DOCSLIB.ORG

Smile Modeling in the LIBOR Market Model Might Be Especially Closed Form Solutions, E.G

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Smile Modeling in the LIBOR Market Model Might Be Especially Closed Form Solutions, E.G University of Karlsruhe (TH) Chair of Statistics, Econometrics and Mathematical Finance (Professor Rachev) Diploma Thesis SMILE MODELING IN THE LIBOR MARKET MODEL submitted by: supervised by: Markus Meister Dr. Christian Fries Seckbacher Landstraße 45 (Dresdner Bank, Risk Control) 60389 Frankfurt am Main Dr. Christian Menn [email protected] (University of Karlsruhe) Frankfurt am Main, August 20, 2004 Contents List of Figures vii List of Tables viii Abbreviations and Notation ix I The LIBOR Market Model and the Volatility Smile 1 1 Introduction 2 2 The LIBOR Market Model 5 2.1 Yield Curve . 5 2.2 Volatility . 6 2.2.1 Black’s Formula for Caplets . 8 2.2.2 Term Structure of Volatility . 10 2.3 Correlation Matrix . 11 2.3.1 Black’s Formula for Swaptions . 13 i ii 2.3.2 A Closed Form Approximation for Swaptions . 14 2.3.3 Determining the Correlation Matrix . 16 2.3.4 Factor Reduction Techniques . 17 2.4 Deriving the Drift . 19 2.5 Monte Carlo Simulation . 22 2.6 Differences to Spot and Forward Rate Models . 24 3 The Volatility Smile 25 3.1 Reasons for the Smile . 26 3.2 Sample Data . 28 3.3 Requirements for a Good Model . 31 3.4 Calibration Techniques . 33 3.5 Overview over Different Basic Models . 34 II Basic Models 37 4 Local Volatility Models 38 4.1 Displaced Diffusion (DD) . 39 4.2 Constant Elasticity of Variance (CEV) . 42 4.3 Equivalence of DD and CEV . 46 4.4 General Properties . 49 4.5 Mixture of Lognormals . 50 4.6 Comparison of the Different Local Volatility Models . 55 iii 5 Uncertain Volatility Models 56 6 Stochastic Volatility Models 59 6.1 General Characteristics and Problems . 59 6.2 Andersen, Andreasen (2002) . 60 6.3 Joshi, Rebonato (2001) . 64 6.4 Wu, Zhang (2002) . 66 6.5 Comparison of the Different Stochastic Volatility Models . 71 7 Models with Jump Processes 73 7.1 General Characteristics and Problems . 73 7.2 Merton’s Fundament . 74 7.3 Glasserman, Kou (1999) . 77 7.4 Kou (1999) . 79 7.5 Glasserman, Merener (2001) . 84 7.6 Comparison of the Different Models with Jump Processes . 93 III Combined Models and Outlook 95 8 Comparison of the Different Basic Models 96 8.1 Self-Similar Volatility Smiles . 96 8.2 Conclusions from the Different Basic Models . 101 iv 9 Combined Models 104 9.1 Stochastic Volatility with Jump Processes and CEV . 105 9.2 Stochastic Volatility with DD . 110 10 Summary 121 Appendix I A Mathematical Methods II A.1 Determining the Implied Distribution from Market Prices . II A.2 Numerical Integration with Adaptive Step Size . IV A.3 Deriving a Closed-Form Solution to Riccati Equations with Piece- Wise Constant Coefficients . V A.4 Deriving the Partial Differential Equation for Heston’s Stochastic Volatility Model . VII A.5 Drawing the Random Jump Size for Glasserman, Merener (2001) . X A.6 Parameters for Jarrow, Li, Zhao (2002) . XI B Additional Figures XIII C Calibration Tables for SV and DD XX Bibliography XXIV Index XXXI List of Figures 3.1 Caplet Volatility Surface for e and US-$ . 29 3.2 Caplet Volatility Smiles for Different Expiries (e)......... 30 3.3 Swaption Volatility Smiles for Different Tenors (e)........ 31 3.4 Comparison of the Forward Rate Distribution between Market Data and a Flat Volatility Smile . 32 3.5 Model Overview . 36 4.1 Displaced Diffusion - Possible Volatility Smiles . 41 4.2 Market and Displaced Diffusion Implied Volatilities (e)...... 42 4.3 Constant Elasticity of Variance - Possible Volatility Smiles . 46 4.4 Market and Constant Elasticity of Variance Implied Volatilities (e) 47 4.5 Equivalence of DD and CEV . 48 4.6 Market and Mixture of Lognormals Implied Volatilities (e).... 52 4.7 Market and Extended Mixture of Lognormals Implied Volatilities (e).................................. 54 6.1 Market and Andersen/Andreasen’s Stochastic Volatility Model Implied Volatilities (e)....................... 64 6.2 Market and Wu/Zhang’s Stochastic Volatility Model Implied Volatilities (e)............................ 71 v vi 7.1 Market and Glasserman/Kou’s Jump Process Implied Volatilities (e).................................. 78 7.2 Comparison between Different Distributions for the Jump Size . 83 7.3 Market and Kou’s Jump Process Implied Volatilities (e)..... 84 7.4 Market and Glasserman/Merener’s Jump Process Implied Volatili- ties (e)............................... 88 7.5 Market and Glasserman/Merener’s Restricted Jump Process Im- plied Volatilities (e)........................ 93 8.1 Basic Models Calibrated to Sample Market Data . 98 8.2 Future Volatility Smiles Implied by the Local Volatility Model . 99 8.3 Future Volatility Smiles Implied by the Uncertain Volatility Model 100 8.4 Future Volatility Smiles Implied by the Stochastic Volatility Model 101 8.5 Future Volatility Smile Implied by the Jump Model . 102 9.1 Comparison between Different Expansions around the Volatility of Variance . 108 9.2 Market and Jarrow/Li/Zhao’s Combined Model Implied Volatili- ties (e)............................... 109 9.3 Market and SV & DD Model Implied Caplet Volatilities (e)... 112 9.4 Market and SV & DD Model Implied Swaption Volatilities (e).. 113 9.5 The Dependencies between the Swaption Implied Volatilities and the Parameters for the SV & DD Model . 117 9.6 Comparison between Market and Model Implied Volatilities in the SV & DD Model for a Caplet . 120 9.7 Comparison between Market and Model Implied Volatilities in the SV & DD Model for a Swaption . 120 vii B.1 Caplet Volatility Smiles for Different Expiries (US-$) . XIII B.2 Swaption Volatility Smiles for Different Tenors (US-$) . XIV B.3 Market and Displaced Diffusion Implied Volatilities (US-$) . XIV B.4 Market and Constant Elasticity of Variance Implied Volatilities (US-$) . XV B.5 Market and Mixture of Lognormals Implied Volatilities (US-$) . XV B.6 Market and Extended Mixture of Lognormals Implied Volatilities (US-$) . XVI B.7 Market and Andersen/Andreasen’s Stochastic Volatility Model Implied Volatilities (US-$) . XVI B.8 Market and Wu/Zhang’s Stochastic Volatility Model Implied Volatilities (US-$) . XVII B.9 Market and Glasserman/Kou’s Jump Process Implied Volatilities (US-$) . XVII B.10 Market and Kou’s Jump Process Implied Volatilities (US-$) . XVIII B.11 Market and SV & DD Model Implied Caplet Volatilities (US-$) . XVIII B.12 Market and SV & DD Model Implied Swaption Volatilities (US-$) XIX List of Tables 8.1 Comparison of the Basic Models and Their Characteristics . 103 C.1 Volatility and Skew Parameters for SV & DD . XX C.2 Bootstrapped Forward Rate Volatilities for SV & DD . XXI C.3 Optimized Forward Rate Volatilities for SV & DD . XXI C.4 Optimized Forward Rate Skews for SV & DD . XXII C.5 Differences between Volatility and Skew Parameters for the Swap- tion Smile . XXIII viii Abbreviations and Notation A(t) the asset or stock price at time t a the mean of the logarithm of the jump size (ln[J]) b the n × m matrix of the coefficients bik bik(t) the percentage volatility of the forward rate Li(t) coming from factor k as part of the total volatility ln[L/K]+ 1 v2 ln[L/K]− 1 v2 Φ 2 Φ 2 Bl(K,L,v) = L v − K v J the jump size of the forward rate K the strike of an option L(t,T,T + δ) forward rate at time t for the expiry(T)-maturity(T + δ) pair Li(t) L(t,Ti,Ti+1) LN (a,b) the lognormal distribution with parameters a and b for the underlying normal distribution N(a,b) h i ln K M the standardized moneyness of an option M = √Li(t) σi Ti−t m the expected percentage change of the forward rate because of one jump N (a,b) the Gaussian normal distribution with mean a and variance b NT the number of jump events up to time T NP the notional of an option O(xk) the residual error is smaller in absolute value than some constant times xk if x is close enough to 0 pi, j the probability of scenario j for forward rate Li(t) P(t,T + δ) the price of a discount bond at time t with maturity T + δ Payoff(Product)t the (expected) payoff of a derivative at time t ix ABBREVIATIONS AND NOTATION x Sr,s(t) the equilibrium swap rate at time t for a swap with the first reset date in Tr and the last payment in Ts s the volatility of the logarithm of the jump size (ln[J]) Xi the variable in the displaced diffusion approach that is lognormally distributed V the variance of the forward rate w the Brownian motion of the variance in the stochastic volatility models z(k) the Brownian motion of factor k zi the Brownian motion of the forward rate Li(t) with Pm dzi = k=1 bik(t)dz(k) zr,s the Brownian motion of the swap rate Sr,s(t) with σ Pm r,s,k(t) dzr,s = dz k=1 σr,s(t) (k) α the offset of the forward rate in the displaced diffusion approach β the skew parameter in the stochastic volatility models γ the parameter for the forward rate in the constant elasticity of variance approach δ the time distance between Ti and Ti+1 and therefore the tenor of a forward rate ε the volatility of variance κ the reversion speed to the reversion level of the variance λ the Poisson arrival rate of a jump process µi(t) the drift of the forward rate Li(t) ρ the correlation matrix of the forward rates ρi, j(t) the correlation between the forward rates Li(t) and L j(t) ρi,V (t) the correlation between the forward rate Li(t) and the variance V(t) ρ∞ the parameter determining the correlation between the first and the last forward rate σi(t) the volatility of the logarithm of the forward rate Li(t) σik(t) the volatility of the logarithm of the forward rate Li(t) coming from factor k σi(t;Li(t)) the local volatility of the forward rate Li(t) σr,s(t) the volatility of the logarithm of the swap rate Sr,s(t) ABBREVIATIONS AND NOTATION xi σr,s,k(t) the volatility of the logarithm
Load more

Recommended publications
  • Numerical Solution of Jump-Diffusion LIBOR Market Models
    Finance Stochast. 7, 1–27 (2003) c Springer-Verlag 2003 Numerical solution of jump-diffusion LIBOR market models Paul Glasserman1, Nicolas Merener2 1 403 Uris Hall, Graduate School of Business, Columbia University, New York, NY 10027, USA (e-mail: [email protected]) 2 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA (e-mail: [email protected]) Abstract. This paper develops, analyzes, and tests computational procedures for the numerical solution of LIBOR market models with jumps. We consider, in par- ticular, a class of models in which jumps are driven by marked point processes with intensities that depend on the LIBOR rates themselves. While this formulation of- fers some attractive modeling features, it presents a challenge for computational work. As a first step, we therefore show how to reformulate a term structure model driven by marked point processes with suitably bounded state-dependent intensities into one driven by a Poisson random measure. This facilitates the development of discretization schemes because the Poisson random measure can be simulated with- out discretization error. Jumps in LIBOR rates are then thinned from the Poisson random measure using state-dependent thinning probabilities. Because of discon- tinuities inherent to the thinning process, this procedure falls outside the scope of existing convergence results; we provide some theoretical support for our method through a result establishing first and second order convergence of schemes that accommodates thinning but imposes stronger conditions on other problem data. The bias and computational efficiency of various schemes are compared through numerical experiments. Key words: Interest rate models, Monte Carlo simulation, market models, marked point processes JEL Classification: G13, E43 Mathematics Subject Classification (1991): 60G55, 60J75, 65C05, 90A09 The authors thank Professor Steve Kou for helpful comments and discussions.
    [Show full text]
  • 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.
    [Show full text]
  • Sensitivity Estimation of Sabr Model Via Derivative of Random Variables
    Proceedings of the 2011 Winter Simulation Conference S. Jain, R. R. Creasey, J. Himmelspach, K. P. White, and M. Fu, eds. SENSITIVITY ESTIMATION OF SABR MODEL VIA DERIVATIVE OF RANDOM VARIABLES NAN CHEN YANCHU LIU The Chinese University of Hong Kong 709A William Mong Engineering Building Shatin, N. T., HONG KONG ABSTRACT We derive Monte Carlo simulation estimators to compute option price sensitivities under the SABR stochastic volatility model. As a companion to the exact simulation method developed by Cai, Chen and Song (2011), this paper uses the sensitivity of “vol of vol” as a showcase to demonstrate how to use the pathwise method to obtain unbiased estimators to the price sensitivities under SABR. By appropriately conditioning on the path generated by the volatility, the evolution of the forward price can be represented as noncentral chi-square random variables with stochastic parameters. Combined with the technique of derivative of random variables, we can obtain fast and accurate unbiased estimators for the sensitivities. 1 INTRODUCTION The ubiquitous existence of the volatility smile and skew poses a great challenge to the practice of risk management in fixed and foreign exchange trading desks. In foreign currency option markets, the implied volatility is often relatively lower for at-the-money options and becomes gradually higher as the strike price moves either into the money or out of the money. Traders refer to this stylized pattern as the volatility smile. In the equity and interest rate markets, a typical aspect of the implied volatility, which is also known as the volatility skew, is that it decreases as the strike price increases.
    [Show full text]
  • Option Pricing with Jump Diffusion Models
    UNIVERSITY OF PIRAEUS DEPARTMENT OF BANKING AND FINANCIAL MANAGEMENT M. Sc in FINANCIAL ANALYSIS FOR EXECUTIVES Option pricing with jump diffusion models MASTER DISSERTATION BY: SIDERI KALLIOPI: MXAN 1134 SUPERVISOR: SKIADOPOULOS GEORGE EXAMINATION COMMITTEE: CHRISTINA CHRISTOU PITTIS NIKITAS PIRAEUS, FEBRUARY 2013 2 TABLE OF CONTENTS ς ABSTRACT .............................................................................................................................. 3 SECTION 1 INTRODUCTION ................................................................................................ 4 ώ SECTION 2 LITERATURE REVIEW: .................................................................................... 9 ι SECTION 3 DATASET .......................................................................................................... 14 SECTION 4 .............................................................................................................................α 16 4.1 PRICING AND HEDGING IN INCOMPLETE MARKETS ....................................... 16 4.2 CHANGE OF MEASURE ............................................................................................ρ 17 SECTION 5 MERTON’S MODEL .........................................................................................ι 21 5.1 THE FORMULA ........................................................................................................... 21 5.2 PDE APPROACH BY HEDGING ................................................................ε ............... 24 5.3 EFFECT
    [Show full text]
  • CURRICULUM VITAE Anatoliy SWISHCHUK Department Of
    CURRICULUM VITAE Anatoliy SWISHCHUK Department of Mathematics & Statistics, University of Calgary 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 Office: MS552 E-mails: [email protected] Tel: +1 (403) 220-3274 (office) home page: http://www.math.ucalgary.ca/~aswish/ Education: • Doctor of Phys. & Math. Sci. (1992, Doctorate, Institute of Mathematics, National Academy of Sciences of Ukraine (NASU), Kiev, Ukraine) • Ph.D. (1984, Institute of Mathematics, NASU, Kiev, Ukraine) • M.Sc., B.Sc. (1974-1979, Kiev State University, Faculty of Mathematics & Mechanics, Probability Theory & Mathematical Statistics Department, Kiev, Ukraine) Work Experience: • Full Professor, Department of Mathematics and Statistics, University of Calgary, Calgary, Canada (April 2012-present) • Co-Director, Mathematical and Computational Finance Laboratory, Department of Math- ematics and Statistics, University of Calgary, Calgary, Canada (October 2004-present) • Associate Professor, Department of Mathematics and Statistics, University of Calgary, Calgary, Canada (July 2006-March 2012) • Assistant Professor, Department of Mathematics and Statistics, University of Calgary, Calgary, Canada (August 2004-June 2006) • Course Director, Department of Mathematics & Statistics, York University, Toronto, ON, Canada (January 2003-June 2004) • Research Associate, Laboratory for Industrial & Applied Mathematics, Department of Mathematics & Statistics, York University, Toronto, ON, Canada (November 1, 2001-July 2004) • Professor, Probability Theory & Mathematical Statistics
    [Show full text]
  • 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.
    [Show full text]
  • Lévy Finance *[0.5Cm] Models and Results
    Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Stochastic Calculus for L´evy Processes L´evy-Process Driven Financial Market Models Jump-Diffusion Models Merton-Model Kou-Model General L´evy Models Variance-Gamma model CGMY model GH models Variance-mean mixtures European Style Options Equivalent Martingale Measure Jump-Diffusion Models Variance-Gamma Model NIG Model Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Stochastic Integral for L´evyProcesses Let (Xt ) be a L´evy process with L´evy-Khintchine triplet (α, σ, ν(dx)). By the L´evy-It´odecomposition we know X = X (1) + X (2) + X (3), where the X (i) are independent L´evyprocesses. X (1) is a Brownian motion with drift, X (2) is a compound Poisson process with jump (3) distributed concentrated on R/(−1, 1) and X is a square-integrable martingale (which can be viewed as a limit of compensated compound Poisson processes with small jumps). We know how to define the stochastic integral with respect to any of these processes! Professor Dr. R¨udigerKiesel L´evyFinance Stochastic Calculus for L´evyProcesses L´evy-Process Driven Financial Market Models Jump-Diffusion Models General L´evyModels European Style Options Canonical Decomposition From the L´evy-It´odecomposition we deduce the canonical decomposition (useful for applying the general semi-martingale theory) Z t Z X (t) = αt + σW (t) + x µX − νX (ds, dx), 0 R where Z t Z X xµX (ds, dx) = ∆X (s) 0 R 0<s≤t and Z t Z Z t Z Z X X E xµ (ds, dx) = xν (ds, dx) = t xν(dx).
    [Show full text]
  • Pricing Bermudan Swaptions on the LIBOR Market Model Using the Stochastic Grid Bundling Method
    Pricing Bermudan Swaptions on the LIBOR Market Model using the Stochastic Grid Bundling Method Stef Maree,∗ Jacques du Toity Abstract We examine using the Stochastic Grid Bundling Method (SGBM) to price a Bermu- dan swaption driven by a one-factor LIBOR Market Model (LMM). Using a well- known approximation formula from the finance literature, we implement SGBM with one basis function and show that it is around six times faster than the equivalent Longstaff–Schwartz method. The two methods agree in price to one basis point, and the SGBM path estimator gives better (higher) prices than the Longstaff–Schwartz prices. A closer examination shows that inaccuracies in the approximation formula introduce a small bias into the SGBM direct estimator. 1 Introduction The LIBOR Market Model (LMM) is an interest rate market model. It owes much of its popularity to the fact that it is consistent with Black’s pricing formulas for simple interest rate derivatives such as caps and swaptions, see, for example, [3]. Under LMM all forward rates are modelled as individual processes, which typically results in a high dimensional model. Consequently when pricing more complex instruments under LMM, Monte Carlo simulation is usually used. When pricing products with early-exercise features, such as Bermudan swaptions, a method is required to determine the early-exercise policy. The most popular approach to this is the Longstaff–Schwartz Method (LSM) [8]. When time is discrete (as is usually the case in numerical schemes) Bellman’s backward induction principle says that the price of the option at any exercise time is the maximum of the spot payoff and the so-called contin- uation value.
    [Show full text]
  • Approximations to the Lévy LIBOR Model By
    Approximations to the Lévy LIBOR Model by Hassana Al-Hassan ([email protected]) Supervised by: Dr. Sure Mataramvura Co-Supervised by: Emeritus Professor Ronnie Becker Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town, South Africa A thesis submitted for the degree of Master of Science University of Cape Town May 27, 2014 The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or non- commercial research purposes only. Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University of Cape Town Plagiarism Declaration “I, Hassana AL-Hassan, know the meaning of plagiarism and declare that all of the work in the thesis, save for that which is properly acknowledged, is my own”. SIGNED Hassana Al-Hassan, May 27, 2014 i Declaration of Authorship I, Hassana AL-Hassan, declare that this thesis titled, “The LIBOR Market Model Versus the Levy LIBOR Market Model” and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other quali- fication at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed.
    [Show full text]
  • Interest Rate Models: Paradigm Shifts in Recent Years
    Interest Rate Models: Paradigm shifts in recent years Damiano Brigo Q-SCI, Managing Director and Global Head DerivativeFitch, 101 Finsbury Pavement, London Columbia University Seminar, New York, November 5, 2007 This presentation is based on the book "Interest Rate Models: Theory and Practice - with Smile, Inflation and Credit" by D. Brigo and F. Mercurio, Springer-Verlag, 2001 (2nd ed. 2006) http://www.damianobrigo.it/book.html Damiano Brigo, Q-SCI, DerivativeFitch, London Columbia University Seminar, November 5, 2007 Overview ² No arbitrage and derivatives pricing. ² Modeling suggested by no-arbitrage discounting. 1977: Endogenous short-rate term structure models ² Reproducing the initial market interest-rate curve exactly. 1990: Exogenous short rate models ² A general framework for no-arbitrage rates dynamics. 1990: HJM - modeling instantaneous forward rates ² Moving closer to the market and consistency with market formulas 1997: Fwd market-rates models calibration and diagnostics power ² 2002: Volatility smile extensions of Forward market-rates models Interest rate models: Paradigms shifts in recent years 1 Damiano Brigo, Q-SCI, DerivativeFitch, London Columbia University Seminar, November 5, 2007 No arbitrage and Risk neutral valuation Recall shortly the risk-neutral valuation paradigm of Harrison and Pliska's (1983), characterizing no-arbitrage theory: A future stochastic payo®, built on an underlying fundamental ¯nancial asset, paid at a future time T and satisfying some technical conditions, has as unique price at current time t
    [Show full text]
  • Essays in Quantitative Finance Karlsson, Patrik
    Essays in Quantitative Finance Karlsson, Patrik 2016 Document Version: Publisher's PDF, also known as Version of record Link to publication Citation for published version (APA): Karlsson, P. (2016). Essays in Quantitative Finance. Total number of authors: 1 General rights Unless other specific re-use rights are stated the following general rights apply: Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Read more about Creative commons licenses: https://creativecommons.org/licenses/ Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. LUND UNIVERSITY PO Box 117 221 00 Lund +46 46-222 00 00 Patrik Karlsson Essays in Quantitative Finance Karlsson Essays in Quantitative Patrik Essays in Quantitative Finance Patrik Karlsson 9 789177 Lund Lund University Economic 530602 Department of Economics ISBN 978-91-7753-060-2 Studies ISSN 0460-0029 199 Number 199 Essays in Quantitative Finance Patrik Karlsson DOCTORAL DISSERTATION by due permission of the School of Economics and Management, Lund University, Sweden.
    [Show full text]
  • Estimating Single Factor Jump Diffusion Interest Rate Models
    ESTIMATING SINGLE FACTOR JUMP DIFFUSION INTEREST RATE MODELS Ghulam Sorwar1 University of Nottingham Nottingham University Business School Jubilee Campus Wallaton Road, Nottingham NG8 1BB Email: [email protected] Tele: +44(0)115 9515487 ABSTRACT Recent empirical studies have demonstrated that behaviour of interest rate processes can be better explained if standard diffusion processes are augmented with jumps in the interest rate process. In this paper we examine the performance of both linear and non- linear one factor CKLS model in the presence of jumps. We conclude that empirical features of interest rates not captured by standard diffusion processes are captured by models with jumps and that the linear CKLS model provides sufficient explanation of the data. Keywords: term structure, jumps, Bayesian, MCMC JEL: C11, C13, C15, C32 1 Corresponding author. Email [email protected] 1. INTRODUCTION Accurate valuation of fixed income derivative securities is dependent on the correct specification of the underlying interest rate process driving it. This underlying interest rate is modelled as stochastic process comprising of two components. The first component is the drift associated with the interest rates. This drift incorporates the mean reversion that is observed in interest rates. The second component consists of a gaussian term. To incorporate heteroskedasticity observed in interest rates, the gaussian term is multiplied with the short term interest rate raised to a particular power. Different values of this power leads to different interest rate models. Setting this power to zero yields the Vasicek model (1977), setting this power to half yields the Cox-Ingersoll-Ross model (1985b), setting this power to one and half yields the model proposed by Ahn and Gao (1999).
    [Show full text]