# MTH6121 Introduction to Mathematical Finance Lesson 5

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif Geomeric Brownian moion Convergence of random walks o Brownian moions Convergence o Geomeric Brownian Moions Example Le W be he Wiener process. Find a E[W 2 + W 3]; b Cov W 2, W 3 ; c Var W 2 + W 3. Soluion. a Noe ha W 2 N 0, 2 and W 3 N 0, 3. Thus E[W 2 + W 3] = E[W 2] + E[W 3] = = 0. b By Lemma 2.8. CovW 2, W 3 = min2, 3 = 2. c Noe ha W 2 and W 3 are dependen, so we need o use a Proposiion from he previous chaper regarding he variance of he sum of random variables, Var W 2+W 3 = VarW 2+VarW 3+2Cov W 2, W 3 = = 9. When performing concree calculaions wih he Wiener process i is useful o remember ha, since W N 0,, we have 1 W N 0, 1. Example Le W be he Wiener process. Wha is PW 4 > 5? Soluion. Since W 4 N 0, 4, we have W 4 PW 4 > 5 = P > 5 = P 4 4 Z > 5 4 = 1 Φ 5 = =

2 Brownian moion wih drif In his shor secion we will discuss a generalizaion of he Wiener process ha will urn ou o be useful for our second model of sock marke prices. Definiion Brownian moion wih drif having drif parameer R and volailiy parameer > 0 is he sochasic process Y given by where W is he Wiener process. Y = + W, Noe ha if Y is a Brownian moion wih drif parameer and volailiy parameer, hen for every > 0, we have while E[Y ] = E[ + W ] = + E[W ] =, VarY = Var + W = VarW = 2 VarW = 2. Bu since for every > 0 he random variable Y is Normal, we have Y N, 2 > 0. Noe ha Y j Y j 1 = j j 1 + W j W j 1 for all j = 1,..., n. Using he independen incremens of he Brownian Moion see propery d in Theorem for he characerisaion of he Wiener process, we can conclude ha he random variables Y 1, Y 2 Y 1,..., Y n Y n 1 are independen. Calculaions for Brownian moion wih drif can be performed in he same way as for he Wiener process iself. Example Le Y be Brownian moion wih drif parameer = 0.2 and volailiy parameer = 0.1. Find E[Y 9], Var[Y 9] and PY 9 > 1. Soluion. Using he above resuls, we obain and Finally, we have E[Y 9] = 9 = 1.8, VarY 9 = 9 2 = PY 9 > 1 = P9 + W 9 > 1 = P W 9 > 1 9 = PW 9 > 8 W 9 = P > = P Z > 8 9 = 1 Φ 8 8 = Φ 3 3 =

3 Geomeric Brownian moion Brownian moion wih drif has a drawback, namely is propery ha i can become negaive. This feaure rules i ou as a suiable model for he movemens of he sock marke. The following varian does no suffer from his shorcoming, and provides a decen model of sock marke prices. Definiion Geomeric Brownian moion having drif parameer R, volailiy parameer > 0, and saring value = S is he sochasic process S given by where W is he Wiener process. S = S exp + W, I is easily verified from he above equaion ha = S exp0 + 0 = S which also explains he erminology saring value S of S. Moreover, so where Thus S S S S S log S = exp + W = + W + W N, 2. S LogNormal, 2. Observe also ha for any 0 1 < 2 < < n he successive raios S 2 S 1 = exp W 2 W 1 S 3 S 2 = exp W 3 W 2 S n S n 1 = exp n n 1 + W n W n 1 are all independen since Brownian moion has independen incremens. raios are all lognormally disribued. Thus, Furhermore, he Geomeric Brownian moion is an exension of he IID lognormal model o a coninuous ime parameer. Example Suppose ha he weekly price of chewing gum evolves according o Geomeric Brownian moion wih drif parameer = and volailiy parameer = 0.1. Wha is he probabiliy ha a he price of chewing gum is higher in 2 weeks and 5 days han i is now?

4 29 b he price of chewing gum rises from one day o he nex for he nex 3 days? Soluion. Le S denoe he price of chewing gum a ime, where is measured in weeks. We are old ha S = S exp + W, where W denoes he Wiener process, S is he saring parameer, = and = 0.1. a Since a week has 7 days, we need o find he probabiliy ha S2 + 5/7 >. Now S/7 P S/7 > = P > 1 = P log S/7 > 0. Bu so log S/7 P S/7 > = P log S/7 = P Z > = 1 Φ 7 N, 7 2, > 0 = P = Φ S/7 log 7 > = Φ0.07 = Thus, he probabiliy ha he price of chewing gum is higher in 2 weeks and 5 days han oday is 52.8%. b We shall firs calculae he probabiliy p ha he price of chewing gum is higher a he end of a day compared o he beginning. Now p = PS1/7 > = P log S1/7 > 0. Bu so p = P log S1/7 Bu since he random variables log S1/7 N 1 7, 1 2, S1/7 log 1 7 > 0 = P 1 > 7 1 = P Z > = 1 Φ = Φ = Φ0.02 = S1/7, S2/7 S1/7, S3/7 S2/7, are all independen and have he same disribuion, he probabiliy ha he price of chewing gum rises from one day o he nex for he nex 3 days is p 3 = 0.13 = 13%.

5 Convergence of random walks o Brownian moions. I urns ou ha he Wiener process is, in he sense o be made precise shorly, a re-scaled version of a random walk. Recall ha Y n is a random walk if Y n = n X i, where he X i s are independen and idenically disribued random variables wih PX i = 1 = PX i = 1 = 1 2. Definiion Sochasic processes Y h converge in disribuion o a sochasic process Y as h 0, if for any 1 < 2 < < k, and x 1, x 2,... x k, for some k 1, lim PY h 1 x 1,..., Y h k x k = PY 1 x 1,..., Y k x k. 1 h 0 Recall ha, if 1 holds hen we wrie Y d = lim h 0 Y h. In order o sae he main resul of his subsecion, we also need he following definiion. Definiion Given a real number x, he floor of x, denoed by x, is he larges ineger less han or equal o x. Thus, for example, 1 = 1, 3.2 = 3, 5.9 = 5. We are now able o formulae he following ineresing resul. Theorem 2. Convergence of he random walk o he Wiener process. Le X 1, X 2,... be a sequence of independen and idenically disribued random variables wih common disribuion given by PX i = 1 = PX i = 1 = 1/2 for all i. For > 0, define he processes Y h = /h h X i. Then, here exiss a Wiener process Y, such ha Y d = lim Y h. h 0, h>0 Proof. We sar by showing ha Y N 0,. The proof follows from he Cenral Limi Theorem CLT. To see his, se n = /h = /h δ, for some 0 δ < 1 by he definiion of he floor funcion. Then h = /n + δ and hus Y h = /h h X i = n + δ n X i = n n + δ Since EX i = = 0 and VarX i = 2 = 1, we see ha X i = X i n n n X i n.

6 31 has he same form as he relevan expressions in he CLT. Noe ha, when h 0, we have n and hence CLT implies ha X i n d N 0, 1 as h 0. Also observe ha n n + δ 1 as h 0 which finally implies ha Y h = n n + δ X i n d N 0, 1 = N 0, as h 0. Therefore, we can se Y N 0,. Similarly, i is possible o show ha for any 0 1 < 2 Y 2 Y 1 d = lim h h 0,h>0 2 /h i= 1 /h +1 X i N 0, 2 1. Furhermore, i is possible o show ha for any 0 1 < 2 < < n he random variables Y 1, Y 2 Y 1, Y 3 Y 2,..., Y n Y n 1 are independen. By he Theorem 2.7 for he characerizaion of he Wiener process W, we mus have Y = W for every Convergence o Geomeric Brownian Moions We have seen ha he Wiener process can be obained as he limi of a random walk Theorem 2.. I urns ou ha a similar resul holds for he Geomeric Brownian moion, obained as he limi of a random walk-like process. The laer is a muliplicaive equivalen o he random walk. Denoe by S h he price of a financial asse a imes = 0, h, 2h,... h > 0, which sars from a value S h 0 = S and evolves according o S h i + 1h = Sh ih Y i, for all i = 0, 1,... where Y i is a sequence of independen idenically disribued random variables aking one of he wo possible values: { u = e h, wih probabiliy p = h Y i = d = e h, wih probabiliy 1 p = h The above means ha afer each ime inerval of lengh h he price S h ih of he financial produc eiher goes up by a facor u or down by a facor d. Theorem As h 0 he muliplicaive random walk model described above converges o a Geomeric Brownian moion wih drif parameer, volailiy parameer, and sars from he value S. Proof. The proof relies on he applicaion of he CLT, wih similar argumens as in he proof of Theorem 2. above.

### Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

### Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

### A general decomposition formula for derivative prices in stochastic volatility models

A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 5-7 85 Barcelona Absrac We see ha he price of an european call opion

### The option pricing framework

Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

### 17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

### An Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price

An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor

### INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

### The Transport Equation

The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

### A Generalized Bivariate Ornstein-Uhlenbeck Model for Financial Assets

A Generalized Bivariae Ornsein-Uhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical

### Present Value Methodology

Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

### DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

### Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs

Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edge-dijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.

### ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

### 2.5 Life tables, force of mortality and standard life insurance products

Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n

### Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619

econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;

### Optimal Time to Sell in Real Estate Portfolio Management

Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and Jean-Luc Prigen hema, Universiy of Cergy-Ponoise, Cergy-Ponoise, France E-mails: fabricebarhelemy@u-cergyfr; jean-lucprigen@u-cergyfr

### Efficient Risk Sharing with Limited Commitment and Hidden Storage

Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing

### Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

### Module 3 Design for Strength. Version 2 ME, IIT Kharagpur

Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress

### Stochastic Calculus and Option Pricing

Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 Black-Scholes

### Optimal Investment and Consumption Decision of Family with Life Insurance

Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

### An accurate analytical approximation for the price of a European-style arithmetic Asian option

An accurae analyical approximaion for he price of a European-syle arihmeic Asian opion David Vyncke 1, Marc Goovaers 2, Jan Dhaene 2 Absrac For discree arihmeic Asian opions he payoff depends on he price

### Return Calculation of U.S. Treasury Constant Maturity Indices

Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

### Conditional Default Probability and Density

Condiional Defaul Probabiliy and Densiy N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari Absrac This paper proposes differen mehods o consruc condiional survival processes, i.e, families of maringales decreasing

### HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES?

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and Black-Meron-Scholes

### Jump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach

ump-diffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,

### LIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b

LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

### Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate

Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his

### Niche Market or Mass Market?

Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.

### PRICING and STATIC REPLICATION of FX QUANTO OPTIONS

PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of

### Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

### European option prices are a good sanity check when analysing bonds with exotic embedded options.

European opion prices are a good saniy check when analysing bonds wih exoic embedded opions. I s an old exam quesion. Arbirage-free economy where ZCB prices are driven 1-D BM, i.e. dp (, T ) = r()p (,

### 1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,

Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..

### OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTI-DIMENSIONAL DIFFUSIVE TERMS

OPTIMAL LIFE INSURANCE PURCHASE, CONSUMPTION AND INVESTMENT ON A FINANCIAL MARKET WITH MULTI-DIMENSIONAL DIFFUSIVE TERMS I. DUARTE, D. PINHEIRO, A. A. PINTO, AND S. R. PLISKA Absrac. We inroduce an exension

### Modeling a distribution of mortgage credit losses Petr Gapko 1, Martin Šmíd 2

Modeling a disribuion of morgage credi losses Per Gapko 1, Marin Šmíd 2 1 Inroducion Absrac. One of he bigges risks arising from financial operaions is he risk of counerpary defaul, commonly known as a

### Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

### Individual Health Insurance April 30, 2008 Pages 167-170

Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

### A Re-examination of the Joint Mortality Functions

Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali

### Inductance and Transient Circuits

Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

### ARCH 2013.1 Proceedings

Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

### Pricing Black-Scholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension

Pricing Black-choles Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zu-lang Liao a, and Hsing-Hua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor

### Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work

### On the degrees of irreducible factors of higher order Bernoulli polynomials

ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on

### Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

### Pricing Dynamic Insurance Risks Using the Principle of Equivalent Utility

Scand. Acuarial J. 00; 4: 46 79 ORIGINAL ARTICLE Pricing Dynamic Insurance Risks Using he Principle of Equivalen Uiliy VIRGINIA R. YOUNG and THALEIA ZARIPHOPOULOU Young VR, Zariphopoulou T. Pricing dynamic

### Keldysh Formalism: Non-equilibrium Green s Function

Keldysh Formalism: Non-equilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Non-equilibrium

### The Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.

The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz- und Akuarwissenschafen, Helmholzsraße 22, 89081

### IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **

IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include so-called implici or embedded opions.

### Analysis of Tailored Base-Surge Policies in Dual Sourcing Inventory Systems

Analysis of Tailored Base-Surge Policies in Dual Sourcing Invenory Sysems Ganesh Janakiraman, 1 Sridhar Seshadri, 2, Anshul Sheopuri. 3 Absrac We sudy a model of a firm managing is invenory of a single

### Motion Along a Straight Line

Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his

### SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, E-mail: toronj333@yahoo.

SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.

### 12. Market LIBOR Models

12. Marke LIBOR Models As was menioned already, he acronym LIBOR sands for he London Inerbank Offered Rae. I is he rae of ineres offered by banks on deposis from oher banks in eurocurrency markes. Also,

### A general first-passage-time model for multivariate credit spreads and a note on barrier option pricing. Inaugural-Dissertation

A general firs-passage-ime model for mulivariae credi spreads and a noe on barrier opion pricing Inaugural-Disseraion zur Erlangung des Dokorgrades an den Naurwissenschaflichen Fachbereichen Mahemaik der

### New Pricing Framework: Options and Bonds

arxiv:1407.445v [q-fin.pr] 14 Oc 014 New Pricing Framework: Opions and Bonds Nick Laskin TopQuark Inc. Torono, ON, M6P P Absrac A unified analyical pricing framework wih involvemen of he sho noise random

### On Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations

On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen

### Credit risk. T. Bielecki, M. Jeanblanc and M. Rutkowski. Lecture of M. Jeanblanc. Preliminary Version LISBONN JUNE 2006

i Credi risk T. Bielecki, M. Jeanblanc and M. Rukowski Lecure of M. Jeanblanc Preliminary Version LISBONN JUNE 26 ii Conens Noaion vii 1 Srucural Approach 3 1.1 Basic Assumpions.....................................

### Modelling of Forward Libor and Swap Rates

Modelling of Forward Libor and Swap Raes Marek Rukowski Faculy of Mahemaics and Informaion Science Warsaw Universiy of Technology, -661 Warszawa, Poland Conens 1 Inroducion 2 2 Modelling of Forward Libor

### Time-inhomogeneous Lévy Processes in Cross-Currency Market Models

Time-inhomogeneous Lévy Processes in Cross-Currency Marke Models Disseraion zur Erlangung des Dokorgrades der Mahemaischen Fakulä der Alber-Ludwigs-Universiä Freiburg i. Brsg. vorgeleg von Naaliya Koval

### Cash-Lock Comparison of Portfolio Insurance Strategies

Cash-Lock Comparison of Porfolio Insurance Sraegies Sven Balder Anje B. Mahayni This version: May 3, 28 Deparmen of Banking and Finance, Universiy of Bonn, Adenauerallee 24 42, 533 Bonn. E-mail: sven.balder@uni-bonn.de

### ACTUARIAL FUNCTIONS 1_05

ACTUARIAL FUNCTIONS _05 User Guide for MS Office 2007 or laer CONTENT Inroducion... 3 2 Insallaion procedure... 3 3 Demo Version and Acivaion... 5 4 Using formulas and synax... 7 5 Using he help... 6 Noaion...

### TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS

TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.

### T ϕ t ds t + ψ t db t,

16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in

### Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension

Markov Chain Modeling of Policy Holder Behavior in Life Insurance and Pension Lars Frederik Brand Henriksen 1, Jeppe Woemann Nielsen 2, Mogens Seffensen 1, and Chrisian Svensson 2 1 Deparmen of Mahemaical

### Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor

pimal Annui Purchasing Virginia R. Young and Moshe A. Milevsk Version: 6 Januar 3 Young is an Associae Professor a he chool of Business, Universi of Wisconsin-Madison, Madison, Wisconsin, 5376, UA. he

### SHB Gas Oil. Index Rules v1.3 Version as of 1 January 2013

SHB Gas Oil Index Rules v1.3 Version as of 1 January 2013 1. Index Descripions The SHB Gasoil index (he Index ) measures he reurn from changes in he price of fuures conracs, which are rolled on a regular

### Time Consistency in Portfolio Management

1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen

### Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...

Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The Black-Scholes Shl Ml Moel... pricing opions an calculaing

### Dependent Interest and Transition Rates in Life Insurance

Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

### Valuation of Credit Default Swaptions and Credit Default Index Swaptions

Credi Defaul Swapions Valuaion of Credi Defaul Swapions and Marek Rukowski School of Mahemaics and Saisics Universiy of New Souh Wales Sydney, Ausralia Recen Advances in he Theory and Pracice of Credi

### Unstructured Experiments

Chaper 2 Unsrucured Experimens 2. Compleely randomized designs If here is no reason o group he plos ino blocks hen we say ha Ω is unsrucured. Suppose ha reamen i is applied o plos, in oher words ha i is

### Making a Faster Cryptanalytic Time-Memory Trade-Off

Making a Faser Crypanalyic Time-Memory Trade-Off Philippe Oechslin Laboraoire de Securié e de Crypographie (LASEC) Ecole Polyechnique Fédérale de Lausanne Faculé I&C, 1015 Lausanne, Swizerland philippe.oechslin@epfl.ch

### A Simple Approach to CAPM, Option Pricing and Asset Valuation

A imple Approach o CAPM, Opion Pricing and Asse Valuaion Riccardo Cesari (*) Universià di Bologna, Dip. Maemaes, viale Filopani, 5 406 Bologna, Ialy E-mail: rcesari@economia.unibo.i Carlo D Adda Universià

### AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

### PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM

PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu

### OptimalCompensationwithHiddenAction and Lump-Sum Payment in a Continuous-Time Model

Appl Mah Opim (9) 59: 99 46 DOI.7/s45-8-95- OpimalCompensaionwihHiddenAcion and Lump-Sum Paymen in a Coninuous-Time Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business

### Morningstar Investor Return

Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

### Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

### Analyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective

Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Alexander Bohner, Nadine Gazer Working Paper Chair for Insurance Economics Friedrich-Alexander-Universiy

### CALCULATION OF OMX TALLINN

CALCULATION OF OMX TALLINN CALCULATION OF OMX TALLINN 1. OMX Tallinn index...3 2. Terms in use...3 3. Comuaion rules of OMX Tallinn...3 3.1. Oening, real-ime and closing value of he Index...3 3.2. Index

### A Production-Inventory System with Markovian Capacity and Outsourcing Option

OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp. 328 349 issn 0030-364X eissn 1526-5463 05 5302 0328 informs doi 10.1287/opre.1040.0165 2005 INFORMS A Producion-Invenory Sysem wih Markovian Capaciy

### Optimal Investment, Consumption and Life Insurance under Mean-Reverting Returns: The Complete Market Solution

Opimal Invesmen, Consumpion and Life Insurance under Mean-Revering Reurns: The Complee Marke Soluion Traian A. Pirvu Dep of Mahemaics & Saisics McMaser Universiy 180 Main Sree Wes Hamilon, ON, L8S 4K1

### DETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUN-SHAN WU

Yugoslav Journal of Operaions Research 2 (22), Number, 6-7 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUN-SHAN WU Deparmen of Bussines Adminisraion

### THE RETURN ON INVESTMENT FROM PROPORTIONAL PORTFOLIO STRATEGIES

THE RETURN ON INVESTMENT FROM PROPORTIONAL PORTFOLIO STRATEGIES Si Browne Columbia Universiy Final Version: November 11, 1996 Appeare in: Avances in Applie Probabiliy, 30, 216-238, 1998 Absrac Dynamic

### On the Role of the Growth Optimal Portfolio in Finance

QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 1441-8010 www.qfrc.us.edu.au

### µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ

Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high

### A One-Sector Neoclassical Growth Model with Endogenous Retirement. By Kiminori Matsuyama. Final Manuscript. Abstract

A One-Secor Neoclassical Growh Model wih Endogenous Reiremen By Kiminori Masuyama Final Manuscrip Absrac This paper exends Diamond s OG model by allowing he agens o make he reiremen decision. Earning a

### OPERATION MANUAL. Indoor unit for air to water heat pump system and options EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1

OPERAION MANUAL Indoor uni for air o waer hea pump sysem and opions EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1 EKHBRD011ABY1 EKHBRD014ABY1 EKHBRD016ABY1 EKHBRD011ACV1 EKHBRD014ACV1 EKHBRD016ACV1 EKHBRD011ACY1

### Signal Processing and Linear Systems I

Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

### INVESTMENT GUARANTEES IN UNIT-LINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE

INVESMEN UARANEES IN UNI-LINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN

### adaptive control; stochastic systems; certainty equivalence principle; long-term

COMMUICATIOS I IFORMATIO AD SYSTEMS c 2006 Inernaional Press Vol. 6, o. 4, pp. 299-320, 2006 003 ADAPTIVE COTROL OF LIEAR TIME IVARIAT SYSTEMS: THE BET O THE BEST PRICIPLE S. BITTATI AD M. C. CAMPI Absrac.

### Monte Carlo Observer for a Stochastic Model of Bioreactors

Mone Carlo Observer for a Sochasic Model of Bioreacors Marc Joannides, Irène Larramendy Valverde, and Vivien Rossi 2 Insiu de Mahémaiques e Modélisaion de Monpellier (I3M UMR 549 CNRS Place Eugène Baaillon

### C Fast-Dealing Property Trading Game C

AGES 8+ C Fas-Dealing Propery Trading Game C Y Collecor s Ediion Original MONOPOLY Game Rules plus Special Rules for his Ediion. CONTENTS Game board, 6 Collecible okens, 28 Tile Deed cards, 16 Wha he Deuce?

### Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach

Working Paper 5-81 Business Economics Series 21 January 25 Deparameno de Economía de la Empresa Universidad Carlos III de Madrid Calle Madrid, 126 2893 Geafe (Spain) Fax (34) 91 624 968 Opion-Pricing in

### A rank-dependent utility model of uncertain lifetime, time consistency and life insurance

A rank-dependen uiliy model of uncerain lifeime, ime consisency and life insurance Nicolas Drouhin To cie his version: Nicolas Drouhin. A rank-dependen uiliy model of uncerain lifeime, ime consisency and

### SEASONAL ADJUSTMENT. 1 Introduction. 2 Methodology. 3 X-11-ARIMA and X-12-ARIMA Methods

SEASONAL ADJUSTMENT 1 Inroducion 2 Mehodology 2.1 Time Series and Is Componens 2.1.1 Seasonaliy 2.1.2 Trend-Cycle 2.1.3 Irregulariy 2.1.4 Trading Day and Fesival Effecs 3 X-11-ARIMA and X-12-ARIMA Mehods