DOCSLIB.ORG

December 28, 2000 15:0 WSPC/104-IJTAF 00087 IMPLIED

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
December 28, 2000 15:0 WSPC/104-IJTAF 00087 IMPLIED December 28, 2000 15:0 WSPC/104-IJTAF 00087 International Journal of Theoretical and Applied Finance Vol. 4, No. 1 (2001) 45–89 c World Scientific Publishing Company IMPLIED AND LOCAL VOLATILITIES UNDER STOCHASTIC VOLATILITY ROGER W. LEE∗ Department of Mathematics, Stanford University, Stanford, CA 94305, USA Received 8 February 1999 Accepted 4 June 1999 For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some na¨ıve pricing and hedging schemes. We begin by reviewing option pricing under stochastic volatility and representing option prices and local volatilities in terms of expectations. In the case that fluctuations in price and volatility have zero correlation, the expectations formula shows that local volatility (like implied volatility) as a function of log-moneyness has the shape of a sym- metric smile. In the case of non-zero correlation, we extend Sircar and Papanicolaou’s asymptotic expansion of implied volatilities under slowly-varying stochastic volatility. An asymptotic expansion of local volatilities then verifies the rule of thumb that local volatility has the shape of a skew with roughly twice the slope of the implied volatil- ity skew. Also we compare the slow-variation asymptotics against what we call small- variation asymptotics, and against Fouque, Papanicolaou, and Sircar’s rapid-variation asymptotics. We apply the slow-variation asymptotics to approximate the biases of two na¨ıve pricing strategies. These approximations shed some light on the signs and the relative magnitudes of the biases empirically observed in out-of-sample pricing tests of implied- volatility and local-volatility schemes. Similarly, we examine the biases of three different strategies for hedging under stochastic volatility, and we propose ways to implement these strategies without having to specify or estimate any particular stochastic volatility model. Our approximations suggest that a number of the empirical pricing and hedging biases may be explained by a positive premium for the portion of volatility risk that is uncorrelated with asset risk. Keywords: Stochastic volatility, local volatility, implied volatility, volatility risk pre- mium, skew, smile. 1. Introduction 46 2. Models 48 2.1. Implied and local volatilities . .................... 48 2.2. Review of option pricing under stochastic volatility ........... 49 3. Prices and Volatilities via Expectations 53 3.1.Pricesviaexpectations....................... 53 3.2. Local volatilities via expectations . ................ 54 ∗E-mail: [email protected] 45 December 28, 2000 15:0 WSPC/104-IJTAF 00087 46 R. W. Lee 4. Local and Implied Volatility Smiles 55 4.1.Smilesymmetry:Non-parametriccharacterizations........... 55 4.2. Smile symmetry under stochastic volatility .............. 56 5. Asymptotic Expansions for Implied and Local Volatility 57 5.1.Slow-variationasymptotics..................... 57 5.2.Small-variationasymptotics..................... 60 5.3.Discussion............................ 61 5.4.Figures.............................. 62 5.5.Slowvariationversusrapidvariation................. 62 6. Asymptotic Valuation Errors 72 6.1. Implied volatilities: The Ad-Hoc,“StickyStrike”method........ 72 6.2. Local volatilities: The DVF,“StickyImpliedTree”method....... 75 6.3. Two notions of volatility risk premia . ............... 77 6.4.Comparisons........................... 77 7. Asymptotic Hedging Errors 78 7.1. Hedge at the stochastic-volatility delta . ............... 79 7.2.Hedgetominimizeinstantaneousvariance............... 80 7.3.HedgeattheBlack–Scholesdelta.................. 83 7.4. Two notions of volatility risk premia, revisited . .......... 84 7.5.Comparisons........................... 84 8. Further Research 86 Acknowledgments 86 References 86 1. Introduction For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium- based explanations of the biases in some na¨ıve pricing and hedging schemes. Section 2 reviews implied (Black–Scholes) volatilities, local volatilities and option pricing under stochastic volatility, taking care to distinguish among three different notions of “volatility risk premium.” Section 3 treats these ideas proba- bilistically, representing option prices and local volatilities in terms of expectations. Section 4 deals with the symmetric implied and local volatility smiles that arise in the case where fluctuations in price and volatility are uncorrelated. It begins by showing the general equivalence of symmetry in Black–Scholes implied volatility and symmetry in local volatility. Then it applies the expectations formula to show that local volatility as a function of log-moneyness has the shape of a symmetric smile. In the correlated case, Sircar and Papanicolaou [61] derive for implied volati- lity an approximation linear in time to maturity and in log-moneyness, under the assumption that volatility is a slowly-varying stochastic process. Section 5 extends to the next order this asymptotic expansion, finding a term that is quadratic in December 28, 2000 15:0 WSPC/104-IJTAF 00087 Implied and Local Volatilities Under Stochastic Volatility 47 time-to-maturity and log-moneyness. We expand local volatilities analogously, ver- ifying the rule of thumb that the local volatility skew has roughly twice the slope of the implied volatility skew. Changing the volatility assumption from slow-variation to small-variation, we obtain some alternative expansions for implied and local volatility. The slow and small variation asymptotics are closely related. For example, in both expansions, the local volatility skew, as a function of log-moneyness, has approximately twice the slope of the Black–Scholes implied volatility skew; likewise, the local volatility term structure also has approximately twice the slope of the implied volatility term structure, in both expansions. In contrast, the slow-variation development and the rapid-variation development of Fouque, Papanicolaou and Sircar [33] are in many ways diametrically opposed. We highlight these points of opposition; but we also discuss the possibility of reconciling the two approaches — a project which can be viewed in terms of bridging the gap between the short-horizon domain of validity of the slow-variation asymptotics, and the long-horizon domain of validity of the rapid-variation asymptotics. Turning our attention to applications, we approximate in Sec. 6 the biases of na¨ıve pricing strategies, assuming that slowly-varying stochastic volatility is the true description of the underlying price process. Each na¨ıve model attempts to price options using a local or implied volatility function calibrated to old (say, one week old) market prices. The sticky strike or Ad Hoc (AH) model assumes that implied volatility as a function of strike and maturity has not changed. The sticky implied tree or Deterministic Volatility Function (DV F)modelassumesthatlocal volatility as a function of strike and maturity has not changed. Our calculations offer theoretical explanations for empirical observations, by Dumas, Fleming and Whaley [28], and/or by Rosenberg [58], that the AH and DV F valuations are biased downward, and the DV F bias is worse. The signs and relative magnitudes of the valuation biases appear to be consistent with a positive risk premium λ(2) for the portion of volatility risk that is uncorrelated with asset risk. As another application, we approximate in Sec. 7 the biases of various strategies that attempt to hedge a call under stochastic volatility, using only the underlying stock and the riskless security. The strategies are to hedge at the stochastic volatility delta, to hedge at the ratio that minimizes instantaneous variance of the portfolio, or to hedge at the Black–Scholes delta evaluated at the implied volatility. We also propose how to implement these strategies without specifying or estimating any particular stochastic volatility model. Our calculations indicate that Bakshi, Cao and Chen’s [3] empirical observa- tions, of an upward bias in the minimal instantaneous variance strategy, can be explained again by a positive λ(2). Moreover, under the defensible assumptions of a positive λ(2) and a negative premium λV for a second notion of volatility risk, the slow-variation and small-variation expansions suggest that the stochastic-volatility delta hedge is biased low, the minimal instantaneous variance hedge is biased high, and the Black–Scholes delta hedge is somewhere in between. December 28, 2000 15:0 WSPC/104-IJTAF 00087 48 R. W. Lee 2. Models 2.1. Implied and local volatilities Throughout this paper we will for convenience assume a riskless rate of 0. Generali- zation to non-zero but deterministic interest rates is straightforward; for example we could take the riskless security as numeraire, or express prices as forward prices. If some asset price X follows a diffusion (1) dXt = µtXtdt + σXtdBt with constant volatility σ, then the time-t no-arbitrage price of a European-style call struck at K and expiring at T is given by the Black–Scholes
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Applications of Realized Volatility, Local Volatility and Implied Volatility Surface in Accuracy Enhancement of Derivative Pricing Model
    APPLICATIONS OF REALIZED VOLATILITY, LOCAL VOLATILITY AND IMPLIED VOLATILITY SURFACE IN ACCURACY ENHANCEMENT OF DERIVATIVE PRICING MODEL by Keyang Yan B. E. in Financial Engineering, South-Central University for Nationalities, 2015 and Feifan Zhang B. Comm. in Information Management, Guangdong University of Finance, 2016 B. E. in Finance, Guangdong University of Finance, 2016 PROJECT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN FINANCE In the Master of Science in Finance Program of the Faculty of Business Administration © Keyang Yan, Feifan Zhang, 2017 SIMON FRASER UNIVERSITY Term Fall 2017 All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. Approval Name: Keyang Yan, Feifan Zhang Degree: Master of Science in Finance Title of Project: Applications of Realized Volatility, Local Volatility and Implied Volatility Surface in Accuracy Enhancement of Derivative Pricing Model Supervisory Committee: Associate Professor Christina Atanasova _____ Professor Andrey Pavlov ____________ Date Approved: ___________________________________________ ii Abstract In this research paper, a pricing method on derivatives, here taking European options on Dow Jones index as an example, is put forth with higher level of precision. This method is able to price options with a narrower deviation scope from intrinsic value of options. The finding of this pricing method starts with testing the features of implied volatility surface. Two of three axles in constructed three-dimensional surface are respectively dynamic strike price at a given time point and the decreasing time to maturity within the life duration of one strike-specified option.
    [Show full text]
  • Local Volatility Surface - Market Models - Arbitrage-Free Term Structure Dynamics - Heathjarrowmorton Theory
    1 2 LOCAL VOLATILITY DYNAMIC MODELS RENE´ CARMONA AND SERGEY NADTOCHIY BENDHEIM CENTER FOR FINANCE, ORFE PRINCETON UNIVERSITY PRINCETON, NJ 08544 [email protected] & [email protected] ABSTRACT. This paper is concerned with the characterization of arbitrage free dynamic stochastic models for the equity markets when Itoˆ stochastic differential equations are used to model the dynam- ics of a set of basic instruments including, but not limited to, the underliers. We study these market models in the framework of the HJM philosophy originally articulated for Treasury bond markets. The approach to dynamic equity models which we follow was originally advocated by Derman and Kani in a rather informal way. The present paper can be viewed as a rigorous development of this program, with explicit formulae, rigorous proofs and numerical examples. Keywords Implied volatilty surface - Local Volatility surface - Market models - Arbitrage-free term structure dynamics - HeathJarrowMorton theory. Mathematics Subject Classification (2000) 91B24 JEL Classification (2000) G13 1. INTRODUCTION AND NOTATION Most financial market models introduced for the purpose of pricing and hedging derivatives con- centrate on the dynamics of the underlying stocks, or underlying instruments on which the derivatives are written. This is clearly the case in the Black-Scholes theory where the focus is on the dynamics of the underlying stocks, whether they are assumed to be given by geometric Brownian motions or more general non-negative diffusions, or even semi-martingales with jumps. In contrast, the focus of the present paper is on the simultaneous dynamics of all the liquidly traded derivative instruments written on the underlying stocks. For the sake of simplicity, we limit ourselves to a single underlying index or stock on which all the derivatives under consideration are written.
    [Show full text]
  • Local Volatility, Stochastic Volatility and Jump-Diffusion Models
    IEOR E4707: Financial Engineering: Continuous-Time Models Fall 2013 ⃝c 2013 by Martin Haugh Local Volatility, Stochastic Volatility and Jump-Diffusion Models These notes provide a brief introduction to local and stochastic volatility models as well as jump-diffusion models. These models extend the geometric Brownian motion model and are often used in practice to price exotic derivative securities. It is worth emphasizing that the prices of exotics and other non-liquid securities are generally not available in the market-place and so models are needed in order to both price them and calculate their Greeks. This is in contrast to vanilla options where prices are available and easily seen in the market. For these more liquid options, we only need a model, i.e. Black-Scholes, and the volatility surface to calculate the Greeks and perform other risk-management tasks. In addition to describing some of these models, we will also provide an introduction to a commonly used fourier transform method for pricing vanilla options when analytic solutions are not available. This transform method is quite general and can also be used in any model where the characteristic function of the log-stock price is available. Transform methods now play a key role in the numerical pricing of derivative securities. 1 Local Volatility Models The GBM model for stock prices states that dSt = µSt dt + σSt dWt where µ and σ are constants. Moreover, when pricing derivative securities with the cash account as numeraire, we know that µ = r − q where r is the risk-free interest rate and q is the dividend yield.
    [Show full text]
  • Dissertation
    DISSERTATION Titel der Dissertation Consistent dynamic equity market code-books from a practical point of view Verfasser Sara Karlsson angestrebter akademischer Grad Doktorin der Naturwissenschaften (Dr.rer.nat) Wien, Mai 2011 Studienkennzahl lt. Studienblatt: A 791 405 Dissertationsgebiet lt. Studienblatt: Mathematik Betreuer: Univ.-Prof. Mag. Dr. Walter Schachermayer Abstract Gebr¨auchliche Aktienpreis- bzw. Martingalmodelle beschreiben die Dy- namik des Preises einer Aktie unter einem Martingalmaß; grundlegendes Beispiel ist das Black-Scholes Modell. Im Gegensatz dazu zielt ein Markt- modell darauf ab die Dynamik eines ganzen Marktes (d.h. Aktienpreis plus abgeleitete Optionen) zu beschreiben. In der vorliegenden Arbeit besch¨aftigen wir uns mit arbitragefreien Modellen die die Dynamik eines Aktienpreises sowie flussig¨ gehandelter Derivate beschreiben. (\equity market models" respektive \market models for stock options"). Die Motivation derartige Modelle zu betrachten, liegt darin, dass eu- rop¨aische Optionen flussig¨ gehandelt werden und daher Ruckschl¨ usse¨ auf die zugrundeliegende Stochastik erlauben. Marktmodelle werden auch fur¨ Anleihen angewandt; wir verweisen auf Heath, Jarrow, and Morton[1992]. In j ungerer¨ Zeit beeinflußt dieser Zugang auch Aktienpreismodelle (siehe beispielsweise Derman and Kani [1998], Dupire[1996]). Derman und Kani schlagen vor, die dynamische Entwicklung von M¨arkten mittels Differentialgleichungen fur¨ Aktienpreis und volatility surface zu modellieren. Ein anderer Zugang wird von Sch¨onbucher[1999] gew ¨ahlt; hier ist der Ausgangspunkt die gemeinsame Dynamik von Aktienpreis und impliziten Black-Scholes Volatilit¨aten. Carmona and Nadtochiy[2009] schlagen ein Marktmodell vor, in dem der Aktienpreis als exponentieller Levy-Prozeß gegeben ist. In diesem Fall wird eine zeitinhomogene Levy-Dichte verwen- det um die zus¨atzlichen, durch Optionspreise gegebenen Informationen miteinzubeziehen.
    [Show full text]
  • FX Derivatives: Stochastic-Local-Volatility Model
    Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary FX Derivatives: Stochastic-Local-Volatility Model Uwe Wystup Fr´ed´ericBossens, Andreas Weber MathFinance AG uwe.wystup@mathfinance.com July 2020 FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 1 / 37 Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary Agenda 1 Review Vanna Volga Early Days and Drawbacks Versions of Vanna-Volga Design and Consistency Issues 2 Stochastic-Local-Volatility LV and SV vanilla smile fit SLV Step by Step SLV Pricing / Validation 3 Mixture Local Volatility Model MLV Main Features Vol Process Comparison MLV vs. SLV Granular Model Marking 4 Summary Product/Model Matrix Key Take-Aways FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 2 / 37 Review Vanna Volga Stochastic-Local-Volatility Mixture Local Volatility Model Summary Agenda 1 Review Vanna Volga Early Days and Drawbacks Versions of Vanna-Volga Design and Consistency Issues 2 Stochastic-Local-Volatility LV and SV vanilla smile fit SLV Step by Step SLV Pricing / Validation 3 Mixture Local Volatility Model MLV Main Features Vol Process Comparison MLV vs. SLV Granular Model Marking 4 Summary Product/Model Matrix Key Take-Aways FX Derivatives: Stochastic-Local-Volatility Model c by MathFinance AG uwe.wystup@mathfinance.com 3 / 37 2 For a long time it matched market prices for barrier options and touch contracts quite good. 3 Market fit at least better than the alternatively available Local Volatility (LV) or Stochastic Volatility (SV) models. Review Vanna Volga Early Days and Drawbacks Stochastic-Local-Volatility Versions of Vanna-Volga Mixture Local Volatility Model Design and Consistency Issues Summary Review Vanna Volga: Early Days 1 Pricing in Vanna Volga (VV) model is popular in FX as it still allows to compute prices nearly as fast as in the BS model, using analytical formulas.
    [Show full text]
  • Swaption Volatility Surface Construction Tutorial | Finpricing
    How to Construct Swaption Volatility Surfaces Michael Taylor FinPricing https://finpricing.com/lib/IrVolIntroduction.html Swaption Volatility An implied volatility is the volatility implied by the market price of an option based on the Black-Scholes option pricing model. An interest rate swaption volatility surface is a four-dimensional plot of the implied volatility of a swaption as a function of strike and expiry and tenor. The term structures of implied volatilities which provide indications of the market’s near- and long-term uncertainty about future short- and long-term swap rates. A crucial property of the implied volatility surface is the absence of arbitrage. Swaption Volatility Summary ▪ Swaption Volatility Surface Introduction ▪ The Summary of Volatility Surface Construction Approaches ▪ Arbitrage Free Conditions ▪ The SABR Model ▪ Swaption Volatility Surface Construction via The SABR Model Swaption Volatility Swaption Volatility Surface Introduction ▪ An implied volatility is the volatility implied by the market price of an option based on the Black-Scholes option pricing model. ▪ An swaption volatility surface is a four-dimensional plot of the implied volatility of a swaption as a function of strike and expiry and tenor. ▪ The term structures of implied volatilities provide indications of the market’s near- and long-term uncertainty about future short- and long- term swap rates. ▪ Vol skew or smile pattern is directly related to the conditional non- nomality of the underlying return risk-neutral distribution. ▪ In particular, a smile reflects fat tails in the return distribution whereas a skew indicates return distribution asymmetry. ▪ A crucial property of the implied volatility surface is the absence of arbitrage.
    [Show full text]
  • Recombining Trinomial Tree for Real Option Valuation with Changing Volatility
    Recombining Trinomial Tree for Real Option Valuation with Changing Volatility Tero Haahtela Helsinki University of Technology, P.O. Box 5500, 02015 TKK, Finland +358 50 577 1690 [email protected] Abstract This paper presents a recombining trinomial tree for valuing real options with changing volatility. The trinomial tree presented in this paper is constructed by simultaneously choosing such a parameterization that sets a judicious state space while having sensible transition probabilities between the nodes. The volatility changes are modeled with the changing transition probabilities while the state space of the trinomial tree is regular and has a fixed number of time and underlying asset price levels. The presented trinomial lattice can be extended to follow a displaced diffusion process with changing volatility, allowing also taking into account the level of the underlying asset price. The lattice can also be easily parameterized based on a cash flow simulation, using ordinary least squares regression method for volatility estimation. Therefore, the presented recombining trinomial tree with changing volatility is more flexible and robust for practice use than common lattice models while maintaining their intuitive appeal. JEL Classification: G31, G13, D81 Keywords: Real options, trinomial tree, valuation under uncertainty 1 1. Introduction Volatility estimation is often the most difficult task in financial option valuation. It is even more challenging with real options, as there is not always a tractable underlying asset with a known process and the volatility does not remain the same during the investment period. Volatility tends to decline over the time during many investment projects as new information and knowledge is gathered.
    [Show full text]
  • Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Pricing
    Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Pricing Leif Andersen and Jesper Andreasen General Re Financial Products First Version: November 24, 1998 This Version: May 6, 2000 Summary This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly in the dynamics of the implied volatility surface. The paper derives a forward PIDE (Partial Integro-Differential Equation) and demonstrates how this equation can be used to fit the model to European option prices. For numerical pricing of general contingent claims, we develop an ADI finite difference method that is shown to be unconditionally stable and, if combined with Fast Fourier Transform methods, computationally efficient. The paper contains several detailed examples from the S&P500 market. Key Words: Jump-Diffusion Process, Local Time, Forward Equation, Volatility Smile, ADI Finite Difference Method, Fast Fourier Transform JEL Classification: G13, C14, C63, D52 MSC Classifiation: 60J60, 60J75, 60J55, 45K05, 42A99, 65N06 1. Introduction. The standard Black-Scholes (1973) assumption of log-normal stock diffusion with constant volatility is, as all market participants are keenly aware of, flawed. To equate the Black-Scholes formula with quoted prices of European calls and puts, it is thus generally necessary to use different volatilities (so-called implied volatilities) for different option strikes (K) and maturities (T). The phenomenon is often referred to as the volatility skew or smile (depending on the shape of the mapping of implied volatility as a function of K and T) and exists in all major stock index markets today.
    [Show full text]
  • Deep Local Volatility †
    risks Article Deep Local Volatility † Marc Chataigner 1,‡, Stéphane Crépey 1,* and Matthew Dixon 2,§ 1 Department of Mathematics, University of Evry, Paris Saclay, 91100 Essonne, France; [email protected] 2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616-3793, USA; [email protected] * Correspondence: [email protected] † A Python notebook, compatible with Google Colab, and accompanying data are available in https://github.com/mChataign/DupireNN. Due to file size constraints, the notebook must be run to reproduce the figures and results in this article. ‡ Ph.D. student under the supervision of S. Crépey. The Ph.D. thesis of Marc Chataigner is co-funded by the Research Initiative “Modélisation des marchés actions, obligations et dérivés”, financed by HSBC France under the aegis of the Europlace Institute of Finance, and by a public grant as part of investissement d’avenir project, reference ANR-11-LABX-0056-LLH LabEx LMH. The views and opinions expressed in this paper are those of the authors alone and do not necessarily reflect the views or policies of HSBC Investment Bank, its subsidiaries or affiliates. § The research of Matthew Dixon benefited from the support of Intel Corp. Received: 26 June 2020; Accepted: 22 July 2020; Published: 3 August 2020 Abstract: Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities.
    [Show full text]
  • Marko LALIC Technical University of Kosice Faculty of Economics, Department of Finance Kosice, Slovakia E-Mail: [email protected] Zsuzsanna K
    Marko LALIC Technical University of Kosice Faculty of Economics, Department of Finance Kosice, Slovakia E-mail: [email protected] Zsuzsanna K. SZABO E-mail: [email protected] Petru Maior University Faculty of Economics, Law and Administrative Sciences Tîrgu Mureş HEDGING OF DOWNSIDE RISK WITH PUT OPTIONS IN VARIOUS STOCHASTIC VOLATILITY ENVIRONMENT: HESTON MODEL APPROACH WITH ZERO CORRELATION AND ZERO MARKET PRICE OF VOLATILITY Abstract. The aim of this article is to find optimal strategy of proposed hedging strategy with put options in different stochastic volatility environment. Stochastic volatility and price movements follow processes described by Heston model. We simulate stock price paths with combinations of different stochastic volatility parameters and find optimal strategy for every environment. Market environment is defined with speed of reversion parameter, long run variance (volatility of volatility). Correlation between Brownian motions is set to zero. For evaluation of strategies and solution of optimization problem we use differences between proposed hedging strategy with active buying put options and basic hedging strategy with put option. Results show that optimal proposed strategy can bring additional positive expected return in strong bull market, but price for this additive return is smaller return of proposed hedging strategy in bear market in comparison with basic strategy. The results of the paper can be interpreted in practice as alternative to basic hedging strategy with minimal price evaluated with estimation of negative differences between proposed and basic strategy. Keywords: hedging, stochastic volatility, Heston model, simulation, put options. AMS Subject Classification: 60H35, 91B25, 91G80, 97M30 1. Introduction Derivatives such as options are very useful tool for hedging financial risks in wide range of financial and non financial markets.
    [Show full text]