Ordinary Mixed Life Insurance and Mortality-Linked Insurance Contracts
|
|
- Nelson Jesse Chambers
- 5 years ago
- Views:
Transcription
1 Ordinary Mixed Life Insurance and Mortality-Linked Insurance Contracts M.Sghairi M.Kouki February 16, 2007 Abstract Ordinary mixed life insurance is a mix between temporary deathinsurance and pure endowment. Its duration is at least T years or more and in case of death the capital/pension will be paid to the inheritors and if the insured are alive, they will receive the payment. Thus, a single premium is paid by the policy-holders and placed by the insurance company on the incomplete financial market; but the difficulty lies in a fair tariffing. Our method is based primarily on two principles: equivalent martingale measure and the equality between the retrospective and the prospective market reserves. Then we obtain a condition on the discounting of the investment rate. This result generalizes the results obtained by M.Dahl [1] for a life insurance contract, in case of survival. Keywords: Life insurance; Stochastic mortality intensity; Incomplete financial markets; Lévy process; Equivalent measure principle 1 Introduction Traditionally, actuaries calculate premiums and market reserves using a deterministic mortality intensity that depends on age only, and a constant interest rate. But, in reality, neither the interest rate nor the mortality intensity are deterministic. It s well known, that we deal with two deferents life insurance contract : pure endowment and temporary death-insurance contracts. In this paper we address the case of an ordinary mixed insurance contract for only one x-year-old policy-holder, with stochastic mortality intensity, and an investment on the incomplete financial market. The ordinary 1
2 mixed insurance contract is a mix between temporary death insurance and pure endowment: - upon the policy-holder s death, a capital C0, 1 is paid to a designated recipient in the form of l benefits, in the event of death before the expiration date T of the contract, - a capital KT 2 is paid in full to the ensured person in the event of life at the expiry date (T) of the contract. This contract is paid by a single premium Π l 0 at time 0. In this paper we address the problem of fair pricing, which will be solved in two stages. First, by changing the initial probability measure in an equivalent measure, and then by expecting the retrospective and prospective market reserves. Thus, we obtain a necessary condition to the equity contract that relies on the discounting of the accumulation rate. In section 2 we present related to the financial market and those related the stochastic mortality. In section 3, we solve the problem by equalizing the time T retrospective and prospective market reserves. In section 4 we provide same Monte Carlo simulation experience, and section 5 concludes. 2 Assumptions and notations 2.1 The financial market We work on a space of probability (Ω, F,P) where filtration F = (F t ) t describes the information available on the incomplete financial market. We consider a financial market consisting of two traded assets : a risky asset with price process U and a locally risk-free asset with price process B (see T.Chan [4]). The stochastic process (U t ) t describes the accumulation rate. The dynamics of U and B may be given, for example, by : du t = α U (t,u t ) U t dt + σ U (t,u t ) U t dy t, > 0 db t = r t B t dt, B 0 = 1 (2.1) where r t 0 is the risk-free interest rate, α U (t,u t ) describes the expected behavior of the asset s instantaneous price, σ U (t,u t ) is uniformly bounded away from 0. It describes the sensitivity of the instantaneous price of the assets compared to the general trends in the 2
3 M.Sghairi, M.Kouki financial market. and (Y t ) t, a Lévy process on [0,T], is the index which describes the general trends in the financial market. For analytical purpose, we introduce the probability measure Q, equivalent to P, under which the discounted price process of the asset is a Q-martingale. Indeed, the expected discounted value is given by : p(t,t) = E Q [e R T t r udu F t ] so the financial annuity (see R.Cobbaut [3]) of an unity, paid in l periods, is given by l 1 a l = p(0,u). 2.2 The mortality u=0 The mortality is the second random component of the development of the ordinary mixed life insurance contract. It is linked to an F-adapted rightcontinuous Markov process Z = (Z t ) t [0,T], on a finite state space J = {0, 1} which describes the state of the insured person : 0 = alive, 1 = dead (see M.Dahl [1]). Only the transition from state 0 to state 1 is possible. On the one hand, under the initial probability P, the intensity of transition of Z is simply given at time t by the mortality intensity µ [x]+t, where, (µ [x]+t ) t is an adapted stochastic process. On the other hand, under the probability Q, the intensity of transition is multiplied by (1 + g t ), where (g t ) t is an adapted process, strictly higher than 1 Q-almost surely. Then we introduce the survival probability of an age [x]+t person from time t to T. It is given by G Q (t,µ [x]+t,t) = E Q [e R T t (1+gu)µ [x]+udu µ [x]+t ] where E Q indicates the expectation under Q. Then the death probability estimated at time t under Q is given by Ḡ Q (t,µ [x]+t,t) = 1 G Q (t,µ [x]+t,t). We denote by (K 2 t ) t T the adapted stochastic process, equal to the ensured sum at time t for pure endowment in the event of survival. In particular K 2 T indicates the sum to be paid at time T in the survival event at age x + T. 3
4 3 Solving the problem The equivalent measure Q principle yields the premium Π l 0 for the ordinary mixed life insurance contract, with sum C0 1 as the ensured sum in case of death, and KT 2 as the ensured sum in case of life. Let us notice that on model is an extension of the Norberg [2] model. Thus, the expected discounted value under Q of the actual payments having to be of null Q-mean, we have T Π l 0 = K0p(0,T)G 2 Q (0,µ [x],t) C1 0 l a l p(0,s) s G Q (0,µ [x],s)ds. (3.1) 0 Note that K 1,l 0 = C1 0 l a l the actual annuity value in l terms and the instantaneous risk of death is s G Q (0,µ [x],s). Proposition 3.1: Let the actual payments Π l 0 by given by (3.1) Then E Q [exp( T r 0 udu) U T ] = 1 + K1,l 0 [p(0,t)ḡq (0,µ [x],t) Π l 0 + T 0 (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds] (3.2) Proof : Let V i,l,retro t be the retrospective reserves conditioned at time t [0, T], given that the policy-holder is in the Z t = i state, is defined by: V i,l,retro t = Π l 0 U t K 1,l 0 1 {i=1} 1 {t<t } K 2 T 1 {i=0} 1 {t=t } (3.3) where K 1,l 0 = C1 0 l a l indicates the actual annuity value in l terms and i is person s state at time t. Let V i,l,pro t be the prospective reserves at time t, given that the policy-holder is in the Z t = i state. The sum insured in the event of life is Kt 2 and we consider that the sum insured in the event of death is constant in a course 4
5 M.Sghairi, M.Kouki of time equal to K 1,l 0. Therefore V i,l,pro t = E Q [K 1,l T 0 (1 + g t s)µ [x]+s e R s t ru+(1+gu)µ [x]+udu ds + e R T t r udu e R T 0 (1+gu)µ[x]+udu Kt 2 Z t = i, I t G t ], 0 t < T (3.4) To determine the adapted advantage, Kt 2, we use the following criterion: E Q [V Zt,l,pro t I t G t ] = E Q [V Zt,l,retro t I t G t ] (3.5) It is to be noted that in the above equality, the expectation relates only to the values of Z t. For the contract to be fair the expected discounted value under Q of the actual payments should be 0, i.e. E Q [Π l 0 + K 1,l 0 T 0 (1 + g s)µ [x]+s e R s t ru+(1+gu)µ [x]+udu ds K 2 T e R T 0 ru+(1+gu)µ [x]+udu ] = 0. (3.6) This leads to a condition on the process of accumulation U. Let us express K 2 T in terms of U, by the fact that the prospective reserve at time T must be null: V 0,l,pro T = V 1,l,pro T = 0. According to (3.3) we have : V 0,l,retro T = Π l U T 0 KT 2 K 1,l 0. Thus, criterion (3.5), applied at time T, gives : et V 1,l,retro T = Π l 0 U T KT 2 = Π l U T 1 0 G Q (0,µ [x],t) Ḡ Q (0,µ [x],t) K1,l 0 G Q (0,µ [x],t) with ḠQ (0,µ [x],t) = 1 G Q (0,µ [x],t) as the death probability from time 0 to time T. While inserting this in (3.6), one notes that the process U must check : E Q [e R T 0 rudu U T ] = 1 + K1,l 0 Π l [p(0,t)ḡq (0,µ [x],t) 0 + T 0 (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds]. (3.7) 5
6 In particular, it is to be noted that the expected discounted value of accumulation factor U T from 0 to T, E Q [exp( T r 0 udu) U T ], is higher than 1 whereas under the modeling of M.Dahl [1] it is equal to 1. After solving the problem of a fair contract we addressed the development of the benefits. Criterion (3.5) at time t < T is written as : Π l 0 Ut = e R t 0 (1+gu)µ [x]+udu K 2 t p(t,t)g Q (t,µ [x]+t,t) + K 1,l 0 T t (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds (3.8) Using the expression of premium (3.1), we express the advantages at time t according to the advantages at time 0 R t0 Kt 2 (1+gu)µ e [x]+u du = p(t,t)g Q (t,µ [x]+t,t) [K2 0p(0,T)G Q (0,µ [x],t) Ut + K 1,l 0 [ Ut T 0 (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds (3.9) T t (1 + g s)µ [x]+s p(0,s)g Q (0,µ [x],s)ds]. 4 Monte Carlo study In this section we simulate the loss risk. To get a realistic portfolio of the insured, it would be necessary to introduce policy-holders of different ages/cohorts, different horizons, and different dates of subscription. However, in this study, we started with simulations of homogeneous portfolios. We simulated the risk of loss of 1000 ordinary mixed life insurance contracts, concerning x-year insured people, x varying in somme interval I, and of fixed horizon T years. We used the mortality intensity of Gompertz-Makeham µ x = a + bc x of parameter a = , b = et c = Example: If the insured have different ages/cohorts, then the loss risk of the associated portfolio at T can be obtained as a weighted mean of the individuel risks. The left figure represent the dynamics of assets price is of the form du t = αu t dt + σu t dy t, where α = 0.6, and σ = 0.2, and Y t is a Lévy process of the 6
7 M.Sghairi, M.Kouki form dy t = dw t + dn t where W is the Brownian motion and N the Poisson process of 10% intensity, and that him of right-hand side the dynamics of assets price is of the form du t = αu t dt + σu t dw t. We calculated V Zt,x,retro t t [0,T], x I, given by formula (3.3) for 1000 contracts with C0 1 = 1, KT 2 = 0.5. We simulated the loser contract rate at time t [0,T] for N insured persons with ages x I using the approximation : t [0,T], x I : 1 N N i=1 1 {V x t <0} P(V x t < 0) Figure 1 : The left figure represent the risk of loss of contracts of 1000 insured persons, x [30, 60] years and horizon T = 30 years, with the dynamics of assets price is of the form du t = αu t dt + σu t dy t and that him of right-hand side, with the dynamics of assets price is of the form du t = αu t dt + σu t dw t. Figure 2 : The left figure represent the risk of loss of contracts of 1000 insured persons, x [40, 60] years and horizon T = 20 years, with the dynamics of assets price is of the form du t = αu t dt + σu t dy t and that him of right-hand side, with the dynamics of assets price is of the form du t = αu t dt + σu t dw t. Figure 3 :The left figure represent the risk of loss of contracts of 1000 insured persons, x [35, 55] years and horizon T = 20 years, with the dynamics of assets price is of the form du t = αu t dt + σu t dy t and that him of right-hand side, with the dynamics of assets price is of the form du t = αu t dt + σu t dw t. 7
8 Figure 1: Risk of loss Figure 2: Risk of loss 8
9 M.Sghairi, M.Kouki Figure 3: Risk of loss 5 Conclusion Our objective in this study was to find a tool to determine the premium that policy-holders have to pay to insurers according to the terms of the contracts. However the contracts mention the expiry date, as well as the mode of payment of the annuities due by the policy-holder and the two parties operate in a context of information asymmetry. In this context each individual tries to maximize his/her utility function, in other terms his/her future benefit. The uncertainty that prevails in all incomplete contracts requires an estimate of the risk induced by incomplete information. From this point of view we tried to design a mathematical model in order to help the insurer take his/her decision, estimate the risks, and determine the premium of the contract while taking into account unquestionable variables: mortality intensity rate, the characteristics of each insurance contract(temporary death-insurance, pure endowment). Certain assumptions were put forward to design a model that considers the determination of the premium as well as the risk. An empirical study was undertaken to highlight the reliability of our model. Indeed the model shows that in the event of non-diversification of the insurer s portfolios, the risk of loss caused varies according to the expiry date of the contract. However these results must not be considered as the limits of our model, since the data used to test this model were subjective data which do not reflect 9
10 the reality. Collecting data in this field remains complex and it is the major problem for the validation of our model. References [1] M. Dahl. Stochastic mortality in life insurance:market reserves and mortality-linked insurance contracts. Insurance:Mathematics and Economics 35., pages , [2] R. Norberg. Reserves in life and pension insurance. Scandinavian Actuarial Journal 1, pages 3 24, [3] R.Cobbaut. Théorie Financière. Economica, Paris, [4] T.Chan. Pricing contingent claims on stocks driven by lévy processes. The Annals of Applied Probability., pages , 1999, Vol.9. M.Sghairi M.Kouki Université PARIS DESCARTES Ecole Supérieure de la Statistique LABORATOIRE MAP5 et de l Analyse de l Information 45, RUE DES SAINTS-PÈRES 6, rue des mètiers PARIS CEDEX Charguia 2 Ariana FRANCE TUNISIE sghairi@math-info.univ-paris5.fr mokhtar.kouki@essai.rnu.tn 10
Hedging with Life and General Insurance Products
Hedging with Life and General Insurance Products June 2016 2 Hedging with Life and General Insurance Products Jungmin Choi Department of Mathematics East Carolina University Abstract In this study, a hybrid
More informationinduced by the Solvency II project
Asset Les normes allocation IFRS : new en constraints assurance induced by the Solvency II project 36 th International ASTIN Colloquium Zürich September 005 Frédéric PLANCHET Pierre THÉROND ISFA Université
More informationManaging Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives
Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives Simon Man Chung Fung, Katja Ignatieva and Michael Sherris School of Risk & Actuarial Studies University of
More informationBasic Concepts and Examples in Finance
Basic Concepts and Examples in Finance Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 5, 2017 ICMAT / UC3M The Financial Market The Financial Market We assume there are
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
More informationSPDE and portfolio choice (joint work with M. Musiela) Princeton University. Thaleia Zariphopoulou The University of Texas at Austin
SPDE and portfolio choice (joint work with M. Musiela) Princeton University November 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 Performance measurement of investment strategies 2 Market
More informationStochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models
Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete
More information1. For a special whole life insurance on (x), payable at the moment of death:
**BEGINNING OF EXAMINATION** 1. For a special whole life insurance on (x), payable at the moment of death: µ () t = 0.05, t > 0 (ii) δ = 0.08 x (iii) (iv) The death benefit at time t is bt 0.06t = e, t
More informationIndifference fee rate 1
Indifference fee rate 1 for variable annuities Ricardo ROMO ROMERO Etienne CHEVALIER and Thomas LIM Université d Évry Val d Essonne, Laboratoire de Mathématiques et Modélisation d Evry Second Young researchers
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationVALUATION OF FLEXIBLE INSURANCE CONTRACTS
Teor Imov r.tamatem.statist. Theor. Probability and Math. Statist. Vip. 73, 005 No. 73, 006, Pages 109 115 S 0094-90000700685-0 Article electronically published on January 17, 007 UDC 519.1 VALUATION OF
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationInterest rate models in continuous time
slides for the course Interest rate theory, University of Ljubljana, 2012-13/I, part IV József Gáll University of Debrecen Nov. 2012 Jan. 2013, Ljubljana Continuous time markets General assumptions, notations
More informationParameter sensitivity of CIR process
Parameter sensitivity of CIR process Sidi Mohamed Ould Aly To cite this version: Sidi Mohamed Ould Aly. Parameter sensitivity of CIR process. Electronic Communications in Probability, Institute of Mathematical
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationOptimal trading strategies under arbitrage
Optimal trading strategies under arbitrage Johannes Ruf Columbia University, Department of Statistics The Third Western Conference in Mathematical Finance November 14, 2009 How should an investor trade
More informationValuation of derivative assets Lecture 6
Valuation of derivative assets Lecture 6 Magnus Wiktorsson September 14, 2017 Magnus Wiktorsson L6 September 14, 2017 1 / 13 Feynman-Kac representation This is the link between a class of Partial Differential
More informationValuation of derivative assets Lecture 8
Valuation of derivative assets Lecture 8 Magnus Wiktorsson September 27, 2018 Magnus Wiktorsson L8 September 27, 2018 1 / 14 The risk neutral valuation formula Let X be contingent claim with maturity T.
More information1 Cash-flows, discounting, interest rates and yields
Assignment 1 SB4a Actuarial Science Oxford MT 2016 1 1 Cash-flows, discounting, interest rates and yields Please hand in your answers to questions 3, 4, 5, 8, 11 and 12 for marking. The rest are for further
More informationA. 11 B. 15 C. 19 D. 23 E. 27. Solution. Let us write s for the policy year. Then the mortality rate during year s is q 30+s 1.
Solutions to the Spring 213 Course MLC Examination by Krzysztof Ostaszewski, http://wwwkrzysionet, krzysio@krzysionet Copyright 213 by Krzysztof Ostaszewski All rights reserved No reproduction in any form
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationOPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF FINITE
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 005 Seville, Spain, December 1-15, 005 WeA11.6 OPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF
More informationInvestigation of Dependency between Short Rate and Transition Rate on Pension Buy-outs. Arık, A. 1 Yolcu-Okur, Y. 2 Uğur Ö. 2
Investigation of Dependency between Short Rate and Transition Rate on Pension Buy-outs Arık, A. 1 Yolcu-Okur, Y. 2 Uğur Ö. 2 1 Hacettepe University Department of Actuarial Sciences 06800, TURKEY 2 Middle
More informationLongevity risk: past, present and future
Longevity risk: past, present and future Xiaoming Liu Department of Statistical & Actuarial Sciences Western University Longevity risk: past, present and future Xiaoming Liu Department of Statistical &
More informationMATH/STAT 4720, Life Contingencies II Fall 2015 Toby Kenney
MATH/STAT 4720, Life Contingencies II Fall 2015 Toby Kenney In Class Examples () September 2, 2016 1 / 145 8 Multiple State Models Definition A Multiple State model has several different states into which
More information1. The force of mortality at age x is given by 10 µ(x) = 103 x, 0 x < 103. Compute E(T(81) 2 ]. a. 7. b. 22. c. 23. d. 20
1 of 17 1/4/2008 12:01 PM 1. The force of mortality at age x is given by 10 µ(x) = 103 x, 0 x < 103. Compute E(T(81) 2 ]. a. 7 b. 22 3 c. 23 3 d. 20 3 e. 8 2. Suppose 1 for 0 x 1 s(x) = 1 ex 100 for 1
More informationChapter 15: Jump Processes and Incomplete Markets. 1 Jumps as One Explanation of Incomplete Markets
Chapter 5: Jump Processes and Incomplete Markets Jumps as One Explanation of Incomplete Markets It is easy to argue that Brownian motion paths cannot model actual stock price movements properly in reality,
More informationOptimal stopping problems for a Brownian motion with a disorder on a finite interval
Optimal stopping problems for a Brownian motion with a disorder on a finite interval A. N. Shiryaev M. V. Zhitlukhin arxiv:1212.379v1 [math.st] 15 Dec 212 December 18, 212 Abstract We consider optimal
More informationLimit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies
Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies George Tauchen Duke University Viktor Todorov Northwestern University 2013 Motivation
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationEquivalence between Semimartingales and Itô Processes
International Journal of Mathematical Analysis Vol. 9, 215, no. 16, 787-791 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.411358 Equivalence between Semimartingales and Itô Processes
More information8.5 Numerical Evaluation of Probabilities
8.5 Numerical Evaluation of Probabilities 1 Density of event individual became disabled at time t is so probability is tp 7µ 1 7+t 16 tp 11 7+t 16.3e.4t e.16 t dt.3e.3 16 Density of event individual became
More informationAsymmetric information in trading against disorderly liquidation of a large position.
Asymmetric information in trading against disorderly liquidation of a large position. Caroline Hillairet 1 Cody Hyndman 2 Ying Jiao 3 Renjie Wang 2 1 ENSAE ParisTech Crest, France 2 Concordia University,
More informationSTEX s valuation analysis, version 0.0
SMART TOKEN EXCHANGE STEX s valuation analysis, version. Paulo Finardi, Olivia Saa, Serguei Popov November, 7 ABSTRACT In this paper we evaluate an investment consisting of paying an given amount (the
More informationAsset Pricing Models with Underlying Time-varying Lévy Processes
Asset Pricing Models with Underlying Time-varying Lévy Processes Stochastics & Computational Finance 2015 Xuecan CUI Jang SCHILTZ University of Luxembourg July 9, 2015 Xuecan CUI, Jang SCHILTZ University
More informationIntroduction Credit risk
A structural credit risk model with a reduced-form default trigger Applications to finance and insurance Mathieu Boudreault, M.Sc.,., F.S.A. Ph.D. Candidate, HEC Montréal Montréal, Québec Introduction
More informationRough volatility models: When population processes become a new tool for trading and risk management
Rough volatility models: When population processes become a new tool for trading and risk management Omar El Euch and Mathieu Rosenbaum École Polytechnique 4 October 2017 Omar El Euch and Mathieu Rosenbaum
More informationConstructing Markov models for barrier options
Constructing Markov models for barrier options Gerard Brunick joint work with Steven Shreve Department of Mathematics University of Texas at Austin Nov. 14 th, 2009 3 rd Western Conference on Mathematical
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More informationLIBOR models, multi-curve extensions, and the pricing of callable structured derivatives
Weierstrass Institute for Applied Analysis and Stochastics LIBOR models, multi-curve extensions, and the pricing of callable structured derivatives John Schoenmakers 9th Summer School in Mathematical Finance
More informationAnnuities. Lecture: Weeks 8-9. Lecture: Weeks 8-9 (Math 3630) Annuities Fall Valdez 1 / 41
Annuities Lecture: Weeks 8-9 Lecture: Weeks 8-9 (Math 3630) Annuities Fall 2017 - Valdez 1 / 41 What are annuities? What are annuities? An annuity is a series of payments that could vary according to:
More informationCredit Risk Models with Filtered Market Information
Credit Risk Models with Filtered Market Information Rüdiger Frey Universität Leipzig Bressanone, July 2007 ruediger.frey@math.uni-leipzig.de www.math.uni-leipzig.de/~frey joint with Abdel Gabih and Thorsten
More informationEnlargement of filtration
Enlargement of filtration Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 6, 2017 ICMAT / UC3M Enlargement of Filtration Enlargement of Filtration ([1] 5.9) If G is a
More informationNumerical schemes for SDEs
Lecture 5 Numerical schemes for SDEs Lecture Notes by Jan Palczewski Computational Finance p. 1 A Stochastic Differential Equation (SDE) is an object of the following type dx t = a(t,x t )dt + b(t,x t
More informationMATH 3630 Actuarial Mathematics I Class Test 2 - Section 1/2 Wednesday, 14 November 2012, 8:30-9:30 PM Time Allowed: 1 hour Total Marks: 100 points
MATH 3630 Actuarial Mathematics I Class Test 2 - Section 1/2 Wednesday, 14 November 2012, 8:30-9:30 PM Time Allowed: 1 hour Total Marks: 100 points Please write your name and student number at the spaces
More informationRisk Neutral Measures
CHPTER 4 Risk Neutral Measures Our aim in this section is to show how risk neutral measures can be used to price derivative securities. The key advantage is that under a risk neutral measure the discounted
More informationCredit Risk using Time Changed Brownian Motions
Credit Risk using Time Changed Brownian Motions Tom Hurd Mathematics and Statistics McMaster University Joint work with Alexey Kuznetsov (New Brunswick) and Zhuowei Zhou (Mac) 2nd Princeton Credit Conference
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationParameters Estimation in Stochastic Process Model
Parameters Estimation in Stochastic Process Model A Quasi-Likelihood Approach Ziliang Li University of Maryland, College Park GEE RIT, Spring 28 Outline 1 Model Review The Big Model in Mind: Signal + Noise
More informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
More informationNovember 2012 Course MLC Examination, Problem No. 1 For two lives, (80) and (90), with independent future lifetimes, you are given: k p 80+k
Solutions to the November 202 Course MLC Examination by Krzysztof Ostaszewski, http://www.krzysio.net, krzysio@krzysio.net Copyright 202 by Krzysztof Ostaszewski All rights reserved. No reproduction in
More informationExact Sampling of Jump-Diffusion Processes
1 Exact Sampling of Jump-Diffusion Processes and Dmitry Smelov Management Science & Engineering Stanford University Exact Sampling of Jump-Diffusion Processes 2 Jump-Diffusion Processes Ubiquitous in finance
More informationTEST OF BOUNDED LOG-NORMAL PROCESS FOR OPTIONS PRICING
TEST OF BOUNDED LOG-NORMAL PROCESS FOR OPTIONS PRICING Semih Yön 1, Cafer Erhan Bozdağ 2 1,2 Department of Industrial Engineering, Istanbul Technical University, Macka Besiktas, 34367 Turkey Abstract.
More informationA Proper Derivation of the 7 Most Important Equations for Your Retirement
A Proper Derivation of the 7 Most Important Equations for Your Retirement Moshe A. Milevsky Version: August 13, 2012 Abstract In a recent book, Milevsky (2012) proposes seven key equations that are central
More informationHedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework
Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework Kathrin Glau, Nele Vandaele, Michèle Vanmaele Bachelier Finance Society World Congress 2010 June 22-26, 2010 Nele Vandaele Hedging of
More informationAn overview of some financial models using BSDE with enlarged filtrations
An overview of some financial models using BSDE with enlarged filtrations Anne EYRAUD-LOISEL Workshop : Enlargement of Filtrations and Applications to Finance and Insurance May 31st - June 4th, 2010, Jena
More informationThe Forward PDE for American Puts in the Dupire Model
The Forward PDE for American Puts in the Dupire Model Peter Carr Ali Hirsa Courant Institute Morgan Stanley New York University 750 Seventh Avenue 51 Mercer Street New York, NY 10036 1 60-3765 (1) 76-988
More informationSurvival models. F x (t) = Pr[T x t].
2 Survival models 2.1 Summary In this chapter we represent the future lifetime of an individual as a random variable, and show how probabilities of death or survival can be calculated under this framework.
More informationTHE MARTINGALE METHOD DEMYSTIFIED
THE MARTINGALE METHOD DEMYSTIFIED SIMON ELLERSGAARD NIELSEN Abstract. We consider the nitty gritty of the martingale approach to option pricing. These notes are largely based upon Björk s Arbitrage Theory
More informationPricing in markets modeled by general processes with independent increments
Pricing in markets modeled by general processes with independent increments Tom Hurd Financial Mathematics at McMaster www.phimac.org Thanks to Tahir Choulli and Shui Feng Financial Mathematics Seminar
More information1 Consumption and saving under uncertainty
1 Consumption and saving under uncertainty 1.1 Modelling uncertainty As in the deterministic case, we keep assuming that agents live for two periods. The novelty here is that their earnings in the second
More informationSensitivity of American Option Prices with Different Strikes, Maturities and Volatilities
Applied Mathematical Sciences, Vol. 6, 2012, no. 112, 5597-5602 Sensitivity of American Option Prices with Different Strikes, Maturities and Volatilities Nasir Rehman Department of Mathematics and Statistics
More informationMay 2012 Course MLC Examination, Problem No. 1 For a 2-year select and ultimate mortality model, you are given:
Solutions to the May 2012 Course MLC Examination by Krzysztof Ostaszewski, http://www.krzysio.net, krzysio@krzysio.net Copyright 2012 by Krzysztof Ostaszewski All rights reserved. No reproduction in any
More informationOption Pricing under Delay Geometric Brownian Motion with Regime Switching
Science Journal of Applied Mathematics and Statistics 2016; 4(6): 263-268 http://www.sciencepublishinggroup.com/j/sjams doi: 10.11648/j.sjams.20160406.13 ISSN: 2376-9491 (Print); ISSN: 2376-9513 (Online)
More informationEMPIRICAL EVIDENCE ON ARBITRAGE BY CHANGING THE STOCK EXCHANGE
Advances and Applications in Statistics Volume, Number, This paper is available online at http://www.pphmj.com 9 Pushpa Publishing House EMPIRICAL EVIDENCE ON ARBITRAGE BY CHANGING THE STOCK EXCHANGE JOSÉ
More informationFunctional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs.
Functional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs Andrea Cosso LPMA, Université Paris Diderot joint work with Francesco Russo ENSTA,
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationAn Introduction to Point Processes. from a. Martingale Point of View
An Introduction to Point Processes from a Martingale Point of View Tomas Björk KTH, 211 Preliminary, incomplete, and probably with lots of typos 2 Contents I The Mathematics of Counting Processes 5 1 Counting
More informationLocal Volatility Dynamic Models
René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Columbia November 9, 27 Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding
More informationExponential utility maximization under partial information
Exponential utility maximization under partial information Marina Santacroce Politecnico di Torino Joint work with M. Mania AMaMeF 5-1 May, 28 Pitesti, May 1th, 28 Outline Expected utility maximization
More informationConsistently modeling unisex mortality rates. Dr. Peter Hieber, Longevity 14, University of Ulm, Germany
Consistently modeling unisex mortality rates Dr. Peter Hieber, Longevity 14, 20.09.2018 University of Ulm, Germany Seite 1 Peter Hieber Consistently modeling unisex mortality rates 2018 Motivation European
More informationDEFERRED ANNUITY CONTRACTS UNDER STOCHASTIC MORTALITY AND INTEREST RATES: PRICING AND MODEL RISK ASSESSMENT
DEFERRED ANNUITY CONTRACTS UNDER STOCHASTIC MORTALITY AND INTEREST RATES: PRICING AND MODEL RISK ASSESSMENT DENIS TOPLEK WORKING PAPERS ON RISK MANAGEMENT AND INSURANCE NO. 41 EDITED BY HATO SCHMEISER
More informationConditional Density Method in the Computation of the Delta with Application to Power Market
Conditional Density Method in the Computation of the Delta with Application to Power Market Asma Khedher Centre of Mathematics for Applications Department of Mathematics University of Oslo A joint work
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationDeterministic Income under a Stochastic Interest Rate
Deterministic Income under a Stochastic Interest Rate Julia Eisenberg, TU Vienna Scientic Day, 1 Agenda 1 Classical Problem: Maximizing Discounted Dividends in a Brownian Risk Model 2 Maximizing Discounted
More informationThe Capital Asset Pricing Model as a corollary of the Black Scholes model
he Capital Asset Pricing Model as a corollary of the Black Scholes model Vladimir Vovk he Game-heoretic Probability and Finance Project Working Paper #39 September 6, 011 Project web site: http://www.probabilityandfinance.com
More informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
More informationRisk Neutral Pricing. to government bonds (provided that the government is reliable).
Risk Neutral Pricing 1 Introduction and History A classical problem, coming up frequently in practical business, is the valuation of future cash flows which are somewhat risky. By the term risky we mean
More informationINSTITUTE AND FACULTY OF ACTUARIES. Curriculum 2019 SPECIMEN SOLUTIONS
INSTITUTE AND FACULTY OF ACTUARIES Curriculum 2019 SPECIMEN SOLUTIONS Subject CM1A Actuarial Mathematics Institute and Faculty of Actuaries 1 ( 91 ( 91 365 1 0.08 1 i = + 365 ( 91 365 0.980055 = 1+ i 1+
More informationOptimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing
Optimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing Prof. Chuan-Ju Wang Department of Computer Science University of Taipei Joint work with Prof. Ming-Yang Kao March 28, 2014
More informationPortability, salary and asset price risk: a continuous-time expected utility comparison of DB and DC pension plans
Portability, salary and asset price risk: a continuous-time expected utility comparison of DB and DC pension plans An Chen University of Ulm joint with Filip Uzelac (University of Bonn) Seminar at SWUFE,
More informationDrunken Birds, Brownian Motion, and Other Random Fun
Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales Outline Review of Basic Probability
More information3.4 Copula approach for modeling default dependency. Two aspects of modeling the default times of several obligors
3.4 Copula approach for modeling default dependency Two aspects of modeling the default times of several obligors 1. Default dynamics of a single obligor. 2. Model the dependence structure of defaults
More informationMartingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
More informationPAPER 211 ADVANCED FINANCIAL MODELS
MATHEMATICAL TRIPOS Part III Friday, 27 May, 2016 1:30 pm to 4:30 pm PAPER 211 ADVANCED FINANCIAL MODELS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry equal
More informationOperational Risk. Robert Jarrow. September 2006
1 Operational Risk Robert Jarrow September 2006 2 Introduction Risk management considers four risks: market (equities, interest rates, fx, commodities) credit (default) liquidity (selling pressure) operational
More informationErrata and Updates for ASM Exam MLC (Fifteenth Edition Third Printing) Sorted by Date
Errata for ASM Exam MLC Study Manual (Fifteenth Edition Third Printing) Sorted by Date 1 Errata and Updates for ASM Exam MLC (Fifteenth Edition Third Printing) Sorted by Date [1/25/218] On page 258, two
More informationStochastic Calculus, Application of Real Analysis in Finance
, Application of Real Analysis in Finance Workshop for Young Mathematicians in Korea Seungkyu Lee Pohang University of Science and Technology August 4th, 2010 Contents 1 BINOMIAL ASSET PRICING MODEL Contents
More informationStochastic Partial Differential Equations and Portfolio Choice. Crete, May Thaleia Zariphopoulou
Stochastic Partial Differential Equations and Portfolio Choice Crete, May 2011 Thaleia Zariphopoulou Oxford-Man Institute and Mathematical Institute University of Oxford and Mathematics and IROM, The University
More informationHow to hedge Asian options in fractional Black-Scholes model
How to hedge Asian options in fractional Black-Scholes model Heikki ikanmäki Jena, March 29, 211 Fractional Lévy processes 1/36 Outline of the talk 1. Introduction 2. Main results 3. Methodology 4. Conclusions
More informationMultiple Life Models. Lecture: Weeks Lecture: Weeks 9-10 (STT 456) Multiple Life Models Spring Valdez 1 / 38
Multiple Life Models Lecture: Weeks 9-1 Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring 215 - Valdez 1 / 38 Chapter summary Chapter summary Approaches to studying multiple life models: define
More informationLast Time. Martingale inequalities Martingale convergence theorem Uniformly integrable martingales. Today s lecture: Sections 4.4.1, 5.
MATH136/STAT219 Lecture 21, November 12, 2008 p. 1/11 Last Time Martingale inequalities Martingale convergence theorem Uniformly integrable martingales Today s lecture: Sections 4.4.1, 5.3 MATH136/STAT219
More informationCHAPTER 2: STANDARD PRICING RESULTS UNDER DETERMINISTIC AND STOCHASTIC INTEREST RATES
CHAPTER 2: STANDARD PRICING RESULTS UNDER DETERMINISTIC AND STOCHASTIC INTEREST RATES Along with providing the way uncertainty is formalized in the considered economy, we establish in this chapter the
More informationTWO-STAGE NEWSBOY MODEL WITH BACKORDERS AND INITIAL INVENTORY
TWO-STAGE NEWSBOY MODEL WITH BACKORDERS AND INITIAL INVENTORY Ali Cheaitou, Christian van Delft, Yves Dallery and Zied Jemai Laboratoire Génie Industriel, Ecole Centrale Paris, Grande Voie des Vignes,
More informationContinuous Time Finance. Tomas Björk
Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying
More informationUnified Credit-Equity Modeling
Unified Credit-Equity Modeling Rafael Mendoza-Arriaga Based on joint research with: Vadim Linetsky and Peter Carr The University of Texas at Austin McCombs School of Business (IROM) Recent Advancements
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 14 Lecture 14 November 15, 2017 Derivation of the
More informationPAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS
MATHEMATICAL TRIPOS Part III Thursday, 5 June, 214 1:3 pm to 4:3 pm PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry
More informationPortfolio Optimization using Conditional Sharpe Ratio
International Letters of Chemistry, Physics and Astronomy Online: 2015-07-01 ISSN: 2299-3843, Vol. 53, pp 130-136 doi:10.18052/www.scipress.com/ilcpa.53.130 2015 SciPress Ltd., Switzerland Portfolio Optimization
More information