A spot price model feasible for electricity forward pricing Part II
|
|
- Esmond Flowers
- 8 years ago
- Views:
Transcription
1 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
2 Plan of this lecture 1. Some preliminaries on Lévy processes 2. A multi-factor arithmetic model 3. Electricity forward pricing and option pricing
3 Some preliminaries on Lévy processes
4 L(t) is a Lévy process if Lévy processes 1. The increments L(t) L(s) are stationary 2. The increments L(t) L(s) are independent 3. RCLL paths Example 1: Brownian motion Increments are normally distributed, B(t) B(s) N (0, t s) Continuous paths Example 2: Poisson process Increments are Poisson distributed, N(t) N(s) Poiss(λ(t s)) Pure-jump process Inhomogeneous Lévy process: Remove stationarity condition Also called independent increment (II) process
5 Poisson random measure associated to a Lévy process N(A (0, t]) = 1( L(s) A) 0<s t L(s) = L(s) L(s ) Borel set A R\{0} N(A (0, t]) counts number of jumps of size A up to time t The compensator measure is defined as l(a (0, t]) = E [N(A (0, t])]
6 For a Lévy process L(t) l(a (0, t]) = t l(a) l(a) is called the Lévy measure The process t N(A (0, t]) tl(a) is a (local) martingale
7 Examples of Lévy processes Compound Poisson process N(t) X n L(t) = n=1 N Poisson process with intensity λ {X n } iid random variables Lévy measure is l(dz) = λf X (dz) Paths of finite variation Process of finite activity (finitely many jumps in a neighborhood around zero)
8 Subordinators (nondecreasing Lévy processes) Only positive jumps Finite variation paths Process of infinite activity (infinitely many jumps in a neighborhood around zero) Condition on the Lévy measure for subordinators 0 max(1, z) l(dz) <
9 The Lévy-Kintchine formula ln E [exp(ixl(t))] = ψ(x) t ψ(x) = iγx 1 2 x 2 σ 2 + {e ixz 1 ixz1( z < 1)} l(dz) R 0 The characteristic function of L(t) γ is the drift, σ the volatility of the Brownian motion part Brownian motion: l(dz) = 0, γ = 0 Pure-jump Lévy process: σ = 0
10 A Lévy process consists of three parts a drift a Brownian motion a pure-jump process General condition on the Lévy measure R 0 max(1, z 2 ) l(dz) <
11 The Ito Formula for jump processes X (t) a semimartingale and f (t, x) C 1,2 dx (t) = U(t) dt + V (t) dl(t) f (t, X (t)) f (0, X (0)) = t 0 f t (s, X (s)) ds + t 0 f x (s, X (s )) dx (s) + 1 t 2 σ2 f xx (s, X (s))v 2 (s) ds 0 + f (s, X (s)) f (s, X (s )) f x (s, X (s )) X (s) 0<s t Jumps X (s) = X (s) X (s ), no initial jump
12 Example: Solution of the SDE dx (t) = αx (t) dt + dl(t) L(t) a Lévy process Use f (t, x) = x exp(αt) in Ito
13 t t e αt X (t) X (0) = α e αs X (s) ds + e αs 1 dx (s) + 1 t σ2 0 ds 0 + e αs X (s) e αs X (s ) e α 1 X (s) = α = 0<s t t 0 t 0 e αs X (s) ds α e αs dl(s) t t X (t) = e αt X (0) + e α(t s) dl(s) 0 0 t e αs X (s) ds + e αs dl(s) 0
14 A multi-factor arithmetic model
15 The model and properties The spot price as a sum of non-gaussian OU-processes B., Kallsen and Meyer-Brandis (2007) S(t) = Λ(t) n Y i (t) i=1 dy i (t) = α i Y i (t) dt + dl i (t) Λ(t) deterministic seasonality function L i (t) are independent subordinators Possibly time-inhomogeneous processes (II processes)
16 A simulation of S(t) fitted to EEX electricity data Calibration in B., Kiesel and Nazarova (2010) Top: simulated, bottom: EEX prices
17 Dynamics of S(t) ds(t) = { ( ) } X (t) α n Λ (t) S(t) dt + Λ(t) d L(t) Λ(t) AR(1)-process, with stochastic mean and seasonality Mean-reversion to stochastic base level n 1 X (t) = Λ(t) (α n α i )Y i (t) Seasonal speed of mean-reversion α n Λ (t)/λ(t) Seasonal jumps, where d L(t) = n i=1 dl i(t), dependent on the stochastic mean i=1
18 L i (t) jumps only upwards Jump size is a positive random variable Y i will mean-revert to zero However, Y i is always positive Ensures that S(t) is positive No Brownian motion component in the factors would give a probability for S(t) becoming negative EEX have several occurences of negative prices Hence, reasonable with a Brownian component in this case Calibration becomes simpler
19 Simulation of two processes Y i Spike process Y 1 with fast speed of reversion Normal variation Y 2 with slow mean-reversion Y 2 can be thought of as the stochastic mean Y 2 visually like a Brownian motion driven AR(1)-process
20 Y i (t) is stationary Recall from Ito s Formula Y i (u) = Y i (t)e α i (u t) + u t e α i (u s) dl i (s) The log-characteristic function of Y i (t) is, when t [ u ] ln E [exp(ixy i (u)] = iy i (0)e α i u + ln E exp(ix e α i (u s) dl i (s)) = iy i (0)e α i u + 0 u 0 ψ i (xe α i s ) ds Y (t) has a stationary distribution. 0 ψ i (xe α i (u s) ) ds
21 Autocorrelation function for S(t) := S(t)/Λ(t) We calculate, Cov( S(t + τ), S(t)) n n = Cov( Y i (t + τ), Y i (t)) = = i=1 i=1 n Cov(Y i (t + τ), Y i (t)) i=1 n Var(Y i (t)) e α i τ i=1
22 In conclusion ρ(t, τ) = corr[ S(t), S(t + τ)] = n ω i (t, τ)e α i τ i=1 If Y i are stationary, ω i (t, τ) = ω i The weights ω i sum to 1 The theoretical ACF can be used in practice as follows: 1. Find the number of factors n required 2. Find the speeds of mean-reversion by calibration to empirical ACF
23 Forward pricing
24 Definition of the electricity forward price with constant interest rate Delivery over [T 1, T 2 ]. Assuming financial settlement at maturity T 2 F (t, T 1, T 2 ) = E Q [ 1 T 2 T 1 T2 T 1 S(u) du F t Any Q P a risk-neutral probability/pricing measure ]
25 By commuting expectation and integration F (t, T 1, T 2 ) = 1 T 2 T 1 T2 T 1 f (t, u) du Here, f (t, u) is the price of a forward with fixed-delivery time at u, f (t, u) = E Q [S(u) F t ]
26 Restrict to a subclass of measures Q: the Esscher transform Common choice in energy when models have jumps Idea: structure preserving measures In the further analysis, let us for simplicity assume n = 1...i.e. only one factor Y 1 (t), denoted Y (t) no seasonlity Λ(u) = 1
27 The Esscher transform Define a martingale Z as θ is a constant Z(t) = exp (θl(t) φ(θ) t) φ(x) ψ( ix) is the log-mgf of L(1) Radon-Nikodym derivative for measure change: dq = Z(t) dp Ft
28 Effect of measure change on L: ψ θ (x) ln E θ [exp (ixl(1))] = ln E [exp ((ix + θ)l(1))] exp ( ψ( iθ)) = ψ(x iθ) ψ( iθ) = i(x iθ) + = ixγ + 0 i( iθ)γ {e i(x iθ)z 1} l(dz) L Lévy process under Q, with drift γ and Lévy measure exp(θz)l(dz) 0 0 {e i( iθ)z 1} l(dz) {e ixz 1} e θz l(dz)
29 To study Q-dynamics of Y, define Lθ (t) L(t) E θ [L(t)] = L(t) φ (θ)t Lθ is a Q-martingale dy (t) = ( φ (θ) αy (t) ) dt + d L θ (t) When θ 0, we have a change in the level of Y (t) φ (0) φ (θ) θ often called the market price of risk
30 Derivation of the forward price Calculate first f (t, u) using the Q-dynamics of Y Y (u) = Y (t)e α(u t) + φ (θ) u α (1 e α(u t) )+ e α(t s) d L θ (s) t Independent increment property of L θ yields f (t, u) = E θ [Y (u) F t ] = Y (t)e α(u t) + φ (θ) α (1 e α(u t) )
31 Integrating over the delivery period [T 1, T 2 ] yields the electricity forward price F (t, T 1, T 2 ) = Y (t)α(t, T 1, T 2 ) + φ (θ) α (1 α(t, T 1, T 2 )) α(t, T 1, T 2 ) = 1 (e ) α(t1 t) e α(t 2 t) α(t 2 T 1 ) Note that when T 1, T 2 are large, F (t, T 1, T 2 ) φ (θ)/α In the long end, forward prices vary very little
32 Dynamics of the forward price F (t, T 1, T 2 ) is a martingale (under Q) Q-dynamics of Y then yields, df (t, T 1, T 2 ) = α(t, T 1, T 2 ) d L θ (t)
33 α is the average of exp( α(u t)) over u [T 1, T 2 ] α i (t, T 1, T 2 ) = 1 T 2 T 1 T2 T 1 e α i (u t) du Hence, we have an average Samuelson effect exp( α(u t)) increasing when time to maturity u t goes to zero Volatility goes up as we approach delivery at time u Delivery over a period, so we average
34 Pricing of options on forwards Let g be the payoff of an option E.g, a put option g(x) = max(k x, 0) Call options require a damping factor in what follows (or one can use the put-call parity) Option price is p(t, T ; T 1, T 2 ) = e r(t t) E θ [max (K F (T, T 1, T 2 ), 0) F t ] Calculate this using Fourier transformation
35 The Fourier transform ĝ(y) = g(x) exp( ixy) dx The inverse Fourier transform: g(x) = 1 ĝ(y) exp(ixy) dy 2π From the dynamics of F we have F (T, T 1, T 2 ) = F (t, T 1, T 2 ) φ (θ) T T + α(s, T 1, T 2 ) dl(s) t t α(s, T 1, T 2 ) ds
36 By the independent increment property E θ [g(f (T, T 1, T 2 )) F t ] = 1 2π = 1 2π = 1 2π ĝ(y)e θ [ e iyf (T,T 1,T 2 ) F t ] dy ĝ(y)e iy(f (t,t 1,T 2 ) φ (θ) R [ T t α(s,t 1,T 2 ) ds) E θ e iy R ] T t α(s,t 1,T 2 ) L(s) dy ĝ(y)e iy(f (t,t 1,T 2 ) φ (θ) R T t α(s,t 1,T 2 ) ds) R T e t ψ θ (yα(s,t 1,T 2 )) ds dy
37 Fourier expression for option price ( the convolution product) p(t, T ; T 1, T 2 ) = e r(t t) (g Φ t,t ) (F (t, T 1, T 2 ) φ (θ) where T ( T ) Φ t,t (y) = exp ψ θ (y α(s, T 1, T 2 )) ds t t α(s, T 1, T 2 ) ds) Implementable using FFT techniques
38 Conclusions so far... A small introduction to the basics for Lévy processes Defined a multi-factor spot price model additive positive prices ensured possible to calibrate to data Pricing of electricity forwards on the spot model Explicit price and dynamics Possible to price options by transform methods
39 References Barndorff-Nielsen and Shephard (2001). Non-Gaussian OU based models and some of their uses in financial economics. J. Royal Statist. Soc. B, 63. Benth, Kallsen and Meyer-Brandis (2007). A non-gaussian OU process for electricity spot price modelling and derivatives pricing. Appl. Math. Finance, 14. Benth, Kiesel and Nazarova (2012). A critical empirical study of two electricity spot price models. To appear in Energy Economics
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 informationLecture 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 informationPricing 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 informationMonte 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 informationModelling 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 informationExplicit 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 informationMathematical 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 informationMath 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 informationOption 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 informationThe 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 informationMarkov modeling of Gas Futures
Markov modeling of Gas Futures p.1/31 Markov modeling of Gas Futures Leif Andersen Banc of America Securities February 2008 Agenda Markov modeling of Gas Futures p.2/31 This talk is based on a working
More informationOn the Valuation of Power-Reverse Duals and Equity-Rates Hybrids
On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids Oliver Caps oliver.caps@dkib.com RMT Model Validation Rates Dresdner Bank Examples of Hybrid Products Pricing of Hybrid Products using a
More informationSensitivity 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 informationIntroduction 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 informationNotes on Black-Scholes Option Pricing Formula
. Notes on Black-Scholes Option Pricing Formula by De-Xing Guan March 2006 These notes are a brief introduction to the Black-Scholes formula, which prices the European call options. The essential reading
More informationMarshall-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 informationSensitivity 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 information2 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 informationOn exponentially ane martingales. Johannes Muhle-Karbe
On exponentially ane martingales AMAMEF 2007, Bedlewo Johannes Muhle-Karbe Joint work with Jan Kallsen HVB-Institut für Finanzmathematik, Technische Universität München 1 Outline 1. Semimartingale characteristics
More informationA 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 informationELECTRICITY REAL OPTIONS VALUATION
Vol. 37 (6) ACTA PHYSICA POLONICA B No 11 ELECTRICITY REAL OPTIONS VALUATION Ewa Broszkiewicz-Suwaj Hugo Steinhaus Center, Institute of Mathematics and Computer Science Wrocław University of Technology
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationCaps and Floors. John Crosby
Caps and Floors John Crosby Glasgow University My website is: http://www.john-crosby.co.uk If you spot any typos or errors, please email me. My email address is on my website Lecture given 19th February
More information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 informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationPrivate 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 informationA spot market model for pricing derivatives in electricity markets
A spot market model for pricing derivatives in electricity markets Markus Burger, Bernhard Klar, Alfred Müller* and Gero Schindlmayr EnBW Gesellschaft für Stromhandel, Risk Controlling, Durlacher Allee
More informationMaster s Thesis. Pricing Constant Maturity Swap Derivatives
Master s Thesis Pricing Constant Maturity Swap Derivatives Thesis submitted in partial fulfilment of the requirements for the Master of Science degree in Stochastics and Financial Mathematics by Noemi
More informationStock Price Dynamics, Dividends and Option Prices with Volatility Feedback
Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback Juho Kanniainen Tampere University of Technology New Thinking in Finance 12 Feb. 2014, London Based on J. Kanniainen and R. Piche,
More informationwhere N is the standard normal distribution function,
The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at
More informationOn Black-Scholes Equation, Black- Scholes Formula and Binary Option Price
On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.
More informationTHE 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 informationAsian 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 informationPricing 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 informationDisability 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 informationFrom Binomial Trees to the Black-Scholes Option Pricing Formulas
Lecture 4 From Binomial Trees to the Black-Scholes Option Pricing Formulas In this lecture, we will extend the example in Lecture 2 to a general setting of binomial trees, as an important model for a single
More informationCOMPLETE 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 informationELECTRICITY SPOT PRICE MODELLING WITH A VIEW TOWARDS EXTREME SPIKE RISK
ELECTRICITY SPOT PRICE MODELLING WITH A VIEW TOWARDS EXTREME SPIKE RISK CLAUDIA KLÜPPELBERG, THILO MEYER-BRANDIS, AND ANDREA SCHMIDT Abstract. Sums of Lévy-driven Ornstein-Uhlenbeck processes seem appropriate
More informationSimulating Stochastic Differential Equations
Monte Carlo Simulation: IEOR E473 Fall 24 c 24 by Martin Haugh Simulating Stochastic Differential Equations 1 Brief Review of Stochastic Calculus and Itô s Lemma Let S t be the time t price of a particular
More informationQUANTIZED 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 informationARBITRAGE-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 informationPricing 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 informationVolatility 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 informationCreating Customer Value in Participating
in Participating Life Insurance Nadine Gatzert, Ines Holzmüller, and Hato Schmeiser Page 2 1. Introduction Participating life insurance contracts contain numerous guarantees and options: - E.g., minimum
More informationEstimating 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 informationGuaranteed Annuity Options
Guaranteed Annuity Options Hansjörg Furrer Market-consistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options Contents A. Guaranteed Annuity Options B. Valuation and Risk
More informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More informationOption 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 informationThe Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees
The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees Torsten Kleinow Heriot-Watt University, Edinburgh (joint work with Mark Willder) Market-consistent
More information4. Option pricing models under the Black- Scholes framework
4. Option pricing models under the Black- Scholes framework Riskless hedging principle Writer of a call option hedges his exposure by holding certain units of the underlying asset in order to create a
More informationJung-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 informationPricing Frameworks for Securitization of Mortality Risk
1 Pricing Frameworks for Securitization of Mortality Risk Andrew Cairns Heriot-Watt University, Edinburgh Joint work with David Blake & Kevin Dowd Updated version at: http://www.ma.hw.ac.uk/ andrewc 2
More informationBarrier Options. Peter Carr
Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?
More informationThe 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 informationA BOUND ON LIBOR FUTURES PRICES FOR HJM YIELD CURVE MODELS
A BOUND ON LIBOR FUTURES PRICES FOR HJM YIELD CURVE MODELS VLADIMIR POZDNYAKOV AND J. MICHAEL STEELE Abstract. We prove that for a large class of widely used term structure models there is a simple theoretical
More informationHedging 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 informationLecture. 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 information1 The Black-Scholes model: extensions and hedging
1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More informationOPTION PRICING FOR WEIGHTED AVERAGE OF ASSET PRICES
OPTION PRICING FOR WEIGHTED AVERAGE OF ASSET PRICES Hiroshi Inoue 1, Masatoshi Miyake 2, Satoru Takahashi 1 1 School of Management, T okyo University of Science, Kuki-shi Saitama 346-8512, Japan 2 Department
More informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationBlack-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 informationMore Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options
More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein
More informationValuation 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 informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: Discrete-Time Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationFour Derivations of the Black-Scholes Formula by Fabrice Douglas Rouah www.frouah.com www.volopta.com
Four Derivations of the Black-Scholes Formula by Fabrice Douglas Rouah www.frouah.com www.volota.com In this note we derive in four searate ways the well-known result of Black and Scholes that under certain
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationHedging 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 informationAsymptotics of discounted aggregate claims for renewal risk model with risky investment
Appl. Math. J. Chinese Univ. 21, 25(2: 29-216 Asymptotics of discounted aggregate claims for renewal risk model with risky investment JIANG Tao Abstract. Under the assumption that the claim size is subexponentially
More informationAnalytic Approximations for Multi-Asset Option Pricing
Analytic Approximations for Multi-Asset Option Pricing Carol Alexander ICMA Centre, University of Reading Aanand Venkatramanan ICMA Centre, University of Reading First Version: March 2008 This Version:
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon
More informationLife-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 informationExam MFE/3F Sample Questions and Solutions #1 to #76
Exam MFE/3F Sample Questions and Solutions #1 to #76 In this version, standard normal distribution values are obtained by using the Cumulative Normal Distribution Calculator and Inverse CDF Calculator
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationDiusion 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 informationForward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.
Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero
More informationRafał Weron. Hugo Steinhaus Center Wrocław University of Technology
Rafał Weron Hugo Steinhaus Center Wrocław University of Technology Options trading at Nord Pool commenced on October 29, 1999 with two types of contracts European-style Electric/Power Options (EEO/EPO)
More informationLecture 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 informationIN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED- RATE ANNUITIES in annual
W ORKSHOP B Y H A N G S U C K L E E Pricing Equity-Indexed Annuities Embedded with Exotic Options IN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED- RATE ANNUITIES in annual sales has declined from
More informationMarkovian 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 informationContents. 5 Numerical results 27 5.1 Single policies... 27 5.2 Portfolio of policies... 29
Abstract The capital requirements for insurance companies in the Solvency I framework are based on the premium and claim expenditure. This approach does not take the individual risk of the insurer into
More informationThe 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 informationOption 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 informatione.g. arrival of a customer to a service station or breakdown of a component in some system.
Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationHPCFinance: New Thinking in Finance. Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation
HPCFinance: New Thinking in Finance Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation Dr. Mark Cathcart, Standard Life February 14, 2014 0 / 58 Outline Outline of Presentation
More informationArbitrage-Free Pricing Models
Arbitrage-Free Pricing Models Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Arbitrage-Free Pricing Models 15.450, Fall 2010 1 / 48 Outline 1 Introduction 2 Arbitrage and SPD 3
More informationEssential Topic: Continuous cash flows
Essential Topic: Continuous cash flows Chapters 2 and 3 The Mathematics of Finance: A Deterministic Approach by S. J. Garrett CONTENTS PAGE MATERIAL Continuous payment streams Example Continuously paid
More informationOption Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)
Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.
More informationLECTURE 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 informationBlack-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 informationSemi-Markov model for market microstructure and HF trading
Semi-Markov model for market microstructure and HF trading LPMA, University Paris Diderot and JVN Institute, VNU, Ho-Chi-Minh City NUS-UTokyo Workshop on Quantitative Finance Singapore, 26-27 september
More informationOptimal 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 informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationMerton-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 informationLectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week
Lectures Sergei Fedotov 20912 - Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics
More informationPricing 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 informationMaximum likelihood estimation of mean reverting processes
Maximum likelihood estimation of mean reverting processes José Carlos García Franco Onward, Inc. jcpollo@onwardinc.com Abstract Mean reverting processes are frequently used models in real options. For
More informationFrom Exotic Options to Exotic Underlyings: Electricity, Weather and Catastrophe Derivatives
From Exotic Options to Exotic Underlyings: Electricity, Weather and Catastrophe Derivatives Dr. Svetlana Borovkova Vrije Universiteit Amsterdam History of derivatives Derivative: a financial contract whose
More informationLecture 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