Hedging Variable Annuity Guarantees

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Hedging Variable Annuity Guarantees"

Transcription

1 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.

2 p. 2/4 Outline Introduction Dealing with embedded options Hedging: Some practical spects The Guaranteed Minimum Withdrawal Benefit(GMWB) Basic features Valuation of basic GMWB contract Semi static hedging Concluding remarks

3 p. 3/4 Background: Variable Annuities Investors like to have possibility of upside appreciation Investors also concerned about downside risk Many insurance and annuity products that combine both features Structured products, Equity Indexed Annuities and Variable Annuities(VA) These products contain different types of exotic options

4 p. 4/4 New product: Sequence of Steps Aim: Profitable return on capital Design the contract Manage the risks Risk management Reserving hedging the embedded options Pricing

5 p. 5/4 Ways to Hedge Buy options from another institution Can be expensive, Lapses Farm out hedging to a third party Set up managed account. Hedge in house Keeps control. Need resources. Hedging program can become politicized

6 p. 6/4 Hedging Program: Issues Need very clear objectives P and L should break even. Not a profit centre Educate senior management B of Directors may not fully understand hedging Communicating hedging performance Important Attribution of performance Which metrics to use.

7 p. 7/4 Dynamic Hedging (DH) Calibrate model to market options Value embedded options using calibrated model Construct portfolio of traded assets with same Greeks as liability portfolio Repeat this procedure periodically Rebalance the asset portfolio each time Update for lapses and new business

8 p. 8/4 Features of DH Does not protect against model risk Suppose market jumps. Gives rise to gap risk DH does not work if markets are closed for some reason eg post nine eleven. DH can be a challenge if gamma changes sign frequently (cliquet options) Dynamic hedging assumes you know the model, continuous prices, that markets are open and there is enough liquidity.

9 p. 9/4 Semi Static Hedging (SSH) SSH overcomes many of these problems. Use example Hedging a ten year European option with a portfolio of one year standard options Construct a portfolio of the short term options that replicates the long term option There is an exact expression for this. Assume t < τ < T. S t stock price at t

10 p. 10/4 SSH Formula Let V t (S t, T) be market price at time t, of long option to be hedged. We have (Carr Wu) (1) V t (S t, T) = k=0 g(k)p t (S t, t,τ, k)dk where P t (S t, t, τ, k) is the European put price at time t with maturity τ and strike k.

11 p. 11/4 SSH Formula The coefficients g are given by g(k) = 2 V (S, τ, T) S 2 S=k These correspond to the gammas of the long term derivative at the horizon time τ with stock price equal to k. Can approximate this integral with a finite sum.

12 p. 12/4 GMWB GMWB adds a guaranteed floor of withdrawal benefits to a VA Provides a guaranteed level of income to the policyholder. Suppose investor puts $100,000 in a VA Policyholder can withdraw a certain fixed percentage every year until the initial premium is withdrawn

13 p. 13/4 GMWB Assume the withdrawal rate is 7% per annum. Our policyholder could withdraw $7,000 each year until the total withdrawals reach $100,000. This takes years. Note the policyholder can withdraw the funds irrespective of how the investment account performs Here is an example. Market does well at first and then collapses.

14 p. 14/4 Example Year Rate Fund before Fund after Amount Balance on fund withdrawal withdrawal withdrawn remaining 1 10% 110, ,000 7,000 93, % 113, ,300 7,000 86, % 42,520 35,520 7,000 79, % 14,208 7, 208 7,000 72, % 7,000 Zero 7,000 65,000 6 r% 0 0 7,000 58, r% 0 0 7,

15 p. 15/4 The GMWB The GMWB guarantees a fixed level of income no matter what happens to the market Fee for the GMWB is expressed as a percentage (say fifty basis points) of either The investment account or The outstanding guaranteed withdrawal benefit.

16 p. 16/4 Assumptions Perfect frictionless market Ignore lapses, partial withdrawals mortality etc. Assume max amount taken each year Fixed term contract over [0, T]. Index investment fund dynamics di t = µi t dt + σi t db t where B t is a Brownian motion under P µ is the drift σ is the volatility.

17 p. 17/4 Investor s account Let A t be the value of the investors account at time t. A has an absorbing barrier at zero. Suppose first time it hits zero is τ. Dynamics of A for 0 < t < τ are da t = [(µ q)a t g]dt + σa t db t where q is the fee and g is the withdrawal rate. If the initial investment amount is A 0 then g = A 0 T.

18 p. 18/4 Pricing the contract Put Option decomposition Investment often in mutual fund. Insurer guarantees to pay remaining withdrawal benefits if A(t) reaches zero in [0, T]. Guarantee provided by the insurance is a put option with random exercise time. If policyholder s investment account stays positive there is no payment under the put option. The option is exercised automatically when the account balance first becomes zero.

19 p. 19/4 Put Option decomposition When this happens the insurer agrees to pay the remaining stream of withdrawal benefits of g per annum. T τ g e ru du Put option has a random exercise time τ. Put is funded by the fee payable until time τ. Guarantees backed solely by the claim paying ability of XYZ insurance Co.

20 p. 20/4 A reaches zero in year Account value Exercise Time

21 p. 21/4 Valuation of GMWB Use numerical methods to value the contract (Monte Carlo or pde) Here we just value a very simple contract assuming full utilization. We ignore lapses and mortality and utilization choice. (Could include them.) Assume simple lognormal model for convenience and deterministic interest rates.

22 p. 22/4 Numerical Example Benchmark contract, fifteen year term. No lapses no deaths. All policyholders start to withdraw funds at max rate from the outset. Input parameters are Parameter Symbol Benchmark value Initial investment A Contract term T 15 years Withdrawal rate g Volatility σ 0.20 Riskfree rate r 0.05 Now compute pv contributions and put prices for different values of q.

23 p. 23/4 Put values for basic GMWB Value of q Present value of Put option basis points Contributions

24 p. 24/4 Contributions and put option

25 p. 25/4 No arbitrage Values For this example the no arbitrage Value of q is q = Contributions and Put values when q = Entity Value (sd) Value of contributions (0.0003) Value of put option (0.0008)

26 p. 26/4 Sensitivity to interest rate Suppose we have benchmark contract. Fix q = Explore sensitivity of put value and present value of contributions to inputs. First vary the interest rate Interest Assumption Present Value of Put Option Contributions

27 p. 27/4 Sensitivity to interest rate Contribution & put values Interest rate

28 p. 28/4 Sensitivity to volatility Now vary the volatility Volatility Present Value of Put Option Assumption Contributions

29 p. 29/4 Sensitivity to volatility 8 7 Contribution & put values Volatility

30 p. 30/4 Sensitivity to Account Value Vary A 0 and keep g fixed at = A 0 Present Value of Put Option Contributions Conclusion: Moneyness matters

31 p. 31/4 Semi static Hedging SSH Hedge the risk with a portfolio of one year options Re hedge after one year This overcomes some of the disadvantages of delta hedging SSH hedging is more robust than delta hedging For GMWB there is mild path dependence

32 p. 32/4 Distribution of Index in one year

33 p. 33/4 Evolution of A over year Pick one sample path. Withdraw per quarter. Time t I t A t A 1 = for up down path.

34 p. 34/4 Evolution of A over year Pick another sample path. Withdraw per quarter. Time t I t A t A 1 = for down up path.

35 p. 35/4 Distribution of Liability in one year For each value of the index at time one I 1 there is a range of A 1. Conditional on I 1 the future GMWB liability V 1 (I 1 ) takes a range of values. There is a different value of V 1 (I 1 ) for each A 1. So we can write V 1 = V 1 (I 1, A 1 ) Very messy simulation to get them all Two ways. First way use Brownian bridge construction

36 p. 36/4 Ten paths of a Brownian bridge Time in quarters

37 p. 37/4 Liability Calculation Fix I 1 Use Brownian bridge to get distribution of A 1 I 1 For each pair (A 1, I 1 ) find the GMWB liability at time one Find average value of liability given I 1 (2) V (I 1 ) = E A1 [V 1 (I 1, A 1 )] Use put options with different strikes to construct SS Hedge based on V (I 1 ).

38 p. 38/4 Distribution of A 1 Fix a value of the index at time one I 1. Conditional on I 1 find the distribution of A 1 I 1. We have seen that A 1 is path dependent. Next graph assumes I 1 = 100 and gives distribution of A 1

39 p. 39/4 Distribution of A 1 I 1 =

40 p. 40/4 Distribution of A 1 We can get all the moments of A 1 I 1 in closed form It turns out that A 1 I 1 is almost normal. We fix I 1 and write A 1 = Ā1 + σ A z where Ā1 = E[A 1 ] and where z has mean zero and variance one.

41 p. 41/4 V 1 as a function of A 1 Fix I 1. We have V (I 1, A 1 ) = V (I 1, Ā1 + σ A z) Using Taylor series and taking expectations we have (3) E A1 [V (I 1, A 1 )] = V (I 1,Ā1) + (σ2 A ) 2 V (I 1,Ā1)

42 p. 42/4 SS Hedging Two expressions for E A1 [V (I 1, A 1 )] The last one equation (3) is much more convenient. Find portfolio of put options to replicate average liability E A1 [V (I 1, A 1 )]. Optimization procedure or apply equation (1). Repeat after one year

43 p. 43/4 Profile of average net liability Green line net liability Index value after one year

44 p. 44/4 Summary and future work We discussed Semi static hedging Application to GWMB Future work deal with path dependent structure Stochastic volatility Other products Natural hedges

Financial Engineering g and Actuarial Science In the Life Insurance Industry

Financial Engineering g and Actuarial Science In the Life Insurance Industry Financial Engineering g and Actuarial Science In the Life Insurance Industry Presentation at USC October 31, 2013 Frank Zhang, CFA, FRM, FSA, MSCF, PRM Vice President, Risk Management Pacific Life Insurance

More information

Risk-Neutral Valuation of Participating Life Insurance Contracts

Risk-Neutral Valuation of Participating Life Insurance Contracts Risk-Neutral Valuation of Participating Life Insurance Contracts DANIEL BAUER with R. Kiesel, A. Kling, J. Russ, and K. Zaglauer ULM UNIVERSITY RTG 1100 AND INSTITUT FÜR FINANZ- UND AKTUARWISSENSCHAFTEN

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

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

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

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

Financial Options: Pricing and Hedging

Financial Options: Pricing and Hedging Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equity-linked securities requires an understanding of financial

More information

Traditional, investment, and risk management actuaries in the life insurance industry

Traditional, investment, and risk management actuaries in the life insurance industry Traditional, investment, and risk management actuaries in the life insurance industry Presentation at California Actuarial Student Conference University of California, Santa Barbara April 4, 2015 Frank

More information

Variable Annuities Risk Management

Variable Annuities Risk Management Variable Annuities Risk Management Michele Bergantino Risk and Investment Conference Leeds - June 28, 2012 1 Contents VA Key Features VA Risk Management Conclusions Appendix A - VA Option Valuation, an

More information

Management of Asian and Cliquet Option Exposures for Insurance Companies: SPVA applications (I)

Management of Asian and Cliquet Option Exposures for Insurance Companies: SPVA applications (I) Management of Asian and Cliquet Option Exposures for Insurance Companies: SPVA applications (I) Pin Chung and Rachid Lassoued 5th September, 2014, Wicklow, Ireland 0 Agenda 1. Introduction 2. Review of

More information

Variable Annuities. Society of Actuaries in Ireland Evening Meeting September 17, 2008

Variable Annuities. Society of Actuaries in Ireland Evening Meeting September 17, 2008 Variable Annuities Society of Actuaries in Ireland Evening Meeting September 17, 2008 Variable Annuities Working Party Presented paper to Faculty and Institute in March 2008 Working Party members Colin

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

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

Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation

Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Yoon W. Kwon CIMS 1, Math. Finance Suzanne A. Lewis CIMS, Math. Finance May 9, 000 1 Courant Institue of Mathematical Science,

More information

JANUARY 2016 EXAMINATIONS. Life Insurance I

JANUARY 2016 EXAMINATIONS. Life Insurance I PAPER CODE NO. MATH 273 EXAMINER: Dr. C. Boado-Penas TEL.NO. 44026 DEPARTMENT: Mathematical Sciences JANUARY 2016 EXAMINATIONS Life Insurance I Time allowed: Two and a half hours INSTRUCTIONS TO CANDIDATES:

More information

Ratchet Equity Indexed Annuities

Ratchet Equity Indexed Annuities Ratchet Equity Indexed Annuities Mary R Hardy University of Waterloo Waterloo Ontario N2L 3G1 Canada Tel: 1-519-888-4567 Fax: 1-519-746-1875 email: mrhardy@uwaterloo.ca Abstract The Equity Indexed Annuity

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

VALUATION IN DERIVATIVES MARKETS

VALUATION IN DERIVATIVES MARKETS VALUATION IN DERIVATIVES MARKETS September 2005 Rawle Parris ABN AMRO Property Derivatives What is a Derivative? A contract that specifies the rights and obligations between two parties to receive or deliver

More information

Hedging at Your Insurance Company

Hedging at Your Insurance Company Hedging at Your Insurance Company SEAC Spring 2007 Meeting Winter Liu, FSA, MAAA, CFA June 2007 2006 Towers Perrin Primary Benefits and Motives of Establishing Hedging Programs Hedging can mitigate some

More information

Variable Annuities and Policyholder Behaviour

Variable Annuities and Policyholder Behaviour Variable Annuities and Policyholder Behaviour Prof Dr Michael Koller, ETH Zürich Risk Day, 1192015 Aim To understand what a Variable Annuity is, To understand the different product features and how they

More information

Asset Liability Management for Life Insurance: a Dynamic Approach

Asset Liability Management for Life Insurance: a Dynamic Approach Risk Consulting CAPITAL MANAGEMENT ADVISORS srl Asset Liability Management for Life Insurance: a Dynamic Approach Dr Gabriele Susinno & Thierry Bochud Quantitative Strategies Capital Management Advisors

More information

Featured article: Evaluating the Cost of Longevity in Variable Annuity Living Benefits

Featured article: Evaluating the Cost of Longevity in Variable Annuity Living Benefits Featured article: Evaluating the Cost of Longevity in Variable Annuity Living Benefits By Stuart Silverman and Dan Theodore This is a follow-up to a previous article Considering the Cost of Longevity Volatility

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

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in

More information

AXA EQUITABLE VARIABLE ANNUITY GUARANTEED BENEFITS DYNAMIC HEDGING CONSIDERATIONS

AXA EQUITABLE VARIABLE ANNUITY GUARANTEED BENEFITS DYNAMIC HEDGING CONSIDERATIONS AXA EQUITABLE VARIABLE ANNUITY GUARANTEED BENEFITS DYNAMIC HEDGING CONSIDERATIONS STAN TULIN VICE CHAIRMAN AND CHIEF FINANCIAL OFFICER AXA FINANCIAL MARCH 21, 2005 Cautionary Statements Concerning Forward-looking

More information

Pricing variable annuity product in Hong Kong -- a starting point for practitioners

Pricing variable annuity product in Hong Kong -- a starting point for practitioners Pricing variable annuity product in Hong Kong -- a starting point for practitioners Abstract With regard to its rapid emergence in Hong Kong, this paper aims to perform a pricing exercise for a sample

More information

Hedging Guarantees in Variable Annuities. Under Both Equity and Interest Rate Risks

Hedging Guarantees in Variable Annuities. Under Both Equity and Interest Rate Risks Hedging Guarantees in Variable Annuities Under Both Equity and Interest Rate Risks Thomas F. Coleman a,,yuyingli a, Maria-Cristina Patron b,1 a Department of Computer Science, Cornell University, Ithaca,

More information

FAIR VALUATION OF THE SURRENDER OPTION EMBEDDED IN A GUARANTEED LIFE INSURANCE PARTICIPATING POLICY. Anna Rita Bacinello

FAIR VALUATION OF THE SURRENDER OPTION EMBEDDED IN A GUARANTEED LIFE INSURANCE PARTICIPATING POLICY. Anna Rita Bacinello FAIR VALUATION OF THE SURRENDER OPTION EMBEDDED IN A GUARANTEED LIFE INSURANCE PARTICIPATING POLICY Anna Rita Bacinello Dipartimento di Matematica Applicata alle Scienze Economiche, Statistiche ed Attuariali

More information

Hedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/)

Hedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Hedging Barriers Liuren Wu Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Based on joint work with Peter Carr (Bloomberg) Modeling and Hedging Using FX Options, March

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

INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION

INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION 14 April 2016 (pm) Subject ST6 Finance and Investment Specialist Technical B Time allowed: Three hours 1. Enter all the candidate and examination details

More information

VIX for Variable Annuities

VIX for Variable Annuities White Paper VIX for Variable Annuities A study considering the advantages of tying a Variable Annuity fee to VIX March 2013 VIX for Variable Annuities A study considering the advantages of tying a Variable

More information

ACTUARIAL MATHEMATICS FOR LIFE CONTINGENT RISKS

ACTUARIAL MATHEMATICS FOR LIFE CONTINGENT RISKS ACTUARIAL MATHEMATICS FOR LIFE CONTINGENT RISKS DAVID C. M. DICKSON University of Melbourne MARY R. HARDY University of Waterloo, Ontario V HOWARD R. WATERS Heriot-Watt University, Edinburgh CAMBRIDGE

More information

Working Paper. The Variable Annuities Dilemma: How to redesign hedging strategies without deterring rising demand.

Working Paper. The Variable Annuities Dilemma: How to redesign hedging strategies without deterring rising demand. The Variable Annuities Dilemma: How to redesign hedging strategies without deterring rising demand. Dr. Bernhard Brunner Senior Vice President Mikhail Krayzler Analyst September 2009 risklab GmbH Seidlstraße

More information

Exam 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 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 information

Facilitating On-Demand Risk and Actuarial Analysis in MATLAB. Timo Salminen, CFA, FRM Model IT

Facilitating On-Demand Risk and Actuarial Analysis in MATLAB. Timo Salminen, CFA, FRM Model IT Facilitating On-Demand Risk and Actuarial Analysis in MATLAB Timo Salminen, CFA, FRM Model IT Introduction It is common that insurance companies can valuate their liabilities only quarterly Sufficient

More information

S 1 S 2. Options and Other Derivatives

S 1 S 2. Options and Other Derivatives Options and Other Derivatives The One-Period Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating

More information

TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13

TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:

More information

Financial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan

Financial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction

More information

Autumn Investor Seminar. Workshops. Managing Variable Annuity Risk

Autumn Investor Seminar. Workshops. Managing Variable Annuity Risk Autumn Investor Seminar Workshops Managing Variable Annuity Risk Jean-Christophe Menioux Kevin Byrne Denis Duverne Group CRO CIO AXA Equitable Chief Financial Officer Paris November 25, 2008 Cautionary

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

Overview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies

Overview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies No-arbitrage bounds on option prices Binomial option pricing Black-Scholes-Merton

More information

Basic Black-Scholes: Option Pricing and Trading

Basic Black-Scholes: Option Pricing and Trading Basic Black-Scholes: Option Pricing and Trading BSc (HONS 1 st Timothy Falcon Crack Class), PGDipCom, MCom, PhD (MIT), IMC Contents Preface Tables Figures ix xi xiii 1 Introduction to Options 1 1.1 Hedging,

More information

Important Exam Information:

Important Exam Information: Important Exam Information: Exam Registration Candidates may register online or with an application. Order Study Notes Study notes are part of the required syllabus and are not available electronically

More information

Research on Option Trading Strategies

Research on Option Trading Strategies Research on Option Trading Strategies An Interactive Qualifying Project Report: Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree

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

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

More information

Annuity Decisions with Systematic Longevity Risk. Ralph Stevens

Annuity Decisions with Systematic Longevity Risk. Ralph Stevens Annuity Decisions with Systematic Longevity Risk Ralph Stevens Netspar, CentER, Tilburg University The Netherlands Annuity Decisions with Systematic Longevity Risk 1 / 34 Contribution Annuity menu Literature

More information

Robustly Hedging Variable Annuities with Guarantees Under Jump and Volatility Risks

Robustly Hedging Variable Annuities with Guarantees Under Jump and Volatility Risks Robustly Hedging Variable Annuities with Guarantees Under Jump and Volatility Risks T. F. Coleman, Y. Kim, Y. Li, and M. Patron CTC Computational Finance Group Cornell Theory Center, www.tc.cornell.edu

More information

Pricing Exotics under the Smile 1

Pricing Exotics under the Smile 1 Introduction Pricing Exotics under the Smile 1 The volatility implied from the market prices of vanilla options, using the Black Scholes formula, is seen to vary with both maturity and strike price. This

More information

Mathematical Finance: Stochastic Modeling of Asset Prices

Mathematical Finance: Stochastic Modeling of Asset Prices Mathematical Finance: Stochastic Modeling of Asset Prices José E. Figueroa-López 1 1 Department of Statistics Purdue University Worcester Polytechnic Institute Department of Mathematical Sciences December

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

Equity-Based Insurance Guarantees Conference November 1-2, 2010. New York, NY. Operational Risks

Equity-Based Insurance Guarantees Conference November 1-2, 2010. New York, NY. Operational Risks Equity-Based Insurance Guarantees Conference November -, 00 New York, NY Operational Risks Peter Phillips Operational Risk Associated with Running a VA Hedging Program Annuity Solutions Group Aon Benfield

More information

ANALYSIS OF PARTICIPATING LIFE INSURANCE CONTRACTS: A UNIFICATION APPROACH

ANALYSIS OF PARTICIPATING LIFE INSURANCE CONTRACTS: A UNIFICATION APPROACH ANALYSIS OF PARTICIPATING LIFE INSURANCE CONTRACTS: A UNIFICATION APPROACH NADINE GATZERT ALEXANDER KLING WORKING PAPERS ON RISK MANAGEMENT AND INSURANCE NO. 18 EDITED BY HATO SCHMEISER CHAIR FOR RISK

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

GN47: Stochastic Modelling of Economic Risks in Life Insurance

GN47: Stochastic Modelling of Economic Risks in Life Insurance GN47: Stochastic Modelling of Economic Risks in Life Insurance Classification Recommended Practice MEMBERS ARE REMINDED THAT THEY MUST ALWAYS COMPLY WITH THE PROFESSIONAL CONDUCT STANDARDS (PCS) AND THAT

More information

Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Products

Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Products Financial Institutions Center Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Products by Knut K. Aase Svein-Arne Persson 96-20 THE WHARTON FINANCIAL INSTITUTIONS CENTER The Wharton

More information

IAA PAPER VALUATION OF RISK ADJUSTED CASH FLOWS AND THE SETTING OF DISCOUNT RATES THEORY AND PRACTICE

IAA PAPER VALUATION OF RISK ADJUSTED CASH FLOWS AND THE SETTING OF DISCOUNT RATES THEORY AND PRACTICE Introduction This document refers to sub-issue 11G of the IASC Insurance Issues paper and proposes a method to value risk-adjusted cash flows (refer to the IAA paper INSURANCE LIABILITIES - VALUATION &

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

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Institutional 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 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

Option Valuation. Chapter 21

Option Valuation. Chapter 21 Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

More information

Matching Investment Strategies in General Insurance Is it Worth It? Aim of Presentation. Background 34TH ANNUAL GIRO CONVENTION

Matching Investment Strategies in General Insurance Is it Worth It? Aim of Presentation. Background 34TH ANNUAL GIRO CONVENTION Matching Investment Strategies in General Insurance Is it Worth It? 34TH ANNUAL GIRO CONVENTION CELTIC MANOR RESORT, NEWPORT, WALES Aim of Presentation To answer a key question: What are the benefit of

More information

The IASB Insurance Project for life insurance contracts: Impact on reserving methods and solvency requirements

The IASB Insurance Project for life insurance contracts: Impact on reserving methods and solvency requirements Insurance: Mathematics and Economics 39 (2006) 356 375 www.elsevier.com/locate/ime The IASB Insurance Project for life insurance contracts: Impact on reserving methods and solvency requirements Laura Ballotta

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

INSTITUTE OF ACTUARIES OF INDIA

INSTITUTE OF ACTUARIES OF INDIA INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 17 th November 2011 Subject CT5 General Insurance, Life and Health Contingencies Time allowed: Three Hours (10.00 13.00 Hrs) Total Marks: 100 INSTRUCTIONS TO

More information

Pricing Variable Annuity With Embedded Guarantees. - a case study. David Wang, FSA, MAAA May 21, 2008 at ASHK

Pricing Variable Annuity With Embedded Guarantees. - a case study. David Wang, FSA, MAAA May 21, 2008 at ASHK Pricing Variable Annuity With Embedded Guarantees - a case study David Wang, FSA, MAAA May 21, 2008 at ASHK Set The Stage Peter is the pricing actuary of company LifeGoesOn and LifeGoesOn wishes to launch

More information

Options/1. Prof. Ian Giddy

Options/1. Prof. Ian Giddy Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2

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

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

Lecture 11: Risk-Neutral Valuation Steven Skiena. skiena

Lecture 11: Risk-Neutral Valuation Steven Skiena.  skiena Lecture 11: Risk-Neutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Risk-Neutral Probabilities We can

More information

Stochastic Analysis of Long-Term Multiple-Decrement Contracts

Stochastic Analysis of Long-Term Multiple-Decrement Contracts Stochastic Analysis of Long-Term Multiple-Decrement Contracts Matthew Clark, FSA, MAAA, and Chad Runchey, FSA, MAAA Ernst & Young LLP Published in the July 2008 issue of the Actuarial Practice Forum Copyright

More information

Projection of the With-Profits Balance Sheet under ICA+ John Lim (KPMG) & Richard Taylor (AEGON) 11 November 2013

Projection of the With-Profits Balance Sheet under ICA+ John Lim (KPMG) & Richard Taylor (AEGON) 11 November 2013 Projection of the With-Profits Balance Sheet under ICA+ John Lim (KPMG) & Richard Taylor (AEGON) 11 November 2013 Introduction Projecting the with-profits business explicitly is already carried out by

More information

One Period Binomial Model

One Period Binomial Model FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing

More information

Guarantees and Target Volatility Funds

Guarantees and Target Volatility Funds SEPTEMBER 0 ENTERPRISE RISK SOLUTIONS B&H RESEARCH E SEPTEMBER 0 DOCUMENTATION PACK Steven Morrison PhD Laura Tadrowski PhD Moody's Analytics Research Contact Us Americas +..55.658 clientservices@moodys.com

More information

Variable Annuities The new complexity of insurance products

Variable Annuities The new complexity of insurance products Variable Annuities The new complexity of insurance products Dr. Ulrich Nögel, DEVnet Member of Executive Committee, Partner, Quantitative Finance u.noegel@devnet.de DEVnet Facts and Figures - The Group

More information

Variable Annuities in Australia: Managing the Risks. Jeff Gebler & Warren Manners

Variable Annuities in Australia: Managing the Risks. Jeff Gebler & Warren Manners Variable Annuities in Australia: Managing the Risks Jeff Gebler & Warren Manners Agenda VA Risks: Lessons learned from the past Dynamic Hedging: Ingredients for an optimal hedging strategy VA Risks: Lessons

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

The Intuition Behind Option Valuation: A Teaching Note

The Intuition Behind Option Valuation: A Teaching Note The Intuition Behind Option Valuation: A Teaching Note Thomas Grossman Haskayne School of Business University of Calgary Steve Powell Tuck School of Business Dartmouth College Kent L Womack Tuck School

More information

Betting on Volatility: A Delta Hedging Approach. Liang Zhong

Betting on Volatility: A Delta Hedging Approach. Liang Zhong Betting on Volatility: A Delta Hedging Approach Liang Zhong Department of Mathematics, KTH, Stockholm, Sweden April, 211 Abstract In the financial market, investors prefer to estimate the stock price

More information

Schonbucher Chapter 9: Firm Value and Share Priced-Based Models Updated 07-30-2007

Schonbucher Chapter 9: Firm Value and Share Priced-Based Models Updated 07-30-2007 Schonbucher Chapter 9: Firm alue and Share Priced-Based Models Updated 07-30-2007 (References sited are listed in the book s bibliography, except Miller 1988) For Intensity and spread-based models of default

More information

Caput Derivatives: October 30, 2003

Caput Derivatives: October 30, 2003 Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor

More information

On Simulation Method of Small Life Insurance Portfolios By Shamita Dutta Gupta Department of Mathematics Pace University New York, NY 10038

On Simulation Method of Small Life Insurance Portfolios By Shamita Dutta Gupta Department of Mathematics Pace University New York, NY 10038 On Simulation Method of Small Life Insurance Portfolios By Shamita Dutta Gupta Department of Mathematics Pace University New York, NY 10038 Abstract A new simulation method is developed for actuarial applications

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

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

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

A SNOWBALL CURRENCY OPTION

A SNOWBALL CURRENCY OPTION J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA E-mail address: gshim@ajou.ac.kr ABSTRACT. I introduce

More information

Equity-Based Insurance Guarantees Conference November 18-19, 2013. Atlanta, GA. Development of Managed Risk Funds in the VA Market

Equity-Based Insurance Guarantees Conference November 18-19, 2013. Atlanta, GA. Development of Managed Risk Funds in the VA Market Equity-Based Insurance Guarantees Conference November 18-19, 2013 Atlanta, GA Development of Managed Risk Funds in the VA Market Chad Schuster DEVELOPMENT OF MANAGED RISK FUNDS IN THE VA MARKET CHAD SCHUSTER,

More information

ASSESSING THE RISK POTENTIAL OF PREMIUM PAYMENT OPTIONS IN PARTICIPATING LIFE INSURANCE CONTRACTS

ASSESSING THE RISK POTENTIAL OF PREMIUM PAYMENT OPTIONS IN PARTICIPATING LIFE INSURANCE CONTRACTS ASSESSING THE RISK POTENTIAL OF PREMIUM PAYMENT OPTIONS IN PARTICIPATING LIFE INSURANCE CONTRACTS NADINE GATZERT HATO SCHMEISER WORKING PAPERS ON RISK MANAGEMENT AND INSURANCE NO. 22 EDITED BY HATO SCHMEISER

More information

American Academy of Actuaries Life Financial Reporting Committee. Draft

American Academy of Actuaries Life Financial Reporting Committee. Draft American Academy of Actuaries Life Financial Reporting Committee Draft 6-13-02 Frequently Asked Questions Regarding the Valuation of Embedded Derivatives in Life Insurance and Annuity Contracts in Accordance

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

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

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

Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER

Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER INTRODUCTION Having been exposed to a variety of applications of Monte Carlo

More information

Digital Options. and d 1 = d 2 + σ τ, P int = e rτ[ KN( d 2) FN( d 1) ], with d 2 = ln(f/k) σ2 τ/2

Digital Options. and d 1 = d 2 + σ τ, P int = e rτ[ KN( d 2) FN( d 1) ], with d 2 = ln(f/k) σ2 τ/2 Digital Options The manager of a proprietary hedge fund studied the German yield curve and noticed that it used to be quite steep. At the time of the study, the overnight rate was approximately 3%. The

More information

Monte Carlo Simulation in the Pricing of Derivatives. Cara M. Marshall. Fordham University. Queens College of the City University of New York

Monte Carlo Simulation in the Pricing of Derivatives. Cara M. Marshall. Fordham University. Queens College of the City University of New York Monte Carlo Simulation in the Pricing of Derivatives Cara M. Marshall Fordham University Queens College of the City University of New York 1 Monte Carlo Simulation in the Pricing of Derivatives Of all

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

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

Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time Arbitrage Theory in Continuous Time THIRD EDITION TOMAS BJORK Stockholm School of Economics OXTORD UNIVERSITY PRESS 1 Introduction 1 1.1 Problem Formulation i 1 v. 2 The Binomial Model 5 2.1 The One Period

More information