MANAGING SMILE RISK 1. Introduction. European Options Are
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Martingale Methods in Financial Modelling
Marek Musiela Marek Rutkowski Martingale Methods in Financial Modelling Second Edition Springer Table of Contents Preface to the First Edition V Preface to the Second Edition VII Part I. Spot and Futures Markets 1. An Introduction to Financial Derivatives 3 1.1 Options 3 1.2 Futures Contracts and Options 6 1.3 Forward Contracts 7 1.4 Call and Put Spot Options 8 1.4.1 One-period Spot Market 10 1.4.2 Replicating Portfolios 11 1.4.3 Maxtingale Measure for a Spot Market 12 1.4.4 Absence of Arbitrage 14 1.4.5 Optimality of Replication 15 1.4.6 Put Option 18 1.5 Futures Call and Put Options 19 1.5.1 Futures Contracts and Futures Prices 20 1.5.2 One-period Futures Market 20 1.5.3 Maxtingale Measure for a Futures Market 22 1.5.4 Absence of Arbitrage 22 1.5.5 One-period Spot/Futures Market 24 1.6 Forward Contracts 25 1.6.1 Forward Price 25 1.7 Options of American Style 27 1.8 Universal No-arbitrage Inequalities 32 2. Discrete-time Security Markets 35 2.1 The Cox-Ross-Rubinstein Model 36 2.1.1 Binomial Lattice for the Stock Price 36 2.1.2 Recursive Pricing Procedure 38 2.1.3 CRR Option Pricing Formula 43 X Table of Contents 2.2 Martingale Properties of the CRR Model 46 2.2.1 Martingale Measures 47 2.2.2 Risk-neutral Valuation Formula 50 2.3 The Black-Scholes Option Pricing Formula 51 2.4 Valuation of American Options 56 2.4.1 American Call Options 56 2.4.2 American Put Options 58 2.4.3 American Claim 60 2.5 Options on a Dividend-paying Stock 61 2.6 Finite Spot Markets 63 2.6.1 Self-financing Trading Strategies 63 2.6.2 Arbitrage Opportunities 65 2.6.3 Arbitrage Price 66 2.6.4 Risk-neutral Valuation Formula 67 2.6.5 Price Systems 70 2.6.6 Completeness of a Finite Market 73 2.6.7 Change of a Numeraire 74 2.7 Finite Futures Markets 75 2.7.1 Self-financing Futures Strategies 75 2.7.2 Martingale Measures for a Futures Market 77 2.7.3 Risk-neutral Valuation Formula 79 2.8 Futures Prices Versus Forward Prices 79 2.9 Discrete-time Models with Infinite State Space 82 3. -
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]. -
An Empirical Investigation of the Black- Scholes Call
International Journal of BRIC Business Research (IJBBR) Volume 6, Number 2, May 2017 AN EMPIRICAL INVESTIGATION OF THE BLACK - SCHOLES CALL OPTION PRICING MODEL WITH REFERENCE TO NSE Rajesh Kumar 1 and Rachna Agrawal 2 1Research Scholar, Department of Management Studies 2Associate Professor, Department of Management Studies YMCA University of Science and Technology, Faridabad, Haryana, India ABSTRACT This paper investigates the efficiency of Black-Scholes model used for valuation of call option contracts written on Eight Indian stocks quoted on NSE. It has been generally observed that the B & S Model misprices options considerably on several occasions and the volatilities are ‘high for options which are highly overpriced. Mispriced worsen with the increased in volatility of the underlying stocks. In most of cases options are also highly underpriced by the model. In this research paper, the theoretical options prices of Nifty stock call options are calculated under the B & S Model. These theoretical prices are compared with the actual quoted prices in the market to gauge the pricing accuracy. KEYWORDS Black-Scholes Model, standard deviation, Mean Error, Root Mean Squared Error, Thiel’s Inequality coefficient etc. 1. INTRODUCTION An effective security market provides three principal opportunities- trading equities, debt securities and derivative products. For the purpose of risk management and trading, the pricing theories of stock options have occupied important place in derivative market. These theories range from relatively undemanding binomial model to more complex B & S Model (1973). The Black-Scholes option pricing model is a landmark in the history of Derivative. This preferred model provides a closed analytical view for the valuation of European style options. -
Introduction to Black's Model for Interest Rate
INTRODUCTION TO BLACK'S MODEL FOR INTEREST RATE DERIVATIVES GRAEME WEST AND LYDIA WEST, FINANCIAL MODELLING AGENCY© Contents 1. Introduction 2 2. European Bond Options2 2.1. Different volatility measures3 3. Caplets and Floorlets3 4. Caps and Floors4 4.1. A call/put on rates is a put/call on a bond4 4.2. Greeks 5 5. Stripping Black caps into caplets7 6. Swaptions 10 6.1. Valuation 11 6.2. Greeks 12 7. Why Black is useless for exotics 13 8. Exercises 13 Date: July 11, 2011. 1 2 GRAEME WEST AND LYDIA WEST, FINANCIAL MODELLING AGENCY© Bibliography 15 1. Introduction We consider the Black Model for futures/forwards which is the market standard for quoting prices (via implied volatilities). Black[1976] considered the problem of writing options on commodity futures and this was the first \natural" extension of the Black-Scholes model. This model also is used to price options on interest rates and interest rate sensitive instruments such as bonds. Since the Black-Scholes analysis assumes constant (or deterministic) interest rates, and so forward interest rates are realised, it is difficult initially to see how this model applies to interest rate dependent derivatives. However, if f is a forward interest rate, it can be shown that it is consistent to assume that • The discounting process can be taken to be the existing yield curve. • The forward rates are stochastic and log-normally distributed. The forward rates will be log-normally distributed in what is called the T -forward measure, where T is the pay date of the option. -
Forward Contracts and Futures a Forward Is an Agreement Between Two Parties to Buy Or Sell an Asset at a Pre-Determined Future Time for a Certain Price
Forward contracts and futures A forward is an agreement between two parties to buy or sell an asset at a pre-determined future time for a certain price. Goal To hedge against the price fluctuation of commodity. • Intension of purchase decided earlier, actual transaction done later. • The forward contract needs to specify the delivery price, amount, quality, delivery date, means of delivery, etc. Potential default of either party: writer or holder. Terminal payoff from forward contract payoff payoff K − ST ST − K K ST ST K long position short position K = delivery price, ST = asset price at maturity Zero-sum game between the writer (short position) and owner (long position). Since it costs nothing to enter into a forward contract, the terminal payoff is the investor’s total gain or loss from the contract. Forward price for a forward contract is defined as the delivery price which make the value of the contract at initiation be zero. Question Does it relate to the expected value of the commodity on the delivery date? Forward price = spot price + cost of fund + storage cost cost of carry Example • Spot price of one ton of wood is $10,000 • 6-month interest income from $10,000 is $400 • storage cost of one ton of wood is $300 6-month forward price of one ton of wood = $10,000 + 400 + $300 = $10,700. Explanation Suppose the forward price deviates too much from $10,700, the construction firm would prefer to buy the wood now and store that for 6 months (though the cost of storage may be higher). -
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. -
Annual Accounts and Report
Annual Accounts and Report as at 30 June 2018 WorldReginfo - 5b649ee5-9497-487a-9fdb-e60cdfc3179f LIMITED COMPANY SHARE CAPITAL € 443,521,470 HEAD OFFICE: PIAZZETTA ENRICO CUCCIA 1, MILAN, ITALY REGISTERED AS A BANK. PARENT COMPANY OF THE MEDIOBANCA BANKING GROUP. REGISTERED AS A BANKING GROUP Annual General Meeting 27 October 2018 WorldReginfo - 5b649ee5-9497-487a-9fdb-e60cdfc3179f www.mediobanca.com translation from the Italian original which remains the definitive version WorldReginfo - 5b649ee5-9497-487a-9fdb-e60cdfc3179f BOARD OF DIRECTORS Term expires Renato Pagliaro Chairman 2020 * Maurizia Angelo Comneno Deputy Chairman 2020 Alberto Pecci Deputy Chairman 2020 * Alberto Nagel Chief Executive Officer 2020 * Francesco Saverio Vinci General Manager 2020 Marie Bolloré Director 2020 Maurizio Carfagna Director 2020 Maurizio Costa Director 2020 Angela Gamba Director 2020 Valérie Hortefeux Director 2020 Maximo Ibarra Director 2020 Alberto Lupoi Director 2020 Elisabetta Magistretti Director 2020 Vittorio Pignatti Morano Director 2020 * Gabriele Villa Director 2020 * Member of Executive Committee STATUTORY AUDIT COMMITTEE Natale Freddi Chairman 2020 Francesco Di Carlo Standing Auditor 2020 Laura Gualtieri Standing Auditor 2020 Alessandro Trotter Alternate Auditor 2020 Barbara Negri Alternate Auditor 2020 Stefano Sarubbi Alternate Auditor 2020 * * * Massimo Bertolini Secretary to the Board of Directors www.mediobanca.com translation from the Italian original which remains the definitive version WorldReginfo - 5b649ee5-9497-487a-9fdb-e60cdfc3179f -
Option Pricing: a Review
Option Pricing: A Review Rangarajan K. Sundaram Introduction Option Pricing: A Review Pricing Options by Replication The Option Rangarajan K. Sundaram Delta Option Pricing Stern School of Business using Risk-Neutral New York University Probabilities The Invesco Great Wall Fund Management Co. Black-Scholes Model Shenzhen: June 14, 2008 Implied Volatility Outline Option Pricing: A Review Rangarajan K. 1 Introduction Sundaram Introduction 2 Pricing Options by Replication Pricing Options by Replication 3 The Option Delta The Option Delta Option Pricing 4 Option Pricing using Risk-Neutral Probabilities using Risk-Neutral Probabilities The 5 The Black-Scholes Model Black-Scholes Model Implied 6 Implied Volatility Volatility Introduction Option Pricing: A Review Rangarajan K. Sundaram These notes review the principles underlying option pricing and Introduction some of the key concepts. Pricing Options by One objective is to highlight the factors that affect option Replication prices, and to see how and why they matter. The Option Delta We also discuss important concepts such as the option delta Option Pricing and its properties, implied volatility and the volatility skew. using Risk-Neutral Probabilities For the most part, we focus on the Black-Scholes model, but as The motivation and illustration, we also briefly examine the binomial Black-Scholes Model model. Implied Volatility Outline of Presentation Option Pricing: A Review Rangarajan K. Sundaram The material that follows is divided into six (unequal) parts: Introduction Pricing Options: Definitions, importance of volatility. Options by Replication Pricing of options by replication: Main ideas, a binomial The Option example. Delta The option delta: Definition, importance, behavior. Option Pricing Pricing of options using risk-neutral probabilities. -
Pricing of Index Options Using Black's Model
Global Journal of Management and Business Research Volume 12 Issue 3 Version 1.0 March 2012 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4588 & Print ISSN: 0975-5853 Pricing of Index Options Using Black’s Model By Dr. S. K. Mitra Institute of Management Technology Nagpur Abstract - Stock index futures sometimes suffer from ‘a negative cost-of-carry’ bias, as future prices of stock index frequently trade less than their theoretical value that include carrying costs. Since commencement of Nifty future trading in India, Nifty future always traded below the theoretical prices. This distortion of future prices also spills over to option pricing and increase difference between actual price of Nifty options and the prices calculated using the famous Black-Scholes formula. Fisher Black tried to address the negative cost of carry effect by using forward prices in the option pricing model instead of spot prices. Black’s model is found useful for valuing options on physical commodities where discounted value of future price was found to be a better substitute of spot prices as an input to value options. In this study the theoretical prices of Nifty options using both Black Formula and Black-Scholes Formula were compared with actual prices in the market. It was observed that for valuing Nifty Options, Black Formula had given better result compared to Black-Scholes. Keywords : Options Pricing, Cost of carry, Black-Scholes model, Black’s model. GJMBR - B Classification : FOR Code:150507, 150504, JEL Code: G12 , G13, M31 PricingofIndexOptionsUsingBlacksModel Strictly as per the compliance and regulations of: © 2012. -
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 ................................................................................................