Affine Rough Models
The goal of this survey article is to explain and elucidate the affine structure of recent models appearing in the rough volatility literature, and show how it leads to exponential-affine transform formulas.
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arXiv subjects
Martin Keller-Ressel
Publications and source records attributed to Martin Keller-Ressel.
Affine forward variance models
We introduce the class of affine forward variance (AFV) models of which both the conventional Heston model and the rough Heston model are special cases. We show that AFV models can be characterized by the affine form of their cumulant generating function, which can be obtained as solution of a convolution Riccati equation. We further introduce the class of affine forward order flow intensity (AFI) models, which are structurally similar to AFV models, but driven by jump processes, and which include Hawkes-type models. We show that the cumulant generating function of an AFI model satisfies a generalized convolution Riccati equation and that a high-frequency limit of AFI models converges in distribution to the AFV model.
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Detecting independence of random vectors: generalized distance covariance and Gaussian covariance
Distance covariance is a quantity to measure the dependence of two random vectors. We show that the original concept introduced and developed by Székely, Rizzo and Bakirov can be embedded into a more general framework based on symmetric Lévy measures and the corresponding real-valued continuous negative definite functions. The Lévy measures replace the weight functions used in the original definition of distance covariance. All essential properties of distance covariance are preserved in this new framework. From a practical point of view this allows less restrictive moment conditions on the underlying random variables and one can use other distance functions than Euclidean distance, e.g. Minkowski distance. Most importantly, it serves as the basic building block for distance multivariance, a quantity to measure and estimate dependence of multiple random vectors, which is introduced in a follow-up paper [Distance Multivariance: New dependence measures for random vectors (submitted). Revised version of arXiv: 1711.07775v1] to the present article.
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Distance multivariance: New dependence measures for random vectors
We introduce two new measures for the dependence of $n \ge 2$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted $L^2$-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to $n$-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of $n$ random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.
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Erratum to: `Yield curve shapes and the asymptotic short rate distribution in affine one-factor models'
This paper corrects an error in [Keller-Ressel, M. and Steiner T. "Yield curve shapes and the asymptotic short rate distribution in affine one-factor models." Finance and Stochastics 12.2 (2008): 149-172]. The error concerns the correct expression for the boundary between normal and humped yield curve behavior in affine one-factor short-rate models.
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Forward-Invariance and Wong-Zakai Approximation for Stochastic Moving Boundary Problems
We discuss a class of stochastic second-order PDEs in one space-dimension with an inner boundary moving according to a possibly non-linear, Stefan-type condition. We show that proper separation of phases is attained, i.e., the solution remains negative on one side and positive on the other side of the moving interface, when started with the appropriate initial conditions. To extend results from deterministic settings to the stochastic case, we establish a Wong-Zakai type approximation. After a coordinate transformation the problems are reformulated and analysed in terms of stochastic evolution equations on domains of fractional powers of linear operators.
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Semi-Static and Sparse Variance-Optimal Hedging
We consider hedging of a contingent claim by a 'semi-static' strategy composed of a dynamic position in one asset and static (buy-and-hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance-optimality and provide tractable formulas using Fourier-integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semi-static hedging strategy, i.e. a strategy which only uses a small subset of available hedging assets. The developed methods are illustrated in an extended numerical example where we compute a sparse semi-static hedge for a variance swap using European options as static hedging assets.
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Semi-Static Variance-Optimal Hedging in Stochastic Volatility Models with Fourier Representation
In a financial market model, we consider the variance-optimal semi-static hedging of a given contingent claim, a generalization of the classic variance-optimal hedging. To obtain a tractable formula for the expected squared hedging error and the optimal hedging strategy, we use a Fourier approach in a general multidimensional semimartingale factor model. As a special case, we recover existing results for variance-optimal hedging in affine stochastic volatility models. We apply the theory to set up a variance-optimal semi-static hedging strategy for a variance swap in both the Heston and the 3/2-model, the latter of which is a non-affine stochastic volatility model.
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A remark on Gatheral's 'most-likely path approximation' of implied volatility
We give a new proof of the representation of implied volatility as a time-average of weighted expectations of local or stochastic volatility. With this proof we clarify the question of existence of 'forward implied variance' in the original derivation of Gatheral, who introduced this representation in his book 'The Volatility Surface'.
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Implied volatility in strict local martingale models
We consider implied volatilities in asset pricing models, where the discounted underlying is a strict local martingale under the pricing measure. Our main result gives an asymptotic expansion of the right wing of the implied volatility smile and shows that the strict local martingale property can be determined from this expansion. This result complements the well-known asymptotic results of Lee and Benaim-Friz, which apply only to true martingales. This also shows that `price bubbles' in the sense of strict local martingale behaviour can in principle be detected by an analysis of implied volatility. Finally we relate our results to left-wing expansions of implied volatilities in models with mass at zero by a duality method based on an absolutely continuous measure change.
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A Stefan-type stochastic moving boundary problem
Motivated by applications in economics and finance, in particular to the modeling of limit order books, we study a class of stochastic second-order PDEs with non-linear Stefan-type boundary interaction. To solve the equation we transform the problem from a moving boundary problem into a stochastic evolution equation with fixed boundary conditions. Using results from interpolation theory we obtain existence and uniqueness of local strong solutions, extending results of Kim, Zheng and Sowers. In addition, we formulate conditions for existence of global solutions and provide a refined analysis of possible blow-up behavior in finite time.
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Simple examples of pure-jump strict local martingales
We present simple new examples of pure-jump strict local martingales. The examples are constructed as exponentials of self-exciting affine Markov processes. We characterize the strict local martingale property of these processes by an integral criterion and by non-uniqueness of an associated ordinary differential equation. Finally we show an alternative construction for our examples by an absolutely continuous measure change in the spirit of (Delbaen and Schachermayer, PTRF 1995).
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Exponential moments of affine processes
We investigate the maximal domain of the moment generating function of affine processes in the sense of Duffie, Filipović and Schachermayer [Ann. Appl. Probab. 13 (2003) 984-1053], and we show the validity of the affine transform formula that connects exponential moments with the solution of a generalized Riccati differential equation. Our result extends and unifies those preceding it (e.g., Glasserman and Kim [Math. Finance 20 (2010) 1-33], Filipović and Mayerhofer [Radon Ser. Comput. Appl. Math. 8 (2009) 1-40] and Kallsen and Muhle-Karbe [Stochastic Process Appl. 120 (2010) 163-181]) in that it allows processes with very general jump behavior, applies to any convex state space and provides both sufficient and necessary conditions for finiteness of exponential moments.
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Geometric Asian Option Pricing in General Affine Stochastic Volatility Models with Jumps
In this paper we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain close-form solutions for some relevant affine model classes.
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Convex order of discrete, continuous and predictable quadratic variation & applications to options on variance
We consider a square-integrable semimartingale and investigate the convex order relations between its discrete, continuous and predictable quadratic variation. As the main results, we show that if the semimartingale has conditionally independent increments and symmetric jump measure, then its discrete realized variance dominates its quadratic variation in increasing convex order. The results have immediate applications to the pricing of options on realized variance. For a class of models including time-changed Levy models and Sato processes with symmetric jumps our results show that options on variance are typically underpriced, if quadratic variation is substituted for the discretely sampled realized variance.
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Regularity of affine processes on general state spaces
We consider a stochastically continuous, affine Markov process in the sense of Duffie, Filipovic and Schachermayer, with cadlag paths, on a general state space D, i.e. an arbitrary Borel subset of R^d. We show that such a process is always regular, meaning that its Fourier-Laplace transform is differentiable in time, with derivatives that are continuous in the transform variable. As a consequence, we show that generalized Riccati equations and Levy-Khintchine parameters for the process can be derived, as in the case of $D = R_+^m \times R^n$ studied in Duffie, Filipovic and Schachermayer (2003). Moreover, we show that when the killing rate is zero, the affine process is a semi-martingale with absolutely continuous characteristics up to its time of explosion. Our results generalize the results of Keller-Ressel, Schachermayer and Teichmann (2011) for the state space $R_+^m \times R^n$ and provide a new probabilistic approach to regularity.
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On the Limit Distributions of Continuous-State Branching Processes with Immigration
We consider the class of continuous-state branching processes with immigration (CBI-processes), introduced by Kawazu and Watanabe (1971) and their limit distributions as time tends to infinity. We determine the Levy-Khintchine triplet of the limit distribution and give an explicit description in terms of the characteristic triplets of the Levy subordinator and the spectrally positive Levy process, which describe the immigration resp. branching mechanism of the CBI-process. This representation allows us to describe the support of the limit distribution and characterise its absolute continuity and asymptotic behavior at the boundary of the support, generalizing several known results on self-decomposable distributions.
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Polynomial processes and their applications to mathematical Finance
We introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order $m$ only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Lévy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.
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