The Equivalent CEV Volatility of the SABR Model up to O(T ) Is Given By
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A Tree-Based Method to Price American Options in the Heston Model
The Journal of Computational Finance (1–21) Volume 13/Number 1, Fall 2009 A tree-based method to price American options in the Heston model Michel Vellekoop Financial Engineering Laboratory, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands; email: [email protected] Hans Nieuwenhuis University of Groningen, Faculty of Economics, PO Box 800, 9700 AV Groningen, The Netherlands; email: [email protected] We develop an algorithm to price American options on assets that follow the stochastic volatility model defined by Heston. We use an approach which is based on a modification of a combined tree for stock prices and volatilities, where the number of nodes grows quadratically in the number of time steps. We show in a number of numerical tests that we get accurate results in a fast manner, and that features which are essential for the practical use of stock option pricing algorithms, such as the incorporation of cash dividends and a term structure of interest rates, can easily be incorporated. 1 INTRODUCTION One of the most popular models for equity option pricing under stochastic volatility is the one defined by Heston (1993): 1 dSt = µSt dt + Vt St dW (1.1) t = − + 2 dVt κ(θ Vt ) dt ω Vt dWt (1.2) In this model for the stock price process S and squared volatility process V the processes W 1 and W 2 are standard Brownian motions that may have a non-zero correlation coefficient ρ, while µ, κ, θ and ω are known strictly positive parameters. -
Local Volatility Modelling
LOCAL VOLATILITY MODELLING Roel van der Kamp July 13, 2009 A DISSERTATION SUBMITTED FOR THE DEGREE OF Master of Science in Applied Mathematics (Financial Engineering) I have to understand the world, you see. - Richard Philips Feynman Foreword This report serves as a dissertation for the completion of the Master programme in Applied Math- ematics (Financial Engineering) from the University of Twente. The project was devised from the collaboration of the University of Twente with Saen Options BV (during the course of the project Saen Options BV was integrated into AllOptions BV) at whose facilities the project was performed over a period of six months. This research project could not have been performed without the help of others. Most notably I would like to extend my gratitude towards my supervisors: Michel Vellekoop of the University of Twente, Julien Gosme of AllOptions BV and Fran¸coisMyburg of AllOptions BV. They provided me with the theoretical and practical knowledge necessary to perform this research. Their constant guidance, involvement and availability were an essential part of this project. My thanks goes out to Irakli Khomasuridze, who worked beside me for six months on his own project for the same degree. The many discussions I had with him greatly facilitated my progress and made the whole experience much more enjoyable. Finally I would like to thank AllOptions and their staff for making use of their facilities, getting access to their data and assisting me with all practical issues. RvdK Abstract Many different models exist that describe the behaviour of stock prices and are used to value op- tions on such an underlying asset. -
THE BLACK-SCHOLES EQUATION in STOCHASTIC VOLATILITY MODELS 1. Introduction in Financial Mathematics There Are Two Main Approache
THE BLACK-SCHOLES EQUATION IN STOCHASTIC VOLATILITY MODELS ERIK EKSTROM¨ 1,2 AND JOHAN TYSK2 Abstract. We study the Black-Scholes equation in stochastic volatility models. In particular, we show that the option price is the unique classi- cal solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is at- tainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate function spaces. On the other hand, if the boundary is non-attainable, then the boundary behaviour is not needed to guarantee uniqueness, but is nevertheless very useful for instance from a numerical perspective. 1. Introduction In financial mathematics there are two main approaches to the calculation of option prices. Either the price of an option is viewed as a risk-neutral expected value, or it is obtained by solving the Black-Scholes partial differ- ential equation. The connection between these approaches is furnished by the classical Feynman-Kac theorem, which states that a classical solution to a linear parabolic PDE has a stochastic representation in terms of an ex- pected value. In the standard Black-Scholes model, a standard logarithmic change of variables transforms the Black-Scholes equation into an equation with constant coefficients. Since such an equation is covered by standard PDE theory, the existence of a unique classical solution is guaranteed. Con- sequently, the option price given by the risk-neutral expected value is the unique classical solution to the Black-Scholes equation. However, in many situations outside the standard Black-Scholes setting, the pricing equation has degenerate, or too fast growing, coefficients and standard PDE theory does not apply. -
MERTON JUMP-DIFFUSION MODEL VERSUS the BLACK and SCHOLES APPROACH for the LOG-RETURNS and VOLATILITY SMILE FITTING Nicola Gugole
International Journal of Pure and Applied Mathematics Volume 109 No. 3 2016, 719-736 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v109i3.19 ijpam.eu MERTON JUMP-DIFFUSION MODEL VERSUS THE BLACK AND SCHOLES APPROACH FOR THE LOG-RETURNS AND VOLATILITY SMILE FITTING Nicola Gugole Department of Computer Science University of Verona Strada le Grazie, 15-37134, Verona, ITALY Abstract: In the present paper we perform a comparison between the standard Black and Scholes model and the Merton jump-diffusion one, from the point of view of the study of the leptokurtic feature of log-returns and also concerning the volatility smile fitting. Provided results are obtained by calibrating on market data and by mean of numerical simulations which clearly show how the jump-diffusion model outperforms the classical geometric Brownian motion approach. AMS Subject Classification: 60H15, 60H35, 91B60, 91G20, 91G60 Key Words: Black and Scholes model, Merton model, stochastic differential equations, log-returns, volatility smile 1. Introduction In the early 1970’s the world of option pricing experienced a great contribu- tion given by the work of Fischer Black and Myron Scholes. They developed a new mathematical model to treat certain financial quantities publishing re- lated results in the article The Pricing of Options and Corporate Liabilities, see [1]. The latter work became soon a reference point in the financial scenario. Received: August 3, 2016 c 2016 Academic Publications, Ltd. Revised: September 16, 2016 url: www.acadpubl.eu Published: September 30, 2016 720 N. Gugole Nowadays, many traders still use the Black and Scholes (BS) model to price as well as to hedge various types of contingent claims. -
Implied Volatility Modeling
Implied Volatility Modeling Sarves Verma, Gunhan Mehmet Ertosun, Wei Wang, Benjamin Ambruster, Kay Giesecke I Introduction Although Black-Scholes formula is very popular among market practitioners, when applied to call and put options, it often reduces to a means of quoting options in terms of another parameter, the implied volatility. Further, the function σ BS TK ),(: ⎯⎯→ σ BS TK ),( t t ………………………………(1) is called the implied volatility surface. Two significant features of the surface is worth mentioning”: a) the non-flat profile of the surface which is often called the ‘smile’or the ‘skew’ suggests that the Black-Scholes formula is inefficient to price options b) the level of implied volatilities changes with time thus deforming it continuously. Since, the black- scholes model fails to model volatility, modeling implied volatility has become an active area of research. At present, volatility is modeled in primarily four different ways which are : a) The stochastic volatility model which assumes a stochastic nature of volatility [1]. The problem with this approach often lies in finding the market price of volatility risk which can’t be observed in the market. b) The deterministic volatility function (DVF) which assumes that volatility is a function of time alone and is completely deterministic [2,3]. This fails because as mentioned before the implied volatility surface changes with time continuously and is unpredictable at a given point of time. Ergo, the lattice model [2] & the Dupire approach [3] often fail[4] c) a factor based approach which assumes that implied volatility can be constructed by forming basis vectors. Further, one can use implied volatility as a mean reverting Ornstein-Ulhenbeck process for estimating implied volatility[5]. -
The Equity Option Volatility Smile: an Implicit Finite-Difference Approach 5
The equity option volatility smile: an implicit finite-difference approach 5 The equity option volatility smile: an implicit ®nite-dierence approach Leif B. G. Andersen and Rupert Brotherton-Ratcliffe This paper illustrates how to construct an unconditionally stable finite-difference lattice consistent with the equity option volatility smile. In particular, the paper shows how to extend the method of forward induction on Arrow±Debreu securities to generate local instantaneous volatilities in implicit and semi-implicit (Crank±Nicholson) lattices. The technique developed in the paper provides a highly accurate fit to the entire volatility smile and offers excellent convergence properties and high flexibility of asset- and time-space partitioning. Contrary to standard algorithms based on binomial trees, our approach is well suited to price options with discontinuous payouts (e.g. knock-out and barrier options) and does not suffer from problems arising from negative branch- ing probabilities. 1. INTRODUCTION The Black±Scholes option pricing formula (Black and Scholes 1973, Merton 1973) expresses the value of a European call option on a stock in terms of seven parameters: current time t, current stock price St, option maturity T, strike K, interest rate r, dividend rate , and volatility1 . As the Black±Scholes formula is based on an assumption of stock prices following geometric Brownian motion with constant process parameters, the parameters r, , and are all considered constants independent of the particular terms of the option contract. Of the seven parameters in the Black±Scholes formula, all but the volatility are, in principle, directly observable in the ®nancial market. The volatility can be estimated from historical data or, as is more common, by numerically inverting the Black±Scholes formula to back out the level of Ðthe implied volatilityÐthat is consistent with observed market prices of European options. -
Quantification of the Model Risk in Finance and Related Problems Ismail Laachir
Quantification of the model risk in finance and related problems Ismail Laachir To cite this version: Ismail Laachir. Quantification of the model risk in finance and related problems. Risk Management [q-fin.RM]. Université de Bretagne Sud, 2015. English. NNT : 2015LORIS375. tel-01305545 HAL Id: tel-01305545 https://tel.archives-ouvertes.fr/tel-01305545 Submitted on 21 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ` ´ THESE / UNIVERSITE DE BRETAGNE SUD present´ ee´ par UFR Sciences et Sciences de l’Ing´enieur sous le sceau de l’Universit´eEurop´eennede Bretagne Ismail LAACHIR Pour obtenir le grade de : DOCTEUR DE L’UNIVERSITE´ DE BRETAGNE SUD Unite´ de Mathematiques´ Appliques´ (ENSTA ParisTech) / Mention : STIC Ecole´ Doctorale SICMA Lab-STICC (UBS) Th`esesoutenue le 02 Juillet 2015, Quantification of the model risk in devant la commission d’examen composee´ de : Mme. Monique JEANBLANC Professeur, Universite´ d’Evry´ Val d’Essonne, France / Presidente´ finance and related problems. M. Stefan ANKIRCHNER Professeur, University of Jena, Germany / Rapporteur Mme. Delphine LAUTIER Professeur, Universite´ de Paris Dauphine, France / Rapporteur M. Patrick HENAFF´ Maˆıtre de conferences,´ IAE Paris, France / Examinateur M. -
Empirical Performance of Alternative Option Pricing Models For
Empirical Performance of Alternative Option Pricing Models for Commodity Futures Options (Very draft: Please do not quote) Gang Chen, Matthew C. Roberts, and Brian Roe¤ Department of Agricultural, Environmental, and Development Economics The Ohio State University 2120 Fy®e Road Columbus, Ohio 43210 Contact: [email protected] Selected Paper prepared for presentation at the American Agricultural Economics Association Annual Meeting, Providence, Rhode Island, July 24-27, 2005 Copyright 2005 by Gang Chen, Matthew C. Roberts, and Brian Boe. All rights reserved. Readers may make verbatim copies of this document for non-commercial purposes by any means, provided that this copyright notice appears on all such copies. ¤Graduate Research Associate, Assistant Professor, and Associate Professor, Department of Agri- cultural, Environmental, and Development Economics, The Ohio State University, Columbus, OH 43210. Empirical Performance of Alternative Option Pricing Models for Commodity Futures Options Abstract The central part of pricing agricultural commodity futures options is to ¯nd appro- priate stochastic process of the underlying assets. The Black's (1976) futures option pricing model laid the foundation for a new era of futures option valuation theory. The geometric Brownian motion assumption girding the Black's model, however, has been regarded as unrealistic in numerous empirical studies. Option pricing models incor- porating discrete jumps and stochastic volatility have been studied extensively in the literature. This study tests the performance of major alternative option pricing models and attempts to ¯nd the appropriate model for pricing commodity futures options. Keywords: futures options, jump-di®usion, option pricing, stochastic volatility, sea- sonality Introduction Proper model for pricing agricultural commodity futures options is crucial to estimating implied volatility and e®ectively hedging in agricultural ¯nancial markets. -
Stochastic Volatility and Black – Scholes Model Evidence of Amman Stock Exchange
R M B www.irmbrjournal.com June 2017 R International Review of Management and Business Research Vol. 6 Issue.2 I Stochastic Volatility and Black – Scholes Model Evidence of Amman Stock Exchange MOHAMMAD. M. ALALAYA Associate Prof in Economic Methods Al-Hussein Bin Talal University, Collage of Administrative Management and Economics, Ma‟an, Jordan. Email: [email protected] SULIMAN ALKHATAB Professer in Marketing, Al-Hussein Bin Talal University, Collage of Administrative Management and Economics, Ma‟an, Jordan. AHMAD ALMUHTASEB Assistant Prof in Marketing, Al-Hussein Bin Talal University, Collage of Administrative Management and Economics, Ma‟an, Jordan. JIHAD ALAFARJAT Assistant Prof in Management, Al-Hussein Bin Talal University, Collage of Administrative Management and Economics, Ma‟an, Jordan. Abstract This paper makes an attempt to decompose the Black – Scholes into components in Garch option model, and to examine the path of dependence in the terminal stock price distribution of Amman Stock Exchange (ASE), such as Black – Scholes’, the leverage effect in this paper which represents the result of analysis is important to determine the direction of the model bias, a time varying risk, may give fruitful help in explaining the under pricing of traded stock sheers and traded options in ASE. Generally, this study considers various pricing bias related to warrant of strike prices, and observes time to time maturity. The Garch option price does not seem overly sensitive to (a, B1) parameters, or to the time risk premium variance persistence parameter, Ω = a1+B1, heaving on the magnitude of Black –Scholes’ bias of the result of analysis, where the conditional variance bias does not improve in accuracy to justify the model to ASE data. -
Managed Futures and Long Volatility by Anders Kulp, Daniel Djupsjöbacka, Martin Estlander, Er Capital Management Research
Disclaimer: This article appeared in the AIMA Journal (Feb 2005), which is published by The Alternative Investment Management Association Limited (AIMA). No quotation or reproduction is permitted without the express written permission of The Alternative Investment Management Association Limited (AIMA) and the author. The content of this article does not necessarily reflect the opinions of the AIMA Membership and AIMA does not accept responsibility for any statements herein. Managed Futures and Long Volatility By Anders Kulp, Daniel Djupsjöbacka, Martin Estlander, er Capital Management Research Introduction and background The buyer of an option straddle pays the implied volatility to get exposure to realized volatility during the lifetime of the option straddle. Fung and Hsieh (1997b) found that the return of trend following strategies show option like characteristics, because the returns tend to be large and positive during the best and worst performing months of the world equity markets. Fung and Hsieh used lookback straddles to replicate a trend following strategy. However, the standard way to obtain exposure to volatility is to buy an at-the-money straddle. Here, we will use standard option straddles with a dynamic updating methodology and flexible maturity rollover rules to match the characteristics of a trend follower. The lookback straddle captures only one trend during its maturity, but with an effective systematic updating method the simple option straddle captures more than one trend. It is generally accepted that when implementing an options strategy, it is crucial to minimize trading, because the liquidity in the options markets are far from perfect. Compared to the lookback straddle model, the trading frequency for the simple straddle model is lower. -
White Paper Volatility: Instruments and Strategies
White Paper Volatility: Instruments and Strategies Clemens H. Glaffig Panathea Capital Partners GmbH & Co. KG, Freiburg, Germany July 30, 2019 This paper is for informational purposes only and is not intended to constitute an offer, advice, or recommendation in any way. Panathea assumes no responsibility for the content or completeness of the information contained in this document. Table of Contents 0. Introduction ......................................................................................................................................... 1 1. Ihe VIX Index .................................................................................................................................... 2 1.1 General Comments and Performance ......................................................................................... 2 What Does it mean to have a VIX of 20% .......................................................................... 2 A nerdy side note ................................................................................................................. 2 1.2 The Calculation of the VIX Index ............................................................................................. 4 1.3 Mathematical formalism: How to derive the VIX valuation formula ...................................... 5 1.4 VIX Futures .............................................................................................................................. 6 The Pricing of VIX Futures ................................................................................................ -
Some Mathematical Aspects of Market Impact Modeling by Alexander Schied and Alla Slynko
EMS Series of Congress Reports EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowro´nski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes Jochen Blath Peter Imkeller Sylvie Rœlly Editors Editors: Jochen Blath Peter Imkeller Sylvie Rœlly Institut für Mathematik Institut für Mathematik Institut für Mathematik der Technische Universität Berlin Humboldt-Universität zu Berlin Universität Potsdam Straße des 17. Juni 136 Unter den Linden 6 Am Neuen Palais, 10 10623 Berlin 10099 Berlin 14469 Potsdam Germany Germany Germany [email protected] [email protected] [email protected] 2010 Mathematics Subject Classification: Primary: 60-06, Secondary 60Gxx, 60Jxx Key words: Stochastic processes, stochastic finance, stochastic analysis,statistical physics, stochastic differential equations ISBN 978-3-03719-072-2 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks.