average and the proposed measure using maximum likelihood analysis. The statistical properties of the proposed approach is discussed in section 5 together with graphical illustration of the inferences based on the joint probability distribution on portfolio level. The proposed alternative works well with current industry risk management frameworks and procedures. Our proposal address several flaws that comes with the ordinary simple average approach and reconfirms the common understanding that the credit default are of Bernoulli nature with non-identical distributions.

In this paper, we derive an approximate unconditional distribution for normal GARCH(1, 1) model by application of perturbation theory. We compute the characteristic function and distribution up to the fourth order approximation. Additionally, we derive an integral equation for the characteristic function. By application of Dyson equation, we develop a new method for calculating distribution from the characteristic function. Utilizing self-similarity transformation in parametric space, we improve the accuracy of our results, especially in the neighborhood of α + β = 1. Kullback-Leibler divergence, computed against simulated data shows good performance of the approximation.

We propose an extension of the Gamma-OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the Gamma-OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the Gamma-OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the Gamma-OU Barndorff-Nielsen and Shephard model.

In this paper, we derive explicit expressions for certain joint moments of stock prices and option prices within a generic affine stochastic volatility model. Evaluation of each moment requires weighted inverse Fourier transformation of a function that is determined by the risk-neutral and real-world characteristic functions of the state vector. Explicit availability of such moment expressions allows to devise a novel GMM approach to jointly estimate real-world and risk-neutral parameters of affine stochastic volatility models using observed individual option prices. Moreover, the moment expressions may be used to include option price information into other existing moment-based estimation approaches.

We propose a new stochastic volatility model for pricing options on assets that exhibit seasonal trends in volatility. Such assets are prevalent among commodities, with futures on grains and energy being an example. The model is based on the 3/2 stochastic volatility model, but includes a cyclical long-run volatility component. The model yields a closed-form characteristic function which can be used to rapidly calibrate the model to benchmark options. We test the model on market data and show that the model proposed here allows for significantly better empirical fit to option prices on corn and wheat futures than the unmodified 3/2 model.

In this paper, we analyze Trolle-Schwartz stochastic volatility model and we provide an analytical solution to its characteristic function, which can be used for a fast pricing of plain vanilla options on commodities. Then we explore some technical issues of the model, like the Heston trap, and, after testing, we provide the solutions. All in all the framework developed by Trolle and Schwartz, coupled with the pricing formula described in this article, will have the potential for a rapid calibration and a reliable pricing of complex commodity derivatives.

The Heston stochastic volatility model is a standard model for valuing financial derivatives, since it can be calibrated using semi-analytical formulas and captures the most basic structure of the market for financial derivatives with simple structure in time-direction. However, extending the model to the case of time-dependent parameters, which would allow for a parametrization of the market at multiple timepoints, proves more challenging. We present a simple and numerically efficient approach to the calibration of the Heston stochastic volatility model with piecewise constant parameters. We show that semi-analytical formulas can also be derived in this more complex case and combine them with recent advances in computational techniques for the Heston model. Our numerical scheme is based on the calculation of the characteristic function using Gauss–Kronrod quadrature with an additional control variate that stabilizes the numerical integrals. We use our method to calibrate the Heston model with piecewise constant parameters to the foreign exchange (FX) options market. Finally, we demonstrate improvements of the Heston model with piecewise constant parameters upon the standard Heston model in selected cases.

We use the Ito Decomposition Formula (see Alos (2012)) to express certain conditional expectations as exponentials of iterated integrals. As one application, we compute an exact formal expression for the leverage swap for any stochastic volatility model expressed in forward variance form. As another, we show how to extend the Bergomi Guyon expansion to all orders in volatility of volatility. Finally, we compute exact expressions under rough volatility, obtaining in particular the fractional Riccati equation for the rough Heston characteristic function. As a corollary, we compute a closed-form expression for the leverage swap in the rough Heston model which can be used for fast calibration.

In this paper we introduce a novel pricing methodology for a broad class of models for which the characteristic function of the log-asset price can be efficiently computed. The new method avoids the numerical integration required by the Fourier-based approaches and reveals to be fast and accurate, to the point that we can calibrate the models on real data. Our approach allows to price also American- style options, as it is possible to compute the transition probabilities for the underlying. This is accomplished through an efficient multinomial lattice discretization of the asset price based on a new quantization procedure which exploits the knowledge of the Fourier transform of the process at a given time. As a motivating example, we price an American Put option in a Tempered Stable model, with constitutes the first application of quantization to a pure jump process.

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale. This class of models allows for a local volatility, local default intensity and a locally dependent Levy measure. We present a pricing method for Bermudan options based on an analytical approximation of the characteristic function combined with the COS method. Due to a special form of the obtained characteristic function the price can be computed using a Fast Fourier Transform-based algorithm resulting in a fast and accurate calculation. The Greeks can be computed at almost no additional computational cost. Error bounds for the approximation of the characteristic function as well as for the total option price are given.

Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FBSDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local Levy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.

In this paper we propose a closed-form pricing formula for European basket and spread options. Our approach is based on approximating the risk-neutral probability density function of the terminal value of the basket using a Gauss-Hermite series expansion around the Gaussian density. The new method is quite general as it can be applied for a basket with a large number of assets and for all dynamics where the joint characteristic function of log-returns is known in closed form. We provide a simulation study to show the accuracy and the speed of our methodology.

This paper provides an efficient and accurate approximation for the distorted expectation of a risk factor when its density function or characteristic function is given in an analytical form. The fast Fourier transform (FFT) algorithm is used to set up an approximation for the distorted density function with a truncated sum of its Fourier-cosine series expansion on a finite interval. The resulting truncation approximation leads to an analytical approximation for the distorted expectation. The proposed approach can be used in different branches of applications of distorted expectations, such as in risk management, portfolio optimization and conic finance.

The aims of this paper are twofold. Firstly, we present an approximating formula for pricing basket and multi-asset spread options, which genuinely extends Caldana and Fusai (2013) two-asset spread options formula. Secondly, under the lognormal setting, we show that our formula becomes a Black and Scholes type formula, extending Bjerksund and Stensland (2011). Numerical experiments and comparison with Monte Carlo simulations and other methods available in the literature are discussed. The main contribution of this paper is to provide practitioners with a pricing formula, which can be used for pricing basket and multi-asset spread options, even under a non-Gaussian framework.

We develop three numerical methods to solve coupled forward-backward stochastic differential equations. We propose three different discretization techniques for the forward stochastic differential equation. A theta-discretization of the time-integrands is used to arrive at schemes with conditional expectations. These conditional expectations are approximated by using the COS method, which relies on the availability of the conditional characteristic function of the discrete forward process. The numerical methods are applied to different problems, including a financial problem. Richardson extrapolation is used to obtain more accurate results, resulting in the observation of second-order convergence in the number of time steps. Advantages and disadvantages of each method are compared against each other.

This paper proposes new bounds on the prices of European-style swaptions for affine and quadratic interest rate models. These bounds are computable whenever the joint characteristic function of the state variables is known. In particular, our lower bound involves the computation of a one dimensional Fourier transform independently of the swap length. In addition, we control the error of our method by providing a new upper bound on swaption price applicable to all considered models. We test our bounds on different affine models and on a quadratic Gaussian model. We also apply our procedure to the multiple curve framework. The bounds are found to be accurate and computationally efficient.

A model with a stochastic interest rate process correlated to a stochastic volatility process is needed to accurately price long-dated contingent claims. Such a model should also price claims efficiently in order to allow for fast calibration. This dissertation explores the approximations for the characteristic function of the Heston-Hull & White model introduced by Grzelak and Oosterlee (2011). Fourier-Cosine expansion pricing is then used to price contingent claims under this model, which is implemented in MATLAB (Fang and Oosterlee, 2008). We find that the model is efficient, accurate and has a relatively simple calibration procedure. In back-tests, it is determined that the Heston-Hull & White model produces better hedging profit and loss results than a Heston (1993) or a Black and Scholes (1973) model.

This paper proposes a generalization of the Barndorff-Nielsen and Shephard model, in which the log return on an asset is governed by a Lévy process with stochastic volatility modeled by a non-Gaussian Ornstein–Uhlenbeck process. Under the generalized model, we derive a closed-form expression of the multivariate characteristic function of the intertemporal joint distribution of the underlying log return. Then, we also investigate asymptotic behavior of the log return and its variance. Moreover, we evaluate discretely monitored path-dependent derivatives such as geometric Asian, forward start, barrier, fade-in, and lookback options as well as European options.

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known in closed-form. Moreover, the basket value is not required to be positive. We test our new bounds on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms, hence they do not suffer from the curse of dimensionality. In particular, our new lower bound turns out to be fast and accurate.

We analyze cooperative Cournot games with boundedly rational firms. Due to cogni- tive constraints, the members of a coalition cannot accurately predict the coalitional structure of the non-members. Thus, they compute their value using simple heuris- tics. In particular, they assign various non-equilibrium probability distributions over the outsiders' set of partitions. We construct the characteristic function of a coalition in such an environment and we analyze the core of the corresponding games. We show that the core is non-empty provided the number of firms in the market is sufficiently large. Moreover, we show that if two distributions over the set of partitions are related via first-order dominance, then the core of the game under the dominated distribution is a subset of the core under the dominant distribution., Draft version of the published ANOR paper, Annals of Operations Research, May 2014

We study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models, and explain that, in many cases, a calibration procedure based on such a realization will be able to find a "correct parameter set" only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time consuming. We explain how to construct more accurate and faster pricing and calibration procedures. Efficient realizations require analytic continuation of the characteristic function not only into a strip around the real axis (the problem of finding the maximal strip of analyticity has been extensively studied in the literature under the name "moment explosion") but into much wider regions of the complex plane, and an appropriate conformal deformation of the contour of integration. In the paper, we study analytic continuation of the characteristic functions in wide classes of affine models, models with jumps including. An important ingredient of our method is the study of the associated system of generalized Riccati equations. As a byproduct, we show that the straightforward application of the Runge-Kutta method may lead to sizable errors, and suggest certain remedies. The methodology of the paper can be applied to other models (e.g. quadratic term structure models, Wishart dynamics, 3/2-model), where European options and other options can be priced using the Fourier transform techniques.

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram-Charlier series expansion, known as the Gauss-Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

We develop a Fourier method to solve backward stochastic differential equations (BSDEs). General theta-discretization of the time-integrands leads to an induction scheme with conditional expectations. These are approximated by using Fourier-cosine series expansions, relying on the availability of a characteristic function. The method is applied to BSDEs with jumps. Numerical experiments demonstrate the applicability of BSDEs in financial and economic problems and show fast convergence of our efficient probabilistic numerical method.

We derive a new, efficient closed-form formula approximating the price of discrete look-back options, whose underlying asset price is driven by an exponential semi-martingale process including (jump) diffusions, Levy models, affine processes and other models. The derivation of our pricing formula is based on inverting the Fourier transform using B-spline approximation theory. We give an error bound for our formula and establish its fast rate of convergence to the true price. Our method provides look-back option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods.We provide an alternative proof to the Spitzer formula for the characteristic function of the maximum of a discretely observed stochastic process, which yields a numerically efficient algorithm based on convolutions. This is an important result which could have a wide range of applications where the Spitzer formula is utilized. We illustrate the numerical efficiency of our algorithm by applying it in pricing fixed and floating discrete look-back options under Brownian motion, jump diffusion models, and the variance gamma process.A new efficient and robust methodology is presented for pricing discrete look-back options whose underlying asset price is driven by an exponential semi-martingale process and no analytical pricing formula exists. Using B-spline interpolation we obtain an accurate closed-form representation of the look-back option price under an inverse generalized Fourier transform. This provides look-back option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods.We derive an explicit representation for the characteristic function of the maximum of a discretely observed stochastic process, which provides a significant improvement in terms of numerical efficiency over the Spitzer recurrence formula. This is of fundamental importance and could have a wide range of applications where the Spitzer formula is utilized. Examples considered include pricing fixed and floating discrete look-back options under Brownian motion, jump diffusion models, and the Variance Gamma process.

This article considers effective Empirical Characteristic Function (ECF) methods for estimating parameters of affine diffusions with unobserved stochastic variance and their application to some problems of continuous-time Finance. We develop a new approach utilizing the Realized Variance (RV) in the ECF for the quadratic variation of the underlying stochastic process, which is then fitted by the corresponding unconditional characteristic function for the integrated variance available in the closed form. Based on the results of Barndorff-Nielsen and Shephard, the bias correction is derived for the RV-based ECF, as well as for the Ditlevsen--Sorensen Method of Moments for integrated diffusion processes, that significantly improves accuracy of the estimation. We also investigate an effective numerical implementation of the unconditional ECF method based on the increments of state variables through the special configuration of grid points in the frequency domain. This approach provides better accuracy and stability for the parameter estimates than some methods of a joint unconditional characteristic function for the original state variables on multiple intervals. Finally, we propose a new very general version for the Singleton's method of "conditional" ECF estimation that requires knowledge only of the conditional characteristic function and is equivalent to some non-linear regression. The article presents numerical results for S&P 500 Index, VIX Index and a rigorous Monte-Carlo study for the CIR model and Heston stochastic volatility model.

The constant proportion portfolio insurance is analyzed by assuming that the risky asset price follows a regime switching exponential Levy process. Analytical forms of the shortfall probability, expected shortfall and expected gain are derived. The characteristic function of the gap risk is also obtained for further exploration on its distribution. The specific implementation is discussed under some popular L evy models including the Merton’s jump- diffusion, Kou’s jump-diffusion, variance gamma and normal inverse Gaus- sian models. Finally, a numerical example is presented to demonstrate the implication of the established results.

For any strictly positive martingale S=e^X for which X has an analytically tractable characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in log(K/S 0 ). We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Levy model (Merton), one infinite activity exponential Levy model (Variance Gamma), and one stochastic volatility model (Heston). We show how this technique can be extended to compute approximate forward implied volatilities and we implement this extension in the Heston setting. Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

We present new approximation formulas for local stochastic volatility models, possibly including Levy jumps. Our main result is an expansion of the characteristic function which is worked out in the Fourier space. Combined with standard Fourier methods, our result provides efficient and accurate formulas for the prices and the Greeks of plain vanilla options. We finally provide numerical results to illustrate the accuracy with real market data.

We examine the pricing of variance swaps and some generalizations and variants such as self-quantoed variance swaps, gamma swaps, skewness swaps and proportional variance swaps.We consider the pricing of both discretely monitored and continuously monitored versions of these swaps when the dynamics of the log of the underlying stock price are driven by (possibly, multiple) time-changed Levy processes. We derive exact and easily implementable formulae for the prices of these swaps in terms of, essentially, the characteristic function of the log of the stock price and its derivatives. We consider the convergence of the prices of discretely monitored versions of these swaps to the prices of their continuously monitored counterparts as the number N of monitoring times is allowed to tend to infi nity. We signi ficantly extend results in Broadie and Jain (2008a) by showing that the prices of discretely monitored variance swaps and all of the generalizations and variants listed above all converge to the prices of their continuously monitored counterparts as O(1/N). We generalize results in Carr and Lee (2009) by relating the prices of variance swaps and the generalizations and variants listed above to the prices of log-forward-contracts and entropy-forward-contracts. Carr and Lee (2009) show that, under an independence assumption, discretely monitored variance swaps are worth at least as much as their continuously monitored counterparts. We extend this result in two directions, by dropping the independence assumption and by proving analogous results for some of the other swaps listed above.

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Levy models.This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile.In passing we (i) show that the forward-start date has to be rescaled in order to obtain non-trivial small-maturity asymptotics,(ii) prove that the forward-start date may influence the large-maturity behaviour of the forward smile,and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.

We consider the problem of pricing options on a leveraged ETF (LETF) and the underlying ETF in a consistent manner. We show that if the underlying ETF has Heston dynamics then the LETF also has Heston dynamics so that options on both the ETF and the LETF can be priced analytically using standard transform methods. If the underlying ETF has tractable jump-diffusion dynamics then the dynamics of the corresponding LETF are generally intractable in that we cannot compute a closed-form expression for the characteristic function of the log-LETF price. In that event we propose tractable approximations based on either moment-matching techniques or saddlepoint approximations to the LETF price dynamics under which the characteristic function of the log-LETF price can be found in closed form. In a series of numerical experiments including both low and high volatility regimes, we show that the resulting LETF option price approximations are very close to the true prices which we calculate via Monte-Carlo. Because approximate LETF option prices can be computed very quickly our methodology should be useful in practice for pricing and risk-managing portfolios that contain options on both ETFs and related LETFs. Our numerical results also demonstrate the model-dependency of LETF option prices and this is particularly noticeable in high-volatility environments.

Non-linearity is an important consideration in many problems of finance and economics, such as pricing securities and solving equilibrium models. This paper provides analytical treatment of a general class of nonlinear transforms for processes with tractable conditional characteristic functions, which extends existing results on characteristic function based transforms to a substantially wider class of nonlinear functions while maintaining low dimensionality by avoiding the need to compute the density function. We illustrate the applications of the generalized transform in pricing defaultable bonds with stochastic recovery. We also use the method to analytically solve a class of general equilibrium models with multiple goods and apply this model to study the effects of time-varying labor income risk on the equity premium.

This paper develops a fast method for the computation of option prices for models whose characteristic function is time-consuming to compute due to the need to solve ordinary differential equations or difference equations numerically, which is the case for a wide class of models of stocks, bonds, or currencies, including the general affine jump-diffusion models, quadratic-Gaussian models, and the GARCH option pricing models. This paper shows that approximating the model's cumulant generating function as a rational function of polynomials (Pade approximant form) within the saddlepoint approach of Lugannani and Rice leads to a fast and accurate computation of option prices. For most practical purposes, 2~3 evaluations of the cumulant generating function (characteristic function) turn out to be sufficient to get an accurate approximation of the cumulative distribution function that appears in the option price formula.

We propose a novel method for the analytical approximation in local volatility models with Levy jumps. The main result is an expansion of the characteristic function in a local Levy 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 efficient 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.

Compound options are not only sensitive to future movements of the underlying asset price, but also to future changes in volatility levels. Because the Black-Scholes analytical valuation formula for compound options is not able to incorporate the sensitivity to volatility, the aim of this paper is to develop a numerical pricing procedure for this type of option in stochastic volatility models, specifically focusing on the model of Heston.For this, the compound option value is represented as the difference of its exercise probabilities, which depend on three random variables through a complex functional form. Then the joint distribution of these random variables is uniquely determined by their characteristic function and therefore the probabilities can each be expressed as a multiple inverse Fourier transform. Solving the inverse Fourier transform with respect to volatility, we can reduce the pricing problem from three to two dimensions. This reduced dimensionality simplifes the application of the fast Fourier transform method developed by Dempster and Hong when transfered to our stochastic volatility framework. After combining their approach with a new extension of the fractional fast Fourier transform technique for option pricing to the two-dimensional case, it is possible to obtain good approximations to the exercise probabilities. The resulting upper and lower bounds are then compared with other numerical methods such as Monte Carlo simulations and show promising results.

This paper introduces a multivariate pure-jump Lévy process which allows for skewness and excess kurtosis of single asset returns and for asymptotic tail dependence in the multivariate setting. It is termed Variance Compound Gamma (VCG). The novelty of my approach is that, by applying a two-stage stochastic time change to Brownian motions, I derive a hierarchical structure with different properties of inter- and intra-sector dependence. I investigate the properties of the implied static copula families and come to the conclusion that they are ordered with respect to their parameters and that the lower-tail dependence of the intra-sector copula is increasing in the absolute values of skewness parameters. Furthermore, I show that the joint characteristic function of the VCG asset returns can be explicitly given as a nested Archimedean copula of their marginal characteristic functions. Applied to credit portfolio modelling, the framework introduced results in a more conservative tail risk assessment than a Gaussian framework with the same linear correlation structure, as I show in a simulation study. To foster the simulation efficiency, I provide an Importance Sampling algorithm for the VCG portfolio setting.

This paper proposes a characteristic function-based method to estimate the time-changed Levy models, which take into account both stochastic volatility and infinite activity jumps. The method facilitates computation and overcomes problems related to the discretization error and to the non-tractable probability density. Estimation results and option pricing performance indicate that the infinite activity model in general performs better than the finite activity one. We also provide an extension to investigate the double-jump model by introducing a jump component in the volatility process.

In an affine term structure framework with stochastic volatility, we derive the characteristic function of the log swap rate. Having the characteristic function, we employ Fast Fourier Techniques (FFT) to price swaptions. Using ten years of swap rates and swaption premiums, model parameters are estimated using square-root unscented Kalman filter. We investigate the relationship between model premiums and interest rate factors, as well as market premiums and interest factors to conclude that long-dated swaptions are highly correlated to the shape of the curve.

In an affine term structure framework we propose a discrete time stochastic volatility model. We derive the characteristic function of the log swap rate under swap measure. Having the characteristic function, we employ the Fourier cosine (COS) technique to price swaptions. Using data on tweleve years of swap rates and swaption premiums, model parameters are estimated using an unscented Kalman filter algorithm.

Plain vanilla options have a single underlying asset and a single condition on the payoff at the expiration date. For this class of options, a well-known result of Duffie, Pan and Singleton (2000) shows how to invert the characteristic function to obtain a closed-form formula for their prices. However, multiple-asset and multiple-condition derivatives such as rainbow options cannot be priced within this framework. Utilizing inversion of the Fourier transform – and resorting to neither the Black-Scholes framework nor the affine models settings – the authors provide an analytical solution for options whose payoffs depend on two or more conditions. Numerical experiments based on the multiple-asset and multiple-condition derivatives are provided to illustrate the usefulness of the proposed approach.