Linear-Rational Term-Structure Models
|
|
- Rosanna Dennis
- 5 years ago
- Views:
Transcription
1 Linear-Rational Term-Structure Models Anders Trolle (joint with Damir Filipović and Martin Larsson) Ecole Polytechnique Fédérale de Lausanne Swiss Finance Institute AMaMeF and Swissquote Conference, September 9, 215
2 Near-zero short-term interest rates 2/23
3 Contribution Existing models that respect zero lower bound (ZLB) on interest rates face limitations: Shadow-rate models do not capture volatility dynamics Multi-factor CIR and quadratic models do not easily accommodate unspanned factors and swaption pricing We develop a new class of linear-rational term structure models Respects ZLB on interest rates Easily accommodates unspanned factors affecting volatility and risk premia Admits semi-analytical solutions to swaptions Extensive empirical analysis Parsimonious model specification has very good fit to interest rate swaps and swaptions since 1997 Captures many features of term structure, volatility, and risk premia dynamics. 3/23
4 Contribution Existing models that respect zero lower bound (ZLB) on interest rates face limitations: Shadow-rate models do not capture volatility dynamics Multi-factor CIR and quadratic models do not easily accommodate unspanned factors and swaption pricing We develop a new class of linear-rational term structure models Respects ZLB on interest rates Easily accommodates unspanned factors affecting volatility and risk premia Admits semi-analytical solutions to swaptions Extensive empirical analysis Parsimonious model specification has very good fit to interest rate swaps and swaptions since 1997 Captures many features of term structure, volatility, and risk premia dynamics. 3/23
5 Contribution Existing models that respect zero lower bound (ZLB) on interest rates face limitations: Shadow-rate models do not capture volatility dynamics Multi-factor CIR and quadratic models do not easily accommodate unspanned factors and swaption pricing We develop a new class of linear-rational term structure models Respects ZLB on interest rates Easily accommodates unspanned factors affecting volatility and risk premia Admits semi-analytical solutions to swaptions Extensive empirical analysis Parsimonious model specification has very good fit to interest rate swaps and swaptions since 1997 Captures many features of term structure, volatility, and risk premia dynamics. 3/23
6 Outline The linear-rational framework The Linear-Rational Square-Root (LRSQ) model Empirical analysis 4/23
7 Outline The linear-rational framework The Linear-Rational Square-Root (LRSQ) model Empirical analysis The linear-rational framework 5/23
8 Linear-rational framework and bond pricing State-price density, ζ t Π(t, T ) = 1 ζ t E t [ζ T C T ] m-dimensional factor process, Z t, with linear drift given by dz t = κ(θ Z t )dt + dm t, for some κ R m m, θ R m, and some martingale M t ζ t given by ζ t = e αt ( φ + ψ Z t ), for some φ R and ψ R m such that φ + ψ z > for all z E, and some α R Conditional expectations: Price of zero-coupon bond: E t [Z T ] = θ + e κ(t t) (Z t θ) P(t, t + τ) = (φ + ψ θ)e ατ + ψ e (α+κ)τ (Z t θ) The linear-rational framework φ + ψ Z 6/23 t
9 Linear-rational framework and bond pricing State-price density, ζ t Π(t, T ) = 1 ζ t E t [ζ T C T ] m-dimensional factor process, Z t, with linear drift given by dz t = κ(θ Z t )dt + dm t, for some κ R m m, θ R m, and some martingale M t ζ t given by ζ t = e αt ( φ + ψ Z t ), for some φ R and ψ R m such that φ + ψ z > for all z E, and some α R Conditional expectations: Price of zero-coupon bond: E t [Z T ] = θ + e κ(t t) (Z t θ) P(t, t + τ) = (φ + ψ θ)e ατ + ψ e (α+κ)τ (Z t θ) The linear-rational framework φ + ψ Z 6/23 t
10 Linear-rational framework and bond pricing State-price density, ζ t Π(t, T ) = 1 ζ t E t [ζ T C T ] m-dimensional factor process, Z t, with linear drift given by dz t = κ(θ Z t )dt + dm t, for some κ R m m, θ R m, and some martingale M t ζ t given by ζ t = e αt ( φ + ψ Z t ), for some φ R and ψ R m such that φ + ψ z > for all z E, and some α R Conditional expectations: Price of zero-coupon bond: E t [Z T ] = θ + e κ(t t) (Z t θ) P(t, t + τ) = (φ + ψ θ)e ατ + ψ e (α+κ)τ (Z t θ) The linear-rational framework φ + ψ Z 6/23 t
11 Interest rates and the zero lower bound Short rate: r t = T log P(t, T ) T =t = α ψ κ(θ Z t ) φ + ψ Z t Define α = sup z ψ κ(θ z) φ + ψ z and α = inf z ψ κ(θ z) φ + ψ z Set α = α so that r t [, α α ] α and α are finite if z R d + and all components of ψ are strictly positive Range is parameter dependent, verify that range is wide enough If eigenvalues of κ have nonnegative real part then α is the infinite-maturity ZCB yield The linear-rational framework 7/23
12 Interest rates and the zero lower bound Short rate: r t = T log P(t, T ) T =t = α ψ κ(θ Z t ) φ + ψ Z t Define α = sup z ψ κ(θ z) φ + ψ z and α = inf z ψ κ(θ z) φ + ψ z Set α = α so that r t [, α α ] α and α are finite if z R d + and all components of ψ are strictly positive Range is parameter dependent, verify that range is wide enough If eigenvalues of κ have nonnegative real part then α is the infinite-maturity ZCB yield The linear-rational framework 7/23
13 Interest rates and the zero lower bound Short rate: r t = T log P(t, T ) T =t = α ψ κ(θ Z t ) φ + ψ Z t Define α = sup z ψ κ(θ z) φ + ψ z and α = inf z ψ κ(θ z) φ + ψ z Set α = α so that r t [, α α ] α and α are finite if z R d + and all components of ψ are strictly positive Range is parameter dependent, verify that range is wide enough If eigenvalues of κ have nonnegative real part then α is the infinite-maturity ZCB yield The linear-rational framework 7/23
14 Interest rate swaps Exchange a stream of fixed-rate for floating-rate payments Consider a tenor structure T < T 1 < < T n, T i T i 1 At T i, i = 1... n: pay k, for fixed rate k 1 receive floating LIBOR L(T i 1, T i ) = P(T i 1,T i ) 1 Value of payer swap at t T n t = P(t, T ) P(t, T n ) k P(t, T i ) }{{} i=1 floating leg }{{} fixed leg Π swap Forward swap rate S t = P(t,T) P(t,Tn) n i=1 P(t,T i ) The linear-rational framework 8/23
15 Swaptions Payer swaption = option to enter the swap at T paying fixed, receiving floating Payoff at expiry T of the form C T = ( Π swap T ) + = ( n ) + c i P(T, T i ) = 1 p swap (Z T ) + ζ T i= for the explicit linear function p swap (z) = n c i e αt i i= Swaption price at t T is given by ( ) φ + ψ θ + ψ e κ(t i T ) (z θ) Π swaption t = 1 ζ t E[ζ T C T F t ] = 1 ζ t E t [ pswap (Z T ) +] Efficient swaption pricing via Fourier transform...! The linear-rational framework 9/23
16 Swaptions Payer swaption = option to enter the swap at T paying fixed, receiving floating Payoff at expiry T of the form C T = ( Π swap T ) + = ( n ) + c i P(T, T i ) = 1 p swap (Z T ) + ζ T i= for the explicit linear function p swap (z) = n c i e αt i i= Swaption price at t T is given by ( ) φ + ψ θ + ψ e κ(t i T ) (z θ) Π swaption t = 1 ζ t E[ζ T C T F t ] = 1 ζ t E t [ pswap (Z T ) +] Efficient swaption pricing via Fourier transform...! The linear-rational framework 9/23
17 Fourier transform Define q(x) = E t [exp (x p swap (Z T ))] for every x C such that the conditional expectation is well-defined Then Π swaption t = 1 ζ t π for any µ > with q(µ) < Re ] [ q(µ + iλ) (µ + iλ) 2 dλ q(x) has semi-analytical solution in LRSQ model The linear-rational framework 1/23
18 Outline The linear-rational framework The Linear-Rational Square-Root (LRSQ) model Empirical analysis The Linear-Rational Square-Root (LRSQ) model 11/23
19 Linear-Rational Square-Root (LRSQ) model Objective: A model with joint factor process (Z t, U t ), where Zt : m term structure factors U t : n m USV factors Denoted LRSQ(m,n) Based on a (m + n)-dimensional square-root diffusion process X t taking values in R m+n + of the form ) dx t = (b βx t ) dt + Diag (σ 1 X1t,..., σ m+n Xm+n,t db t, Define (Z t, U t ) = SX t as linear transform of X t with state space E = S(R m+n + ) Need to specify a (m + n) (m + n)-matrix S such that the implied term structure state space is E = R m + the drift of Z t does not depend on U t, while U t feeds into the martingale part of Z t The Linear-Rational Square-Root (LRSQ) model 12/23
20 Linear-Rational Square-Root (LRSQ) model (cont.) S given by S = ( ) Idm A Id n with A = ( Idn ). β chosen upper block-triangular of the form ( ) ( ) κ β = S 1 κ κa AA A S = κa κa A κa for some κ R m m b given by b = βs 1 ( θ θ U ) = for some θ R m and θ U R n. ( ) κθ AA κaθ U A κaθ U The Linear-Rational Square-Root (LRSQ) model 13/23
21 Linear-Rational Square-Root (LRSQ) model (cont.) Resulting joint factor process (Z t, U t ): dz t = κ (θ Z t) dt + σ(z t, U t)db t ) du t = A κa (θ U U t) dt + Diag (σ m+1 U1t db m+1,t,..., σ m+n Unt db m+n,t, with dispersion function of Z t given by σ(z, u) = (Id m, A) Diag ( σ 1 z1 u 1,..., σ m+n un ). Example: LRSQ(1,1) dz 1t = κ 11 (θ 1 + θ 2 Z 1t ) dt + σ 1 Z1t U 1t db 1t + σ 2 U1t db 2t du 1t = κ 22 (θ 2 U 1t ) dt + σ 2 U1t db 2t The Linear-Rational Square-Root (LRSQ) model 14/23
22 Linear-rational vs. exponential-affine framework Exponential-affine Linear-rational Short rate affine LR ZCB price exponential-affine LR ZCB yield affine log of LR Coupon bond price sum of exponential-affines LR Swap rate ratio of sums of exponential-affines LR ZLB ( ) USV ( ) Cap/floor valuation semi-analytical semi-analytical Swaption valuation approximate semi-analytical Linear state inversion ZCB yields bond prices or swap rates Table 1: Comparison of exponential-affine and linear-rational frameworks. In the exponential-affine framework, respecting the zero lower bound (ZLB) on interest rates is only possible if all factors are of the square-root type, and accommodating unspanned stochastic volatility (USV) is only possible if at least one factor is conditionally Gaussian. The Linear-Rational Square-Root (LRSQ) model 15/23
23 Outline The linear-rational framework The Linear-Rational Square-Root (LRSQ) model Empirical analysis Empirical analysis 16/23
24 Data and estimation approach Panel data set of swaps and swaptions Swap maturities: 1Y, 2Y, 3Y, 5Y, 7Y, 1Y Swaptions expiries: 3M, 1Y, 2Y, 5Y 866 weekly observations, Jan 29, 1997 Aug 28, 213 Estimation approach: Quasi-maximum likelihood in conjunction with the unscented Kalman Filter.8 Panel A1: Swap data 25 Panel B1: Swaption data Jan97 Jan1 Jan5 Jan9 Jan13 Jan97 Jan1 Jan5 Jan9 Jan13 Empirical analysis 17/23 Panel A2: Swap fit, LRSQ(3,3) Panel B2: Swaption fit, LRSQ(3,3)
25 Model specifications Model specifications (always 3 term structure factors) LRSQ(3,1): volatility of Z 1t containing an unspanned component LRSQ(3,2): volatility of Z1t and Z 2t containing unspanned components LRSQ(3,3): volatility of term structure factors containing unspanned components α = α and range of r t : LRSQ(3,1) LRSQ(3,2) LRSQ(3,3) Long ZCB yield α 7.46% 6.88% 5.66% Upper bound on r t 2% 146% 72% Empirical analysis 18/23
26 Level-dependence in factor volatilities Volatility of Z it with USV: Volatility of Z it without USV: σ i Zit σ 2 i Z it + (σ 2 i+3 σ2 i )U it Vol. of Z1,t LRSQ(3,1) LRSQ(3,2) LRSQ(3,3) Vol. of Z2,t Vol. of Z3,t Empirical analysis /
27 Volatility dynamics near the ZLB Level-dependence in volatility, 3M/1Y swaption IV vs. 1Y swap rate M normal implied volatility, basis points Empirical analysis 1Y swap rate 2/23 Figure 1: Level-dependence in volatility of 1-year swap rate
28 Level-dependence in volatility Regress weekly changes in the 3M swaption IV on weekly changes in the underlying swap rate σ N,t = β + β 1 S t + ɛ t 1 yr 2 yrs 3 yrs 5 yrs 7 yrs 1 yrs Mean Panel A: ˆβ 1 All (2.38) (2.88) (3.31) (4.12) (4.59) (4.97) %-1% (8.3) (8.79) (8.19) (7.83) 1%-2% (2.7) (6.21) (6.77) (5.2) (5.23) (8.24) 2%-3% (3.1) (1.97) (3.77) (5.8) (5.62) (4.93) 3%-4% (.22) (1.21) (.92) (.8) (1.82) (1.96) 4%-5%.4 (.31).7 (.82).1 (.8).8 (1.59).7 (1.76).7.3 (1.65) Panel B: R 2 All %-1% %-2% %-3% %-4% %-5% Empirical analysis 21/23 Table 4: Level-dependence in volatility.
29 Level-dependence in volatility, LRSQ(3,3).8 Panel A: ˆβ1 in data.8 Panel B: Model-implied ˆβ All %-1% 1%-2% 2%-3% 3%-4% 4%-5% All %-1% All %-1% 1%-2% 2%-3% 3%-4% 4%-5% All %-1% 1%-2% 2%-3% 3%-4% 4%-5% 1%-2% 2%-3% 3%-4% 4%-5% Panel C: R 2 in data Panel D: Model-implied R Empirical analysis 22/23
30 Conclusion Key features of framework: Respects ZLB on interest rates Easily accommodates unspanned factors affecting volatility and risk premia Admits semi-analytical solutions to swaptions Extensive empirical analysis: Parsimonious model specification has very good fit to interest rate swaps and swaptions since 1997 Captures many features of term structure, volatility, and risk premia dynamics. Conclusion 23/23
Polynomial Models in Finance
Polynomial Models in Finance Martin Larsson Department of Mathematics, ETH Zürich based on joint work with Damir Filipović, Anders Trolle, Tony Ware Risk Day Zurich, 11 September 2015 Flexibility Tractability
More information7 th General AMaMeF and Swissquote Conference 2015
Linear Credit Damien Ackerer Damir Filipović Swiss Finance Institute École Polytechnique Fédérale de Lausanne 7 th General AMaMeF and Swissquote Conference 2015 Overview 1 2 3 4 5 Credit Risk(s) Default
More informationLinear-Rational Term Structure Models
THE JOURNAL OF FINANCE VOL. LXXII, NO. 2 APRIL 217 Linear-Rational Term Structure Models DAMIR FILIPOVIĆ, MARTIN LARSSON, and ANDERS B. TROLLE ABSTRACT We introduce the class of linear-rational term structure
More informationA Term-Structure Model for Dividends and Interest Rates
A Term-Structure Model for Dividends and Interest Rates Sander Willems Joint work with Damir Filipović School and Workshop on Dynamical Models in Finance May 24th 2017 Sander Willems (SFI@EPFL) A TSM for
More informationLinearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing
Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing Liuren Wu, Baruch College Joint work with Peter Carr and Xavier Gabaix at New York University Board of
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Vienna, 16 November 2007 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November
More informationLinear-Rational Term Structure Models
Linear-Rational Term Structure Models Damir Filipović Martin Larsson Anders Trolle EPFL and Swiss Finance Institute September 4, 214 Abstract We introduce the class of linear-rational term structure models,
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationModeling Yields at the Zero Lower Bound: Are Shadow Rates the Solution?
Modeling Yields at the Zero Lower Bound: Are Shadow Rates the Solution? Jens H. E. Christensen & Glenn D. Rudebusch Federal Reserve Bank of San Francisco Term Structure Modeling and the Lower Bound Problem
More informationmodels Dipartimento di studi per l economia e l impresa University of Piemonte Orientale Faculty of Finance Cass Business School
A.M. Gambaro R. G. Fusai Dipartimento di studi per l economia e l impresa University of Piemonte Orientale Faculty of Finance Cass Business School Dipartimento di Statistica e Metodi Quantitativi University
More informationLIBOR models, multi-curve extensions, and the pricing of callable structured derivatives
Weierstrass Institute for Applied Analysis and Stochastics LIBOR models, multi-curve extensions, and the pricing of callable structured derivatives John Schoenmakers 9th Summer School in Mathematical Finance
More informationHedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework
Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework Kathrin Glau, Nele Vandaele, Michèle Vanmaele Bachelier Finance Society World Congress 2010 June 22-26, 2010 Nele Vandaele Hedging of
More informationValuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model
Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 1(23) Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility
More informationOption Pricing Under a Stressed-Beta Model
Option Pricing Under a Stressed-Beta Model Adam Tashman in collaboration with Jean-Pierre Fouque University of California, Santa Barbara Department of Statistics and Applied Probability Center for Research
More informationDecomposing swap spreads
Decomposing swap spreads Peter Feldhütter Copenhagen Business School David Lando Copenhagen Business School (visiting Princeton University) Stanford, Financial Mathematics Seminar March 3, 2006 1 Recall
More informationThe Lognormal Interest Rate Model and Eurodollar Futures
GLOBAL RESEARCH ANALYTICS The Lognormal Interest Rate Model and Eurodollar Futures CITICORP SECURITIES,INC. 399 Park Avenue New York, NY 143 Keith Weintraub Director, Analytics 1-559-97 Michael Hogan Ex
More informationPolicy iterated lower bounds and linear MC upper bounds for Bermudan style derivatives
Finance Winterschool 2007, Lunteren NL Policy iterated lower bounds and linear MC upper bounds for Bermudan style derivatives Pricing complex structured products Mohrenstr 39 10117 Berlin schoenma@wias-berlin.de
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Haindorf, 7 Februar 2008 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar
More informationInterest rate models and Solvency II
www.nr.no Outline Desired properties of interest rate models in a Solvency II setting. A review of three well-known interest rate models A real example from a Norwegian insurance company 2 Interest rate
More informationModern Methods of Option Pricing
Modern Methods of Option Pricing Denis Belomestny Weierstraß Institute Berlin Motzen, 14 June 2007 Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 1 / 30 Overview 1 Introduction
More informationA Multifrequency Theory of the Interest Rate Term Structure
A Multifrequency Theory of the Interest Rate Term Structure Laurent Calvet, Adlai Fisher, and Liuren Wu HEC, UBC, & Baruch College Chicago University February 26, 2010 Liuren Wu (Baruch) Cascade Dynamics
More informationShort-time asymptotics for ATM option prices under tempered stable processes
Short-time asymptotics for ATM option prices under tempered stable processes José E. Figueroa-López 1 1 Department of Statistics Purdue University Probability Seminar Purdue University Oct. 30, 2012 Joint
More informationTerm Structure Models with Negative Interest Rates
Term Structure Models with Negative Interest Rates Yoichi Ueno Bank of Japan Summer Workshop on Economic Theory August 6, 2016 NOTE: Views expressed in this paper are those of author and do not necessarily
More informationEstimation of dynamic term structure models
Estimation of dynamic term structure models Greg Duffee Haas School of Business, UC-Berkeley Joint with Richard Stanton, Haas School Presentation at IMA Workshop, May 2004 (full paper at http://faculty.haas.berkeley.edu/duffee)
More informationForecasting Interest Rates and Exchange Rates under Multi-Currency Quadratic Models
Forecasting Interest Rates and Exchange Rates under Multi-Currency Quadratic Models Markus Leippold Swiss Banking Institute, University of Zurich Liuren Wu Graduate School of Business, Fordham University
More informationParameters Estimation in Stochastic Process Model
Parameters Estimation in Stochastic Process Model A Quasi-Likelihood Approach Ziliang Li University of Maryland, College Park GEE RIT, Spring 28 Outline 1 Model Review The Big Model in Mind: Signal + Noise
More informationUnified Credit-Equity Modeling
Unified Credit-Equity Modeling Rafael Mendoza-Arriaga Based on joint research with: Vadim Linetsky and Peter Carr The University of Texas at Austin McCombs School of Business (IROM) Recent Advancements
More informationEuropean option pricing under parameter uncertainty
European option pricing under parameter uncertainty Martin Jönsson (joint work with Samuel Cohen) University of Oxford Workshop on BSDEs, SPDEs and their Applications July 4, 2017 Introduction 2/29 Introduction
More informationParametric Inference and Dynamic State Recovery from Option Panels. Torben G. Andersen
Parametric Inference and Dynamic State Recovery from Option Panels Torben G. Andersen Joint work with Nicola Fusari and Viktor Todorov The Third International Conference High-Frequency Data Analysis in
More informationNonlinear Filtering in Models for Interest-Rate and Credit Risk
Nonlinear Filtering in Models for Interest-Rate and Credit Risk Rüdiger Frey 1 and Wolfgang Runggaldier 2 June 23, 29 3 Abstract We consider filtering problems that arise in Markovian factor models for
More informationGeneralized Affine Transform Formulae and Exact Simulation of the WMSV Model
On of Affine Processes on S + d Generalized Affine and Exact Simulation of the WMSV Model Department of Mathematical Science, KAIST, Republic of Korea 2012 SIAM Financial Math and Engineering joint work
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationDYNAMIC CDO TERM STRUCTURE MODELLING
DYNAMIC CDO TERM STRUCTURE MODELLING Damir Filipović (joint with Ludger Overbeck and Thorsten Schmidt) Vienna Institute of Finance www.vif.ac.at PRisMa 2008 Workshop on Portfolio Risk Management TU Vienna,
More informationModel Estimation. Liuren Wu. Fall, Zicklin School of Business, Baruch College. Liuren Wu Model Estimation Option Pricing, Fall, / 16
Model Estimation Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 Liuren Wu Model Estimation Option Pricing, Fall, 2007 1 / 16 Outline 1 Statistical dynamics 2 Risk-neutral dynamics 3 Joint
More informationA Macro-Finance Model of the Term Structure: the Case for a Quadratic Yield Model
Title page Outline A Macro-Finance Model of the Term Structure: the Case for a 21, June Czech National Bank Structure of the presentation Title page Outline Structure of the presentation: Model Formulation
More informationA new approach to LIBOR modeling
A new approach to LIBOR modeling Antonis Papapantoleon FAM TU Vienna Based on joint work with Martin Keller-Ressel and Josef Teichmann Istanbul Workshop on Mathematical Finance Istanbul, Turkey, 18 May
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationInterest Rate Bermudan Swaption Valuation and Risk
Interest Rate Bermudan Swaption Valuation and Risk Dmitry Popov FinPricing http://www.finpricing.com Summary Bermudan Swaption Definition Bermudan Swaption Payoffs Valuation Model Selection Criteria LGM
More informationMarket Price of Longevity Risk for A Multi-Cohort Mortality Model with Application to Longevity Bond Option Pricing
1/51 Market Price of Longevity Risk for A Multi-Cohort Mortality Model with Application to Longevity Bond Option Pricing Yajing Xu, Michael Sherris and Jonathan Ziveyi School of Risk & Actuarial Studies,
More informationMacroeconomic Announcements and Investor Beliefs at The Zero Lower Bound
Macroeconomic Announcements and Investor Beliefs at The Zero Lower Bound Ben Carlston Marcelo Ochoa [Preliminary and Incomplete] Abstract This paper examines empirically the effect of the zero lower bound
More informationNo arbitrage conditions in HJM multiple curve term structure models
No arbitrage conditions in HJM multiple curve term structure models Zorana Grbac LPMA, Université Paris Diderot Joint work with W. Runggaldier 7th General AMaMeF and Swissquote Conference Lausanne, 7-10
More informationMulti-dimensional Term Structure Models
Multi-dimensional Term Structure Models We will focus on the affine class. But first some motivation. A generic one-dimensional model for zero-coupon yields, y(t; τ), looks like this dy(t; τ) =... dt +
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationAnalytical Option Pricing under an Asymmetrically Displaced Double Gamma Jump-Diffusion Model
Analytical Option Pricing under an Asymmetrically Displaced Double Gamma Jump-Diffusion Model Advances in Computational Economics and Finance Univerity of Zürich, Switzerland Matthias Thul 1 Ally Quan
More informationarxiv: v1 [q-fin.mf] 6 Mar 2018
A Term Structure Model for Dividends and Interest Rates Damir Filipović Sander Willems March 7, 218 arxiv:183.2249v1 [q-fin.mf] 6 Mar 218 Abstract Over the last decade, dividends have become a standalone
More informationPricing Variance Swaps on Time-Changed Lévy Processes
Pricing Variance Swaps on Time-Changed Lévy Processes ICBI Global Derivatives Volatility and Correlation Summit April 27, 2009 Peter Carr Bloomberg/ NYU Courant pcarr4@bloomberg.com Joint with Roger Lee
More informationMulti-Curve Pricing of Non-Standard Tenor Vanilla Options in QuantLib. Sebastian Schlenkrich QuantLib User Meeting, Düsseldorf, December 1, 2015
Multi-Curve Pricing of Non-Standard Tenor Vanilla Options in QuantLib Sebastian Schlenkrich QuantLib User Meeting, Düsseldorf, December 1, 2015 d-fine d-fine All rights All rights reserved reserved 0 Swaption
More informationInvestigation of Dependency between Short Rate and Transition Rate on Pension Buy-outs. Arık, A. 1 Yolcu-Okur, Y. 2 Uğur Ö. 2
Investigation of Dependency between Short Rate and Transition Rate on Pension Buy-outs Arık, A. 1 Yolcu-Okur, Y. 2 Uğur Ö. 2 1 Hacettepe University Department of Actuarial Sciences 06800, TURKEY 2 Middle
More informationEnlargement of filtration
Enlargement of filtration Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 6, 2017 ICMAT / UC3M Enlargement of Filtration Enlargement of Filtration ([1] 5.9) If G is a
More informationPricing and Modelling in Electricity Markets
Pricing and Modelling in Electricity Markets Ben Hambly Mathematical Institute University of Oxford Pricing and Modelling in Electricity Markets p. 1 Electricity prices Over the past 20 years a number
More informationIntroduction. Practitioner Course: Interest Rate Models. John Dodson. February 18, 2009
Practitioner Course: Interest Rate Models February 18, 2009 syllabus text sessions office hours date subject reading 18 Feb introduction BM 1 25 Feb affine models BM 3 4 Mar Gaussian models BM 4 11 Mar
More informationMultiname and Multiscale Default Modeling
Multiname and Multiscale Default Modeling Jean-Pierre Fouque University of California Santa Barbara Joint work with R. Sircar (Princeton) and K. Sølna (UC Irvine) Special Semester on Stochastics with Emphasis
More informationA New Class of Non-linear Term Structure Models. Discussion
A New Class of Non-linear Term Structure Models by Eraker, Wang and Wu Discussion Pietro Veronesi The University of Chicago Booth School of Business Main Contribution and Outline of Discussion Main contribution
More informationLimit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies
Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies George Tauchen Duke University Viktor Todorov Northwestern University 2013 Motivation
More informationThings You Have To Have Heard About (In Double-Quick Time) The LIBOR market model: Björk 27. Swaption pricing too.
Things You Have To Have Heard About (In Double-Quick Time) LIBORs, floating rate bonds, swaps.: Björk 22.3 Caps: Björk 26.8. Fun with caps. The LIBOR market model: Björk 27. Swaption pricing too. 1 Simple
More informationLecture 5: Review of interest rate models
Lecture 5: Review of interest rate models Xiaoguang Wang STAT 598W January 30th, 2014 (STAT 598W) Lecture 5 1 / 46 Outline 1 Bonds and Interest Rates 2 Short Rate Models 3 Forward Rate Models 4 LIBOR and
More informationPredictability of Interest Rates and Interest-Rate Portfolios
Predictability of Interest Rates and Interest-Rate Portfolios Liuren Wu Zicklin School of Business, Baruch College Joint work with Turan Bali and Massoud Heidari July 7, 2007 The Bank of Canada - Rotman
More informationA Two-Factor Model for Low Interest Rate Regimes
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 130 August 2004 A Two-Factor Model for Low Interest Rate Regimes Shane Miller and Eckhard Platen ISSN 1441-8010
More informationChapter 3: Black-Scholes Equation and Its Numerical Evaluation
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Generating Random Variables and Stochastic Processes Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationdt+ ρσ 2 1 ρ2 σ 2 κ i and that A is a rather lengthy expression that we may or may not need. (Brigo & Mercurio Lemma Thm , p. 135.
A 2D Gaussian model (akin to Brigo & Mercurio Section 4.2) Suppose where ( κ1 0 dx(t) = 0 κ 2 r(t) = δ 0 +X 1 (t)+x 2 (t) )( X1 (t) X 2 (t) ) ( σ1 0 dt+ ρσ 2 1 ρ2 σ 2 )( dw Q 1 (t) dw Q 2 (t) ) In this
More informationMgr. Jakub Petrásek 1. May 4, 2009
Dissertation Report - First Steps Petrásek 1 2 1 Department of Probability and Mathematical Statistics, Charles University email:petrasek@karlin.mff.cuni.cz 2 RSJ Invest a.s., Department of Probability
More informationThe Term Structure of Interbank Risk
The Term Structure of Interbank Risk Anders B. Trolle (joint work with Damir Filipović) Ecole Polytechnique Fédérale de Lausanne and Swiss Finance Institute CREDIT 2011, September 30 Objective The recent
More informationOn modelling of electricity spot price
, Rüdiger Kiesel and Fred Espen Benth Institute of Energy Trading and Financial Services University of Duisburg-Essen Centre of Mathematics for Applications, University of Oslo 25. August 2010 Introduction
More informationInterest Rate Cancelable Swap Valuation and Risk
Interest Rate Cancelable Swap Valuation and Risk Dmitry Popov FinPricing http://www.finpricing.com Summary Cancelable Swap Definition Bermudan Swaption Payoffs Valuation Model Selection Criteria LGM Model
More informationTime-changed Brownian motion and option pricing
Time-changed Brownian motion and option pricing Peter Hieber Chair of Mathematical Finance, TU Munich 6th AMaMeF Warsaw, June 13th 2013 Partially joint with Marcos Escobar (RU Toronto), Matthias Scherer
More informationSaddlepoint Approximation Methods for Pricing. Financial Options on Discrete Realized Variance
Saddlepoint Approximation Methods for Pricing Financial Options on Discrete Realized Variance Yue Kuen KWOK Department of Mathematics Hong Kong University of Science and Technology Hong Kong * This is
More informationA Robust Option Pricing Problem
IMA 2003 Workshop, March 12-19, 2003 A Robust Option Pricing Problem Laurent El Ghaoui Department of EECS, UC Berkeley 3 Robust optimization standard form: min x sup u U f 0 (x, u) : u U, f i (x, u) 0,
More information16. Inflation-Indexed Swaps
6. Inflation-Indexed Swaps Given a set of dates T,...,T M, an Inflation-Indexed Swap (IIS) is a swap where, on each payment date, Party A pays Party B the inflation rate over a predefined period, while
More informationExact Sampling of Jump-Diffusion Processes
1 Exact Sampling of Jump-Diffusion Processes and Dmitry Smelov Management Science & Engineering Stanford University Exact Sampling of Jump-Diffusion Processes 2 Jump-Diffusion Processes Ubiquitous in finance
More informationSupplementary Appendix to The Risk Premia Embedded in Index Options
Supplementary Appendix to The Risk Premia Embedded in Index Options Torben G. Andersen Nicola Fusari Viktor Todorov December 214 Contents A The Non-Linear Factor Structure of Option Surfaces 2 B Additional
More informationState Space Estimation of Dynamic Term Structure Models with Forecasts
State Space Estimation of Dynamic Term Structure Models with Forecasts Liuren Wu November 19, 2015 Liuren Wu Estimation and Application November 19, 2015 1 / 39 Outline 1 General setting 2 State space
More informationSPDE and portfolio choice (joint work with M. Musiela) Princeton University. Thaleia Zariphopoulou The University of Texas at Austin
SPDE and portfolio choice (joint work with M. Musiela) Princeton University November 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 Performance measurement of investment strategies 2 Market
More informationNormal Inverse Gaussian (NIG) Process
With Applications in Mathematical Finance The Mathematical and Computational Finance Laboratory - Lunch at the Lab March 26, 2009 1 Limitations of Gaussian Driven Processes Background and Definition IG
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay. Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2012, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (40 points) Answer briefly the following questions. 1. Consider
More informationRisk Premia and the Conditional Tails of Stock Returns
Risk Premia and the Conditional Tails of Stock Returns Bryan Kelly NYU Stern and Chicago Booth Outline Introduction An Economic Framework Econometric Methodology Empirical Findings Conclusions Tail Risk
More informationMLEMVD: A R Package for Maximum Likelihood Estimation of Multivariate Diffusion Models
MLEMVD: A R Package for Maximum Likelihood Estimation of Multivariate Diffusion Models Matthew Dixon and Tao Wu 1 Illinois Institute of Technology May 19th 2017 1 https://papers.ssrn.com/sol3/papers.cfm?abstract
More informationFourier Space Time-stepping Method for Option Pricing with Lévy Processes
FST method Extensions Indifference pricing Fourier Space Time-stepping Method for Option Pricing with Lévy Processes Vladimir Surkov University of Toronto Computational Methods in Finance Conference University
More informationInterest Rate Volatility
Interest Rate Volatility III. Working with SABR Andrew Lesniewski Baruch College and Posnania Inc First Baruch Volatility Workshop New York June 16-18, 2015 Outline Arbitrage free SABR 1 Arbitrage free
More informationModelling Credit Spread Behaviour. FIRST Credit, Insurance and Risk. Angelo Arvanitis, Jon Gregory, Jean-Paul Laurent
Modelling Credit Spread Behaviour Insurance and Angelo Arvanitis, Jon Gregory, Jean-Paul Laurent ICBI Counterparty & Default Forum 29 September 1999, Paris Overview Part I Need for Credit Models Part II
More informationUSC Math. Finance April 22, Path-dependent Option Valuation under Jump-diffusion Processes. Alan L. Lewis
USC Math. Finance April 22, 23 Path-dependent Option Valuation under Jump-diffusion Processes Alan L. Lewis These overheads to be posted at www.optioncity.net (Publications) Topics Why jump-diffusion models?
More informationDynamic Fund Protection. Elias S. W. Shiu The University of Iowa Iowa City U.S.A.
Dynamic Fund Protection Elias S. W. Shiu The University of Iowa Iowa City U.S.A. Presentation based on two papers: Hans U. Gerber and Gerard Pafumi, Pricing Dynamic Investment Fund Protection, North American
More informationEquity correlations implied by index options: estimation and model uncertainty analysis
1/18 : estimation and model analysis, EDHEC Business School (joint work with Rama COT) Modeling and managing financial risks Paris, 10 13 January 2011 2/18 Outline 1 2 of multi-asset models Solution to
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationShort & Long Run impact of volatility on the effect monetary shocks
Short & Long Run impact of volatility on the effect monetary shocks Fernando Alvarez University of Chicago & NBER Inflation: Drivers & Dynamics Conference 218 Cleveland Fed Alvarez Volatility & Monetary
More informationInterest rate modelling: How important is arbitrage free evolution?
Interest rate modelling: How important is arbitrage free evolution? Siobhán Devin 1 Bernard Hanzon 2 Thomas Ribarits 3 1 European Central Bank 2 University College Cork, Ireland 3 European Investment Bank
More informationIntroduction to Affine Processes. Applications to Mathematical Finance
and Its Applications to Mathematical Finance Department of Mathematical Science, KAIST Workshop for Young Mathematicians in Korea, 2010 Outline Motivation 1 Motivation 2 Preliminary : Stochastic Calculus
More informationStochastic Volatility and Jump Modeling in Finance
Stochastic Volatility and Jump Modeling in Finance HPCFinance 1st kick-off meeting Elisa Nicolato Aarhus University Department of Economics and Business January 21, 2013 Elisa Nicolato (Aarhus University
More informationIntroduction Credit risk
A structural credit risk model with a reduced-form default trigger Applications to finance and insurance Mathieu Boudreault, M.Sc.,., F.S.A. Ph.D. Candidate, HEC Montréal Montréal, Québec Introduction
More informationOptimal Hedging of Variance Derivatives. John Crosby. Centre for Economic and Financial Studies, Department of Economics, Glasgow University
Optimal Hedging of Variance Derivatives John Crosby Centre for Economic and Financial Studies, Department of Economics, Glasgow University Presentation at Baruch College, in New York, 16th November 2010
More informationPricing swaps and options on quadratic variation under stochastic time change models
Pricing swaps and options on quadratic variation under stochastic time change models Andrey Itkin Volant Trading LLC & Rutgers University 99 Wall Street, 25 floor, New York, NY 10005 aitkin@volanttrading.com
More informationsymmys.com 3.2 Projection of the invariants to the investment horizon
122 3 Modeling the market In the swaption world the underlying rate (3.57) has a bounded range and thus it does not display the explosive pattern typical of a stock price. Therefore the swaption prices
More informationOptimal trading strategies under arbitrage
Optimal trading strategies under arbitrage Johannes Ruf Columbia University, Department of Statistics The Third Western Conference in Mathematical Finance November 14, 2009 How should an investor trade
More informationQuadratic hedging in affine stochastic volatility models
Quadratic hedging in affine stochastic volatility models Jan Kallsen TU München Pittsburgh, February 20, 2006 (based on joint work with F. Hubalek, L. Krawczyk, A. Pauwels) 1 Hedging problem S t = S 0
More informationAlgorithmic Trading under the Effects of Volume Order Imbalance
Algorithmic Trading under the Effects of Volume Order Imbalance 7 th General Advanced Mathematical Methods in Finance and Swissquote Conference 2015 Lausanne, Switzerland Ryan Donnelly ryan.donnelly@epfl.ch
More informationSensitivity Analysis on Long-term Cash flows
Sensitivity Analysis on Long-term Cash flows Hyungbin Park Worcester Polytechnic Institute 19 March 2016 Eastern Conference on Mathematical Finance Worcester Polytechnic Institute, Worceseter, MA 1 / 49
More informationInflation-indexed Swaps and Swaptions
Inflation-indexed Swaps and Swaptions Mia Hinnerich Aarhus University, Denmark Vienna University of Technology, April 2009 M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009
More informationOption pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard
Option pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard Indifference pricing and the minimal entropy martingale measure Fred Espen Benth Centre of Mathematics for Applications
More informationLarge Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models
Large Deviations and Stochastic Volatility with Jumps: TU Berlin with A. Jaquier and A. Mijatović (Imperial College London) SIAM conference on Financial Mathematics, Minneapolis, MN July 10, 2012 Implied
More informationdt + ρσ 2 1 ρ2 σ 2 B i (τ) = 1 e κ iτ κ i
A 2D Gaussian model (akin to Brigo & Mercurio Section 4.2) Suppose where dx(t) = ( κ1 0 0 κ 2 ) ( X1 (t) X 2 (t) In this case we find (BLACKBOARD) that r(t) = δ 0 + X 1 (t) + X 2 (t) ) ( σ1 0 dt + ρσ 2
More information