Optimal Stopping and Policyholder Behaviour in …

1 Optimal Stopping andPolicyholder Behaviour inLife InsuranceKamille Sofie T gholt GadPhD ThesisThis thesis has been submitted to the PhD School of the Faculty of Science,University of :Jesper Lund Pedersen, University of CopenhagenCo-supervisor: Mogens Steffensen, University of CopenhagenSubmitted:February 13, of Mathematical SciencesFaculty of ScienceUniversity of CopenhagenAuthor:Kamille Sofie T gholt GadStudsgaardsgade 42, 22100 K benhavn Committee: Associate Professor Marcus C. Christiansen,Heriot-Watt University,Edinburgh, ScotlandProfessor Goran Peskir,The University of Manchester,Manchester, United KingdomAssociate Professor Jeffrey F. Collamore,University of Copenhagen,Denmark (Chairman)ISBN:978-87-7078-961-5 PrefaceThis thesis has been prepared in fulfilment of the requirements for the at the Department of Mathematical Sciences, Faculty of Science, Uni-versity of Copenhagen, Denmark.

2 The work has been carried out under thesupervision of Jesper Lund Pedersen and Professor Mogens Steffensen, Uni-versity of Copenhagen in the period from May 1st 2011 to February 13th2015 (including months maternity leave).The main body of the thesis consists of an introduction to the material inthe thesis, and five chapters on different but related topics. The five chaptersare written as individual academic papers, and are thus self-contained andcan be read independently. This final version contains the following changescompared to the version submitted for assessment: Chapter 2 includes trivialproof-reading changes from the journal; Chapter 5 includes added referencesand numerical examples, and suggestions from review led to the inclusion ofthe new Section ; Chapter 6 includes proof-reading changesI would like to thank my two advisors Jesper Lund Pedersen and MogensSteffensen for support, ideas and discussions during the work.

3 A specialthanks to Mogens, Edlund A/S and the IT University for setting up theActulus project and letting me participate. Also I would like to thank Jesper,Mogens, Jeppe Juhl and Jeppe Woetmann Nielsen for co-authoring differentparts of the , I would like to thank David Brillinger and Michael S rensenfor setting up my visit at the Statistics Department at University of Cali-fornia, Berkeley. Also thanks to David and my office and class mates (Kai,Hyo Jeong, Hye Kyung and Dave) at U. C. Berkeley for great discussionsboth academic and non-academic. This extends to my colleges at the Actu-lus project and in the IE group in Copenhagen. Especially my office mates(Andr , Sima, Ninna, Mai-Britt and Rune).Finally, I would like to thank my family for their support. My parentsfor presenting to me the joy of mathematics, and for their support whichmakes our everyday work.

4 My daughter, Filippa, for all her laughs whichcan brighten up the most stressful day. And especially my beloved husband,Esben, for love, proof-reading and for, while holding on to his own dreams,always supporting and engaging in Sofie T gholt GadCopenhagen, April 2015iiiPREFACES ummaryThis thesis consists of an introductory chapter and five papers. The papersare each concerning questions within the topics life insurance , Optimal stop-ping or the interplay between these. Each paper is presented in a chapter,and thus each of the chapters are self-contained and may be read alone. Be-low, I give a brief overview of the results of each of the chapters. A morethorough overview is presented in Chapter Chapter 2 we consider a general geometric L vy process and solvethe non-linear Optimal Stopping problem of maximizing the variance at thestopping time.

5 For solving this problem we solve an auxiliary quadratic op-timal Stopping problem. We show that the solution to maximizing variancedepends on whether randomized Stopping times are included in the set ofstopping times we maximize over. For some problems the inclusion of ran-domized Stopping times increase the value function and for some it doesnot. Even when the value function is not affected by inclusion of randomizedstopping times, a solution may be easier to identify when they Chapter 3 we consider the non-linear Optimal Stopping problem ofmaximizing the mean minus a positive constant times the variance at thestopping time. First we solve the problem for spectrally negative geometricL vy process. We derive both static and dynamic solutions which are excessboundary Stopping times. Afterwards we solve the problem for a Cram r-Lundberg process with exponential upwards jumps.

6 We derive a staticallyoptimal Stopping time which is a hitting time of an interval, and we derivea dynamically Optimal Stopping time which is an excess boundary stoppingtime. Finally, we derive Optimal Stopping times to the Optimal stoppingproblem of minimizing the variance conditioned on a lower bound on Chapter 4 we consider the American put in a Black-Scholes market. Wesuggest a model for irrational exercises. We model the exercise by a stochas-tic intensity which depends on the profitability. Our model contains a singleparameter which express how strongly the exercise intensity is affected bythe profitability. This parameter we denotethe rationality parameter. Wegive sufficient conditions and a probabilistic proof that when the rationalityparameter increases to infinity the corresponding prices converge to to clas-sical arbitrage-free price.

7 We conclude the chapter with partial differentialequations for valuation under irrational exercise, and we discuss relations toiiiivSUMMARYthe penalty Chapter 5 is related to Chapter 4, but in Chapter 5 we consider mod-elling the time of surrender in a classical life insurance model. We suggest amodel where the probability of surrender at any time depends on the prof-itability. We measure the profitability as the difference between the valueof the insurance contract and the surrender value. The value of the insur-ance contract may be determined as a solution to a differential equationmuch similar to the Thiele differential equation. As in Chapter 4 the modelcontains a rationality parameter which express how strongly the surrenderprobability is affected by the profitability. Again we derive a probabilisticproof of the intuitive convergence result that when the rationality parameterincreases to infinity, the value of the life - insurance contract converge to thevalue corresponding to if the Policyholder surrendered at the Optimal Chapter 6 we add stochastic retirement to a classical finite state lifeinsurance model.

8 We do this by splitting theactivestate in apremium pay-ingstate and aretiredstate. We derive formulas for scaling the benefitsreasonably according to the time of retirement. Then we show how to cal-culate the reserves and expected cash. Afterwards we describe a way to addto the model that policyholders might change their benefit structure uponretirement. We determine formulas for calculating reserves and cash flows inthis model too. Finally, we conclude with a numerical investigation of theimplication stochastic retirement has on reserves and cash p danskDenne afhandling best r af et introducerende kapitel og fem artikler. Ar-tiklerne besk ftiger sig med sp rgsm l indenfor emnerne livsforsikring, op-timale stoppetider og samspillet mellem disse. Hver artikel er pr senteret iet kapitel, og kapitlerne kan derfor alle l ses enkeltst ende.

9 Nedenfor giverjeg et meget overordnet overblik over resultaterne fra hvert kapitel. Et meregrundigt overblik pr senteres i Kapitel Kapitel 2 betragter vi en generel geometrisk L vy proces og l ser detikke-line re optimale stoppetidsproblem om at maksimere variansen p stop-petidspunktet. For at l se dette problem l ser vi f rst et hj lpeproblemet er et klassisk, kvadratisk Optimal stoppetidsproblem. Vifinder at l sningen til problemet med at maksimere varians afh nger af hvor-vidt randomiserede stoppetiden er inkluderet i den m ngde af stoppetider vimaksimerer over. For nogle processer vil inklusionen af randomiserede stop-petider h ve v rdifunktionen og for andre processer vil det ikke. Selv n rv rdifunktionen ikke p virkes af at de randomiserede stoppetider er inklu-deret, er problemet nogle gange lettere at l se n r de Kapitel 3 betragter vi det ikke-line re optimale stoppetidsproblem somg r ud p at maksimere middelv rdien minus en konstant gange variansenp stoppetidspunktet.

10 F rst l ser vi problemet for spektralt negative geo-metriske L vy processer. Vi udleder b de statiske og dynamiske l sningersom er givet ved f rste gang processen krydser over en gr nse. Derefter l servi problemet for en Cram r-Lundberg proces med exponentialfordelte springopad. Vi udleder en statisk Optimal stoppetid givet ved f rste gang processenrammer et interval, og vi udleder en dynamisk Optimal stoppetid givet vedf rste gang processen kommer over en gr nse. Til sidst udleder vi optimalestoppetider for det optimale stoppetidsproblem som g r ud p at minimerevariansen givet en nedre gr nse p middelv Kapitel 4 betragter vi en amerikansk put option i et Black-Scholes mar-ked. Vi foresl r en model for irrationel indl sning af optionen. Vi modellererindl sningstidspunktet ved hj lp af en stokastisk intensitet som afh nger afhvor profitabelt det er at indl se.