Volatility Markets Consistent modeling, hedging and practical implementation vorgelegt von Diplom-Mathematiker Hans BÄuhler Berlin FakultÄatII - Mathematik und Naturwissenschaften der Technischen UniversitÄatBerlin zur Erlangung des akademischen Grades Doktor der Natuwissenschaften Dr. rer. nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. rer. nat. Stefan Felsner Berichter/Gutachter: Prof. Dr. rer. nat. Alexander Schied Berichter/Gutachter: Prof. Dr. rer. nat. Peter Imkeller Gutachter: Prof. Dr. rer. nat. Josef Teichmann Tag der wissenschaftlichen Aussprache: 26. Juni 2006 Berlin 2006, D83 1 To my parents, for all their love and patience. In Liebe, Hans. Contents 1 Introduction 5 1.1 Basic Assumptions . 10 1.1.1 Variance Swaps . 10 1.1.2 Options on Variance . 12 1.2 Mathematical Notation . 13 I Consistent Modelling 16 2 Consistent Variance Curve Models 17 2.1 Problem Statements and Overview . 17 2.1.1 Review of the Stochastic Volatility Case . 19 2.2 General Variance Curve Models . 21 2.2.1 The Martingale Property and Explosion of Variance . 24 2.2.2 Fixed Time-to-Maturity . 24 2.2.3 Fitting the Market with Exponential Variance Curve Models . 27 2.3 Consistent Variance Curve Functionals . 29 2.3.1 Markov Variance Curve Market Models . 30 2.3.2 HJM-Conditions for Consistent Parameter Processes . 31 2.3.3 Extensions to Manifolds: When does Z stay in Z ? . 32 2.4 Variance Curve Models in Hilbert Spaces . 35 3 Examples 37 3.1 Exponential-Polynomial Variance Curve Models . 37 3.2 Exponential Curves . 41 3.3 Variance Swap Volatility Curves . 41 3.4 Fitting Models . 43 II Hedging 47 4 Theory of Replication 48 4.1 Problem Statements and Overview . 48 4.2 Hedging in Complete Markets . 51 4.2.1 General Complete Markovian Markets . 51 4.2.2 Pricing with Local Martingales . 58 4.2.3 Hedging with Variance Swaps . 59 2 CONTENTS 3 4.2.4 Hedging in classic Stochastic Volatility Models . 65 5 Hedging in Practice 66 5.1 Model and Market . 66 5.1.1 Additional Market Instruments . 69 5.2 Parameter Hedging in Practise . 73 5.2.1 Constrained Parameter-Hedging in Practise . 74 5.3 Dynamic Arbitrage . 77 5.3.1 Entropy Swaps . 78 III Practical Implementation 82 6 A variance curve model 83 6.1 The Model . 83 6.1.1 Existence, Uniqueness and the Martingale Property . 87 6.2 Pricing . 92 6.2.1 Pricing General Payo®s using an unbiased Milstein Monte-Carlo Scheme . 93 6.2.2 Control Variates . 98 6.2.3 Pricing Options on Variance . 99 6.2.4 Pricing European Options on the Stock . 99 7 Calibration 102 7.1 Notation and Overview . 102 7.2 Numerical Calibration . 104 7.2.1 Intraday Use . 106 7.3 Phase 1: Market Data Adjustment . 107 7.3.1 The Balayage-Order . 107 7.3.2 Upper Pricing Measures . 108 7.3.3 Relative Upper Pricing Measures . 112 7.3.4 Test for Strict Absence of Arbitrage . 114 7.3.5 How to Produce a strictly Arbitrage-free Surface . 115 7.3.6 Summary . 117 7.4 Phase 2: State Calibration . 117 7.5 Phase 3: Parameter Calibration . 118 7.5.1 ATM calibration . 118 7.5.2 Calibration in Steps . 120 7.5.3 Examples . 120 7.5.4 Pricing Options on Variance . 124 8 Conclusions 129 A Variance Swaps, Entropy Swaps and Gamma Swaps 131 A.1 Basic Pricing and Hedging . 131 A.1.1 Variance Swaps . 132 A.1.2 Entropy Swaps . 134 A.1.3 Shadow Options . 135 CONTENTS 4 A.2 Deterministic Interest Rates and Proportional Dividends . 137 A.2.1 European Options . 138 A.2.2 Variance Swaps . 139 A.2.3 Gamma Swaps . 139 B Example Term Sheets 142 C Auxiliary Results 148 C.1 The Impact of incorrect Stochastic Volatility Dynamics . 148 D Transition Densities under Constraints 150 D.1 Expensive Martingales . 150 D.1.1 Construction of a Transition Kernel . 151 D.2 Incorporating Weak Information . 153 D.2.1 Mean-Variance Pricing . 153 D.2.2 Using Forward Started Call Prices . 154 Index 156 Bibliography 158 Chapter 1 Introduction Ever since Black, Scholes and Merton published their famous articles [BS73] and [M73], huge markets of ¯nancial derivatives on a wide range of underlying economic quantities have devel- oped. One of the most visible markets of underlyings is surely the equity market with index level and share price quotes being a common part of today's news programmes. Upon it rest deep exchange-based markets of \vanilla" derivatives on indices and single stocks. This development of course changes the way over-the-counter products (those which are agreed upon on a case-by-case basis between the counter-parties) are evaluated and risk-managed. While Black and Scholes (BS) used only the underlying stock price and the bond to hedge a derivative in their model,1 this cannot be justi¯ed anymore: their model is not able to capture what is today known as the \volatility skew", or \volatility smile", of the implied volatility of traded vanilla options. The root of the discrepancy is that volatility is not, as assumed in BS' model, a deterministic quantity. Rather, it is by itself stochastic. The stochastic nature of the instantaneous variance of the stock price process is particular important if we want to price and hedge heavily volatility-dependent exotic options such as options on realized variance or cliquet-type products.2 Such products cannot be priced correctly in the BS-model since their very risk lies in the movement of volatility (or variance, for that matter) itself. Beyond Black-Scholes There have been many approaches to remedy this problem: the most pragmatic idea is to infer an implied risk-neutral distribution from the observed market prices. To this end, Dupire [D96] has completely solved the problem of ¯nding a one-factor di®usion which reprices a continuum of market prices. His implied local volatility is today a standard tool for evaluating exotic derivatives. However, the resulting stock price dynamics are not overly realistic since the resulting dif- fusion is usually highly inhomogeneous in time. This implies that the model makes predictions about the future which are not matched by past market experience. Most notably, the implied volatility smile inside the model flattens out over time which is in contrast to the persistent presence of this phenomenon in reality. This in turn means that the dynamical behavior of the liquid options is not captured very well. 1In fact, the model is due to Samuelson [S65], but it is common to call it \Black&Scholes model". 2See section 1.1.2 for example payo®s. 5 CHAPTER 1. INTRODUCTION 6 Conceptually quite di®erent from this ¯tting approach are stochastic volatility models. In these models, a parsimonious description of the dynamics of both the stock price and its in- stantaneous variance is the starting point. Such a model is based on \structural" assumptions on the underlying stock price. For example, Heston's popular model [H93] assumes that the instantaneous variance of the stock price is a square-root di®usion whose increments are cor- related to the increments of the return of the stock price. Other popular stochastic volatility models are Hagan et al. [HKLW02], Schoebel/Zhu [SZ99] and Fouque et al. [FPS00] to name but a few. In addition, there are also models which incorporate jump processes (see for example Merton [M76], Carr et al. [CCM98]) or mixtures of both concepts such as Bates [B96]. A good reference on models based on L¶evyprocesses is Cont/Tankov [CT03]. Most of these structural models will lead per se to incomplete market models if only the stock and the cash bond are considered as tradable instruments. As a result, there is no unique fair price for most derivatives. To alleviate this problem in continuous models,3 we have to extend the range of tradable instruments. Broadly speaking, each additional source of randomness requires an additional traded instrument to be able to hedge the resulting risk. This is called \completion of the market" (see also Davis [Da04]). However, it not clear which traded instruments we have to choose to complete our market. Indeed, if we are to use the stock price together with a range of liquid reference options as hedging instruments, a more natural approach would be to model directly the evolution of the stock price and these reference options simultaneously. Such a framework has the advantage that the options are an integral part of the model and that the model yields hedging strategies directly in terms of the traded reference instruments. The most prominent approach has been to model call and put prices via their implied volatil- ities. This has been undertaken by Brace et al. [BGKW01], Cont et al. [CFD02], Fengler et al. [FHM03] and Ha®ner [H04], among others. However, to our knowledge, none of theses stochastic implied volatility models is able to ensure the absence of arbitrage situations (such as negative prices for butterfly trades) throughout the life of the model. For this reason, some authors have focused on the term-structure of implied volatilities for just one ¯xed cash strike. This approach has been pioneered by SchÄonbucher [S99] and has recently been put into a more general framework by Schweizer/Wissel [SW05] who also consider power-type payo®s. This approach is attractive for pricing strike-dependent options such as compound options. However, the dependency on a ¯xed cash strike also implies that if the market moves, the model's ¯xed strike may drift too far out of the money to be suitable for hedging purposes. This is of particular concern if we want to price and hedge mainly volatility- dependent products such as options on realized variance or cliquets.