DOCSLIB.ORG

Calibrating the Black-Derman-Toy Model

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Calibrating the Black-Derman-Toy Model AppliedMathematical Finance 8,27–48 (2001) Calibrating the Black–Derman– Toy model: some theoretical results PHELIM P.BOYLE, KEN SENG TAN andWEIDONG TIAN Universityof Waterloo,Ontario, Canada, N2L 3G1 ReceivedJune 2000 TheBlack– Derman– Toy (BDT) modelis a popularone-factor interest rate model that is widely used by practitioners. Oneof itsadvantages is that the model can be calibratedto both the current market term structure of interestrate and thecurrent term structure of volatilities.The input term structure of volatilitycan be eitherthe short term volatility or theyield volatility. Sandmann and Sondermann derived conditions for the calibration to be feasible when the conditionalshort rate volatility is used.In this paper conditions are investigated under which calibration to the yield volatilityis feasible. Mathematical conditions for this to happen are derived. The restrictions in this case are more complicatedthan when the short rate volatilities are used since the calibration at each time step now involves the solutionof two non-linear equations. The theoretical results are illustrated by showing numerically that in certain situationsthe calibration based on the yield volatility breaks down for apparently plausible inputs. In implementing the calibrationfrom period n to period n +1,thecorresponding yield volatility has to lie within certain bounds. Under certaincircumstances these bounds become very tight. For yield volatilities that violate these bounds, the computed shortrates for the period ( n, n +1)either become negative or else explode and this feature corresponds to the economicintuition behind the breakdown. Keywords: interestrate models, Black– Derman– T oymodel, volatility, short term, yield 1.Introduction Themodern approach to the modelling of stochastic interest rates started with the classic paper by Vasicek(1977). The early models postulated a stochasticdifferential equation for theevolution of theshort rate of interest and, by invoking no-arbitrage arguments, developed expressions for theprices ofpure discount bond and other securities of interest such as options. Other examples of such modelsinclude Brennan and Schwartz (1979) and Cox, Ingersoll and Ross (1985). One of theproblems withthese models was thatthey did not have enough degrees of freedom to match the model prices ofpure discount bonds with the corresponding market prices. This was anuncomfortable situation Theauthors are gratefulto Baoyan Ding, Fred Guan, Houben Huang, George Lai, David Li, Chonghui Liu, and Ken Vetzal foruseful discussion.Phelim P .Boylethanks the Social Science andHumanities Research Councilof Canada andthe Natural Sciences and EngineeringResearch Councilof Canada,for research support.Ken SengTan acknowledges the research supportfrom the Natural Sciences andEngineering Research Councilof Canada.Weidong Tian thanks CITO (Communicationsand Information Technology Ontario)for research support. AppliedMathematical Finance ISSN1350-486X print/ ISSN1466-4313 online # 2001Taylor & FrancisLtd http://www.tandf.co.uk/journals DOI: 10.1080/13504860110062049 28 Boyle,T anand Tian sincethe model could not be described as arbitrage free whenthe model prices differed from thecurrent marketprices. Oneway to resolve this problem is to makethe parameters of theshort term rate time dependent. In the Vasicekcase, the resulting model is known as the extended V asicekmodel. This approach has been popularizedby Hull and White (1993). Another approach is to model the uncertainty by assuming a stochasticprocess for theevolution of the forward rate.Since this approach starts from thecurrent observedforward rate,the market prices of today’ s zero-couponbonds are built directly into the foundationsof themodel. The rst paperto applythis approach was Hoand Lee (1986) in a discrete-time framework.Subsequently Heath, Jarrow andMorton (1990) (HJM) provideda muchmore extensive and rigorousapproach in continuous-timeframework. The HJM modelis quite general; it canbe calibratedto currentbond prices and option prices. In practice, the model parameters are selected by ttingmodel pricesto the current market prices of the most liquid instruments. From atradingperspective, this approachis useful since the model reproduces market prices over for themost liquid traded securities. 1 For manypricing applications, it is convenient to have a simplebinomial type model that tsthe currentterm structure of bondprices and the current term structure of volatilities.In this connection, the Black,Derman and Toy (1990) (BDT) modelis awidelyused model that can be calibratedto matchthe termstructure of zero-couponbond prices and the term structure of volatilities.In the original BDT paper, theauthors used the yield volatilities as the input term structure of volatility. The yield volatility correspondsto the volatility of the yields on long term bonds. In practical applications, it is often more convenientto use the term structure of shortrate volatilities since they can be directly inferred from the marketprices of interestrate caps. Thus,there are two ways to calibrate the BDT model.The rst oneis the short rate volatility method which uses thecurrent term structure of zerocoupon bond prices; theterm structure of futureshort rate volatilities. Thesecond one is the yield rate volatility method which uses thecurrent term structure of zerocoupon bond prices; theterm structure of yieldson zero coupon bonds. For eachof theseapproaches it is of interestto investigate the conditions under which ttingthe BDT model results in a reasonablecalibration .Our conditionsfor areasonablecalibration are quite weak. W ewillrequire thatall the short rates and all the output volatilities 2 inthe calibrated BDT modelare positive. Sandmann and Sondermann(1993) have analysed the case when the BDT calibrationis based on theshort rate volatilities. Theyprovide necessary and suf cient conditions for thisto happen. Their result is: if the current implied forward ratesare all positive (i.e. the pure discount bond price declines as the time to maturity increases) and theshort rate volatilities are all positive, then it is possible to calibrate the BDT model.Their conditions are simplefrom amathematicalperspective and have an intuitive economic interpretation. 1 Thisapproach has disadvantagesfrom an econometric perspective since, by ttingnew parameters ona dailybasis, we are effectivelyassuming a new modelevery day. 2 Theoutput volatility under the rst approachis theyield volatility as computedfrom the calibrated BDT modelsince theinput volatilityis theshort rate volatility.The output volatility under the second approach is theshort rate volatilityas computedfrom the calibratedBDT modelsince theinput volatility is theyield rate volatility. Calibratingthe Black– Derman– Toy model 29 Thispaper investigates the conditions under which we cancalibrate the BDT modelwhen the yield volatilityis used. W ederivethe precise mathematical conditions which the input data must satisfy so that the BDT modelcan be calibrated.These conditions are less elegant than in the case when the short rate volatility isused. However ,we ndthatit isnot possible to calibrate the BDT modelfor seeminglyplausible input termstructures. T oobtainour conditions we useresults from thetheory of polynomialequations. Theoutline of the rest of the paper is as follows. In the next section, we reviewthe details of the procedurethat is used to calibrate a BDT model.W eexamineboth the original and modi ed BDT models whichcalibrate to the term structure of yieldvolatilities and short rate volatilities respectively. W eshow thatthe calibration equations can be reducedto a systemof polynomialequations so that we candraw on resultsfrom theso-called Quanti er Elimination;an algebraic approach that provides conditions for polynomialsto have real roots. In Section 3, we providea detailedanalysis of thecalibration of athree- periodBDT model.W eprovideboth necessary and suf cient conditions for thecalibration to be feasible. Theconditions are quite complicated even in the three period case and it appears dif cult to extend this typeof analysis to the n periodscase. Hence in Section4, we providea sufcient condition for calibration tobe feasible from the nthstep to the ( n +1)thstep given that the calibration was successfulfor the preceding n periods.In Section 5 we provideseveral examples which illustrate the conditions developed inSection 4. The nalsection concludes the paper. 2.Calibration ofthe BDT model Inthis section, we reviewthe procedure used to calibrate a BDT interestrate model. In this model, a recombiningbinomial lattice is constructed so that it matches the current yield curve and the current yield volatilitycurve. W eassumethe calibrated binomial lattice has N periodsand each period is of size t years.Hence the total time horizon of thebinomial lattice is T = N t years.The recombining nature of thebinomial lattice ensures that at time period n, there are n +1states.W elabelthese states as i = 0, 1, . ., n. Let r(n, i)bethe(annualized) one-period short rate at period n and state i.Theshort rate r(n, i) evolves either to r(n + 1, i)(i.e.down-state) or to r(n + 1, i +1)(i.e. up-state) one period hence with equal risk-neutralprobability. Let Y^(0, n) Y^(n)bethecurrent (market) yield on a n-periodzero-coupon bond (i.e. with maturity n t) and ^Y (n)bethe corresponding current yield volatility. Then the price of an n-periodzero-coupon bond, P^(0, n) P^(n), isgiven as n P^(n) 1 Y^(n) t ¡ (1) ˆ ‰ ‡ Š Similarly,let P (n) and Y (n)denotethe model
Load more

Recommended publications
  • An Empirical Comparison of Interest Rate Models for Pricing Zero Coupon Bond Options
    AN EMPIRICAL COMPARISON OF INTEREST RATE MODELS FOR PRICING ZERO COUPON BOND OPTIONS HUSEYÄ IN_ S»ENTURKÄ AUGUST 2008 AN EMPIRICAL COMPARISON OF INTEREST RATE MODELS FOR PRICING ZERO COUPON BOND OPTIONS A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED MATHEMATICS OF THE MIDDLE EAST TECHNICAL UNIVERSITY BY HUSEYÄ IN_ S»ENTURKÄ IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE DEPARTMENT OF FINANCIAL MATHEMATICS AUGUST 2008 Approval of the Graduate School of Applied Mathematics Prof. Dr. Ersan AKYILDIZ Director I certify that this thesis satis¯es all the requirements as a thesis for the degree of Master of Science. Prof. Dr. Ersan AKYILDIZ Head of Department This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Master of Science. Assist. Prof. Dr. Kas³rga Y³ld³rak Assist. Prof. Dr. OmÄurU¸gurÄ Co-advisor Supervisor Examining Committee Members Assist. Prof. Dr. OmÄurU¸gurÄ Assist. Prof. Dr. Kas³rga Y³ld³rak Prof. Dr. Gerhard Wilhelm Weber Assoc. Prof. Dr. Azize Hayfavi Dr. C. Co»skunKÄu»cÄukÄozmen I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last name: HÄuseyinS»ENTURKÄ Signature: iii abstract AN EMPIRICAL COMPARISON OF INTEREST RATE MODELS FOR PRICING ZERO COUPON BOND OPTIONS S»ENTURK,Ä HUSEYÄ IN_ M.Sc., Department of Financial Mathematics Supervisor: Assist.
    [Show full text]
  • Parameter Estimation in Stochastic Volatility Models Via Approximate Bayesian Computing
    Parameter Estimation in Stochastic Volatility Models Via Approximate Bayesian Computing A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Achal Awasthi, B.S. Graduate Program in Department of Statistics The Ohio State University 2018 Master's Examination Committee: Radu Herbei,Ph.D., Advisor Laura S. Kubatko, Ph.D. c Copyright by Achal Awasthi 2018 Abstract In this thesis, we propose a generalized Heston model as a tool to estimate volatil- ity. We have used Approximate Bayesian Computing to estimate the parameters of the generalized Heston model. This model was used to examine the daily closing prices of the Shanghai Stock Exchange and the NIKKEI 225 indices. We found that this model was a good fit for shorter time periods around financial crisis. For longer time periods, this model failed to capture the volatility in detail. ii This is dedicated to my grandmothers, Radhika and Prabha, who have had a significant impact in my life. iii Acknowledgments I would like to thank my thesis supervisor, Dr. Radu Herbei, for his help and his availability all along the development of this project. I am also grateful to Dr. Laura Kubatko for accepting to be part of the defense committee. My gratitude goes to my parents, without their support and education I would not have had the chance to study worldwide. I would also like to express my gratitude towards my uncles, Kuldeep and Tapan, and Mr. Richard Rose for helping me transition smoothly to life in a different country.
    [Show full text]
  • Regularity of Solutions and Parameter Estimation for Spde’S with Space-Time White Noise
    REGULARITY OF SOLUTIONS AND PARAMETER ESTIMATION FOR SPDE’S WITH SPACE-TIME WHITE NOISE by Igor Cialenco A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2007 Copyright 2007 Igor Cialenco Dedication To my wife Angela, and my parents. ii Acknowledgements I would like to acknowledge my academic adviser Prof. Sergey V. Lototsky who introduced me into the Theory of Stochastic Partial Differential Equations, suggested the interesting topics of research and guided me through it. I also wish to thank the members of my committee - Prof. Remigijus Mikulevicius and Prof. Aris Protopapadakis, for their help and support. Last but certainly not least, I want to thank my wife Angela, and my family for their support both during the thesis and before it. iii Table of Contents Dedication ii Acknowledgements iii List of Tables v List of Figures vi Abstract vii Chapter 1: Introduction 1 1.1 Sobolev spaces . 1 1.2 Diffusion processes and absolute continuity of their measures . 4 1.3 Stochastic partial differential equations and their applications . 7 1.4 Ito’sˆ formula in Hilbert space . 14 1.5 Existence and uniqueness of solution . 18 Chapter 2: Regularity of solution 23 2.1 Introduction . 23 2.2 Equations with additive noise . 29 2.2.1 Existence and uniqueness . 29 2.2.2 Regularity in space . 33 2.2.3 Regularity in time . 38 2.3 Equations with multiplicative noise . 41 2.3.1 Existence and uniqueness . 41 2.3.2 Regularity in space and time .
    [Show full text]
  • Cox, Ingersoll and Ross Models of Interest Rates
    RECORD, Volume 28, No. 1* Colorado Springs Spring Meeting May 30-31, 2002 Session 86L Cox, Ingersoll And Ross Models Of Interest Rates Track: Education and Research Moderator: PETER D. TILLEY Lecturer: WOJCIECH SZATZSCHNEIDER† Summary: Our lecturer presents findings from a research project relating to the Cox, Ingersoll & Ross (CIR) model of interest rates. The findings address the extended CIR model and easiest construction thereof; an effective way of pricing general interest rate derivatives within this model; and pricing of bonds using the Laplace transform of functionals of the "elementary process," which is squared Bessell. MR. PETER D. TILLEY: Our guest speaker today is Wojciech Szatzschneider. Dr. Szatzschneider received his Ph.D. from the Polish Academy of Sciences, and he's been at Anahuac University in Mexico City since 1983. He started the graduate program in actuarial science and financial mathematics at that university. He's also been a visiting professor at universities in Canada, Germany, Switzerland, Spain and Poland and has participated in conferences around the world. He's going to be speaking on the Cox, Ingersoll, Ross (CIR) interest rate model and other interest rate models in general; calibration techniques for this model; and he's going to look at the CIR model as part of a financial market. DR. WOJCIECH SZATZSCHNEIDER: Why CIR? I'm a mathematician, and the CIR model is a very difficult model to analyze correctly. I tried to solve some problems. I saw many solutions to these problems, which I couldn't solve, but these solutions were wrong. So there are many, many published results, but they're incorrect results.
    [Show full text]
  • Valuing Interest Rate Swap Contracts in Uncertain Financial Market
    Article Valuing Interest Rate Swap Contracts in Uncertain Financial Market Chen Xiao 1,†, Yi Zhang 2,* and Zongfei Fu 2 1 Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA; [email protected] 2 School of Information, Renmin University, Beijing 100872, China; [email protected] * Correspondence: [email protected] † Current address: School of Finance, Nankai University, Tianjin 300071, China. Academic Editors: Xiang Li, Jian Zhou, Hua Ke, Xiangfeng Yang and Giuseppe Ioppolo Received: 12 September 2016; Accepted: 3 November 2016 ; Published: 18 November 2016 Abstract: Swap is a financial contract between two counterparties who agree to exchange one cash flow stream for another, according to some predetermined rules. When the cash flows are fixed rate interest and floating rate interest, the swap is called an interest rate swap. This paper investigates two valuation models of the interest rate swap contracts in the uncertain financial market. The new models are based on belief degrees, and require relatively less historical data compared to the traditional probability models. The first valuation model is designed for a mean-reversion term structure, while the second is designed for a term structure with hump effect. Explicit solutions are developed by using the Yao–Chen formula. Moreover, a numerical method is designed to calculate the value of the interest rate swap alternatively. Finally, two examples are given to show their applications and comparisons. Keywords: interest rate swap; uncertain process; uncertain differential equation; Yao-Chen formula 1. Introduction Interest rate swap is one of the most popular interest derivatives. The interest rate swap began trading in 1981, and now owns a hundred billion dollar market.
    [Show full text]
  • CHAPTER 7 Interest Rate Models and Bond Pricing
    CHAPTER 7 Interest Rate Models and Bond Pricing The riskless interest rate has been assumed to be constant in most of the pric- ing models discussed in previous chapters. Such an assumption is acceptable when the interest rate is not the dominant state variable that determines the option payoff, and the life of the option is relatively short. In recent decades, we have witnessed a proliferation of new interest rate dependent securities, like bond futures, options on bonds, swaps, bonds with option features, etc., whose payoffs are strongly dependent on the interest rates. Note that interest rates are used for discounting as well as for defining the payoff of the deriva- tive. The values of these interest rate derivative products depend sensibly on the level of interest rates. In the construction of valuation models for these securities, it is crucial to incorporate the stochastic movement of interest rates into consideration. Several approaches for pricing interest rate deriva- tives have been proposed in the literature. Unfortunately, up to this time, no definite consensus has been reached with regard to the best approach for these pricing problems. The correct modelling of the stochastic behaviors of interest rates, or more specifically, the term structure of the interest rate through time is important for the construction of realistic and reliable valuation models for interest rate derivatives. The extension of the Black-Scholes valuation framework to bond options and other bond derivatives is doomed to be difficult because of the pull-to-par phenomenon, where the bond price converges to par at maturity, thus causing the instantaneous rate of return on the bond to be distributed with a diminishing variance through time.
    [Show full text]
  • Lecture Notes: Interest Rate Theory
    Lecture Notes: Interest Rate Theory Lecture Notes: Interest Rate Theory Josef Teichmann ETH Zurich¨ Fall 2010 J. Teichmann Lecture Notes: Interest Rate Theory Lecture Notes: Interest Rate Theory Foreword Mathematical Finance Basics on Interest Rate Modeling Black formulas Affine LIBOR Models Markov Processes The SABR model HJM-models References Catalogue of possible questions for the oral exam 1 / 107 Lecture Notes: Interest Rate Theory Foreword In mathematical Finance we need processes I which can model all stylized facts of volatility surfaces and times series (e.g. tails, stochastic volatility, etc) I which are analytically tractable to perform efficient calibration. I which are numerically tractable to perform efficient pricing and hedging. 2 / 107 Lecture Notes: Interest Rate Theory Foreword Goals I Basic concepts of stochastic modeling in interest rate theory. I "No arbitrage" as concept and through examples. I Concepts of interest rate theory like yield, forward rate curve, short rate. I Spot measure, forward measures, swap measures and Black's formula. I Short rate models I Affine LIBOR models I Fundamentals of the SABR model I HJM model I Consistency and Yield curve estimation 3 / 107 Lecture Notes: Interest Rate Theory Mathematical Finance Modeling of financial markets We are describing models for financial products related to interest rates, so called interest rate models. We are facing several difficulties, some of the specific for interest rates, some of them true for all models in mathematical finance: I stochastic nature: traded prices, e.g.~prices of interest rate related products, are not deterministic! I information is increasing: every day additional information on markets appears and this stream of information should enter into our models.
    [Show full text]
  • Using Path Integrals to Price Interest Rate Derivatives
    Using path integrals to price interest rate derivatives Matthias Otto Institut f¨ur Theoretische Physik, Universit¨at G¨ottingen Bunsenstr. 9, D-37073 G¨ottingen Abstract: We present a new approach for the pricing of interest rate derivatives which allows a direct computation of option premiums without deriving a (Black- Scholes type) partial differential equation and without explicitly solving the stochas- tic process for the underlying variable. The approach is tested by rederiving the prices of a zero bond and a zero bond option for a short rate environment which is governed by Vasicek dynamics. Furthermore, a generalization of the method to general short rate models is outlined. In the case, where analytical solutions are not accessible, numerical implementations of the path integral method in terms of lattice calculations as well as path integral Monte Carlo simulations are possible. arXiv:cond-mat/9812318v2 [cond-mat.stat-mech] 14 Jun 1999 1 1 Introduction The purpose of this article is to present a new approach for the pricing of interest rate derivatives: the path integral formalism. The claim is that interest rate derivatives can be priced without explicitly solving for the stochastic process of the underlying (e.g. a short rate) and without deriving a (Black-Scholes type) partial differential equation (PDE). The mathematical foundation of the PDE approach to derivatives pricing, which is used traditionally, is the Feynman-Kac lemma [1, 2] connecting the solution of a certain type of parabolic PDE to expectation values with respect to stochastic processes. Usually, the original pricing problem is given in terms of an expectation value.
    [Show full text]
  • Term Structure Models
    Term Structure Models 1. Zero-coupon bond prices and yields 2. Vasicek model 3. Cox-Ingersoll-Ross model 4. Multifactor Cox-Ingersoll-Ross models 5. Affine models 6. Completely affine models 7. Bond risk premia 8. Inflation and nominal asset prices 1 Readings and References Back, chapter 17. Duffie, chapter 7. Vasicek, O., 1977, An equilibrium characterization of the term structure, Journal of Financial Economics 5, 177-188. Cox, J., J. Ingersoll, and S. Ross, 1985, A theory of the term structure of interest rates, Econometrica 53, 385-407. Longsta↵, F., and E. Schwartz, 1992, Interest rate volatility and the term structure: A two-factor general equilibrium model, Journal of Finance 47, 1259-1282. Duffie, D., and R. Kan, 1996, A yield-factor model of interest rates, Mathematical Finance 6, 379-406. Du↵ee, G., 2002, Term premia and interest rate forecasts in affine models, Journal of Finance 57, 405-443. Drechsler, I., A. Savov, and P. Schnabl, A Model of Monetary Policy and Risk Premia, Journal of Finance forthcoming. 2 Summary of the Continuous-Time Financial Market dS0,t dSk,t I Security prices satisfy = rt dt and =(µk,t δk,t) dt + σk,tdBt. S0,t Sk,t − I Given tight tr. strat. ⇡t and consumption ct, portfolio value Xt satisfies the WEE: dXt = rtXt dt + ⇡t(µt rt1) dt + ⇡tσt dBt ct dt. • − − No arbitrage if ⇡tσt =0then ⇡t(µt rt1) = 0 ✓t s.t. σt✓t = µt rt1 I ) − )9 − dXt = rtXt dt + ⇡tσt(✓t dt + dBt) ct dt. ) − t 1 t 2 d ✓ dBs ✓s ds ⇤ = = − 0 s0 −2 0 | | I Under emm ⇤ given by dP ZT where Zt e , P P t R R B = Bt + ✓s ds is Brownian motion.
    [Show full text]
  • Monte Carlo Strategies in Option Pricing for Sabr Model
    MONTE CARLO STRATEGIES IN OPTION PRICING FOR SABR MODEL Leicheng Yin A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Statistics and Operations Research. Chapel Hill 2015 Approved by: Chuanshu Ji Vidyadhar Kulkarni Nilay Argon Kai Zhang Serhan Ziya c 2015 Leicheng Yin ALL RIGHTS RESERVED ii ABSTRACT LEICHENG YIN: MONTE CARLO STRATEGIES IN OPTION PRICING FOR SABR MODEL (Under the direction of Chuanshu Ji) Option pricing problems have always been a hot topic in mathematical finance. The SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. To price options under SABR model, there are analytical and probability approaches. The probability approach i.e. the Monte Carlo method suffers from computation inefficiency due to high dimensional state spaces. In this work, we adopt the probability approach for pricing options under the SABR model. The novelty of our contribution lies in reducing the dimensionality of Monte Carlo simulation from the high dimensional state space (time series of the underlying asset) to the 2-D or 3-D random vectors (certain summary statistics of the volatility path). iii To Mom and Dad iv ACKNOWLEDGEMENTS First, I would like to thank my advisor, Professor Chuanshu Ji, who gave me great instruction and advice on my research. As my mentor and friend, Chuanshu also offered me generous help to my career and provided me with great advice about life. Studying from and working with him was a precious experience to me.
    [Show full text]
  • HJM: a Unified Approach to Dynamic Models for Fixed Income, Credit
    HJM: A Unified Approach to Dynamic Models for Fixed Income, Credit and Equity Markets René A. Carmona1 Bendheim Center for Finance Department of Operations Research & Financial Engineering, Princeton University, Princeton, NJ 08544, USA email: [email protected] Summary. The purpose of this paper is to highlight some of the key elements of the HJM approach as originally introduced in the framework of fixed income market models, to explain how the very same philosophy was implemented in the case of credit portfolio derivatives and to show how it can be extended to and used in the case of equity market models. In each case we show how the HJM approach naturally yields a consistency condition and a no-arbitrage conditions in the spirit of the original work of Heath, Jarrow and Morton. Even though the actual computations and the derivation of the drift condition in the case of equity models seems to be new, the paper is intended as a survey of existing results, and as such, it is mostly pedagogical in nature. 1 Introduction The motivation for this papercan be found in the desire to understand recent attempts to implement the HJM philosophy in the valuation of options on credit portfolios. Several proposals appeared almost simultaneously in the literature on credit portfo- lio valuation. They were written independently by N. Bennani [3], J. Sidenius, V. Piterbarg and L. Andersen [26] and P. Shönbucher [41], the latter being most influ- ential in the preparation of the present survey. After a sharp increase in volume and liquidity due to the coming of age of the single tranche synthetic CDOs, markets for these credit portfolios came to a stand still due to the lack of dynamic mod- els needed to price forward starting contracts, options on options, .
    [Show full text]
  • 2 Abstract 1. Introduction
    ABSTRACT This paper illustrates the calibration of a term structure model to market data. The specific application uses Eurodollar futures options prices and the underlying futures contracts to calibrate a one- and two-factor Hull-White term structure model. This paper explains the Hull-White term structure models, Eurodollar futures options contracts, and a numerical methodology (Hull-White trees) that is used to evaluate the American feature of the traded options. When the calibrated models are applied to Eurodollar futures options, the results illustrate that the two-factor model appears to have fewer biases in modeling market prices than a one-factor model. 1. INTRODUCTION A number of term structure models are available to simulate movements of interest rates. Term structure models play an important role in many types of financial analysis including the valuation of interest rate dependent securities, the measurement of interest rate risk, and even strategic planning exercises. Understanding the characteristics of different term structure models can help the user choose the most appropriate model for his or her application. A vital step to using a term structure model is to choose appropriate parameters. Even if the form of the term structure model incorporates all of the desirable historical movements in yields (such as mean reversion), such models will provide poor results if the parameters are carelessly chosen. This paper documents a methodology, called calibration, which chooses appropriate parameters for term structure models. The rationale for the approach is that, when a term structure model is used to price interest rate contingent claims, the model should (approximately) replicate market prices.
    [Show full text]