{"359671":{"#nid":"359671","#data":{"type":"event","title":"PhD Defense by Allen Hoffmeyer","body":[{"value":"\u003Cp\u003E PhD Defense by \u003Cstrong\u003EAllen Hoffmeyer\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003EIam defending my thesis titled \u0022Small-time asyptotics of call prices and implied volatilities for exponential L\u00e9vy models\u0022 on\u0026nbsp;January 6th\u0026nbsp;at 3pm. My committee members are:\u003Cbr \/\u003E\u003Cbr \/\u003EChristian Houdr\u00e9 (Adviser, School of Mathematics)\u003Cbr \/\u003EYuri Bakhtin (Courant Institute of Mathematical Sciences, NYU)\u003Cbr \/\u003EJos\u00e9 Enrique Figueroa-L\u00f3pez (Department of Statistics, Purdue)\u003C\/p\u003EVladimir Koltchinskii (School of Mathematics)Liang Peng (School of Mathematics)\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\u003Cp\u003E\u003Cstrong\u003ESmall-time Asymptotics of Call Prices and Implied Volatilities for Exponential L\u00b4evy\u003C\/strong\u003E\u003Cbr \/\u003E\u003Cstrong\u003EModels\u003C\/strong\u003E\u003Cbr \/\u003E\u003Cbr \/\u003EDirected by Professor Christian Houdr\u00b4e\u003Cbr \/\u003EWe derive call-price and implied volatility asymptotic expansions in time\u003Cbr \/\u003Eto maturity for a selection of exponential L\u00b4evy models. We consider asset-price models\u003Cbr \/\u003Ewhose log returns structure is a L\u00b4evy process. In particular, we consider L\u00b4evy\u003Cbr \/\u003Eprocesses of the form (Lt + \u03c3Wt)\u003Cbr \/\u003Et\u22650 where L = (Lt)\u003Cbr \/\u003Et\u22650\u003Cbr \/\u003Eis a pure-jump L\u00b4evy process\u003Cbr \/\u003Ein the domain of attraction of a stable random variable, W = (Wt)\u003Cbr \/\u003Et\u22650\u003Cbr \/\u003Eis a standard\u003Cbr \/\u003EBrownian motion independent of L, and \u03c3 \u2265 0.\u003Cbr \/\u003ECall-price asymptotics for in-the-money (ITM) and out-of-the-money (OTM) options\u003Cbr \/\u003Eare extensively covered in the literature; however, at-the-money (ATM) callprice\u003Cbr \/\u003Easymptotics under exponential L\u00b4evy models are relatively new.\u003Cbr \/\u003EIn this thesis, we consider two main problems. First, we consider very general\u003Cbr \/\u003EL\u00b4evy models for L. More specifically, L that are in the domain of attraction of a\u003Cbr \/\u003Estable random variable. Under some relatively minor assumptions, we give first-order\u003Cbr \/\u003Ecall-price and implied volatility asymptotics.\u003Cbr \/\u003EInterestingly, in the case where \u03c3 = 0 new orders of convergence are discovered\u003Cbr \/\u003Ewhich show a much richer structure than was previously considered. Concretely, we\u003Cbr \/\u003Eshow that the rate of convergence can be of the form t\u003Cbr \/\u003E1\/\u03b1`(t) where ` is a slowly\u003Cbr \/\u003Evarying function. We also give an example of a L\u00b4evy model which exhibits this new\u003Cbr \/\u003Etype of behavior and has a new order of convergence where ` is not asymptotically\u003Cbr \/\u003Econstant.\u003Cbr \/\u003EIn the case where \u03c3 6= 0, we show that the Brownian component is the dominant\u003Cbr \/\u003Eterm in the asymptotic expansion of the call-price. Under more general conditions on\u003Cbr \/\u003EL (even removing the requirement of L to be in the domain of attraction of a stablerandom variable), we show that the first-order call-price asymptotics are of the order\u003Cbr \/\u003E\u221a\u003Cbr \/\u003Et.\u003Cbr \/\u003EFinally, we investigate the CGMY process. For this process, call-price asymptotics\u003Cbr \/\u003Eare known to third order. Previously, measure transformation and technical\u003Cbr \/\u003Eestimation methods were the only tools available for proving the order of convergence.\u003Cbr \/\u003EIn the last chapter, we give a new method that relies on the Lipton-Lewis\u003Cbr \/\u003E(LL) formula. Using the LL formula guarantees that we can estimate the call-price\u003Cbr \/\u003Easymptotics using only the characteristic function of the L\u00b4evy process. While this\u003Cbr \/\u003Emethod does not provide a less technical approach, it is novel and is promising for\u003Cbr \/\u003Eobtaining second-order call-price asymptotics for ATM options for a more general\u003Cbr \/\u003Eclass of L\u00b4evy processes.\u003C\/p\u003E","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Small-time asyptotics of call prices and implied volatilities for exponential L\u00e9vy models"}],"uid":"28077","created_gmt":"2014-12-31 13:57:27","changed_gmt":"2016-10-08 02:10:54","author":"Danielle Ramirez","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2015-01-06T14:00:00-05:00","event_time_end":"2015-01-06T16:00:00-05:00","event_time_end_last":"2015-01-06T16:00:00-05:00","gmt_time_start":"2015-01-06 19:00:00","gmt_time_end":"2015-01-06 21:00:00","gmt_time_end_last":"2015-01-06 21:00:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"221981","name":"Graduate Studies"}],"categories":[],"keywords":[{"id":"1808","name":"graduate students"},{"id":"100811","name":"Phd Defense"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[{"id":"1788","name":"Other\/Miscellaneous"}],"invited_audience":[{"id":"78771","name":"Public"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}