Please use this identifier to cite or link to this item: http://dx.doi.org/10.14279/depositonce-1411
Main Title: Volatility Markets: Consistent Modeling, Hedging and Practical Implementation
Translated Title: Volatilitätsmärkte: Konsistente Modellierung, Hedging und praktische Umsetzung
Author(s): Bühler, Hans
Advisor(s): Schied, Alexander
Granting Institution: Technische Universität Berlin, Fakultät II - Mathematik und Naturwissenschaften
Type: Doctoral Thesis
Language: English
Language Code: en
Abstract: Diese Arbeit beschäftigt sich mit dem systematischen Aufbau einer Theorie zur Modellierung von Volatilitätsmärkten mittels sogenannter Variance Swap Kurven. Ein Variance Swap auf beispielsweise einen Aktienindex zahlt, unter den Annahmen der vorliegenden Arbeit, die quadratische Variation der relativen Kursentwicklung an den Käufer des Vertrages. Die Frage ist nun, wie wir Modelle solcher Variance-Swap-Märkte entwickeln können, in denen es zum einen einen assoziierten Indexpreisprozess gibt, der mit den Variance Swaps kompatibel ist (in dem Sinne, dass die Variance-Swap-Preise der erwarteten quadratischen Variation der relativen Kursentwicklung entsprechen) und die zum anderen frei von Arbitrage bleiben. Entsprechende Ergebnisse im Geiste der Heath-Jarrow-Merton Theorie für Zinsraten werden entwickelt. Weiterhin zeigen wir, wie solche "Variance-Curve Modelle" durch niedrig-dimensionale Diffusionen getrieben werden können und ethablieren den Begriff eines "konsistent" Modells. Da wir diese Modelle außerdem zum Absichern exotischer Optionen auf Varianz, beispielsweise von Calls benutzen wollen, wenden wir uns außerdem der Frage der Vollständigkeit zu. Wir leiten Bedingungen her, unter welchen allgemeine Markovsche Märkte vollständig sind. Insbesondere wir nicht verlangt, daß die Volatilitätsmatrix der handelbaren Instrumente vollen Rang hat. Im angewandten Teil der Arbeit diskutieren wir die konkrete Implementierung eines Variance-Curve-Modells. Wir weisen nach, dass der Indexpreisprozess im vorgeschlagenen Modell ein echtes Martingal ist und dass der Markt der Optionen auf Varianz vollständig ist. Außerdem zeigen wir, wie der Prozeß effizient mit Monte-Carlo simuliert werden kann und wie diese numerische Simulation zur Kalibrierung des Modells eingesetzt werden kann. Wir diskutieren viele praktische Details wie auch Parameterhedging und die Frage der Vorbereitung von leicht gestörten Eingabedaten. Ein Vergleich einiger Modelle sowie eine Diskussion von Variance Swaps, Gamma Swaps und Entropy Swaps beenden die Arbeit.
In this dissertation, we study the modeling of markets of volatility contracts using variance swaps as basic instruments. Variance swaps essentially promise the payment of the "realized variance" of the returns of the underlying to the holder: their price is the market's expectation of the realized variance of the returns of the stock up to the maturity of the contract. As such, variance swaps are inherently strike-independent and a natural candidate for volatility-hedging of volatility products: for contracts such as calls and puts on realized variance, they are the equivalent of the discounted forward on the underlying. We develop a theoretical framework of general variance swap term structure models by following the ideas of Heath-Jarrow-Morton (HJM) for interest rates: the term structure of variance swaps will play the same role as the role played by term structure of zero bonds there. That means that instead of developing a model for the short variance directly, we describe the dynamics of the entire variance swap price curve. We then construct compatible stock price processes and their corresponding implied short variance dynamics. We tjem specialize the general framework to models which are driven by a low-dimensional Diffusion. The idea is that we first specify a functional form for the implied variance swap price curve and then drive the parameters of this curve in an arbitrage-free "consistent" way. These models are easier to handle and provide a "structural" access to a variance curve model. We also show how we can move from a "structural" variance curve model to a "fitting"-type model which perfectly matches observed market data. Next, we focus on the question of completeness. We provide the theoretical framework to assess when a variance curve model (or, in fact, any general Markov-driven model) generates a complete market. To this end, we show that if we only consider payoffs which are measurable with respect to the information generated by the traded assets (in opposition to the informati generated by the background driving Brownian motion), then a financial model often allows the replication of a payoff, even if the volatility matrix of the tradable instruments with respect to the driving Brownian motion is singular: if a value function of an exotic product can be differentiated at least once in the parameter of the market instruments, then these derivatives provide as expected the desired hedging ratios. We will show that if the value function for each non-negative smooth payoff function whose derivatives have compact support is always continuously differentiable, then the entire market is shown to be complete. We will show that this is the case, for example, if the coefficients of the diffusion which drive the market instruments are continuously differentiable with locally Lipschitz derivatives. In a second part, we will then discuss the impact of the practice of re-calibration to the "meta-model" of the institution, in particular the question whether the real-life price processes which are the result of this recalibration remain local martingales. We will show that this is for example not the case if the speed of mean-reversion or the product of "volatility of variance" and "correlation" in Heston's model are not kept constant. Similar results are shown for other mean-reversion type models. In the course of the discussion we also introduce what we will call "entropy swaps". They are closely related to another product, called "gamma swaps" or "weighted variance swaps". An appendix is devoted to the latter structures. The third part of this thesis is the application of the first two parts: we discuss the implementation of a double mean-reverting variance curve model. It is shown that the proposed model is well-defined and that the associated stock price process is a true martingale. We then proceed and discuss a Monte-Carlo implementation which allows efficient evaluation of exotic products. The resulting engine is finally used to calibrate the model in a multi-phase calibration routine. Ev though the routine is based on Monte-Carlo, it is still relatively fast and yields good results for most major indices. We also employ efficient algorithms to detect arbitrage in European option markets and show how market data which violate arbitrage-conditions can be fixed.
URI: urn:nbn:de:kobv:83-opus-13390
http://depositonce.tu-berlin.de/handle/11303/1708
http://dx.doi.org/10.14279/depositonce-1411
Exam Date: 26-Jun-2006
Issue Date: 10-Aug-2006
Date Available: 10-Aug-2006
DDC Class: 510 Mathematik
Subject(s): HJM
Varianz Swap
Vollständigkeit
Arbitrage
Complete Market
HJM
S tochastic Volatility
Variance Swap
Usage rights: Terms of German Copyright Law
Appears in Collections:Technische Universität Berlin » Fakultäten & Zentralinstitute » Fakultät 2 Mathematik und Naturwissenschaften » Institut für Mathematik » Publications

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