On exponentially ane martingales. Johannes Muhle-Karbe

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "On exponentially ane martingales. Johannes Muhle-Karbe"

Transcription

1 On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes Muhle-Karbe Joint work with Jan Kallsen HVB-Institut für Finanzmathematik, Technische Universität München 1

2 Outline 1. Semimartingale characteristics 2. Ane processes 3. Exponentially ane martingales 4. Applications HVB-Institut für Finanzmathematik, Technische Universität München 2

3 Semimartingale characteristics Idea: Local characterization of semimartingales Deterministic: Linear functions characterized by constant increments Derivative: local approximation by linear functions Stochastic analogon: Lévy processes characterized by independent, stationary increments Semimartingale characteristics: local approximation by Lévy processes HVB-Institut für Finanzmathematik, Technische Universität München 3

4 Semimartingale characteristics Linear function determined by slope b R Distribution of Lévy process (X t ) t R+ on R d determined by Lévy-Khintchine triplet (b, c, F ) (Dierential) Semimartingale characteristics: X t := (b t (ω), c t (ω), F t (ω, )) Local Lévy-Khintchine triplet, time-dependent and random Connection to (integral) characteristics from Jacod & Shiryaev: B t = t 0 b s ds, C t = t 0 c s ds, ν([0, t] G) = t 0 F s (G)ds HVB-Institut für Finanzmathematik, Technische Universität München 4

5 Semimartingale characteristics Idea: Modeling through local dynamics Deterministic: Ordinary dierential equation d dt X t = f(t, X t ) Stochastic analogon: Martingale problem, X = (b, c, F ) with β, γ, ϕ constant: X Lévy process b t (ω) = β(t, X t (ω)) c t (ω) = γ(t, X t (ω)) F t (ω, G) = ϕ(t, G, X t (ω)) HVB-Institut für Finanzmathematik, Technische Universität München 5

6 Ane processes Ane process: Characteristics X = (b, c, F ) ane in X : b t (ω) = β 0 + c t (ω) = γ 0 + d Xt (ω)β j j j=1 m Xt (ω)γ j j j=1 F t (G, ω) = ϕ 0 (G) + m Xt (ω)ϕ j j (G) j=1 (β j, γ j, ϕ j ) given Lévy-Khintchine triplets Due, Filipovi & Schachermayer (2003): Admissibility conditions for existence and uniqueness, Filipovi (2005): time-inhomogeneous triplets HVB-Institut für Finanzmathematik, Technische Universität München 6

7 Ane processes Example: Stochastic volatility model of Heston (1993) S t = E (X) t asset price, v t volatility, where (v, X) solves dv t = (κ λv t )dt + σ v t dz t dx t = (µ + δv t )dt + v t dw t W, Z Brownian motions with constant correlation ϱ Characteristics: ( v X) ane in v = v t = (( ) ( ) ) κ λvt σ, 2 v t σϱv t, 0 µ δv t σϱv t v t HVB-Institut für Finanzmathematik, Technische Universität München 7

8 Ane processes Example: Model of Carr, Geman, Madan & Yor (2001) X t = X 0 + L Vt dv t = v t dt, dv t = λv t dt + dz t L, Z independent Lévy processes with Lévy-Khintchine triplets (b L, c L, F L ) and (b Z, 0, F Z ) Characteristics: (( ) ( ) b Z λv t 0 0 b L, v t 0 c L, v t 1 G (y, 0)F Z (dy) + 1 G (0, x)f L (dx)v t ) ane in v HVB-Institut für Finanzmathematik, Technische Universität München 8

9 Exponentially ane martingales X ane process, h truncation function Goal: Conditions for E (X i ) to be a martingale Jacod & Shiryaev (2003) E (X i ) is a local martingale, i h i (x i )x i ϕ j (dx) <, 0 j d βj i + (x i h i (x i ))ϕ j (dx) = 0, 0 j d Continuous case: Local martingale, i βj i = 0, 0 j d Condition on X, easy to check in applications HVB-Institut für Finanzmathematik, Technische Universität München 9

10 Exponentially ane martingales Criterion for true martingale property? For Lévy process X: E (X) local martingale, E (X) 0 E (X)martingale Does NOT hold in general for ane processes! Some general criteria either hard to check (e.g. Wong & Heyde (2004), Jacod & Shiryaev (2003), Kallsen & Shiryaev (2002)) Others not even necessary in the Lévy case (e.g. Lepingle & Memin (1978)) On the other hand: powerful theory of ane processes HVB-Institut für Finanzmathematik, Technische Universität München 10

11 Exponentially ane martingales Theorem: X ane relative to admissible Lévy-Khintchine triplets (β j, γ j, ϕ j ), 0 j d. Then E (X i ) is a martingale, if 1. E (X i ) 0, 2. h i (x i )x i ϕ j (dx) <, 0 j d 3. βj i + (x i h i (x))ϕ j (dx) = 0, 0 j d 4. { x k >1} xk 1 + x i ϕ j (dx) <, 1 k, j d Similar criterion for exp(x i ) instead of E (X i ) Extension to time-inhomogeneous ane processes possible HVB-Institut für Finanzmathematik, Technische Universität München 11

12 Exponentially ane martingales Continuous case (e.g. Heston (1993)): E (X) positive local martingale martingale Also holds for CGMY asset price S S positive, if F L ((, 1)) = 0 S local martingale, if (yh(y))f L (dy) <, b L + (y h(y))f L (dy) = 0 For PII X: E (X) positive local martingale martingale HVB-Institut für Finanzmathematik, Technische Universität München 12

13 Application 1: Absolutely continuous change of measure X, Y semimartingales with ane characteristics Goal: Criterion for P Y loc P X Application: X model for asset price under physical, Y under risk neutral measure. Equivalence for arbitrage theory Idea: Dene appropriate candidate Z for density process Show: Z exponentially ane, local martingale martingale Dene Q loc P X via density process Z Q = P Y by Girsanov and uniqueness of ane martingale problems HVB-Institut für Finanzmathematik, Technische Universität München 13

14 Application 1: Absolutely continuous change of measure Similar results by Cheridito, Filipovi and Yor (2005) in more general setup Here, the moment conditions are often less restrictive though Example: Esscher change of measure Condition in Cheridito, Filipovi, & Yor (2005): (H x)e H x ϕ j (dx) <, 0 j d { x >1} Condition here: { x >1} e H x ϕ j (dx) <, 0 j d HVB-Institut für Finanzmathematik, Technische Universität München 14

15 Application 2: Exponential moments X R d -valued ane process Due, Filipovi and Schachermayer (2003): conditional characteristic function given by E(e iu X T F t ) = exp(φ(t t, iu) + Ψ(T t, iu) X t ), Φ and Ψ solve integro-dierential equations with initial values 0, iu If analytic extension to open set U exists, E(e p X T F t ) = exp(φ(t t, p) + Ψ(T t, p) X t ), p U Problem: construction of analytic extension is often tedious HVB-Institut für Finanzmathematik, Technische Universität München 15

16 Application 2: Exponential moments Alternative approach: Assume solutions Φ and Ψ to integro-dierential equations with initial values 0 and p R d exist Dene N t := Φ(T t) + Ψ(T t)x t Show: (X, N) is ane, exp(n) is a local martingale Results on time-inhomogeneous exponentially ane martingales, (mild) condition on the big jumps of X martingale Martingale property yields E(e p X T F t ) = E(e N T F t ) = e N t = exp(φ(t t) + Ψ(T t) X t ) HVB-Institut für Finanzmathematik, Technische Universität München 16

17 Application 3: Portfolio optimization Goal: Find trading strategy ϕ, such that E(u(V T (ϕ))) E(u(V T (ψ))), ψ u utility function Example: Power utility, i.e. u(x) = x 1 p /(1 p) ϕ, ψ admissible, i.e. V (ϕ), V (ψ) 0 Asset price modeled as an ane process, e.g. Heston (1993) or Carr, Geman, Madan & Yor (2001) HVB-Institut für Finanzmathematik, Technische Universität München 17

18 Application 3: Portfolio optimization Sucient criterion for optimality: If there exists a positive martingale Z, such that 1. (ZS) T is a local martingale 2. Z T = u (V T (ϕ)) 3. (ZV (ϕ)) T is a martingale we have E(u(V T (ϕ))) E(u(V T (ψ))), ψ Observation: S, V (ϕ) and u (V (ϕ)) exponentially ane HVB-Institut für Finanzmathematik, Technische Universität München 18

19 Application 3: Portfolio optimization Idea: Make an exponentially ane ansatz for Z as well! Computation of ansatz functions through drift conditions Verication of the candidate processes with results on exponentially ane martingales Approach works for the models proposed by Heston (1993) and Carr, Geman, Madan & Yor (2001) among others HVB-Institut für Finanzmathematik, Technische Universität München 19

20 References Carr, Geman, Madan & Yor (2001): Stochastic volatility for Lévy processes, Mathematical Finance 13(3), Cheridito, Filipovi & Yor (2005): Equivalent and absolutely continuous measure changes for jump-diusion processes, The Annals of Applied Probability 15(3), Due, Filipovi & Schachermayer (2003): Ane processes and applications in nance, The Annals of Applied Probability, 13(3), Filipovi (2005): Time-inhomogeneous ane processes, Stochastic Processes and their Applications 115(4), Heston (1993): A closed-form solution for options with stochastic volatilities with applications to bond and currency options, The Review of Financial Studies, 6(2), HVB-Institut für Finanzmathematik, Technische Universität München 20

21 References Jacod & Shiryaev (2003): Limit theorems for stochastic processes, Springer Verlag, Berlin, second edition Kallsen & Shiryaev (2002): The cumulant process and Esscher's change of measure, Finance and Stochastics, 6(4), Lépingle & Mémin (1978): Sur l'intégrabilité uniforme des martingales exponentielles, Zur Wahrscheinlichkeit Verwandte Gebiete, 42(3), Wong & Heyde (2004): On the martingale property of stochastic exponentials, Journal of Applied Probability, 41(3), HVB-Institut für Finanzmathematik, Technische Universität München 21

A Novel Fourier Transform B-spline Method for Option Pricing*

A Novel Fourier Transform B-spline Method for Option Pricing* A Novel Fourier Transform B-spline Method for Option Pricing* Paper available from SSRN: http://ssrn.com/abstract=2269370 Gareth G. Haslip, FIA PhD Cass Business School, City University London October

More information

Simple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University

Simple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University Simple Arbitrage Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila Saarland University December, 8, 2011 Problem Setting Financial market with two assets (for simplicity) on

More information

Introduction to Arbitrage-Free Pricing: Fundamental Theorems

Introduction to Arbitrage-Free Pricing: Fundamental Theorems Introduction to Arbitrage-Free Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 8-10, 2015 1 / 24 Outline Financial market

More information

MATHEMATICAL FINANCE and Derivatives (part II, PhD)

MATHEMATICAL FINANCE and Derivatives (part II, PhD) MATHEMATICAL FINANCE and Derivatives (part II, PhD) Lecturer: Prof. Dr. Marc CHESNEY Location: Time: Mon. 08.00 09.45 Uhr First lecture: 18.02.2008 Language: English Contents: Stochastic volatility models

More information

A spot price model feasible for electricity forward pricing Part II

A spot price model feasible for electricity forward pricing Part II A spot price model feasible for electricity forward pricing Part II Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Wolfgang Pauli Institute, Wien January 17-18

More information

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic mean-variance problems in

More information

Pricing of catastrophe insurance options written on a loss index with reestimation

Pricing of catastrophe insurance options written on a loss index with reestimation Pricing of catastrophe insurance options written on a loss index with reestimation Francesca Biagini ) Yuliya Bregman ) Thilo Meyer-Brandis 2) April 28, 29 ) Department of Mathematics, LMU, 2) CMA, University

More information

Explicit Option Pricing Formula for a Mean-Reverting Asset in Energy Markets

Explicit Option Pricing Formula for a Mean-Reverting Asset in Energy Markets Explicit Option Pricing Formula for a Mean-Reverting Asset in Energy Markets Anatoliy Swishchuk Mathematical & Computational Finance Lab Dept of Math & Stat, University of Calgary, Calgary, AB, Canada

More information

Stochastic Skew Models for FX Options

Stochastic Skew Models for FX Options Stochastic Skew Models for FX Options Peter Carr Bloomberg LP and Courant Institute, NYU Liuren Wu Zicklin School of Business, Baruch College Special thanks to Bruno Dupire, Harvey Stein, Arun Verma, and

More information

MEAN VARIANCE HEDGING AND OPTIMAL INVESTMENT IN HESTON S MODEL WITH CORRELATION

MEAN VARIANCE HEDGING AND OPTIMAL INVESTMENT IN HESTON S MODEL WITH CORRELATION MEAN VARIANCE HEDGING AND OPTIMAL INVESTMENT IN HESTON S MODEL WITH CORRELATION ALEŠ ČERNÝ AND JAN KALLSEN Abstract. This paper solves the mean variance hedging problem in Heston s model with a stochastic

More information

Risk-minimization for life insurance liabilities

Risk-minimization for life insurance liabilities Risk-minimization for life insurance liabilities Francesca Biagini Mathematisches Institut Ludwig Maximilians Universität München February 24, 2014 Francesca Biagini USC 1/25 Introduction A large number

More information

Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer

Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of Discrete-Time Stochastic

More information

A new Feynman-Kac-formula for option pricing in Lévy models

A new Feynman-Kac-formula for option pricing in Lévy models A new Feynman-Kac-formula for option pricing in Lévy models Kathrin Glau Department of Mathematical Stochastics, Universtity of Freiburg (Joint work with E. Eberlein) 6th World Congress of the Bachelier

More information

Monte Carlo Methods in Finance

Monte Carlo Methods in Finance Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction

More information

Marshall-Olkin distributions and portfolio credit risk

Marshall-Olkin distributions and portfolio credit risk Marshall-Olkin distributions and portfolio credit risk Moderne Finanzmathematik und ihre Anwendungen für Banken und Versicherungen, Fraunhofer ITWM, Kaiserslautern, in Kooperation mit der TU München und

More information

Diusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute

Diusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 Black-Scholes 5 Equity linked life insurance 6 Merton

More information

Stochastic Skew in Currency Options

Stochastic Skew in Currency Options Stochastic Skew in Currency Options PETER CARR Bloomberg LP and Courant Institute, NYU LIUREN WU Zicklin School of Business, Baruch College Citigroup Wednesday, September 22, 2004 Overview There is a huge

More information

Pricing catastrophe options in incomplete market

Pricing catastrophe options in incomplete market Pricing catastrophe options in incomplete market Arthur Charpentier arthur.charpentier@univ-rennes1.fr Actuarial and Financial Mathematics Conference Interplay between finance and insurance, February 2008

More information

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

Sensitivity analysis of utility based prices and risk-tolerance wealth processes

Sensitivity analysis of utility based prices and risk-tolerance wealth processes Sensitivity analysis of utility based prices and risk-tolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,

More information

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for

More information

Online Appendix. Supplemental Material for Insider Trading, Stochastic Liquidity and. Equilibrium Prices. by Pierre Collin-Dufresne and Vyacheslav Fos

Online Appendix. Supplemental Material for Insider Trading, Stochastic Liquidity and. Equilibrium Prices. by Pierre Collin-Dufresne and Vyacheslav Fos Online Appendix Supplemental Material for Insider Trading, Stochastic Liquidity and Equilibrium Prices by Pierre Collin-Dufresne and Vyacheslav Fos 1. Deterministic growth rate of noise trader volatility

More information

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model 1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

More information

Lecture. S t = S t δ[s t ].

Lecture. S t = S t δ[s t ]. Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important

More information

Asian Option Pricing Formula for Uncertain Financial Market

Asian Option Pricing Formula for Uncertain Financial Market Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s4467-15-35-7 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei

More information

Time-inhomogeneous Lévy processes in interest rate and credit risk models

Time-inhomogeneous Lévy processes in interest rate and credit risk models Time-inhomogeneous Lévy processes in interest rate and credit risk models Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg im Breisgau

More information

Probability and Statistics

Probability and Statistics Probability and Statistics Syllabus for the TEMPUS SEE PhD Course (Podgorica, April 4 29, 2011) Franz Kappel 1 Institute for Mathematics and Scientific Computing University of Graz Žaneta Popeska 2 Faculty

More information

On a comparison result for Markov processes

On a comparison result for Markov processes On a comparison result for Markov processes Ludger Rüschendorf University of Freiburg Abstract A comparison theorem is stated for Markov processes in polish state spaces. We consider a general class of

More information

Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space

Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle

More information

The Black-Scholes-Merton Approach to Pricing Options

The Black-Scholes-Merton Approach to Pricing Options he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

More information

Analytically Tractable Stochastic Stock Price Models

Analytically Tractable Stochastic Stock Price Models Archil Gulisashvili Analytically Tractable Stochastic Stock Price Models 4Q Springer Contents 1 Volatility Processes 1 1.1 Brownian Motion 1 1.2 s Geometric Brownian Motion 6 1.3 Long-Time Behavior of

More information

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine Aït-Sahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times

More information

Jung-Soon Hyun and Young-Hee Kim

Jung-Soon Hyun and Young-Hee Kim J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL Jung-Soon Hyun and Young-Hee Kim Abstract. We present two approaches of the stochastic interest

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

PRICING DIGITAL CALL OPTION IN THE HESTON STOCHASTIC VOLATILITY MODEL

PRICING DIGITAL CALL OPTION IN THE HESTON STOCHASTIC VOLATILITY MODEL STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVIII, Number 3, September 003 PRICING DIGITAL CALL OPTION IN THE HESTON STOCHASTIC VOLATILITY MODEL VASILE L. LAZAR Dedicated to Professor Gheorghe Micula

More information

A non-gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing

A non-gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing A non-gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing Thilo Meyer-Brandis Center of Mathematics for Applications / University of Oslo Based on joint work

More information

Option Pricing with Lévy Processes

Option Pricing with Lévy Processes Department of Finance Department of Mathematics Faculty of Sciences Option Pricing with Lévy Processes Jump models for European-style options Rui Monteiro Dissertation Master of Science in Financial Mathematics

More information

Optimal design of prot sharing rates by FFT.

Optimal design of prot sharing rates by FFT. Optimal design of prot sharing rates by FFT. Donatien Hainaut May 5, 9 ESC Rennes, 3565 Rennes, France. Abstract This paper addresses the calculation of a fair prot sharing rate for participating policies

More information

Bubbles and futures contracts in markets with short-selling constraints

Bubbles and futures contracts in markets with short-selling constraints Bubbles and futures contracts in markets with short-selling constraints Sergio Pulido, Cornell University PhD committee: Philip Protter, Robert Jarrow 3 rd WCMF, Santa Barbara, California November 13 th,

More information

Black-Scholes Equation for Option Pricing

Black-Scholes Equation for Option Pricing Black-Scholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there

More information

Brownian Motion and Stochastic Flow Systems. J.M Harrison

Brownian Motion and Stochastic Flow Systems. J.M Harrison Brownian Motion and Stochastic Flow Systems 1 J.M Harrison Report written by Siva K. Gorantla I. INTRODUCTION Brownian motion is the seemingly random movement of particles suspended in a fluid or a mathematical

More information

Modelling electricity market data: the CARMA spot model, forward prices and the risk premium

Modelling electricity market data: the CARMA spot model, forward prices and the risk premium Modelling electricity market data: the CARMA spot model, forward prices and the risk premium Formatvorlage des Untertitelmasters Claudia Klüppelberg durch Klicken bearbeiten Technische Universität München

More information

Markovian projection for volatility calibration

Markovian projection for volatility calibration cutting edge. calibration Markovian projection for volatility calibration Vladimir Piterbarg looks at the Markovian projection method, a way of obtaining closed-form approximations of European-style option

More information

Valuation of Asian Options

Valuation of Asian Options Valuation of Asian Options - with Levy Approximation Master thesis in Economics Jan 2014 Author: Aleksandra Mraovic, Qian Zhang Supervisor: Frederik Lundtofte Department of Economics Abstract Asian options

More information

Option pricing in a quadratic variance swap model

Option pricing in a quadratic variance swap model Elise Gourier School of Economics and Finance Queen Mary University of London Joint work with Damir Filipović and Loriano Mancini Swiss Finance Institute and EPFL Workshop "Mathematical Finance beyond

More information

Option Pricing Formulae using Fourier Transform: Theory and Application

Option Pricing Formulae using Fourier Transform: Theory and Application Option Pricing Formulae using Fourier Transform: Theory and Application Martin Schmelzle * April 2010 Abstract Fourier transform techniques are playing an increasingly important role in Mathematical Finance.

More information

Hedging Exotic Options

Hedging Exotic Options Kai Detlefsen Wolfgang Härdle Center for Applied Statistics and Economics Humboldt-Universität zu Berlin Germany introduction 1-1 Models The Black Scholes model has some shortcomings: - volatility is not

More information

Life-insurance-specific optimal investment: the impact of stochastic interest rate and shortfall constraint

Life-insurance-specific optimal investment: the impact of stochastic interest rate and shortfall constraint Life-insurance-specific optimal investment: the impact of stochastic interest rate and shortfall constraint An Radon Workshop on Financial and Actuarial Mathematics for Young Researchers May 30-31 2007,

More information

Disability insurance: estimation and risk aggregation

Disability insurance: estimation and risk aggregation Disability insurance: estimation and risk aggregation B. Löfdahl Department of Mathematics KTH, Royal Institute of Technology May 2015 Introduction New upcoming regulation for insurance industry: Solvency

More information

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial

More information

Who Should Sell Stocks?

Who Should Sell Stocks? Who Should Sell Stocks? Ren Liu joint work with Paolo Guasoni and Johannes Muhle-Karbe ETH Zürich Imperial-ETH Workshop on Mathematical Finance 2015 1 / 24 Merton s Problem (1969) Frictionless market consisting

More information

Volatility Jumps. April 12, 2010

Volatility Jumps. April 12, 2010 Volatility Jumps Viktor Todorov and George Tauchen April 12, 2010 Abstract The paper undertakes a non-parametric analysis of the high frequency movements in stock market volatility using very finely sampled

More information

Black-Scholes Option Pricing Model

Black-Scholes Option Pricing Model Black-Scholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,

More information

Affine LIBOR models with multiple curves:

Affine LIBOR models with multiple curves: Affine LIBOR models with multiple curves: John Schoenmakers, WIAS Berlin joint with Z. Grbac, A. Papapantoleon & D. Skovmand 9th Summer School in Mathematical Finance Cape Town, 18-20 February 2016 18-20.02.2016

More information

Homework #2 Solutions

Homework #2 Solutions MAT Spring Problems Section.:, 8,, 4, 8 Section.5:,,, 4,, 6 Extra Problem # Homework # Solutions... Sketch likely solution curves through the given slope field for dy dx = x + y...8. Sketch likely solution

More information

Optimisation Problems in Non-Life Insurance

Optimisation Problems in Non-Life Insurance Frankfurt, 6. Juli 2007 1 The de Finetti Problem The Optimal Strategy De Finetti s Example 2 Minimal Ruin Probabilities The Hamilton-Jacobi-Bellman Equation Two Examples 3 Optimal Dividends Dividends in

More information

Math 526: Brownian Motion Notes

Math 526: Brownian Motion Notes Math 526: Brownian Motion Notes Definition. Mike Ludkovski, 27, all rights reserved. A stochastic process (X t ) is called Brownian motion if:. The map t X t (ω) is continuous for every ω. 2. (X t X t

More information

LECTURE 9: A MODEL FOR FOREIGN EXCHANGE

LECTURE 9: A MODEL FOR FOREIGN EXCHANGE LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling

More information

Hedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15

Hedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15 Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.

More information

Essays in Financial Mathematics

Essays in Financial Mathematics Essays in Financial Mathematics Essays in Financial Mathematics Kristoffer Lindensjö Dissertation for the Degree of Doctor of Philosophy, Ph.D. Stockholm School of Economics, 2013. Dissertation title:

More information

Valuation of Equity-Linked Insurance Products and Practical Issues in Equity Modeling. March 12, 2013

Valuation of Equity-Linked Insurance Products and Practical Issues in Equity Modeling. March 12, 2013 Valuation of Equity-Linked Insurance Products and Practical Issues in Equity Modeling March 12, 2013 The University of Hong Kong (A SOA Center of Actuarial Excellence) Session 2 Valuation of Equity-Linked

More information

Applications of semimartingales and Lévy processes in finance: duality and valuation

Applications of semimartingales and Lévy processes in finance: duality and valuation Applications of semimartingales and Lévy processes in finance: duality and valuation Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik und Physik der Albert-Ludwigs-Universität Freiburg

More information

2 The Term Structure of Interest Rates in a Hidden Markov Setting

2 The Term Structure of Interest Rates in a Hidden Markov Setting 2 The Term Structure of Interest Rates in a Hidden Markov Setting Robert J. Elliott 1 and Craig A. Wilson 2 1 Haskayne School of Business University of Calgary Calgary, Alberta, Canada relliott@ucalgary.ca

More information

LOG-TYPE MODELS, HOMOGENEITY OF OPTION PRICES AND CONVEXITY. 1. Introduction

LOG-TYPE MODELS, HOMOGENEITY OF OPTION PRICES AND CONVEXITY. 1. Introduction LOG-TYPE MODELS, HOMOGENEITY OF OPTION PRICES AND CONVEXITY M. S. JOSHI Abstract. It is shown that the properties of convexity of call prices with respect to spot price and homogeneity of call prices as

More information

Pricing of an Exotic Forward Contract

Pricing of an Exotic Forward Contract Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,

More information

The Black-Scholes Formula

The Black-Scholes Formula FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

More information

THE BLACK-SCHOLES MODEL AND EXTENSIONS

THE BLACK-SCHOLES MODEL AND EXTENSIONS THE BLAC-SCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that

More information

Characterizing Option Prices by Linear Programs

Characterizing Option Prices by Linear Programs Contemporary Mathematics Characterizing Option Prices by Linear Programs Richard H. Stockbridge Abstract. The price of various options on a risky asset are characterized via a linear program involving

More information

Modeling the Implied Volatility Surface. Jim Gatheral Stanford Financial Mathematics Seminar February 28, 2003

Modeling the Implied Volatility Surface. Jim Gatheral Stanford Financial Mathematics Seminar February 28, 2003 Modeling the Implied Volatility Surface Jim Gatheral Stanford Financial Mathematics Seminar February 28, 2003 This presentation represents only the personal opinions of the author and not those of Merrill

More information

Small-Time Asymptotics of Option Prices and First Absolute Moments

Small-Time Asymptotics of Option Prices and First Absolute Moments Small-Time Asymptotics of Option Prices and First Absolute Moments Johannes Muhle-Karbe Marcel Nutz First version: June 11, 2010. This version: June 11, 2011. Abstract We study the leading term in the

More information

Nonparametric estimation for a Time-changed Lévy Model

Nonparametric estimation for a Time-changed Lévy Model Nonparametric estimation for a Time-changed Lévy Model José Enrique Figueroa-López 1 1 Department of Statistics Purdue University Research Colloquium Purdue University January 22, 2010 Outline 1 Motivation

More information

Merton-Black-Scholes model for option pricing. Peter Denteneer. 22 oktober 2009

Merton-Black-Scholes model for option pricing. Peter Denteneer. 22 oktober 2009 Merton-Black-Scholes model for option pricing Instituut{Lorentz voor Theoretische Natuurkunde, LION, Universiteit Leiden 22 oktober 2009 With inspiration from: J. Tinbergen, T.C. Koopmans, E. Majorana,

More information

Miscellaneous. Simone Freschi freschi.simone@gmail.com Tommaso Gabriellini tommaso.gabriellini@mpscs.it. Università di Siena

Miscellaneous. Simone Freschi freschi.simone@gmail.com Tommaso Gabriellini tommaso.gabriellini@mpscs.it. Università di Siena Miscellaneous Simone Freschi freschi.simone@gmail.com Tommaso Gabriellini tommaso.gabriellini@mpscs.it Head of Global MPS Capital Services SpA - MPS Group Università di Siena ini A.A. 2014-2015 1 / 1 A

More information

Optimal portfolio allocation with higher moments

Optimal portfolio allocation with higher moments Jakša Cvitanić Vassilis Polimenis Fernando Zapatero Optimal portfolio allocation with higher moments Received: (date)/revised version: (date) Abstract We model the risky asset as driven by a pure jump

More information

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

More information

Pricing of electricity futures The risk premium

Pricing of electricity futures The risk premium Pricing of electricity futures The risk premium Fred Espen Benth In collaboration with Alvaro Cartea, Rüdiger Kiesel and Thilo Meyer-Brandis Centre of Mathematics for Applications (CMA) University of Oslo,

More information

Black-Scholes model under Arithmetic Brownian Motion

Black-Scholes model under Arithmetic Brownian Motion Black-Scholes model under Arithmetic Brownian Motion Marek Kolman Uniersity of Economics Prague December 22 23 Abstract Usually in the Black-Scholes world it is assumed that a stock follows a Geometric

More information

BLACK-SCHOLES GOES HYPERGEOMETRIC

BLACK-SCHOLES GOES HYPERGEOMETRIC BLACK-SCHOLES GOES HYPERGEOMETRIC CLAUDIO ALBANESE, GIUSEPPE CAMPOLIETI, PETER CARR, AND ALEXANDER LIPTON ABSTRACT. We introduce a general pricing formula that extends Black-Scholes and contains as particular

More information

Private Equity Fund Valuation and Systematic Risk

Private Equity Fund Valuation and Systematic Risk An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology

More information

Model Independent Greeks

Model Independent Greeks Model Independent Greeks Mathematical Finance Winter School Lunteren January 2014 Jesper Andreasen Danske Markets, Copenhagen kwant.daddy@danskebank.com Outline Motivation: Wots me Δelδa? The short maturity

More information

Stocks paying discrete dividends: modelling and option pricing

Stocks paying discrete dividends: modelling and option pricing Stocks paying discrete dividends: modelling and option pricing Ralf Korn 1 and L. C. G. Rogers 2 Abstract In the Black-Scholes model, any dividends on stocks are paid continuously, but in reality dividends

More information

Lecture 1: Stochastic Volatility and Local Volatility

Lecture 1: Stochastic Volatility and Local Volatility Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch Case Studies in Financial Modelling Course Notes, Courant Institute of Mathematical Sciences, Fall Term, 2002 Abstract

More information

Rate of convergence towards Hartree dynamics

Rate of convergence towards Hartree dynamics Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson

More information

MEAN-VARIANCE HEDGING WITH RANDOM VOLATILITY JUMPS

MEAN-VARIANCE HEDGING WITH RANDOM VOLATILITY JUMPS MEAN-VARIANCE HEDGING WIH RANDOM VOLAILIY JUMPS PAOLO GUASONI AND FRANCESCA BIAGINI Abstract. We introduce a general framework for stochastic volatility models, with the risky asset dynamics given by:

More information

Option Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013

Option Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013 Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed

More information

Grey Brownian motion and local times

Grey Brownian motion and local times Grey Brownian motion and local times José Luís da Silva 1,2 (Joint work with: M. Erraoui 3 ) 2 CCM - Centro de Ciências Matemáticas, University of Madeira, Portugal 3 University Cadi Ayyad, Faculty of

More information

Modelling Interest Rates for Public Debt Management

Modelling Interest Rates for Public Debt Management Università La Sapienza di Roma Facoltà di Scienze Matematiche, Fisiche e Naturali Dottorato di Ricerca in Matematica (XV ciclo) Modelling Interest Rates for Public Debt Management Adamo Uboldi Contents

More information

Pricing options with VG model using FFT

Pricing options with VG model using FFT Pricing options with VG model using FFT Andrey Itkin itkin@chem.ucla.edu Moscow State Aviation University Department of applied mathematics and physics A.Itkin Pricing options with VG model using FFT.

More information

Risk management of CPPI funds in switching regime markets.

Risk management of CPPI funds in switching regime markets. Risk management of CPPI funds in switching regime markets. Donatien Hainaut October, 1 NSA-CRST. 945 Malako Cedex, France. mail: donatien.hainaut@ensae.fr Abstract The constant proportion portfolio insurance

More information

Operations Research and Financial Engineering. Courses

Operations Research and Financial Engineering. Courses Operations Research and Financial Engineering Courses ORF 504/FIN 504 Financial Econometrics Professor Jianqing Fan This course covers econometric and statistical methods as applied to finance. Topics

More information

European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

More information

Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6

Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static

More information

Cross hedging with stochastic correlation

Cross hedging with stochastic correlation Gregor Heyne (joint work with Stefan Ankirchner) Humboldt Universität Berlin, QPL March 20th, 2009 Finance and Insurance, Jena Motivation Consider a call option on the DAX. How to hedge? Maybe with Futures

More information

Introduction to portfolio insurance. Introduction to portfolio insurance p.1/41

Introduction to portfolio insurance. Introduction to portfolio insurance p.1/41 Introduction to portfolio insurance Introduction to portfolio insurance p.1/41 Portfolio insurance Maintain the portfolio value above a certain predetermined level (floor) while allowing some upside potential.

More information

Option Pricing with Time Varying Volatility

Option Pricing with Time Varying Volatility Corso di Laurea Specialistica in Economia, curriculum Models and Methods of Quantitative Economics per la Gestione dell Impresa Prova finale di Laurea Option Pricing with Time Varying Volatility Relatore

More information

Lecture IV: Option pricing in energy markets

Lecture IV: Option pricing in energy markets Lecture IV: Option pricing in energy markets Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Fields Institute, 19-23 August, 2013 Introduction OTC market huge for

More information

Monte Carlo option pricing for tempered stable (CGMY) processes

Monte Carlo option pricing for tempered stable (CGMY) processes Monte Carlo option pricing for tempered stable (CGMY) processes Jérémy Poirot INRIA Rocquencourt email: jeremy.poirot@inria.fr Peter Tankov Laboratoire de Probabilités et Modèles Aléatoires CNRS, UMR 7599

More information

Jump-adapted discretization schemes for Lévy-driven SDEs

Jump-adapted discretization schemes for Lévy-driven SDEs Jump-adapted discretization schemes for Lévy-driven SDEs Arturo Kohatsu-Higa Osaka University Graduate School of Engineering Sciences Machikaneyama cho -3, Osaka 56-853, Japan E-mail: kohatsu@sigmath.es.osaka-u.ac.jp

More information

The Black-Scholes pricing formulas

The Black-Scholes pricing formulas The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

More information

Option Pricing. Prof. Dr. Svetlozar (Zari) Rachev

Option Pricing. Prof. Dr. Svetlozar (Zari) Rachev Option Pricing Prof. Dr. Svetlozar (Zari) Rachev Frey Family Foundation Chair-Professor, Applied Mathematics and Statistics, Stony Brook University Chief Scientific Officer, FinAnalytica Outline: Option

More information