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Asymptotics of Forward Implied Volatility ASYMPTOTICS OF FORWARD IMPLIED VOLATILITY by Patrick Francois Springfield Roome Department of Mathematics Imperial College London London SW7 2AZ United Kingdom Submitted to Imperial College London for the degree of Doctor of Philosophy 2015 1 Declaration I the undersigned hereby declare that the work presented in this thesis is my own. When mate- rial from other authors has been used, these have been duly acknowledged. This thesis has not previously been presented for this or any other PhD examinations. Patrick Francois Springfield Roome 2 Copyright The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 3 \Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." Niels Hendrik Abel, 1828 Abstract We study asymptotics of forward-start option prices and the forward implied volatility smile using the theory of sharp large deviations (and refinements). In Chapter 1 we give some intu- ition and insight into forward volatility and provide motivation for the study of forward smile asymptotics. We numerically analyse no-arbitrage bounds for the forward smile given calibration to the marginal distributions using (martingale) optimal transport theory. Furthermore, we derive several representations of forward-start option prices, analyse various measure-change symmetries and explore asymptotics of the forward smile for small and large forward-start dates. In Chapter 2 we derive a general closed-form expansion formula (including large-maturity and `diagonal' small-maturity asymptotics) for the forward smile in a large class of models including the Heston and Sch¨obel-Zhu stochastic volatility models and time-changed exponential L´evymodels. In Chapter 3 we prove that the out-of-the-money small-maturity forward smile explodes in the Heston model and a separate model-independent analysis shows that the at-the-money small- maturity limit is well defined for any It^odiffusion. Chapter 4 provides a full characterisation of the large-maturity forward smile in the Heston model. Although the leading-order decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms depends highly on the parameters, and different powers of the maturity come into play. Classical (It^odiffusions) stochastic volatility models are not able to capture the steepness of small-maturity (spot) implied volatility smiles. Models with jumps, exhibiting small-maturity exploding smiles, have historically been proposed as an alternative. A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum [74], who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity. In Chapter 5 we suggest a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential L´evymodels and fractional stochastic volatility models. As a by-product, we make a conjecture on the small- maturity forward smile asymptotics of stochastic volatility models, in exact agreement with the results in Chapter 3 for Heston. 4 I dedicate this thesis to the memory of my father. 5 Acknowledgements First and foremost, I would like to thank my supervisor, Dr. Antoine Jacquier. It has been an honour to be his first PhD student. His guidance, support and encouragement were vital throughout my PhD and his passion and insight into the subject was inspirational. I remember leaving his office numerous times (usually after several hours of discussions) with a renewed sense of motivation and excitement. The Imperial Mathematical Finance department always made PhD students welcome and pro- vided a number of avenues for learning and growth. I was given the opportunity to present my work to members of the group on several occasions. The atmosphere was stimulating and discussions and questions about my work moved me in the right direction. I would like to thank my co-authors, Dr. Stefano De Marco (Ecole Polytechnique) and Mr. Hamza Guennoun (Ecole Polytechnique). I would also like to thank Prof. Archil Gulisashvili (Ohio University) for a stimulating and insightful discussion on my research during one of his Imperial visits. Thanks to Prof. Damiano Brigo and Dr. Claude Martini for agreeing to be my PhD examiners: I very much appreciate the time and effort they will put into my thesis. Before the PhD I was a quant at Bank of America Merrill Lynch. I was very fortunate to have the opportunity to work with Dr. Leif Andersen. He supported and encouraged my decision to do a full-time PhD and I am very grateful for his advice and guidance. I would very much like to thank my parents and family for their support and always encouraging me to follow my dreams. Finally, I would like to thank my fianc´eeDaphne for her endless love, kindness and support. 6 Epigraph \Only that at times like this, when you're directionless in a dark wood, trust to the abstract deductive... Leap like a knight of faith into the arms of Peano, Leibniz, Hilbert, L'H^opital. You will be lifted up." David Foster Wallace 7 Contents 1 Introduction 13 1.1 No-arbitrage bounds for the forward smile given marginals . 15 1.1.1 Problem formulation . 17 1.1.2 No-Arbitrage discretisation of the primal and dual problems . 18 1.1.2.1 Primal and dual formulation . 19 1.1.2.2 Approximation of the dual . 21 1.1.3 Primal solution for the at-the-money case . 21 1.1.3.1 Structure of the transport plan . 21 1.1.3.2 Implementation . 22 1.1.4 Numerical analysis of the no-arbitrage bounds . 23 1.1.5 Numerical analysis of the transport plans . 24 1.2 Large deviations theory and the Laplace method . 28 1.3 Models and forward moment generating functions . 31 1.3.1 Stochastic volatility models . 31 1.3.1.1 Heston . 32 1.3.1.2 Sch¨obel-Zhu . 33 1.3.2 Time-changed exponential L´evymodels . 34 1.4 Pricing forward-start options . 35 1.4.1 Measure-change symmetries . 37 1.4.2 Representations of forward-start option prices . 40 1.4.2.1 Inverse Fourier transform representation . 40 1.4.2.2 Mixing formula in stochastic volatility models . 41 1.4.2.3 Non-stationary representation in stochastic volatility models . 42 1.4.2.4 Random initial variance representation in stochastic volatility models 42 1.5 Small and large forward-start dates . 43 1.5.1 Small forward-start dates . 43 1.5.2 Large forward-start dates . 47 8 1.6 Structure of thesis . 47 2 A general asymptotic formula for the forward smile 49 2.1 Introduction . 49 2.2 General Results . 50 2.2.1 Notations and main theorem . 50 2.2.1.1 Notations and preliminary results . 50 2.2.1.2 Main theorem and corollaries . 52 2.2.2 Forward-start option asymptotics . 53 2.2.2.1 Diagonal small-maturity asymptotics . 53 2.2.2.2 Large-maturity asymptotics . 54 2.2.3 Forward smile asymptotics . 55 2.2.3.1 Diagonal small-maturity forward smile . 56 2.2.3.2 Large-maturity forward smile . 56 2.2.3.3 Type-II forward smile . 57 2.3 Applications . 58 2.3.1 Heston . 58 2.3.1.1 Diagonal Small-Maturity Heston Forward Smile . 58 2.3.1.2 Large-maturity Heston forward smile . 61 2.3.2 Sch¨obel-Zhu . 63 2.3.3 Time-changed exponential L´evy . 64 2.4 Numerics . 67 2.5 Proofs . 68 2.5.1 Proofs of Section 2.2 . 68 2.5.1.1 Proof of Theorem 2.2.4 . 68 2.5.1.2 Proof of Propositions 2.2.10 and 2.2.11 . 78 2.5.2 Proofs of Section 2.3.1 . 79 2.5.2.1 Proofs of Section 2.3.1.1 . 79 2.5.2.2 Proofs of Section 2.3.1.2 . 83 2.5.3 Proofs of Section 2.3.3 . 88 3 The small-maturity Heston forward smile 90 3.1 Introduction . 90 3.2 Forward time-scales . 91 3.3 Small-maturity forward-start option asymptotics . 92 3.4 Small-maturity forward smile asymptotics . 95 3.4.1 Out-of-the-money forward implied volatility . 95 9 3.4.2 At-the-money forward implied volatility . 96 3.5 Numerics . 99 3.6 Proof of Theorems 3.3.1 and 3.4.1 . 99 3.6.1 Heston forward time-scale . 100 3.6.2 Asymptotic time-dependent measure-change . 104 3.6.3 Characteristic function asymptotics . 108 3.6.4 Option price and forward smile asymptotics . 113 4 Large-maturity regimes of the Heston forward smile 114 4.1 Introduction . 114 4.2 Large-maturity regimes . 115 4.3 Forward-start option asymptotics . 116 4.3.1 Connection with large deviations . 119 4.4 Forward smile asymptotics . 120 4.4.1 SVI-type limits . 122 4.5 Numerics . 124 4.6 Proof of Theorems 4.3.1 and 4.4.1 . 124 4.6.1 Forward cumulant generating function (cgf) expansion and limiting domain 127 4.6.2 The strictly convex case .
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