Program Securities List 7 29 2009
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Variance Reduction with Control Variate for Pricing Asian Options in a Geometric Lévy Model
IAENG International Journal of Applied Mathematics, 41:4, IJAM_41_4_05 ______________________________________________________________________________________ Variance Reduction with Control Variate for Pricing Asian Options in a Geometric Levy´ Model Wissem Boughamoura and Faouzi Trabelsi Abstract—This paper is concerned with Monte Carlo simu- use of the path integral formulation. [19] studied a cer- lation of generalized Asian option prices where the underlying tain one-dimensional, degenerate parabolic partial differential asset is modeled by a geometric (finite-activity) Levy´ process. equation with a boundary condition which arises in pricing A control variate technique is proposed to improve standard Monte Carlo method. Some convergence results for plain Monte of Asian options. [25] considered the approximation of Carlo simulation of generalized Asian options are proven, and a the optimal stopping problem associated with ultradiffusion detailed numerical study is provided showing the performance processes in valuating Asian options. The value function is of the variance reduction compared to straightforward Monte characterized as the solution of an ultraparabolic variational Carlo method. inequality. Index Terms—Levy´ process, Asian options, Minimal-entropy martingale measure, Monte Carlo method, Variance Reduction, For discontinuous markets, [8] generalized Laplace trans- Control variate. form for continuously sampled Asian option where under- lying asset is driven by a Levy´ Process, [18] presented a binomial tree method for Asian options under a particular I. INTRODUCTION jump-diffusion model. But such Market may be incomplete. He pricing of Asian options is a delicate and interesting Therefore, it will be a big class of equivalent martingale T topic in quantitative finance, and has been a topic of measures (EMMs). For the choice of suitable one, many attention for many years now. -
Structured Notes, Which Have Embedded Derivatives, Some of Them Very Complex
Structured note markets: products, participants and links to wholesale derivatives markets David Rule and Adrian Garratt, Sterling Markets Division, and Ole Rummel, Foreign Exchange Division, Bank of England Hedging and taking risk are the essence of financial markets. A relatively little known mechanism through which this occurs is the market in structured notes, which have embedded derivatives, some of them very complex. Understanding these instruments can be integral to understanding the underlying derivative markets. In some cases, dealers have used structured notes to bring greater balance to their market risk exposures, by transferring risk elsewhere, including to households, where the risk may be well diversified. But the positions arising from structured notes can sometimes leave dealers ‘the same way around’, potentially giving rise to ‘crowded trades’. In the past that has sometimes been associated with episodes of market stress if the markets proved less liquid than normal when faced with lots of traders exiting at the same time. FOR CENTRAL BANKS, understanding how the modern For issuers, structured notes can be a way of buying financial system fits together is a necessary options to hedge risks in their business. Most, foundation for making sense of market developments, however, swap the cash flows due on the notes with a for understanding how to interpret changes in asset dealer for a more straightforward set of obligations. prices and, therefore, for identifying possible threats In economic terms, the dealer then holds the to stability and comprehending the dynamics of embedded options. Sometimes they may hedge crises. Derivatives are an integral part of this, used existing exposures taken elsewhere in a dealer’s widely for the management of market, credit and business. -
Unified Pricing of Asian Options
Unified Pricing of Asian Options Jan Veˇceˇr∗ ([email protected]) Assistant Professor of Mathematical Finance, Department of Statistics, Columbia University. Visiting Associate Professor, Institute of Economic Research, Kyoto University, Japan. First version: August 31, 2000 This version: April 25, 2002 Abstract. A simple and numerically stable 2-term partial differential equation characterizing the price of any type of arithmetically averaged Asian option is given. The approach includes both continuously and discretely sampled options and it is easily extended to handle continuous or dis- crete dividend yields. In contrast to present methods, this approach does not require to implement jump conditions for sampling or dividend days. Asian options are securities with payoff which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. There is enormous literature devoted to study of this option. A number of approxima- tions that produce closed form expressions have appeared, most recently in Thompson (1999), who provides tight analytical bounds for the Asian option price. Geman and Yor (1993) computed the Laplace transform of the price of continuously sampled Asian option, but numerical inversion remains problematic for low volatility and/or short maturity cases as shown by Fu, Madan and Wang (1998). Very recently, Linetsky (2002) has derived new integral formula for the price of con- tinuously sampled Asian option, which is again slowly convergent for low volatility cases. Monte Carlo simulation works well, but it can be computationally expensive without the enhancement of variance reduction techniques and one must account for the inherent discretization bias result- ing from the approximation of continuous time processes through discrete sampling as shown by Broadie, Glasserman and Kou (1999). -
View Annual Report
ANNUAL REPORT MetLife, Inc. 2013 Chairman’s Letter To my fellow shareholders: When I became CEO in May of 2011, my mandate was clear: Improve the profitability and risk profile of the business. MetLife was the largest life insurer in the United States with a great brand and a nearly 145-year history of keeping its promises to policyholders, but we had taken on too much risk in certain parts of our business and our profitability was in the middle of the pack relative to industry peers. The first order of business for the executive group in 2011 was to develop a strategy to achieve our goal. It is not easy to turn a ship as large as MetLife, especially in an industry where profits emerge slowly over time. The strategy we launched in 2012 was not a one- or two-year strategy. It was a five-year strategy to raise our return on equity, reduce our cost of equity, and return capital to shareholders. Even though we have faced both economic and regulatory headwinds along the way, I am pleased to report that we are ahead of schedule on our plan to increase the profitability of MetLife’s business while also decreasing its level of risk. A Very Good Year Our progress was clearly evident in 2013. Operating earnings increased 11% over the prior year, exceeding our plan.1 Premiums, fees and other revenues increased 2% on a reported basis and 5% on a constant currency basis – solid top-line growth in light of our efforts to improve MetLife’s risk profile. -
A Discrete-Time Approach to Evaluate Path-Dependent Derivatives in a Regime-Switching Risk Model
risks Article A Discrete-Time Approach to Evaluate Path-Dependent Derivatives in a Regime-Switching Risk Model Emilio Russo Department of Economics, Statistics and Finance, University of Calabria, Ponte Bucci cubo 1C, 87036 Rende (CS), Italy; [email protected] Received: 29 November 2019; Accepted: 25 January 2020 ; Published: 29 January 2020 Abstract: This paper provides a discrete-time approach for evaluating financial and actuarial products characterized by path-dependent features in a regime-switching risk model. In each regime, a binomial discretization of the asset value is obtained by modifying the parameters used to generate the lattice in the highest-volatility regime, thus allowing a simultaneous asset description in all the regimes. The path-dependent feature is treated by computing representative values of the path-dependent function on a fixed number of effective trajectories reaching each lattice node. The prices of the analyzed products are calculated as the expected values of their payoffs registered over the lattice branches, invoking a quadratic interpolation technique if the regime changes, and capturing the switches among regimes by using a transition probability matrix. Some numerical applications are provided to support the model, which is also useful to accurately capture the market risk concerning path-dependent financial and actuarial instruments. Keywords: regime-switching risk; market risk; path-dependent derivatives; insurance policies; binomial lattices; discrete-time models 1. Introduction With the aim of providing an accurate evaluation of the risks affecting financial markets, a wide range of empirical research evidences that asset returns show stochastic volatility patterns and fatter tails with respect to the standard normal model. -
VALUATION of CALLABLE BONDS: the SALOMON BROTHERS APPROACH Fernando Daniel Rubio Fernández
VALUATION OF CALLABLE BONDS: THE SALOMON BROTHERS APPROACH Fernando Daniel Rubio Fernández VALUATION OF CALLABLE BONDS: THE SALOMON BROTHERS APPROACH FERNANDO RUBIO1 Director FERNCAPITAL S.A. and Invited Professor at the Graduated Business School Universidad de Valparaíso, Chile. Pasaje La Paz 1302, Viña del Mar, Chile. Phone (56) (32) 507507 EXTRACT This paper explain, analyze and apply in an example the original paper developed by Kopprasch, Boyce, Koenigsberg, Tatevossian, and Yampol (1987) from The Salomon Brothers Inc. Bond Portfolio Analysis Group. Please, be aware. This paper is for educational issues only. There is a Spanish version in EconWPA. JEL Classification: G10, G15, G21, G32. Keywords: Salomon Brothers, bond portfolio, duration and convexity, effective duration, valuation, callable and non callable bond. Originally developed January, 1999 Originally published October, 2004 This update July, 2005 1 This paper was made while I was assisting to the Doctoral Programme in Financial Economics, Universidad Autónoma de Madrid, Spain. Comments and suggestions will be appreciated. Please, send them by e-mail to [email protected] [email protected] 1 VALUATION OF CALLABLE BONDS: THE SALOMON BROTHERS APPROACH Fernando Daniel Rubio Fernández VALUATION OF CALLABLE BONDS: THE SALOMON BROTHERS APPROACH By Professor Dr. © Fernando Rubio 1 DURATION AND CONVEXITY FOR NORMAL (NO CALLABLE) BONDS Bonds are fixed income investments that have a fixed interest rate or coupon, payable on the principal amount. All fixed income investments are evidence of indebtedness which represent a loan or debt between the issuer and the owner or holder of the security. The value of any bond is the present value of its expected cash flows. -
Constructing an Alternative Allocation
DECEMBER 2018 Constructing an Alternative Allocation NOT FDIC INSURED | MAY LOSE VALUE | NO BANK GUARANTEE The opinions referenced are as of the date of the publication, are subject to change due to changes in the market or economic conditions, and may not necessarily come to pass. Information contained herein is for informational purposes only and should not be considered investment advice. Past performance is no guarantee of future results. CONSTRUCTING AN ALTERNATIVE ALLOCATION Important Information Some of the risks associated with investing in alternatives may include hedging risk – hedging activities can reduce investment performance through added costs; derivative risk – derivatives may experience greater price volatility than the underlying securities; short sale risk – investments may incur a loss without limit as a result of a short sale if the market value of the security increases; interest rate risk – loss of value for income securities as interest rates rise; credit risk – risk of the borrower to miss payments; liquidity risk – low trading volume may lead to increased volatility in certain securities; non-U.S. government obligation risk – non-U.S. government obligations may be subject to increased credit risk; portfolio selection risk – investment managers may select securities that fare worse than the overall market. Alternative investments may not be suitable for all investors. An investment in the Fund(s) is subject to risks, and you could lose money on your investment in the Fund(s). There can be no assurance that the Fund(s) will achieve its investment objective. Your investment in the Fund(s) is not a deposit in a bank and is not insured or guaranteed by the Federal Deposit Insurance Corporation (FDIC) or any other government agency. -
Pricing and Hedging of Asian Option Under Jumps
IAENG International Journal of Applied Mathematics, 41:4, IJAM_41_4_04 ______________________________________________________________________________________ Pricing And Hedging of Asian Option Under Jumps Wissem Boughamoura, Anand N. Pandey and Faouzi Trabelsi Abstract—In this paper we study the pricing and hedging To price Asian options in such setting, [12] presents gener- problems of ”generalized” Asian options in a jump-diffusion alized Laplace transform for continuously sampled Asian op- model. We choose the minimal entropy martingale measure tion where underlying asset is driven by a Levy´ Process, [29] (MEMM) as equivalent martingale measure and we derive a partial-integro differential equation for their price. We discuss proposes a binomial tree method under a particular jump- the minimal variance hedging including the optimal hedging diffusion model. For the case of semimartingale models, [36] ratio and the optimal initial endowment. When the Asian payoff shows that the pricing function is satisfying a partial-integro is bounded, we show that for the exponential utility function, differential equation. For the precise numerical computation the utility indifference price goes to the Asian option price of the PIDE using difference schemes we refer interested evaluated under the MEMM as the Arrow-Pratt measure of absolute risk-aversion goes to zero. readers to explore with [11]. In [3] is derived the range of price for Asian option when underlying stock price is Index Terms—Geometric Levy´ process, Generalized Asian following geometric Brownian motion and jump (Poisson). In options, Minimal entropy martingale measure, Partial-integro differential equation, Utility indifference pricing, Exponential [30] is derived bounds for the price of a discretely monitored utility function. -
2009 FINANCIAL SUMMARY Enerplus Has a Plan and Is Transitioning Our Business from an Income Fund to a Competitive Growth and Income-Oriented Oil and Gas Company
Season in Review ENERPLUS RESOURCES 2009 FINANCIAL SUMMARY Enerplus has a plan and is transitioning our business from an income fund to a competitive growth and income-oriented oil and gas company. 1. Add more early-stage resource plays 2. Maintain a strong financial position 3. Focus our portfolio 4. Build our operational capability Enerplus is one of Canada’s oldest and largest independent oil and gas producers with a portfolio of both early-stage resource plays and mature cash-generating properties. We are focused on creating value for our investors through the successful development of our properties and the disciplined management of our balance sheet. Through these activities, we strive to provide investors with a competitive return comprised of both growth and income. 1 Financial and Operating Highlights 4 Management’s Discussion and Analysis 37 Financial Statements 67 Five-Year Detailed Statistical Review 68 Five-Year Operational Statistics 69 Supplemental Information 79 Abbreviations 80 Definitions 82 Board of Directors 83 Officers 84 Corporate Information 2009 SUMMARY Selected Financial and Operating Highlights Three months ended Twelve months ended December 31, December 31, SELECTED FINANCIAL RESULTS (in Canadian dollars) 2009 2008 2009 2008 Financial (000’s) Cash Flow from Operating Activities $ 188,579 $ 258,536 $ 775,786 $ 1,262,782 Cash Distributions to Unitholders(1) 95,550 167,017 368,201 786,138 Excess of Cash Flow Over Cash Distributions 93,029 91,519 407,585 476,644 Net Income 2,718 189,495 89,117 888,892 Debt Outstanding -
Eaton Vance Tax-Advantaged Bond and Option Strategies Fund Notice
Eaton Vance Tax-Advantaged Bond and Option Strategies Fund Notice of Changes to Fund Name, Investment Objective, Fees and Distributions The Board of Trustees of Eaton Vance Tax-Advantaged Bond and Option Strategies Fund (NYSE: EXD) (the “Fund”) has approved changes to the Fund’s name, investment objective and investment policies as described below. In connection with these changes, the portfolio managers of the Fund will change and the Fund’s investment advisory fee rate will be reduced. Each of the foregoing changes will be effective on or about February 8, 2019. Following implementation of the changes to the Fund’s investment objective and policies, the Fund will increase the frequency of its shareholder distributions from quarterly to monthly and raise the distribution rate as described below. Name. The Fund’s name will change to “Eaton Vance Tax-Managed Buy-Write Strategy Fund.” It will continue to be listed on the New York Stock Exchange under the ticker symbol “EXD.” Investment Objectives. As revised, the Fund will have a primary objective to provide current income and gains, with a secondary objective of capital appreciation. In pursuing its investment objectives, the Fund will evaluate returns on an after-tax basis, seeking to minimize and defer shareholder federal income taxes. There can be no assurance that the Fund will achieve its investment objectives. The Fund’s current investment objective is tax-advantaged income and gains. Principal Investment Policies. The Fund currently employs a tax-advantaged short-term bond strategy (“Bond Strategy”) and a rules-based option overlay strategy that consists of writing a series of put and call spreads on the S&P 500 Composite Stock Price Index® (the “S&P 500”) (“Option Overlay Strategy”). -
On the Equivalence of Floating and Fixed-Strike Asian Options
Applied Probability Trust (5 September 2001) ON THE EQUIVALENCE OF FLOATING AND FIXED-STRIKE ASIAN OPTIONS VICKY HENDERSON,∗ University of Oxford RAFALWOJAKOWSKI, ∗∗ Lancaster University Abstract There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating and fixed-strike Asian options. The proof involves a change of num´eraireand time reversal of Brownian motion. Symmetries are very useful in option valuation and in this case, the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options. Keywords: Asian options, floating strike Asian options, put call symmetry, change of num´eraire,time reversal, Brownian motion AMS 2000 Subject Classification: Primary 60G44 Secondary 91B28 1. Introduction The purpose of this paper is to establish a useful symmetry result between floating and fixed-strike Asian options. A change of probability measure or num´eraireand a time reversal argument are used to prove the result for models where the underlying asset follows exponential Brownian motion. There are many known symmetry results in financial option pricing. Such results are useful for transferring knowledge about one type of option to another and may be used to simplify coding of one type of option when the other is already coded. However, one must take care as the transformed option may not exist in the market or have a sensible economic interpretation. These tricks become very useful for exotic options, when perhaps no closed form solution exists, but an equivalence relation holds, together with an accurate compu- tational procedure for the related class. -
Confidential Treatment Requested by Lehman Brothers Holdings, Inc
CONFIDENTIAL TREATMENT REQUESTED BY LEHMAN BROTHERS HOLDINGS, INC. LBEX-WGM 747129 Content Pricing I. Organization 3 Organization Structure 4-5 Staffing 6 II. Pricing and Control Philosophy 7 Mark Review Process 8-9 Products with Fair Value Categories 10 Price Verification Thresholds 11 Complex Derivatives Transaction Review Committee 12-14 Model Approval Process and Control Committee 15-16 Ill. Summary ofPrice Verification Results 17 Executive Summary- Fixed Income Division 18 Price Verification Coverage - Fixed Income Division 19 Price Verification Projects- Fixed Income Division 20 Executive Summary- Equities Division 21-22 Price Verification Coverage - Equities Division 23 Price Verification Projects- Equities Division 24 lY. Examples of Price Verification 25 Interest Rate Products 26-30 Equities 31-34 Credit Products 35-44 V. Appendix 45 Explanation of Significant Variances -Fixed Income Division 46-48 r Explanation of Significant Variances - Equities Division 49-50 OJm ~ ~ G) :s;: -..J 2 ~ LEHMAN BROTHERS -..J ...... (...) CONFIDENTIAL TREATMENT REQUESTED BY 0 LEHMAN BROTHERS HOLDINGS, INC. c: ........,0 ca N CONFIDENTIAL TREATMENT REQUESTED BY LEHMAN BROTHERS HOLDINGS, INC. LBEX-WGM 747131 Organization Structure Pricing I Capital Markets Finance is a group offinance professionals who are independent from the business and report to the CFO ofthe firm. The group is closely aligned with the business units in order to enhance the control process and maximize the flow of communication across the organization. Capital Markets Finance is primarily accountable to: .. Protect the Firm .. Provide daily revenue analysis and reporting .. Provide an independent and documented validation of inventory valuations "' Ensure appropriateness of valuation adjustments that adhere to globally consistent and approved standards ., Ensure, in conjunction with Risk Management, that appropriate models are used to value derivative transactions .