Private Equity Fund Valuation and Systematic Risk

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Private Equity Fund Valuation and Systematic Risk"

Transcription

1 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 3 Passau University

2 Table of Contents I 1 Motivation 2 3 Assumptions Derivation and Result Numerical Illustration 4 Derivation and Result Numerical Illustration 5

3 General Problem The illiquid character of private equity investments poses particular challenges for private equity (PE) research. As PE investments are not traded on secondary markets, liquidity is low and observable market prices are missing. Lack of Liquidity is a critical issue in evaluating PE investments. Therefore, valuation is a crucial task. The optimal allocation of PE in a investment portfolio typically requires the risk-return characteristics of PE as an asset class.

4 Contents Motivation Model PE as an asset class under the simplifying assumption of a frictionless market and knowledge of investor preferences. Derive a first value assessment (upper boundary) assuming no private valuation information is available. Derive the theoretical (boundary) value of a typical private equity fund based on equilibrium asset pricing considerations. Valuation is based on a stochastic model of a typical fund s cash flows, capital drawdowns and distributions.

5 General Assumptions and Notation Assumption Assumption 1: We consider a Private Equity fund with total (initial) commitments denoted by C. Assumption 2: The PE fund has a total (legal) maturity T l and a commitment period T c, where T l T c must hold. Cumulated capital drawdowns up to t are denoted D t, undrawn committed amounts up to time t are U t. Under these assumptions it must hold: D t = C U t, where D = and U = C.

6 Cumulated Capital Drawdowns I Assumption Assumption 3: Capital drawdowns over the commitment period T c occur in continuous-time. The dynamics of the cumulated drawdowns D t can be described by the ordinary differential equation: dd t = δ t U t 1 { t Tc }dt, where δ t denotes the rate of contribution or simply the fund s drawdown rate at time t. 1 { } is an indicator function.

7 Cumulated Capital Drawdowns II The solution of the ordinary differential equation (1) can be derived by substituting the identity dd t = du t, and using the initial condition U = C. It follows: D t = C C exp dd t = δ t C exp t t δ u du, where t T c, δ u du 1 { t Tc }dt.

8 Stochastic Process Drawdown Rate Assumption Assumption 4: The drawdown rate is modeled by a non-negative stochastic process {δ t, t T c }. In particular, the mean-reverting square root process dδ t = κ(θ δ t )dt + σ δ δt db δ,t is applied, where θ > denotes the long-run mean of the drawdown rate, κ > governs the rate of reversion to this mean and σ δ > reflects the volatility of the drawdown rate. B δ,t is a standard Brownian motion.

9 Simulated Capital Drawdowns I.25 1 Capital drawdowns Cumulated capital drawdowns Figure: Simulated Paths of the Capital Drawdowns (Left) and Cumulated Capital Drawdowns (Right); Parameters: C = 1, κ = 2, θ = 1, δ =.1 and σ δ =.2

10 Simulated Capital Drawdowns II.25 1 Capital drawdowns Cumulated capital drawdowns Figure: Simulated Paths of the Capital Drawdowns (Left) and Cumulated Capital Drawdowns (Right); Parameters: C = 1, κ = 2, θ = 1, δ =.1 and σ δ =.5

11 Simulated Capital Drawdowns III.2 Capital drawdowns Cumulated capital drawdowns Figure: Simulated Paths of the Capital Drawdowns (Left) and Cumulated Capital Drawdowns (Right); Parameters: C = 1, κ =.5, θ = 1, δ =.1 and σ δ =.2

12 Consistency Test: Capital Drawdowns I.18 Cumulated capital drawdowns Capital drawdowns Figure: Full Sample of Liquidated Funds (N = 95); Data set from Venture Economics, January 198 to June 23

13 Consistency Test: Capital Drawdowns II.2 Cumulated capital drawdowns Capital drawdowns Figure: VC-Funds of the Sample of Liquidated Funds (N = 47)

14 Consistency Test: Capital Drawdowns III.2 Cumulated capital drawdowns Capital drawdowns Figure: BO-Funds of the Sample of Liquidated Funds (N = 48)

15 General Assumptions and Notation Assumption Assumption 1: (Non-negative) capital distributions of the PE fund to its investors occur in continuous-time over the fund s legal lifetime T l. Assumption 2: Instantaneous capital distributions p t are assumed to be log-normally distributed according to d lnp t = µ t dt + σ P db P,t, where µ t denotes the time dependent drift and σ P the constant volatility of the stochastic process. B P,t is a standard Brownian motion that is uncorrelated with B δ,t, i.e., Cov t [db P,t, db δ,t ] =.

16 Multiple of the Fund Definition: The funds multiple M t is defined by: M t = t p sds/c. Expected instantaneous capital distributions are related to the fund multiple by the relationship: E t [p t ] = E t [dm t /dt]c. Assumption Assumption 3: The funds expected multiple M t is assumed to follow the ordinary differential equation: E s [dm t ] = α t (m E s [M t ])dt, s t, where m is the multiple s long-run mean and α is the constant speed of reversions to this mean.

17 Stochastic Process Capital Distributions From Assumption 2 it follows: [ t ] p t = p s exp µ u du + σ P (B P,t B P,s ). s If the multiple M t satisfies Assumption 3, it can be shown that the stochastic process of the instantaneous capital distributions is given by: { p t = αt(mc P s )exp 1 } 2 {α(t2 s 2 ) + σp(t 2 s)] + σ P (B P,t B P,s ), where (B P,t B P,s ) = ǫ t t s with ǫt N(, 1).

18 Simulated Capital Distributions I Capital distributions Cumulated capital distributions Figure: Simulated Paths of the Capital Distributions (Left) and Cumulated Capital Distributions (Right); Parameters: C = 1, m = 1.5, α =.5 and σ P =.2

19 Simulated Capital Distributions II Capital distributions Cumulated capital distributions Figure: Simulated Paths of the Capital Distributions (Left) and Cumulated Capital Distributions (Right); Parameters: C = 1, m = 1.5, α =.5 and σ P =.4

20 Simulated Capital Distributions III Capital distributions Cumulated capital distributions Figure: Simulated Paths of the Capital Distributions (Left) and Cumulated Capital Distributions (Right); Parameters: C = 1, m = 1.5, α =.2 and σ P =.2

21 Consistency Test: Capital Distributions I Cumulated capital distributions Capital distributions Figure: Full Sample of Liquidated Funds (N = 95); Data set from Venture Economics, January 198 to June 23

22 Consistency Test: Capital Distributions II Cumulated capital distributions Capital distributions Figure: VC-Funds of the Sample of Liquidated Funds (N = 47)

23 Consistency Test: Capital Distributions III Cumulated capital distributions Capital distributions Figure: BO-Funds of the Sample of Liquidated Funds (N = 48)

24 Simulated Fund Cash Flows Net cash flows Cumulated net cash flows Figure: Simulated Paths of the Net Cash Flows (Left) and Cumulated Net Cash Flows (Right); Parameters: C = 1, κ = 2, θ = 1, δ =.1, σ δ =.2, m = 1.5, α =.5 and σ P =.2

25 Assumptions Derivation and Result Numerical Illustration Assumptions Underlying the Valuation Assumption Assumption 1: PE funds are priced under the risk neutral measure as if they were traded in a frictionless capital market in equilibrium (upper price boundary). Assumption 2: There exists a representative investor with log utility such that in equilibrium expected returns of all assets are generated by a specialized version of the ICAPM. Assumption 3: The drawdown Rate δ t is uncorrelated with returns of the market portfolio. Assumption 4: The covariance σ PM between log capital distributions and returns of the market portfolio is constant.

26 Derivation I Assumptions Derivation and Result Numerical Illustration From Assumption 1 the (upper boundary) market value of a fund Vt F is defined as [ ] [ Tl ] Tl Vt F = E Q e r f (τ t) dp τ F t E Q e r f (τ t) dd τ 1 {t Tc } F t = V P t V D t, t where Vt P (Vt D ) is the present value of all outstanding capital distributions (drawdowns) at time t. Discounting by the riskless rate r f is valid, as all expectations are defined under risk-neutral or equivalent martingale measure Q t

27 Derivation II Assumptions Derivation and Result Numerical Illustration Applying Girsanov s Theorem, it follows that the underlying stochastic processes under the Q-measure are given by: dδ t = [κ (θ δ t ) λ δ σ δ δt ] dt + σ δ δt db Q δ,t, d ln p t = (µ t λ P σ P )dt + σ P db Q P,t, where B Q δ,t and BQ P,t are Q-Brownian motions; λ δ and λ P are market prices of risk, defined by: λ δ µ(δ t, t) r f σ(δ t, t), λ P µ(p t, t) r f σ(p t, t).

28 Result Assumptions Derivation and Result Numerical Illustration From Assumptions 2-4 the market prices of risk are given by λ δ = and λ P = σ PM /σ P. The (upper boundary) value of a private equity fund at any time t [, T l ] during its finite lifetime T l can then be stated as: Tl Vt F =α (m C P t ) e r f (τ t) e C(t,τ) dτ t Tl + U t e r f (τ t) (A (t, τ) B (t, τ)δ t )e A(t,τ) B(t,τ)δt dτ1 {t Tc }, t where A(t, T), B(t, T) and C(t, T) are known deterministic functions.

29 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time I 1.5 Value Fund lifetime (in years) Riskless rate.3 Figure: Over the Fund s Lifetime for Varying Values of r f

30 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time II Value m Fund lifetime (in years) Figure: Over the Fund s Lifetime for Varying Values of m

31 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time III 1.5 Value alpha Fund lifetime (in years) 2 Figure: Over the Fund s Lifetime for Varying Values of α

32 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time IV 1.5 Value Fund lifetime (in years) Covariance.1 Figure: Over the Fund s Lifetime for Varying Values of σ PM

33 Assumptions Derivation and Result Numerical Illustration Dynamics of the s over Time V Value Theta Fund lifetime (in years) Figure: Over the Fund s Lifetime for Varying Values of θ

34 Derivation I Derivation and Result Numerical Illustration Same assumptions that were employed to derive the market value (Assumptions 1-4, Section ). The (annualized) instantaneous return R F t of a private equity fund at time t is defined by: R F t = dv F t dt + dpt dt ddt dt Vt F = dv P t dt dv D t dt + dpt dt ddt dt Vt P Vt D. Taking the conditional expectation E P t [ ], the expected instantaneous fund return µ F t is given by: µ F t = E P t [ R F t ] = E P t [ ] dv P t dt Et P E P t [ dv D t dt ] + E P t [ dpt dt [ V P t ] E P t [V D t ] ] [ E P ddt ] t dt.

35 Derivation II Derivation and Result Numerical Illustration From Assumptions 1-4, it must hold that: [ ] [ dv Et P D t = Et P ddt dt dt [ ] [ dv Et P P t = Et P dpt dt dt ] + r f V D t ] + (r f + σ PM )V P t. Remark: Equilibrium where expected returns are generated by the specialized version of the ICAPM (Assumptions 1-2). The first equation holds as drawdowns are assumed to carry zero systematic risk (Assumptions 3). The second equation holds as constant covariance σ PM is assumed for distributions (Assumptions 4).

36 Result I Derivation and Result Numerical Illustration Substituting the previous results, the expected instantaneous fund return is given by: µ F Vt P t = r f + σ PM Vt P Vt D We can also view the expected fund returns from a traditional beta perspective. It follows: µ F t = r f + β F,t (µ M r f ) where µ M is the expected return of the market portfolio and β F,t is the beta coefficient of the fund at time t given by: β F,t = β P V P t Vt P Vt D with β P = σ PM /σ 2 M..

37 Result II Derivation and Result Numerical Illustration Or more generally, when the beta coefficient of the capital drawdowns β D : β F,t = β P V P t Vt P Vt D β D V P V D t t Vt D, i.e., the beta coefficient of the fund returns is the market value weighted average of the betas of the fund s capital distributions and drawdowns. This result implies that the beta coefficient of the fund is non-stationary whenever β P β D holds, i.e., capital distributions and drawdowns carry different levels of systematic risks. Quite similar result to Brennan (1973), Myers and Turnbull (1977) and Turnbull (1977) on the systematic risk of firms.

38 Derivation and Result Numerical Illustration Expected Return and over Time I Expected fund return Figure: Ex-ante Expected Returns Over the Fund s Lifetime Parameters choice: C = 1, T c = T l = 2, r f =.5, κ =.5, θ =.5, σ δ =.1, δ =.5, m = 1.5, α =.25, σ P =.2, σ PM =.5, µ M =.1125 and σ 2 M =.626.

39 Derivation and Result Numerical Illustration Expected Return and over Time II Table: Beta Coefficients Over the Lifetime of the Fund The Table shows the evolution of Vt P, V t P, (V t P/V t P Vt D ) and of the beta coefficient β F,t over the lifetime of the fund. Time t (in years) V P t V D t V P t V P t V D t β F,t

40 Conclusion We perform stochastic modeling and equilibrium pricing of private equity funds as an asset class. The paper provides a solution for the upper boundary market value of private equity funds with typical drawdown and distribution characteristics. Model parameters can be calibrated from a cross-section of historical cash flow data of private equity funds.

41 Outlook Motivation Future work may address the following points: Calibration and estimation for larger (available?) data sets. Pricing results and (short term?) emergence of secondary markets. Incorporate an illiquidity discount in the pricing model.

Modeling the Cash Flow Dynamics of Private Equity Funds Theory and Empirical Evidence

Modeling the Cash Flow Dynamics of Private Equity Funds Theory and Empirical Evidence Modeling the Cash Flow Dynamics of Private Equity Funds Theory and Empirical Evidence Axel Buchner, Christoph Kaserer and Niklas Wagner This Version: February 29 Axel Buchner is at the Technical University

More information

Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback

Stock 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 information

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 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 information

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model

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

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

Hedging Variable Annuity Guarantees

Hedging Variable Annuity Guarantees p. 1/4 Hedging Variable Annuity Guarantees Actuarial Society of Hong Kong Hong Kong, July 30 Phelim P Boyle Wilfrid Laurier University Thanks to Yan Liu and Adam Kolkiewicz for useful discussions. p. 2/4

More information

The Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014

The Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA 1.

More information

WORKING PAPER SERIES

WORKING PAPER SERIES Working Paper No. 2006-02 Stochastic Modeling of Private Equity An Equilibrium Based Approach to Fund Valuation AXEL BUCHNER CHRISTOPH KASERER NIKLAS WAGNER WORKING PAPER SERIES Center for Entrepreneurial

More information

Option Pricing. 1 Introduction. Mrinal K. Ghosh

Option Pricing. 1 Introduction. Mrinal K. Ghosh Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified

More information

Simulating Stochastic Differential Equations

Simulating 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 information

Expected default frequency

Expected default frequency KM Model Expected default frequency Expected default frequency (EDF) is a forward-looking measure of actual probability of default. EDF is firm specific. KM model is based on the structural approach to

More information

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50

More information

Arbitrage-Free Pricing Models

Arbitrage-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 information

News Trading and Speed

News Trading and Speed Preliminary and Incomplete Please do not quote! News Trading and Speed Thierry Foucault Johan Hombert Ioanid Roşu April 1, 1 Abstract Adverse selection occurs in financial markets because certain investors

More information

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On 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 information

Pricing American Options without Expiry Date

Pricing American Options without Expiry Date Pricing American Options without Expiry Date Carisa K. W. Yu Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom, Hong Kong E-mail: carisa.yu@polyu.edu.hk Abstract This paper

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

金融隨機計算 : 第一章. Black-Scholes-Merton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ.

金融隨機計算 : 第一章. Black-Scholes-Merton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. 金融隨機計算 : 第一章 Black-Scholes-Merton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. Tsing-Hua Univ. Derivative Contracts Derivatives, also called contingent claims, are

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

Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk

Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day

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

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails 12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

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

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the

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

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

Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance. Brennan Schwartz (1976,1979) Brennan Schwartz

Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance. Brennan Schwartz (1976,1979) Brennan Schwartz Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance Brennan Schwartz (976,979) Brennan Schwartz 04 2005 6. Introduction Compared to traditional insurance products, one distinguishing

More information

International Stock Market Integration: A Dynamic General Equilibrium Approach

International Stock Market Integration: A Dynamic General Equilibrium Approach International Stock Market Integration: A Dynamic General Equilibrium Approach Harjoat S. Bhamra London Business School 2003 Outline of talk 1 Introduction......................... 1 2 Economy...........................

More information

Fundamental Capital Valuation for IT Companies: A Real Options Approach

Fundamental Capital Valuation for IT Companies: A Real Options Approach Capital Valuation for IT Companies: A Real Options Approach Chung Baek 1 Brice Dupoyet 2 Arun Prakash 3 Abstract This study attempts to estimate the fundamental capital value of a growing firm by combining

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

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

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

Valuation of Lease Contracts In Continuous Time With Stochastic Asset Values

Valuation of Lease Contracts In Continuous Time With Stochastic Asset Values Valuation of Lease Contracts In Continuous Time With Stochastic Asset Values Riaz Hussain *+ * The University of Scranton Abstract A lease is a derivative security the value of which depends upon the value

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

The Black-Scholes Model

The Black-Scholes Model The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

More information

More 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 More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path

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

Barrier Options. Peter Carr

Barrier 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 information

1 The Black-Scholes model: extensions and hedging

1 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 information

The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models

The 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 information

Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing

Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common

More information

Quanto Adjustments in the Presence of Stochastic Volatility

Quanto Adjustments in the Presence of Stochastic Volatility Quanto Adjustments in the Presence of tochastic Volatility Alexander Giese March 14, 01 Abstract This paper considers the pricing of quanto options in the presence of stochastic volatility. While it is

More information

Single period modelling of financial assets

Single period modelling of financial assets Single period modelling of financial assets Pål Lillevold and Dag Svege 17. 10. 2002 Single period modelling of financial assets 1 1 Outline A possible - and common - approach to stochastic modelling of

More information

Tutorial: Structural Models of the Firm

Tutorial: Structural Models of the Firm Tutorial: Structural Models of the Firm Peter Ritchken Case Western Reserve University February 16, 2015 Peter Ritchken, Case Western Reserve University Tutorial: Structural Models of the Firm 1/61 Tutorial:

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

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

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

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

Properties of the SABR model

Properties of the SABR model U.U.D.M. Project Report 2011:11 Properties of the SABR model Nan Zhang Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2011 Department of Mathematics Uppsala University ABSTRACT

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

Options 1 OPTIONS. Introduction

Options 1 OPTIONS. Introduction Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or

More information

Modeling Counterparty Credit Exposure

Modeling Counterparty Credit Exposure Modeling Counterparty Credit Exposure Michael Pykhtin Federal Reserve Board PRMIA Global Risk Seminar Counterparty Credit Risk New York, NY May 14, 2012 The opinions expressed here are my own, and do not

More information

The Alpha and Beta of Private Equity Investments

The Alpha and Beta of Private Equity Investments The Alpha and Beta of Private Equity Investments Axel Buchner University of Passau, Germany October 24, 2014 IwouldliketothankOlegGredil, LudovicPhallippou, andrüdigerstuckeforhelpfulcommentsand discussions.

More information

On ADF Goodness of Fit Tests for Stochastic Processes. Yury A. Kutoyants Université du Maine, Le Mans, FRANCE

On ADF Goodness of Fit Tests for Stochastic Processes. Yury A. Kutoyants Université du Maine, Le Mans, FRANCE On ADF Goodness of Fit Tests for Stochastic Processes Yury A. Kutoyants Université du Maine, Le Mans, FRANCE e-mail: kutoyants@univ-lemans.fr Abstract We present several Goodness of Fit tests in the case

More information

When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment

When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment Siwei Gan, Jin Zheng, Xiaoxia Feng, and Dejun Xie Abstract Refinancing refers to the replacement of an existing debt obligation

More information

How to Manage the Maximum Relative Drawdown

How to Manage the Maximum Relative Drawdown How to Manage the Maximum Relative Drawdown Jan Vecer, Petr Novotny, Libor Pospisil, Columbia University, Department of Statistics, New York, NY 27, USA April 9, 26 Abstract Maximum Relative Drawdown measures

More information

Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price

Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price Exchange Options Consider the Double Index Bull (DIB) note, which is suited to investors who believe that two indices will rally over a given term. The note typically pays no coupons and has a redemption

More information

On Market-Making and Delta-Hedging

On Market-Making and Delta-Hedging On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide

More information

Models Used in Variance Swap Pricing

Models Used in Variance Swap Pricing Models Used in Variance Swap Pricing Final Analysis Report Jason Vinar, Xu Li, Bowen Sun, Jingnan Zhang Qi Zhang, Tianyi Luo, Wensheng Sun, Yiming Wang Financial Modelling Workshop 2011 Presentation Jan

More information

Numerical Methods for Option Pricing

Numerical Methods for Option Pricing Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly

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

Creating Customer Value in Participating

Creating 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 information

Dimitri Vayanos and Pierre-Olivier Weill: A Search-Based Theory of the On-the-Run Phenomenon

Dimitri Vayanos and Pierre-Olivier Weill: A Search-Based Theory of the On-the-Run Phenomenon Dimitri Vayanos and Pierre-Olivier Weill: A Search-Based Theory of the On-the-Run Phenomenon Presented by: András Kiss Economics Department CEU The on-the-run phenomenon Bonds with almost identical cash-flows

More information

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13. Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.

More information

Stock Loans in Incomplete Markets

Stock Loans in Incomplete Markets Applied Mathematical Finance, 2013 Vol. 20, No. 2, 118 136, http://dx.doi.org/10.1080/1350486x.2012.660318 Stock Loans in Incomplete Markets MATHEUS R. GRASSELLI* & CESAR GÓMEZ** *Department of Mathematics

More information

Calibration of Stock Betas from Skews of Implied Volatilities

Calibration of Stock Betas from Skews of Implied Volatilities Calibration of Stock Betas from Skews of Implied Volatilities Jean-Pierre Fouque University of California Santa Barbara Joint work with Eli Kollman (Ph.D. student at UCSB) Joint Seminar: Department of

More information

α α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =

More information

Pricing and Risk Management of Variable Annuity Guaranteed Benefits by Analytical Methods Longevity 10, September 3, 2014

Pricing and Risk Management of Variable Annuity Guaranteed Benefits by Analytical Methods Longevity 10, September 3, 2014 Pricing and Risk Management of Variable Annuity Guaranteed Benefits by Analytical Methods Longevity 1, September 3, 214 Runhuan Feng, University of Illinois at Urbana-Champaign Joint work with Hans W.

More information

The Term Structure of Lease Rates with Endogenous Default Triggers and Tenant Capital Structure: Theory and Evidence

The Term Structure of Lease Rates with Endogenous Default Triggers and Tenant Capital Structure: Theory and Evidence The Term Structure of Lease Rates with Endogenous Default Triggers and Tenant Capital Structure: Theory and Evidence Sumit Agarwal, Brent W. Ambrose Hongming Huang, and Yildiray Yildirim Abstract This

More information

Problem Set 4 Solutions

Problem Set 4 Solutions Chemistry 360 Dr Jean M Standard Problem Set 4 Solutions 1 Two moles of an ideal gas are compressed isothermally and reversibly at 98 K from 1 atm to 00 atm Calculate q, w, ΔU, and ΔH For an isothermal

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

Finite Differences Schemes for Pricing of European and American Options

Finite Differences Schemes for Pricing of European and American Options Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the Black-Scholes

More information

Simple approximations for option pricing under mean reversion and stochastic volatility

Simple approximations for option pricing under mean reversion and stochastic volatility Simple approximations for option pricing under mean reversion and stochastic volatility Christian M. Hafner Econometric Institute Report EI 2003 20 April 2003 Abstract This paper provides simple approximations

More information

ESTIMATING THE VALUE OF DELIVERY OPTIONS IN FUTURES CONTRACTS

ESTIMATING THE VALUE OF DELIVERY OPTIONS IN FUTURES CONTRACTS ESTIMATING THE VALUE OF DELIVEY OPTIONS IN FUTUES CONTACTS Jana Hranaiova U.S. Commodities Futures Trading Commission and University of New Mexico obert A. Jarrow Cornell University and Kamakura Corporation

More information

Martingale Pricing Applied to Options, Forwards and Futures

Martingale 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 information

Vanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013

Vanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013 Vanna-Volga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla

More information

Binomial lattice model for stock prices

Binomial lattice model for stock prices Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }

More information

Stochastic Modelling and Forecasting

Stochastic Modelling and Forecasting Stochastic Modelling and Forecasting Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH RSE/NNSFC Workshop on Management Science and Engineering and Public Policy

More information

ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES

ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES by Xiaofeng Qian Doctor of Philosophy, Boston University, 27 Bachelor of Science, Peking University, 2 a Project

More information

Psychology and Economics (Lecture 17)

Psychology and Economics (Lecture 17) Psychology and Economics (Lecture 17) Xavier Gabaix April 13, 2004 Vast body of experimental evidence, demonstrates that discount rates are higher in the short-run than in the long-run. Consider a final

More information

A CLOSED-FORM OPTION VALUATION FORMULA IN MARKOV JUMP DIFFUSION MODELS

A CLOSED-FORM OPTION VALUATION FORMULA IN MARKOV JUMP DIFFUSION MODELS A CLOSED-FORM OPTION VALUATION FORMULA IN MARKOV JUMP DIFFUSION MODELS CHENG-DER FUH Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, R.O.C. SHIH-KUEI LIN Department of Finance, National

More information

LECTURE 10.1 Default risk in Merton s model

LECTURE 10.1 Default risk in Merton s model LECTURE 10.1 Default risk in Merton s model Adriana Breccia March 12, 2012 1 1 MERTON S MODEL 1.1 Introduction Credit risk is the risk of suffering a financial loss due to the decline in the creditworthiness

More information

4. Option pricing models under the Black- Scholes framework

4. 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 information

Monte Carlo Simulation of Stochastic Processes

Monte Carlo Simulation of Stochastic Processes Monte Carlo Simulation of Stochastic Processes Last update: January 10th, 2004. In this section are presented the steps to perform the simulation of the main stochastic processes used in real options applications,

More information

Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.

Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options. Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rate. Part D Introduction to derivatives. Forwards

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

The Black-Scholes Formula

The Black-Scholes Formula ECO-30004 OPTIONS AND FUTURES SPRING 2008 The Black-Scholes Formula The Black-Scholes Formula We next examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they

More information

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative

More information

Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management

Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management Vincenzo Bochicchio, Niklaus Bühlmann, Stephane Junod and Hans-Fredo List Swiss Reinsurance Company Mythenquai 50/60, CH-8022

More information

Financial Modeling. Class #06B. Financial Modeling MSS 2012 1

Financial Modeling. Class #06B. Financial Modeling MSS 2012 1 Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. Black-Scholes-Merton formula 2. Binomial trees

More information

An exact formula for default swaptions pricing in the SSRJD stochastic intensity model

An exact formula for default swaptions pricing in the SSRJD stochastic intensity model An exact formula for default swaptions pricing in the SSRJD stochastic intensity model Naoufel El-Bachir (joint work with D. Brigo, Banca IMI) Radon Institute, Linz May 31, 2007 ICMA Centre, University

More information

Valuation of commodity derivatives when spot prices revert to a cyclical mean

Valuation of commodity derivatives when spot prices revert to a cyclical mean Valuation of commodity derivatives when spot prices revert to a cyclical mean April, 24 Abstract This paper introduces a new continuous-time model based on the logarithm of the commodity spot price assuming

More information

INTEREST RATES AND FX MODELS

INTEREST RATES AND FX MODELS INTEREST RATES AND FX MODELS 8. Portfolio greeks Andrew Lesniewski Courant Institute of Mathematical Sciences New York University New York March 27, 2013 2 Interest Rates & FX Models Contents 1 Introduction

More information

ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments -

ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments - AARHUS SCHOOL OF BUSINESS AARHUS UNIVERSITY MASTER THESIS ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments - AUTHORS: DANIELA ZABRE GEORGE RARES RADU SIMIAN SUPERVISOR:

More information

Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans

Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Challenges for defined contribution plans While Eastern Europe is a prominent example of the importance of defined

More information

Markov modeling of Gas Futures

Markov 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 information

Pricing and Hedging of Oil Futures - A Unifying Approach -

Pricing and Hedging of Oil Futures - A Unifying Approach - Pricing and Hedging of Oil Futures - A Unifying Approach - Wolfgang Bühler, Olaf Korn, Rainer Schöbel* Contact information: Wolfgang Bühler Olaf Korn Rainer Schöbel University of Mannheim University of

More information

The Fair Valuation of Life Insurance Participating Policies: The Mortality Risk Role

The Fair Valuation of Life Insurance Participating Policies: The Mortality Risk Role The Fair Valuation of Life Insurance Participating Policies: The Mortality Risk Role Massimiliano Politano Department of Mathematics and Statistics University of Naples Federico II Via Cinthia, Monte S.Angelo

More information

The 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 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 information