DOCSLIB.ORG

Act4000, Midterm #2 Advanced Actuarial Topics March 16, 2009 Hal W

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Act4000, Midterm #2 Advanced Actuarial Topics March 16, 2009 Hal W ACT4000, MIDTERM #2 ADVANCED ACTUARIAL TOPICS MARCH 16, 2009 HAL W. PEDERSEN You have 70 minutes to complete this exam. When the invigilator instructs you to stop writing you must do so immediately. If you do not abide by this instruction you will be penalised. All invigilators have full authority to disqualify your paper if, in their judgement, you are found to have violated the code of academic honesty. Each question is worth 10 points. Provide sufficient reasoning to back up your answer but do not write more than necessary. This exam consists of 8 questions. Answer each question on a separate page of the exam book. Write your name and student number on each exam book that you use to answer the questions. Good luck! Question 1. You have written a European call option on a share of stock which expires in six months and has a strike price of $55.00. You are given the following information. • The current price of the stock is S0 = $50.00. • The stock pays no dividends. • The continuous time interest rate r =0.08. • The volatility on the stock is σ =0.25. • The price of the 55-call is $2.37535. • The delta of the 55-strike call you have written is ∆ = 0.41119. • The gamma of the 55-strike call you have written is Γ = 0.04401. You have decided to delta-gamma hedge your position using a European put option on a share of stock which expires in six months and has a strike price of $60.00. You are also given the following information. • The price of the 60-put is $8.77541. • The delta of the 60-strike put is ∆ = −0.76322. • The gamma of the 60-strike put is Γ = 0.03491. • The price of a 55-call with remaining maturity of six months less a day when the underlying stock is trading at $51.00 is $2.79428. • The price of a 60-put with remaining maturity of six months less a day when the underlying stock is trading at $51.00 is $8.03167. (a) [4 points] Compute the position you must take in 60-strike puts to effect the delta-gamma hedge. (b) [6 points] One day after you effect the delta-gamma hedge the stock price has increased by $1.00. What is your profit or less on the hedged portfolio. 1 2 ACT4000 – MIDTERM #2 Question 2. Assume that the Black-Scholes framework holds. The price of a stock that pays no dividends is $30.00. The price of a put option on this stock is $4.00. You are given: • ∆=−0.28, and • Γ=0.20. Using the delta-gamma approximation, determine the price of the put option if the stock price changes to $31.50. Question 3. Consider a model with two stocks. Each stock pays dividends continu- ously at a rate proportional to its price. Sj(t) denotes the price of one share of stock j at time t. Consider a claim maturing at time 3. The payoff of the claim is: Maximum S1(3),S2(3) . You are given the following information • S1(0) = 100 • S2(0) = 200 • Stock 1 pays dividends of amount (0.05)S1(t) dt between time t and time t+dt. • Stock 2 pays dividends of amount (0.10)S2(t) dt between time t and time t+dt. • The price of a European option to exchange Stock 2 for Stock 1 at time 3 is $10.00. Calculate the price of the claim. Question 4. Consider a “forward start option” which, 1 year from today, will give its owner a 1-year European call option with a strike price equal to the stock price at that time. You are given: • The European call option is on a stock that pays no dividends. • The stock’s volatility is 30%. • The forward price for delivery of 1 share of the stock 1 year from today is $100. • The continuously compounded risk-free interest rate is 8%. Under the Black-Scholes framework, determine the price today of the forward start option. ACT4000 – MIDTERM #2 3 Question 5. Let S(t) denote the price at time t of a stock that pays dividends continuously at a rate proportional to its price. Consider a European gap option with expiration date T for T>0. If the stock price at time T is greater than $100, the payoff is S(T ) − 90 otherwise the payoff is 0. You are given: (i) S(0) = $80 (ii) The price of a European call option with expiration date T and strike price $100 is $4. (iii) The delta of the call option in (ii) is 0.2. Calculate the price of the gap option. Question 6. Explain the notion of a compound call option. Be sure to clearly identify what the underlying is and to indicate what the payoff of the compound option is at expiration. Question 7. The stochastic process X follows the SDE dXt =0.14 dt + σdWt Xt where W is a standard Brownian motion. −t 2 Consider the new process Y defined by Yt = f(t, Xt)wheref(t, x)=e x . Apply Itˆo’s lemma to write an expression for dYt. Question 8. An asset price follows the diffusion process defined by St =80· exp 0.08 t +0.25 Wt where W is a standard Brownian motion. The risk-free interest rate is r =0.05. A second asset, denoted X, is available for trade with asset price dynamics dXt = µdt+0.20 dWt Xt and X0 = 20. Assuming that the market is arbitrage-free, compute µ. We- Jl.,L~- jdf"~lI dt.)t.d d"J )d .....r--4 0+ to -t~.... J d-/;" "•..J ~II> ~Ll'" /;Jr,l{llA. 6, ;:. #- ~lJrc...J ~t S.foc-/z. e1...~.If bO-~&.f.) e 1- ~ ~ = 00• Lf/l '1 e7, (I) + ()'l- (- .1~32.1-) , otf'l.e.0 J e'L (.o~LfqJ) [" . At •. 1: J 1 ) (p) P"1 () ~ ss - c~II 1 . 3'7 S 3.5 - I 2-. 1Cf Lf L 8 S-l <> c, Jt- .so S I '.31$,-/ ~ - f~ 8.1'1$'11 t 8'.o~ib7 1. 2b07 ~l\" It A t: c.f -17.31, Ot\~ Vl\i f~L - I (l,"7 9 'f 2 8 - 2.. S 75' .s 5") + J. 37.$1(51->0) + I. If,07(f.o'}/k7-8.?75'11) - 7'7. ), ( e. .,,813'S _ I) - . ()OOI2 . Question 2 The delta-gamma approximation is merely the Taylor series approximation with up to the quadratic term. In terms of the Greek symbols, the first derivative is Δ, and the second derivative is Γ. The approximation formula is 1 2 P(S + ε) ≈ P(S) + ε Δ + ε Γ. (13.2 & 13.5) 2 With P(30) = 4, Δ = −0.28, Γ = 0.10, and ε = 1.50, we have P(31.5) ≈ 4 + (1.5)(−0.28) + 1 (1.5)2(0.1) 2 = 3.6925 ≈ 3.70. Question 3 Because of the identity Maximum( S1(3), S2(3) ) = Maximum( S1(3) – S2(3), 0) + S2(3), the payoff of the claim can be decomposed as the sum of the payoff of the exchange option in statement (v) of the problem and the price of stock 2 at time 3. In a no- arbitrage model, the price of the claim must be equal to the sum of the exchange option price (which is 10) and the prepaid forward price for delivery of stock 2 at −δ2 ×3 time 3 (which is e ×S2(0)). So, the answer is 10 + e−0.1×3×200 = 158.16. Remark: If one buys e−δ2 ×3 share of stock 2 at time 0 and re-invests all dividends, one will have exactly one share of stock 2 at time 3. Question 4 This problem is based on Exercise 14.21 on page 465 of McDonald (2006). Let S1 denote the stock price at the end of one year. Apply the Black-Scholes formula to calculate the price of the at-the-money call one year from today, conditioning on S1. 2 2 d1 = [ln (S1/S1) + (r + σ /2)T]/(σ T ) = (r + σ /2)/σ = 0.417, which turns out to be independent of S1. d2 = d1 − σ T = d1 − σ = 0.117 The value of the forward start option at time 1 is −r C(S1) = S1N(d1) − S1e N(d2) −0.08 = S1[N(0.417) − e N(0.117)] −0.08 ≈ S1[N(0.42) − e N(0.12)] -0.08 = S1[0.6628 − e ×0.5438] = 0.157S1. (Note that, when viewed from time 0, S1 is a random variable.) Thus, the time-0 price of the forward start option must be 0.157 multiplied by the time-0 price of a security that gives S1 as payoff at time 1, i.e., multiplied by the prepaid forward P price F 1,0 (S) . Hence, the time-0 price of the forward start option is P −0.08 −0.08 0.157× F 1,0 (S) = 0.157×e × F 1,0 (S) = 0.157×e ×100 = 14.5 Remark: A key to pricing the forward start option is that d1 and d2 turn out to be independent of the stock price. This is the case if the strike price of the call option will be set as a fixed percentage of the stock price at the issue date of the call option. Question 5 In terms of the notation in Section 14.15, K1 = 90 and K2 = 100.
Load more

Recommended publications
  • The Promise and Peril of Real Options
    1 The Promise and Peril of Real Options Aswath Damodaran Stern School of Business 44 West Fourth Street New York, NY 10012 [email protected] 2 Abstract In recent years, practitioners and academics have made the argument that traditional discounted cash flow models do a poor job of capturing the value of the options embedded in many corporate actions. They have noted that these options need to be not only considered explicitly and valued, but also that the value of these options can be substantial. In fact, many investments and acquisitions that would not be justifiable otherwise will be value enhancing, if the options embedded in them are considered. In this paper, we examine the merits of this argument. While it is certainly true that there are options embedded in many actions, we consider the conditions that have to be met for these options to have value. We also develop a series of applied examples, where we attempt to value these options and consider the effect on investment, financing and valuation decisions. 3 In finance, the discounted cash flow model operates as the basic framework for most analysis. In investment analysis, for instance, the conventional view is that the net present value of a project is the measure of the value that it will add to the firm taking it. Thus, investing in a positive (negative) net present value project will increase (decrease) value. In capital structure decisions, a financing mix that minimizes the cost of capital, without impairing operating cash flows, increases firm value and is therefore viewed as the optimal mix.
    [Show full text]
  • The Evaluation of American Compound Option Prices Under Stochastic Volatility and Stochastic Interest Rates
    THE EVALUATION OF AMERICAN COMPOUND OPTION PRICES UNDER STOCHASTIC VOLATILITY AND STOCHASTIC INTEREST RATES CARL CHIARELLA♯ AND BODA KANG† Abstract. A compound option (the mother option) gives the holder the right, but not obligation to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we consider the problem of pricing American-type compound options when the underlying dynamics follow Heston’s stochastic volatility and with stochastic interest rate driven by Cox-Ingersoll-Ross (CIR) processes. We use a partial differential equation (PDE) approach to obtain a numerical solution. The problem is formulated as the solution to a two-pass free boundary PDE problem which is solved via a sparse grid approach and is found to be accurate and efficient compared with the results from a benchmark solution based on a least-squares Monte Carlo simulation combined with the PSOR. Keywords: American compound option, stochastic volatility, stochastic interest rates, free boundary problem, sparse grid, combination technique, least squares Monte Carlo. JEL Classification: C61, D11. 1. Introduction The compound option goes back to the seminal paper of Black & Scholes (1973). As well as their famous pricing formulae for vanilla European call and put options, they also considered how to evaluate the equity of a company that has coupon bonds outstanding. They argued that the equity can be viewed as a “compound option” because the equity “is an option on an option on an option on the firm”. Geske (1979) developed · · · the first closed-form solution for the price of a vanilla European call on a European call.
    [Show full text]
  • Sequential Compound Options and Investments Valuation
    Sequential compound options and investments valuation Luigi Sereno Dottorato di Ricerca in Economia - XIX Ciclo - Alma Mater Studiorum - Università di Bologna Marzo 2007 Relatore: Prof. ssa Elettra Agliardi Coordinatore: Prof. Luca Lambertini Settore scienti…co-disciplinare: SECS-P/01 Economia Politica ii Contents I Sequential compound options and investments valua- tion 1 1 An overview 3 1.1 Introduction . 3 1.2 Literature review . 6 1.2.1 R&D as real options . 11 1.2.2 Exotic Options . 12 1.3 An example . 17 1.3.1 Value of expansion opportunities . 18 1.3.2 Value with abandonment option . 23 1.3.3 Value with temporary suspension . 26 1.4 Real option modelling with jump processes . 31 1.4.1 Introduction . 31 1.4.2 Merton’sapproach . 33 1.4.3 Further reading . 36 1.5 Real option and game theory . 41 1.5.1 Introduction . 41 1.5.2 Grenadier’smodel . 42 iii iv CONTENTS 1.5.3 Further reading . 45 1.6 Final remark . 48 II The valuation of new ventures 59 2 61 2.1 Introduction . 61 2.2 Literature Review . 63 2.2.1 Flexibility of Multiple Compound Real Options . 65 2.3 Model and Assumptions . 68 2.3.1 Value of the Option to Continuously Shut - Down . 69 2.4 An extension . 74 2.4.1 The mathematical problem and solution . 75 2.5 Implementation of the approach . 80 2.5.1 Numerical results . 82 2.6 Final remarks . 86 III Valuing R&D investments with a jump-di¤usion process 93 3 95 3.1 Introduction .
    [Show full text]
  • Venture Capitalists' Entry-Exit Investment Decisions
    A Dynamic Model for Venture Capitalists' Entry{Exit Investment Decisions∗ Ricardo M. Ferreiray, Paulo J. Pereiraz yFaculdade de Economia, Universidade do Porto, Portugal zCEF.UP and Faculdade de Economia, Universidade do Porto, Portugal. Abstract In this paper we develop a dynamic model to study the entry and the exit decision of a VC facing the opportunity to invest and expand a start-up firm. Two settings are considered. A benchmark setting, where no time constrain for exiting is in place, is compared with the one where, realistically, the VC has a finite time-window for divesting. In both cases we consider the trade sale (M&A) as the exit route. The model returns the entry and the exit triggers, the optimal post-money ownerships, the expected cash multiple for the VC, and also proposes a new time-adjusted version of the cash multiple, useful for measuring, ex ante, the expected performance of the investment. The model aims to guide the VCs when analyzing their investment op- portunities, considering the entire VC's business-cycle (entry{expand{exit). Finally, the model is applied to an hypothetical, but realistic, situation in order to understand the main outcomes. A comparative statics analysis is also performed. Keywords: Finance; Venture Capital; Start-ups; Real Options; Growth Options. JEL codes: G24; G34; L26; M13. ∗We thank Roel Nagy, Miguel Sousa, Miguel Tavares-G¨artner,Lenos Trigeorgis and the participants at the 2019 Annual International Real Options Conference in London. Paulo J. Pereira acknowledge that this research has been financed by Portuguese public funds through FCT - Funda¸c~aopara a Ci^enciae a Tecnologia, I.P., in the framework of the projects UID/ECO/04105/2019.
    [Show full text]
  • FX Options and Structured Products
    FX Options and Structured Products Uwe Wystup www.mathfinance.com 7 April 2006 www.mathfinance.de To Ansua Contents 0 Preface 9 0.1 Scope of this Book ................................ 9 0.2 The Readership ................................. 9 0.3 About the Author ................................ 10 0.4 Acknowledgments ................................ 11 1 Foreign Exchange Options 13 1.1 A Journey through the History Of Options ................... 13 1.2 Technical Issues for Vanilla Options ....................... 15 1.2.1 Value ................................... 16 1.2.2 A Note on the Forward ......................... 18 1.2.3 Greeks .................................. 18 1.2.4 Identities ................................. 20 1.2.5 Homogeneity based Relationships .................... 21 1.2.6 Quotation ................................ 22 1.2.7 Strike in Terms of Delta ......................... 26 1.2.8 Volatility in Terms of Delta ....................... 26 1.2.9 Volatility and Delta for a Given Strike .................. 26 1.2.10 Greeks in Terms of Deltas ........................ 27 1.3 Volatility ..................................... 30 1.3.1 Historic Volatility ............................ 31 1.3.2 Historic Correlation ........................... 34 1.3.3 Volatility Smile .............................. 35 1.3.4 At-The-Money Volatility Interpolation .................. 41 1.3.5 Volatility Smile Conventions ....................... 44 1.3.6 At-The-Money Definition ........................ 44 1.3.7 Interpolation of the Volatility on Maturity
    [Show full text]
  • 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.
    [Show full text]
  • Analytical Finance Volume I
    The Mathematics of Equity Derivatives, Markets, Risk and Valuation ANALYTICAL FINANCE VOLUME I JAN R. M. RÖMAN Analytical Finance: Volume I Jan R. M. Röman Analytical Finance: Volume I The Mathematics of Equity Derivatives, Markets, Risk and Valuation Jan R. M. Röman Västerås, Sweden ISBN 978-3-319-34026-5 ISBN 978-3-319-34027-2 (eBook) DOI 10.1007/978-3-319-34027-2 Library of Congress Control Number: 2016956452 © The Editor(s) (if applicable) and The Author(s) 2017 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Cover image © David Tipling Photo Library / Alamy Printed on acid-free paper This Palgrave Macmillan imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To my soulmate, supporter and love – Jing Fang Preface This book is based upon lecture notes, used and developed for the course Analytical Finance I at Mälardalen University in Sweden.
    [Show full text]
  • Binomial Trees • Stochastic Calculus, Ito’S Rule, Brownian Motion • Black-Scholes Formula and Variations • Hedging • Fixed Income Derivatives
    Pricing Options with Mathematical Models 1. OVERVIEW Some of the content of these slides is based on material from the book Introduction to the Economics and Mathematics of Financial Markets by Jaksa Cvitanic and Fernando Zapatero. • What we want to accomplish: Learn the basics of option pricing so you can: - (i) continue learning on your own, or in more advanced courses; - (ii) prepare for graduate studies on this topic, or for work in industry, or your own business. • The prerequisites we need to know: - (i) Calculus based probability and statistics, for example computing probabilities and expected values related to normal distribution. - (ii) Basic knowledge of differential equations, for example solving a linear ordinary differential equation. - (iii) Basic programming or intermediate knowledge of Excel • A rough outline: - Basic securities: stocks, bonds - Derivative securities, options - Deterministic world: pricing fixed cash flows, spot interest rates, forward rates • A rough outline (continued): - Stochastic world, pricing options: • Pricing by no-arbitrage • Binomial trees • Stochastic Calculus, Ito’s rule, Brownian motion • Black-Scholes formula and variations • Hedging • Fixed income derivatives Pricing Options with Mathematical Models 2. Stocks, Bonds, Forwards Some of the content of these slides is based on material from the book Introduction to the Economics and Mathematics of Financial Markets by Jaksa Cvitanic and Fernando Zapatero. A Classification of Financial Instruments SECURITIES AND CONTRACTS BASIC SECURITIES DERIVATIVES
    [Show full text]
  • On-Line Manual for Successful Trading
    On-Line Manual For Successful Trading CONTENTS Chapter 1. Introduction 7 1.1. Foreign Exchange as a Financial Market 7 1.2. Foreign Exchange in a Historical Perspective 8 1.3. Main Stages of Recent Foreign Exchange Development 9 The Bretton Woods Accord 9 The International Monetary Fund 9 Free-Floating of Currencies 10 The European Monetary Union 11 The European Monetary Cooperation Fund 12 The Euro 12 1.4. Factors Caused Foreign Exchange Volume Growth 13 Interest Rate Volatility 13 Business Internationalization 13 Increasing of Corporate Interest 13 Increasing of Traders Sophistication 13 Developments in Telecommunications 14 Computer and Programming Development 14 FOREX. On-line Manual For Successful Trading ii Chapter 2. Kinds Of Major Currencies and Exchange Systems 15 2.1. Major Currencies 15 The U.S. Dollar 15 The Euro 15 The Japanese Yen 16 The British Pound 16 The Swiss Franc 16 2.2. Kinds of Exchange Systems 17 Trading with Brokers 17 Direct Dealing 18 Dealing Systems 18 Matching Systems 18 2.3. The Federal Reserve System of the USA and Central Banks of the Other G-7 Countries 20 The Federal Reserve System of the USA 20 The Central Banks of the Other G-7 Countries 21 Chapter 3. Kinds of Foreign Exchange Market 23 3.1. Spot Market 23 3.2. Forward Market 26 3.3. Futures Market 27 3.4. Currency Options 28 Delta 30 Gamma 30 Vega 30 Theta 31 FOREX. On-line Manual For Successful Trading iii Chapter 4. Fundamental Analysis 32 4.1. Economic Fundamentals 32 Theories of Exchange Rate Determination 32 Purchasing Power Parity 32 The PPP Relative Version 33 Theory of Elasticities 33 Modern Monetary Theories on Short-term Exchange Rate Volatility 33 The Portfolio-Balance Approach 34 Synthesis of Traditional and Modern Monetary Views 34 4.2.
    [Show full text]
  • Consolidated Policy on Valuation Adjustments Global Capital Markets
    Global Consolidated Policy on Valuation Adjustments Consolidated Policy on Valuation Adjustments Global Capital Markets September 2008 Version Number 2.35 Dilan Abeyratne, Emilie Pons, William Lee, Scott Goswami, Jerry Shi Author(s) Release Date September lOth 2008 Page 1 of98 CONFIDENTIAL TREATMENT REQUESTED BY BARCLAYS LBEX-BARFID 0011765 Global Consolidated Policy on Valuation Adjustments Version Control ............................................................................................................................. 9 4.10.4 Updated Bid-Offer Delta: ABS Credit SpreadDelta................................................................ lO Commodities for YH. Bid offer delta and vega .................................................................................. 10 Updated Muni section ........................................................................................................................... 10 Updated Section 13 ............................................................................................................................... 10 Deleted Section 20 ................................................................................................................................ 10 Added EMG Bid offer and updated London rates for all traded migrated out oflens ....................... 10 Europe Rates update ............................................................................................................................. 10 Europe Rates update continue .............................................................................................................
    [Show full text]
  • Glossary of Financial Derivatives* Paul D. Koch [email protected]
    1 Glossary of Financial Derivatives* Paul D. Koch [email protected] University of Kansas 785-864-7503 Lawrence, KS 66045 *This document draws heavily from several sources: (a) Website of Don Chance, www.fbox.vt.edu/filebox/business/finance/dmc/DRU; (b) Hull, J.C., Fundamentals of Futures & Options Markets, 8th Edition, Prentice-Hall, Inc.: New York, NY, 2014; I. Background. Yield Curve – relation among interest rates paid on securities alike in every respect except maturity. (How interest rates change from short term to long term securities.) Eurodollar - dollar-denominated deposits outside the jurisdiction of the U.S. regulatory authorities. Financial Derivative - financial claim whose value is contingent upon movements in some underlying variable such as a stock or stock index, interest rates, exchange rates, or commodity prices; includes forwards, futures, options, SWAPs, asset-backed securities, structured notes (hybrid debt), & other combinations of these instruments. LIBOR - London Interbank Offer Rate; rate charged on short term Eurodollar deposits; benchmark floating rate for international borrowing/lending in $. Margin - good faith 'collateral' deposit, specified as a percentage of the value of the financial instrument in question; ensures integrity of market. Organized Exchange - centralized location where organized trading is conducted in certain financial instruments under a specific set of rules. The exchange clearinghouse is the counter-party to every transaction; members of the exchange share the responsibility of fulfilling commitments. The exchange: (i) sets standardized terms for all contracts traded, and (ii) often places restrictions on trading (e.g. margin requirements, limits on daily price changes, limits on size of individual positions, ...). Standardization of contracts and other rules make clearing easier, reduce uncertainty about counterparty default risk, and help ensure an orderly market.
    [Show full text]
  • Spread and Basket Option Pricing: an Application to Interconnected Power Markets
    University of Milano-Bicocca Department of Quantitative Methods for Economics and Business Doctoral Thesis in Mathematics for Financial Markets Analysis Spread and Basket Option Pricing: an Application to Interconnected Power Markets Author: Supervisor: Ruggero Caldana Gianluca Fusai Milan, October 2012 Dedicated to my family. Abstract. An interconnector is an asset that gives the owner the right, but not the obligation, to transmit electricity between two locations each hour of the day over a prefixed time period. The financial value of the interconnector is given by a series of options that are written on the price differential between two electricity markets, that is, a strip of European options on an hourly spread. Since the hourly forward price is not directly observable on the market, Chapter 1 proposes a practical procedure to build an hourly forward price curve, fitting both base load and peak load forward quotations. One needs a stochastic model, a valuation formula, and a calibra- tion method to evaluate interconnection capacity contracts. To capture the main features of the electricity price series, we model the energy price log-returns for each hour with a non-Gaussian mean-reverting stochastic process. Unfortunately no explicit solution to the spread option val- uation problem is available. Chapter 2 develops a method for pricing the generic spread option in the non-Gaussian framework by extending the Bjerksund and Stensland (2011) approximation to a Fourier transform framework. We also obtain an upper bound on the estimation error. The method is applicable to models in which the joint characteristic function of the underlying assets is known analytically.
    [Show full text]