Optimal Stopping and Utility in a Simple Model of Unemployment Insurance
aa r X i v : . [ q -f i n . S T ] S e p Optimal Stopping and Utility in a Simple Modelof Unemployment Insurance
Jason S. Anquandah ∗ and Leonid V. Bogachev † Department of Statistics, School of Mathematics, University of Leeds,Leeds LS2 9JT, UK
Abstract
Managing unemployment is one of the key issues in social policies. Unemploymentinsurance schemes are designed to cushion the financial and morale blow of loss ofjob but also to encourage the unemployed to seek new jobs more pro-actively dueto the continuous reduction of benefit payments. In the present paper, a simplemodel of unemployment insurance is proposed with a focus on optimality of theindividual’s entry to the scheme. The corresponding optimal stopping problem issolved, and its similarity and differences with the perpetual American call option arediscussed. Beyond a purely financial point of view, we argue that in the actuarialcontext the optimal decisions should take into account other possible preferencesthrough a suitable utility function. Some examples in this direction are worked out.
Keywords: insurance; unemployment; optimal stopping; geometric Brownian motion;martingale; free-boundary problem; American call option; utility.
MSC 2010:
Primary 97M30; Secondary 60G40, 91B16, 91B30
1. Introduction
Assessing the risk in financial industries often aims at finding optimal choices in decisionmaking. In the insurance sector, optimality considerations are crucial primarily for theinsurers, who have to address monetary issues (such as how to price the insurance policyso as not to run it at a loss but also to keep the product competitive) and time issues(e.g., when to release the product to the market). Less studied but also important areoptimal decisions on behalf of the insured individuals, related to monetary issues (e.g.,how profitable is taking up an insurance policy and the right portion of wealth to invest),consumption decisions (e.g., whether to maximize or optimize own consumption), or time-related decisions (such as when it is best to enter or exit an insurance scheme).In this paper we focus on the particular type of products related to unemploymentinsurance (UI) , whereby an employed individual is covered against the risk of involuntaryunemployment (e.g., due to redundancy). Various UI systems are designed to help cushionthe financial (as well as morale) blow of loss of job and to encourage unemployed workersto find a new job as early as possible in view of the continued reduction of benefits. The ∗ Corresponding author. E-mail: [email protected] † E-mail:
[email protected] P a y r a t e τ τ τ + τ X t Fig. 1: A time chart of the unemployment insurance scheme. The horizontal axis shows(continuous) time; the vertical axis indicates the pay rate (i.e., income receivable per unittime). The origin t = 0 indicates the start of employment. Two pieces of a random path X t depict the dynamics of the individual’s wage whilst in employment. The individual joins theUI scheme at entry time τ (by paying a premium P ). When the current job ends (at time τ > τ ), a benefit proportional to the final wage X τ is payable according to a predefinedschedule (e.g., see Example 2.1), until a new job is found after the unemployment spell ofduration τ .The decision the individual is facing is when (rather than if ) to join the scheme. Whatare the considerations being taken into account when contemplating such a decision? Onthe one hand, delaying the entry may be a good idea in view of the monetary inflation overtime — since the entry premium is fixed, its actual value is decreasing with time. Also, itmay be reasonably expected that the wage is likely to grow with time (e.g., due to inflation2ut also as a reward for improved skills and experience), which may have a potential toincrease the total future benefit (which depends on the final wage). Last but not least, somesavings may be needed before paying the entry premium becomes financially affordable. Onthe other hand, delaying the decision to join the insurance scheme is risky, as the individualremains unprotected against loss of job, with its financial as well as morale impact.Thus, there is a scope for optimizing the decision about the entry time — probably nottoo early but also not too late. Apparently, such a decision should be based on the informa-tion available to date, which of course includes the inflation rate and also the unemploymentand redeployment rates, all of which should, in principle, be available through the publishedstatistical data. Another crucial input for the decision-making is the individual’s wage asa function of time. We prefer to have the situation where this is modelled as a randomprocess, the values of which may go up as well as down. This is the reason why we donot consider salaries (which are in practice piecewise constant and unlikely to decrease),and instead we are talking about wages , which are more responsive to supply and demandand are also subject to “real-wage” adjustments (e.g., through the consumer price index,CPI). Besides, loss of job is more likely in wage-based employments due to the fluidity ofthe job market. For simplicity, we model the wage dynamics using a diffusion process called geometric Brownian motion . To summarize, the optimization problem for our model aims to maximize the expectednet present value of the UI scheme by choosing an optimal entry time τ ∗ . We will showthat this problem can be solved exactly by using the well-developed optimal stopping theory [35, 36, 39]. It turns out that the answer is provided by the hitting time of a suitablethreshold b ∗ , that is, the first time τ b ∗ when the wage process X t will reach this level. Sincethe value of b ∗ is not known in advance, this leads to solving a free-boundary problem forthe differential operator (generator) associated with the diffusion process ( X t ) . In fact, wefirst conjecture the aforementioned structure of the solution and find the value b ∗ , and thenverify that this is indeed the true solution to the optimal stopping problem.In the insurance literature, there has been much interest towards using optimality con-siderations, including optimal stopping problems. From the standpoint of insurer seekingto maximize their expected returns, the optimal stopping time may be interpreted as thetime to suspend the current trading if the situation is unfavourable, and to recalculate pre-miums (see, e.g., [22, 24, 32] and further references therein). Insurance research has alsofocused on optimality from the individual’s perspective. One important direction relevantto the UI context was the investigation of the job seeking processes, especially when re-turning from the unemployed status [4, 29, 43]. This was complemented by a more generalresearch exploring ways to optimize and improve the efficacy of the UI systems (also interms of reducing government expenditure), using incentives such as a decreasing bene-fit throughout the unemployment spell, in conjunction with sanctions and workfare (see[13, 17, 20, 26, 27], to cite but a few). A related strand of research is the study of opti-mal retirement strategies in the presence of involuntary unemployment risks and borrowingconstraints [6, 7, 14, 21, 40].To the best of our knowledge, optimal stopping problems in the UI context (such asthe optimal entry to / exit from a UI policy) have not received sufficient research attention.This issue is important, because knowing the optimal entry strategies is likely to enhance For technical convenience, we choose to work with continuous-time models, but our ideas can also beadapted to discrete time (which may be somewhat more natural, since the wage process is observed by theindividual on a weekly time scale). τ b ∗ may be infinite with positive probability (at least for some valuesof the parameters), and even if it is finite with probability one, the expected waiting timemay be very long.Motivated by this observation, we argue that certain elements of utility should be addedto the analysis, aiming to quantify the individual’s “impatience” as a measure of purposeand satisfaction. We suggest a few simple ideas of how utility might be accommodatedin the UI optimal stopping framework. Despite the simplicity of such examples, in mostcases they lead to much harder optimal stopping problems. Not attempting to solve theseproblems in full generality, we confine ourselves to exploring suboptimal solutions in the classof hitting times, which nonetheless provide useful insight into possible effects of inclusionof utility into the optimal stopping context.The general concept of utility in economics was strongly advocated in the classicalbook by von Neumann and Morgenstern [33], whose aim was in particular to overcomethe idealistic assumption of a strictly rational behaviour of market agents. These ideaswere quickly adopted in insurance, dating back to Borch [3] and soon becoming part ofthe insurance mainstream, culminating in the Expected Utility Theory (see a recent bookby Kaas et al. [23]) routinely used as a standard tool to price insurance products. Inparticular, examples of use of utility in the UI analysis are ubiquitous (see, e.g., [1, 2,13, 17, 19, 20, 26, 27, 28]). There have also been efforts to combine optimal stoppingand utility [5, 6, 18, 24, 32, 43]. However, all such examples were limited to using utilityfunctions to re-calculate wealth, while other important objectives and preferences such asthe desire to buy the policy or to reduce the waiting times have not been considered as yet,as far as we can tell.
Layout.
The rest of the paper is organized as follows. In Section 2, our insurance modelis specified and the optimization problem is set up. In Section 3, the optimal stoppingproblem is solved using a reduction to a suitable free boundary problem, including the iden-tification of the critical threshold b ∗ . This is complemented in Section 4 by an elementaryderivation using explicit information about the distribution of the hitting times for the geo-metric Brownian motion. Section 5 addresses various statistical issues and also provides anumerical example illustrating the optimality of the critical threshold b ∗ . In Section 6, wecarry out the analysis of parametric dependence in our model upon two most significant ex-ogenous parameters, the unemployment rate and the wage drift, and also give an economic Impact of individualistic (not always rational) perception in economics and financial markets is thesubject of the modern behavioural economics (see, e.g., a recent monograph [8]).
Notation.
We use the standard notation a ∧ b := min { a, b } , a ∨ b := max { a, b } , and a + := a ∨ .
2. Optimal stopping problem
Let us describe our model in more detail. Suppose that time t ≥ is continuous and ismeasured (in the units of weeks) starting from the beginning of the individual’s employmentWe assume without loss of generality that the unemployment insurance policy is availableimmediately (although in practice, a qualifying period at work would normally be requiredfor eligibility). Let X t > denote the individual’s wage (i.e., payment per week, paid inarrears) as a function of time t ≥ , such that X = x . We treat X = ( X t , t ≥ as a random process defined on a filtered probability space (Ω , F , ( F t ) , P ) , where Ω is asuitable sample space (e.g., consisting of all possible paths of ( X t ) ), the filtration ( F t ) is an increasing sequence of σ -algebras F t ⊂ F , and P is a probability measure on themeasurable space (Ω , F ) which determines the distribution of various random inputs in themodel, including ( X t ) . It is assumed that the process ( X t ) is adapted to the filtration ( F t ) , that is, X t is F t -measurable for each t ≥ . Intuitively, F t is interpreted as the fullinformation available up to time t , and measurability of X t with respect to F t means thatthis information includes knowledge of the values of the process X t .Furthermore, remembering that X t is positive valued, we use for it a simple model of geometric Brownian motion driven by the stochastic differential equation d X t X t = µ d t + σ d B t , X = x, (2.1)where B t is a standard Brownian motion (i.e., with mean zero, E ( B t ) = 0 , and varianceVar ( B t ) = t , and with continuous sample paths), and µ ∈ R and σ > are the drift andvolatility rates, respectively. The equation (2.1) is well known to have the explicit solution(see, e.g., [39, Ch. III, § X t = x exp (cid:8) ( µ − σ ) t + σB t (cid:9) ( t ≥ . (2.2)Note that E x ( X t ) = x e µt , Var x ( X t ) = x e µt (cid:0) e σ t − (cid:1) , (2.3)where E x and Var x denote expectation and variance with respect to the distribution of X t given the initial value X = x .Let us now specify the unemployment insurance scheme. An individual who is currentlyemployed may join the scheme by paying a fixed one-off premium P > at the pointof entry. If and when the current employment ends (say, at time instant τ ), the benefitproportional to the final wage X τ is payable according to the benefit schedule h ( s ) ; thatis, the payout at time t ≥ τ is given by X τ h ( t − τ ) . However, the payment stops whena new job is found after the unemployment spell of duration τ . For simplicity, we assume5hat both τ and τ have exponential distribution (with parameters λ and λ , respectively);as mentioned in the Introduction, this guarantees a Markovian nature of the correspondingtransitions. These random times are also assumed to be statistically independent of theprocess ( X t ) .Possible transitions in the state space of our insurance model are shown in Fig. 2, wheresymbols “0” and “1” encode the states of being employed and unemployed, respectively,whereas suffixes “+” and “–” indicate whether insurance is in place or not. Note that alltransitions, except from 0– to 0+ (which is subject to optimal control based of observationsover the wage process ( X t ) ), occur in a Markovian fashion; that is, the holding times areexponentially distributed (with parameters λ if in states 0– and 0+, or λ if in states 1–and 1+). ✣✢✤✜
0+ 1+ ✣✢✤✜
0– 1– ✣✢✤✜ ✣✢✤✜ ✲(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✠✻ ③② ( τ < τ )( X t ) τ λ λ λ ( τ ≥ τ ) λ Legend:0 (employed)1 (unemployed)+ (with insurance)– (without insurance)
Fig. 2: Schematic diagram of possible transitions in the unemployment insurance scheme.Here, τ and τ are the (exponential) holding times in states 0 and 1, with parameters λ and λ , respectively, whereas τ is the entry time (i.e., from state 0– to state 0+), which issubject to optimal control based on observations over the wage process ( X t ) .The individual’s decision about a suitable time to join the scheme is based on theinformation available to date. In our model, this information encoded in the filtration ( F t ) is provided by ongoing observations over the wage process ( X t ) . Thus, admissible strategies for choosing τ must be adapted to the filtration ( F t ) ; namely, at any time instant t ≥ itshould be possible to determine whether τ has occurred or not yet, given all the informationin F t . In mathematical terms, this means that τ is a stopping time , whereby for any t ≥ the event { τ > t } belongs to the σ -algebra F t (see, e.g., [44, Ch. 1, §
3, p. 25]).
Remark . In general, a stopping time τ is allowed to take values in [0 , ∞ ] including ∞ ,in which case waiting continues indefinitely and the decision to join the scheme is nevertaken. In practice, it is desirable that the stopping time τ be finite almost surely (a.s.) (i.e.,P x ( τ < ∞ ) = 1 ), but this may not always be the case (see Section 4.1). As was explained informally in the Introduction, there is a scope for optimizing the choiceof the entry time τ , where optimality is measured by maximizing the expected financialgain from the scheme. Our next goal is to obtain an expression for the expected gain underthe contract. First of all, conditional on the final wage X τ , the expected future benefit to6e received under this insurance contract is given by X τ E (cid:18)Z τ e − rs h ( s ) d s (cid:19) = βX τ , (2.4)where r is the inflation rate and β := Z ∞ λ e − λ t H ( t ) d t, H ( t ) := Z t e − rs h ( s ) d s. (2.5)Note that the expectation in formula (2.4) is taken with respect to the (exponential)random waiting time τ (with parameter λ ), and that the expression inside integrationinvolves discounting to the beginning of unemployment at time τ . Example . A specific example of the benefit schedule h ( s ) may be as follows, h ( s ) = ( h , ≤ s ≤ s ,h e − δ ( s − s ) , s ≥ s , (2.6)where < h ≤ , ≤ s ≤ ∞ and δ > . Thus, the insured receives a certain fraction oftheir final wage (i.e., h X τ ) for a grace period s , after which the benefit is falling downexponentially with rate δ . This example is motivated by the declining unemployment com-pensation system in France [25]. Having specified the schedule function, all calculationscan be done explicitly. In particular, the constant β in (2.4) is calculated from (2.5) as β = h (cid:0) − e − ( r + λ ) s (cid:1) r + λ + h e − ( r + λ ) s r + λ + δ . In the extreme cases s = 0 or s = ∞ , this expression simplifies to β = h λ (cid:18) − r + δr + λ + δ (cid:19) , s = 0 ,h λ (cid:18) − rr + λ (cid:19) , s = ∞ . Here, the first factor has a clear meaning as the product of pay per week ( h ) and themean duration of the benefit payment (E ( τ ) = 1 /λ ), whereas the second factor takesinto account the discounting at rates r and δ .Returning to the general case, if the contract is entered immediately (subject to thepayment of premium P ), then the net expected benefit discounted to the entry time t = 0 is given by the gain function g ( x ) := E x (cid:0) e − rτ βX τ (cid:1) − P, (2.7) More specifically, according to the French UI system back in the 1990s (see [25, p. 8]), a workeraged 50 or more, with eight months of insurable employment in the last twelve months, was entitled tofull benefits equal to 57.4% of the final wage payable for the first eight months, thereafter declining by15% every four months; however, the payments continued for no longer than 21 months overall. Thisleads to choosing the following numerical values in (2.6): h = 0 . , s = 8 (52 / . = 34 . (weeks)and δ = − (3 /
52) ln (1 − . . = 0 . . (per week). The restriction of the benefit term by
21 (52 /
12) = 91 weeks can be taken into account in our model by adjusting the parameter λ from thecondition E ( τ ) = 91 , giving λ . = 0 . . A more conservative choice is to use a tail probability condition,for example, P ( τ >
91) = 0 . , yielding λ = − ln (0 . / . = 0 . (with E ( τ ) . = 39 . ). x = X is the starting wage and the symbol E x now indicates expectation with respectto both τ and X τ . Recall that the random time τ is independent of the process ( X t ) andhas the exponential distribution with parameter λ . Using the total expectation formula(see, e.g., [38, § II.7.4, Definition 3, p. 214, and Property G*, p. 216]) and substituting theexpression (2.3), the expectation in (2.7) is computed as follows,E x (cid:0) e − rτ X τ (cid:1) = E x (cid:2) e − rτ E x ( X τ | τ ) (cid:3) = E x (cid:2) e − rτ ( x e µτ ) (cid:3) = x Z ∞ e ( µ − r ) t λ e − λ t d t = λ xr + λ − µ . (2.8)Thus, substituting (2.8) into (2.7) and denoting ˜ r := r + λ , β := β λ ˜ r − µ , (2.9)the gain function is represented explicitly as g ( x ) = β x − P. (2.10)Of course, the computation in (2.8) is only meaningful as long as µ < r + λ = ˜ r. (2.11) Assumption 2.1.
In what follows, we always assume that the condition (2.11) is satisfied.
Remark . In real life applications, the wage growth rate µ is rather small (but maybe either positive or negative). It is unlikely to exceed the inflation rate r , but even if itdoes, then it is hardly possible economically that it is greater than the combined inflation–unemployment rate ˜ r = r + λ . Thus, the condition (2.11) is absolutely realistic.To generalize the expression (2.10), consider a delayed entry time τ > (tacitly as-suming that τ < ∞ ). Discounting first to the entry time τ when the deduction of thepremium P is activated, and then further down to the initial time moment t = 0 , yieldsthe expected net present value of the total gain as a function of the initial wage x ,eNPV ( x ; τ ) := E x (cid:2) e − rτ (cid:0) e − r ( τ − τ ) βX τ − P (cid:1) { τ<τ } (cid:3) , (2.12)where the expectation on the right now also includes averaging with respect to τ , whichis a functional of the path ( X t ) . Note that the indicator function under the expectationspecifies that the entry time τ must occur prior to τ , for otherwise there will be no gain. Remark . The notation (2.12) emphasizes that the expected net present value dependson the specific entry time τ . As was intuitively explained in the Introduction, there is a scopefor optimizing the choice of τ , where optimality is measured by maximizing eNPV ( x ; τ ) .Formula (2.12) indicates that the decision time τ has a finite (random) expiry date τ (using the terminology of financial options). However, the expectation in (2.12) involvesaveraging with respect to τ . Moreover, taking advantage of exponential distribution of τ ,the expression (2.12) can be rewritten without any expiry date (i.e., as a perpetual option ). Lemma 2.1.
The expected net present value defined by formula (2.12) can be expressedin the form eNPV ( x ; τ ) = E x (cid:2) e − ˜ rτ g ( X τ ) { τ< ∞} (cid:3) , (2.13) where the function g ( · ) is defined in (2.7) and ˜ r = r + λ ( see (2.9)) . roof. Since the distribution of τ is exponential, the excess time ˜ τ := τ − τ conditioned on { τ < τ } is again exponentially distributed (with the same parameter λ ) and independentof τ . Hence, conditioning on τ (restricted to the event { τ < ∞} ) and using the totalexpectation formula as before [38, § II.7, Property G*, p. 216]), together with the (strong)Markov property of the process ( X t ) , we get from (2.12)eNPV ( x ; τ ) = E x (cid:0) E x (cid:2) e − rτ (e − r ( τ − τ ) βX τ − P ) { τ >τ } (cid:12)(cid:12) τ (cid:3)(cid:1) = E x (cid:16) e − rτ E x (cid:2) (e − r ˜ τ β X τ +˜ τ − P ) (cid:12)(cid:12) τ (cid:3) · E x (cid:2) { τ >τ } (cid:12)(cid:12) τ (cid:3)(cid:17) = E x (cid:16) e − rτ E X τ (cid:2) (e − r ˜ τ β e X ˜ τ − P ) (cid:3) · P x (cid:0) τ > τ | τ (cid:1)(cid:17) , (2.14)where e X t := X τ + t ( t ≥ ) is a shifted wage process starting at e X = X τ . SubstitutingP x (cid:0) τ > τ | τ (cid:1) = e − λ τ and recalling notation (2.7), formula (2.14) is reduced to (2.13).Finally, without loss we can remove the indicator from the expression (2.13) by definingthe value of the random variable under expectation to be zero on the event { τ = ∞} . Thisdefinition is consistent with the limit at infinity. Indeed, observe using (2.2) and (2.8) that e − ˜ rt g ( X t ) = e − ˜ rt (cid:16) β x e ( µ − σ / t + σB t − P (cid:17) = β x exp (cid:8) − t (cid:0) ˜ r − µ + σ + σ t − B t (cid:1)(cid:9) − P e − ˜ rt . (2.15)Due to the condition (2.11), ˜ r − µ + σ > σ > . In addition, by the (strong) law oflarge numbers for the Brownian motion (see, e.g., [9, Exercise 6.4, p. 265] or [39, Ch. III, § lim t →∞ t − B t = 0 ( P-a.s. ) . Thus, the limit of (2.15) as t → ∞ is zero (P x -a.s.). Hence, the event { τ = ∞} does notcontribute to the expectation (2.13), so that, substituting (2.8), we geteNPV ( x ; τ ) = E x (cid:2) e − ˜ rτ g ( X τ ) (cid:3) . (2.16)To summarize, identification of the optimal entry time τ = τ ∗ , in the sense of maxi-mizing the expected net present value eNPV ( x ; τ ) as a function of strategy τ (see (2.16)),is reduced to solving the following optimal stopping problem , v ( x ) = sup τ E x (cid:2) e − ˜ rτ g ( X τ ) (cid:3) , (2.17)where the function g ( x ) is given by (2.10) and the supremum is taken over the class of alladmissible stopping times τ (i.e., adapted to the filtration ( F t ) ). The supremum v ( x ) in(2.17) is called the value function of the optimal stopping problem. The simple model of unemployment insurance set out in Section 2.1 can be easily extendedto include mortality. Following [31, pp. 399–401], suppose that the individual who contem-plates taking out the unemployment insurance policy may die (say, at a random time τ from zero), independently of employment-related events and subject to a constant force ofmortality λ . That is to say, given that the individual is alive at current age t ≥ , the9esidual lifetime τ − t is an independent random variable exponentially distributed withparameter λ , P ( τ − t > s | τ > t ) = e − λ s ( s ≥ . The necessary modifications to the unemployment insurance model of Section 2.2 startby adjusting the formula for the expected future benefit (see (2.4)). Assuming that deathdoes not occur prior to the time τ of losing the job (i.e., τ > τ , so that ˜ τ := τ − τ is exponentially distributed with parameter λ ), the benefit payments cease at τ ∧ ˜ τ (i.e.,when a new job is found or at death, whichever occurs first). Since τ and ˜ τ are independentand both have exponential distributions, the random variable τ ∧ ˜ τ has the exponentialdistribution with parameter λ + λ . Hence, the constant β from (2.5) is now written as β = Z ∞ ( λ + λ ) e − ( λ + λ ) t H ( t ) d t. Next, we need to take into account the effect of death in service, that is, if τ ≤ τ . To bespecific, it is reasonable to assume that the lump sum to be paid by the employer in thiscase is proportional to the final wage, say a † X τ . Then, separating the cases where deathoccurs after or prior to loss of job, it is easy to see that the definition (2.7) of the gainfunction (i.e., net expected benefit discounted to the policy entry time) takes the form g ( x ) = E x (cid:0) e − rτ β X τ { τ <τ } (cid:1) + E x (cid:0) e − rτ a † X τ { τ ≤ τ } (cid:1) − P. (2.18)The first expectation in (2.18) is computed using conditioning on τ and the totalexpectation formula (cf. (2.8)),E x (cid:0) e − rτ X τ { τ <τ } (cid:1) = E x (cid:2) e − rτ E x (cid:0) X τ { τ <τ } (cid:12)(cid:12) τ (cid:1)(cid:3) = E x (cid:2) e − rτ E x ( X τ | τ ) · P x ( τ > τ | τ ) (cid:3) = E x (cid:0) e − rτ x e µτ e − λ τ (cid:1) = x Z ∞ e ( µ − r − λ ) t λ e − λ t d t = λ xr + λ + λ − µ , (2.19)where in the second line we used conditional independence of X τ and τ given τ . Similarly,by conditioning on τ the second expectation in (2.18) is represented asE x (cid:0) e − rτ X τ { τ ≤ τ } (cid:1) = E x (cid:2) X τ E x (cid:0) e − rτ { τ ≥ τ } (cid:12)(cid:12) τ (cid:1)(cid:3) = E x (cid:20) X τ Z ∞ τ e − rt λ e − λ t d t (cid:21) = λ r + λ E x (cid:0) X τ e − ( r + λ ) τ (cid:1) . (2.20)Again conditioning on τ , the last expectation is computed as follows,E x (cid:0) X τ e − ( r + λ ) τ (cid:1) = E x (cid:2) e − ( r + λ ) τ E x ( X τ | τ ) (cid:3) = E x (cid:0) e − ( r + λ ) τ x e µτ (cid:1) = x Z ∞ e − ( r + λ − µ ) t λ e − λ t d t = λ xr + λ + λ − µ . (2.21)10inally, substituting the expressions (2.19), (2.20) and (2.21) into the definition (2.18),we obtain explicitly g ( x ) = λ xr + λ + λ − µ (cid:18) β + λ a † r + λ (cid:19) − P. This expression has the same form as (2.10) but with the parameters ˜ r and β redefinedas follows (cf. (2.9)), ˜ r := r + λ + λ , β := λ ˜ r − µ (cid:18) β + λ a † r + λ (cid:19) . In addition, the inequality (2.11) of Assumption 2.1 is updated accordingly. Subject to thisreparameterization, all subsequent calculations leading to the optimal stopping problem(2.17) remain unchanged.For the sake of clarity and in order not to distract the reader by unnecessary techni-calities, in the rest of the paper we adhere to the original version of the model (i.e., withno mortality, λ = 0 ); however, see the discussion at the end of Section 6.4 indicating animportant regularizing role of mortality, helping to avoid undesirable inconsistencies of themodel at small unemployment rates λ . The next lemma shows that the optimal stopping problem (2.17) is well posed.
Lemma 2.2.
The value function x v ( x ) of the optimal stopping problem (2.17) hasthe following properties: (i) v (0) = 0 and, moreover, v ( x ) ≥ for all x ≥ ;(ii) v ( x ) < ∞ for all x ≥ .Proof. (i) If x = 0 then, due to (2.2), X t ≡ (P -a.s.) and the stopping problem (2.17)is reduced to v (0) = sup τ E ( − P e − ˜ rτ ) , which has the obvious solution τ = ∞ (P -a.s.), with the corresponding supremum value v (0) = 0 . Furthermore, by considering τ = ∞ (P x -a.s.) it readily follows from (2.17) that v ( x ) ≥ for all x ≥ .(ii) Recalling that µ < ˜ r (see Assumption 2.1), observe that the process e − ˜ rt X t is a supermartingale ; indeed, for ≤ s ≤ t we have, using (2.2) and (2.3),E x (cid:2) e − ˜ rt X t | F s (cid:3) = e − ˜ rt X s E (cid:2) e σ ( B t − B s )+( µ − σ )( t − s ) (cid:3) = e − ˜ rt X s e µ ( t − s ) ≤ e − ˜ rs X s ( P x -a.s. ) . In particular, E x (e − ˜ rt X t ) ≤ E x ( X ) = x. Hence, by Doob’s optional sampling theorem for non-negative, right-continuous super-martingales (see, e.g., [44, Theorem 8.18, pp. 140–141]), for any stopping time τ we haveE x (e − ˜ rτ X τ ) ≤ E x ( X ) = x, and it follows that the supremum in (2.17) is finite.11 .5. The optimal stopping rule For the wage process ( X t ) , consider the hitting time τ b of a threshold b ∈ R , defined by τ b := inf { t ≥ X t ≥ b } ∈ [0 , ∞ ] . (As usual, we make a convention that inf ∅ = ∞ .) Clearly, τ b is a stopping time, that is, { τ ≤ t } ∈ F t for all t ≥ . Since the process X t has a.s.-continuous sample paths, onthe event { τ b < ∞} we have X τ b = b (P x -a.s.). As we will show, the optimal strategy forthe optimal stopping problem (2.17) is to wait until the random process X t hits a certainthreshold b ∗ (see Fig. 3). More precisely, the solution to (2.17) is provided by the followingstopping rule, τ ∗ = ( τ b ∗ if x ∈ [0 , b ∗ ] , if x ∈ [ b ∗ , ∞ ) . (2.22)That is to say, if x ≥ b ∗ then one must stop and buy the policy immediately, or else waituntil the hitting time τ b ∗ ≥ occurs and buy the policy then. (Of course, these two rulescoincide when x = b ∗ .) However, if it happens so that τ b ∗ = ∞ , then, according to theabove rule, one must wait indefinitely and, therefore, never buy the policy. Time t (weeks) W age X t ( eu r o ) b ∗ τ ∗ . . . . . Time t (weeks) Y t = l n X t Fig. 3: Simulated wage process X t (left) and Y t = ln X t (right) according to the geometricBrownian motion model (2.2), with X = 346 (euros) and parameters µ = 0 . and σ = 0 . (see Example 5.2). The dashed horizontal line on the left plot indicates theoptimal threshold b ∗ . = 352 . (euros) first attained in this simulation at τ ∗ = 54 (weeks).The dashed line on the right plot shows the estimated drift of the log-transformed data(see Section 5.2).The specific value of the critical threshold b ∗ is given by b ∗ = P q ∗ β ( q ∗ − , (2.23)where q ∗ = 1 σ (cid:18) − (cid:0) µ − σ (cid:1) + q(cid:0) µ − σ (cid:1) + 2 ˜ rσ (cid:19) . (2.24)12t is straightforward to check, using condition (2.11), that q ∗ > (see also Section 3.2).Finally, the corresponding value function (2.17) is specified as v ( x ) = ( ( β b ∗ − P ) (cid:16) xb ∗ (cid:17) q ∗ , x ∈ [0 , b ∗ ] ,β x − P, x ∈ [ b ∗ , ∞ ) . (2.25)Equivalently, substituting the expression (2.23), formula (2.25) is explicitly rewritten as v ( x ) = Pq ∗ − (cid:18) β ( q ∗ − xP q ∗ (cid:19) q ∗ , ≤ x ≤ P q ∗ β ( q ∗ − ,β x − P, x ≥ P q ∗ β ( q ∗ − . (2.26)In particular, the function x v ( x ) is strictly increasing for x ≥ , with v (0) = 0 (cf.Lemma 2.2).These results will be proved in Section 3. For orientation, it is useful to consider the simple baseline case σ = 0 , where the randomprocess X t (see (2.2)) degenerates to the deterministic function X t = x e µt ( t ≥ . Hence, any stopping time τ is non-random, say τ = t , and the optimal stopping problem(2.17) is reduced to v ( x ) = sup t ≥ (cid:2) e − ˜ rt ( β x e µt − P ) (cid:3) . (2.27)The problem (2.27) is easily solved, with the maximizer t ∗ given by t ∗ = inf (cid:8) t ≥ x e µt ≥ b ∗ (cid:9) ∈ [0 , ∞ ] , (2.28)where b ∗ = P ˜ rβ (˜ r − µ ) , µ > ,Pβ , µ ≤ . (2.29)The expression (2.29) is consistent with the general formula (2.23), noting that, in the limitas σ ↓ , the quantity (2.24) is reduced to (cf. (2.11)) q ∗ = ˜ rµ > , µ > , ∞ , µ ≤ . With this convention, it is easy to check that the value function (2.27) is given by thegeneral formula (2.25). In particular, if µ ≤ and x < b ∗ then, according to (2.28), t ∗ = ∞ and from (2.27) we get v ( x ) = 0 ; indeed, the function t x e µt is non-increasing,so it never attains the required threshold b ∗ > x . In contrast, if x ≥ b ∗ then by (2.28) t ∗ = 0 (for any µ ), and (2.27) readily yields v ( x ) = β x − P .13 . Solving the optimal stopping problem The optimal stopping problem (2.17) involves two tasks: (i) evaluating the value function v ( x ) , and (ii) identifying the maximizer τ = τ ∗ . A standard approach is to try and guess the solution and then to verify that it is correct. Let us look more closely at the nature of the value function v ( x ) that we are trying toidentify. Observe that by picking τ = 0 in (2.17) yields the lower estimate v ( x ) ≥ g ( x ) . (3.1)Clearly, if v ( x ) > g ( x ) then we have not yet achieved the maximum payoff available, sowe should continue to wait. On the other hand, if v ( x ) = g ( x ) then the maximum hasbeen attained and we should stop. This motivates the definition of the two regions, C ( continuation ) and S ( stopping ), C := { x ≥ v ( x ) > g ( x ) } , S := { x ≥ v ( x ) ≤ g ( x ) } . By virtue of the Markov property of the process X t , the same argument can be propa-gated to any time t ≥ , provided that stopping has not yet occurred. Namely, if X t = x ′ (and τ ≥ t ) then the problem (2.17) is updated with the new (residual) stopping time τ ′ = τ − t and with the initial value x replaced by x ′ .Thus, it is natural to expect that the optimal strategy prescribes to continue as long asthe current wage value X t belongs to the region C (i.e., v ( X t ) > g ( X t ) ), but to stop when X t first enters the region S (i.e., v ( X t ) ≤ g ( X t ) ). That is to say, the optimal stoppingtime should be given by τ ∗ = inf { t ≥ X t ∈ S } = inf { t ≥ v ( X t ) ≤ g ( X t ) } ∈ [0 , ∞ ] . (3.2)To clarify the plausible structure of the stopping set S , recall (see the proof of Lemma2.2(i)) that a zero value of the stopping problem (2.17) is achieved by simply using thestrategy τ ≡ ∞ , that is, by never joining the scheme. Thus, if the initial wage X = x is small (e.g., such that g ( x ) = β x − P < ) then, in order to secure a positive payoff,we should wait for a sufficiently high wage X t . This suggests that the stopping rule (3.2)is reduced to the first hitting time for a certain set on the plane { ( t, x ) : t ≥ , x ≥ } .Furthermore, noting that the definition (3.2) is time homogeneous, in that it does notchange in the course of time t , we also hypothesize the simplest situation whereby theregions C and S are determined by a constant threshold y = b ∗ > , C = [ 0 , b ∗ ) , S = [ b ∗ , ∞ ) . (3.3)In other words, the conjectural hitting boundary does not depend on time.Hence, we are led to the reduced optimal stopping problem over the subclass of hittingtimes, u ( x ) = sup b ≥ E x (cid:2) e − ˜ rτ b g ( X τ b ) (cid:3) . (3.4) This conclusion is in accord with the general optimal stopping theory [35, § II.2.2].
14n particular, formula (3.2) specializes to τ b ∗ = inf { t ≥ X t ≥ b ∗ } = inf { t ≥ u ( X t ) ≤ g ( X t ) } ∈ [0 , ∞ ] . (3.5)Our first task is to identify the value function u ( x ) in (3.4) and the corresponding maximizer b = b ∗ by solving the corresponding free-boundary problem (Section 3.2). After that, wewill have to show that this solution is optimal in the general class of stopping times, thatis, u ( x ) = v ( x ) for all x ≥ (Section 3.3). According to general theory of optimal stopping (see, e.g., [35, Ch. IV]), in the continuationregion C = [0 , b ) (see (3.3)) the value function u ( x ) from (3.4) must be harmonic withrespect to the underlying process e X t generated by X t . More precisely, due to the discountingexponential factor in the optimal stopping problem (3.4), the process e X t is obtained from X t by independent killing (or discounting) with rate ˜ r (see [35, §§ b is asuitable threshold and τ b is the corresponding hitting time, then for any x ≥ the followingcondition must hold, E x (cid:2) e − ˜ r ( τ b ∧ t ) u ( X τ b ∧ t ) (cid:3) = u ( x ) ( t ≥ . (3.6)Note that the geometric Brownian motion X t determined by the stochastic differentialequation (2.1) is a diffusion process with the infinitesimal generator L := µx dd x + σ x d d x ( x > . (3.7)The generator of the killed process e X t is then given by (see [35, § ˜ L = L − ˜ rI, (3.8)where I is the identity operator. Then the harmonicity condition (3.6) can be reduced tothe differential equation ˜ Lu = 0 , that is, Lu − ˜ ru = 0 (see (3.8)).On the boundary x = b of the set C = [0 , b ) , due to the stopping rule (3.5) we have u ( b ) = g ( b ) . Moreover, according to the smooth fit principle (see [35, § u ′ ( b ) = g ′ ( b ) . Finally, in view of the equality v (0) = 0 (seeLemma 2.2(i)), we add a Dirichlet boundary condition at zero, u (0+) = lim x ↓ u ( x ) = 0 .Thus, we arrive at the following free-boundary problem , Lu ( x ) − ˜ ru ( x ) = 0 , x ∈ (0 , b ) ,u ( b ) = g ( b ) ,u ′ ( b ) = g ′ ( b ) ,u (0+) = 0 , (3.9)where both b > and u ( x ) are unknown.Substituting (2.10) and (3.7), the problem (3.9) is rewritten explicitly as µx u ′ ( x ) + σ x u ′′ ( x ) − ˜ r u ( x ) = 0 , x ∈ (0 , b ) ,u ( b ) = β b − P,u ′ ( b ) = β ,u (0+) = 0 . (3.10)15et us look for a solution of (3.10) in the form u ( x ) = x q ( x > ), with a suitable parameter q ∈ R . Then the differential equation in (3.10) yields σ q ( q −
1) + µq − ˜ r = 0 . (3.11)This quadratic equation has two distinct roots, q , = 1 σ (cid:18) − (cid:0) µ − σ (cid:1) ± q(cid:0) µ − σ (cid:1) + 2 ˜ rσ (cid:19) , where q < < q = q ∗ (see (2.24)). Also note that, due to the condition (2.11), theleft-hand side of (3.11) is negative at q = 1 , therefore q > . Thus, the general solutionof the differential equation (3.10) is given by u ( x ) = Ax q + B x q , x ∈ (0 , b ) , (3.12)with arbitrary constants A and B . But since q < , the condition u (0+) = 0 implies that B = 0 . Hence, (3.12) is reduced to u ( x ) = Ax q ≡ Ax q ∗ ( < x < b ). Furthermore, theboundary conditions in (3.10) yield ( Ab q ∗ = β b − P,Aq ∗ b q ∗ − = β , whence we find A = β b − Pb q ∗ , b = P q ∗ β ( q ∗ − . (3.13)Thus, the required solution to (3.10) is given by u ( x ) = ( β b − P ) (cid:16) xb (cid:17) q ∗ , x ∈ [0 , b ] ,β x − P, x ∈ [ b, ∞ ) (3.14)where the threshold b is defined in (3.13) and q ∗ > is the positive root of the equa-tion (3.11), given explicitly by formula (2.24). Using (3.13) and (3.14), it is easy to see that u ( x ) = g ( x ) , x ∈ [ b, ∞ ) ,u ( x ) > g ( x ) , x ∈ [0 , b ) , (3.15)in accord with the heuristics outlined in Section 3.1 (see (3.3)). However, there is noneed to check that the function u ( x ) defined in (3.14) solves the reduced optimal stoppingproblem (3.4), because we can prove directly that u ( x ) provides the solution to the originaloptimal stopping problem (2.17), that is, u ( x ) = v ( x ) for all x ≥ . Remark . Since u (0) = 0 by formula (3.14), and v (0) = 0 according to Lemma 2.2(i),in what follows it suffices to assume that x > .The proof of the claim above (commonly referred to as verification theorem ) consistsof two parts. 16i) Let us first show that u ( x ) ≥ v ( x ) ( x > ). If the map x u ( x ) was a C -function(i.e., with continuous second derivative), then the classical Itˆo formula (see, e.g.,[34, Theorem 4.1.2, p. 44]) applied to e − ˜ rt u ( X t ) would yield, on account of (2.1)and (3.7), e − ˜ rt u ( X t ) = u ( x ) + Z t e − ˜ rs (cid:0) Lu ( X s ) − ˜ ru ( X s ) (cid:1) d s + M t ( P x -a.s. ) , (3.16)where M t := Z t e − ˜ rs u ′ ( X s ) σX s d B s ( t ≥ . (3.17)However, for the function u ( x ) given by (3.14), its C -smoothness breaks down at thepoint x = b , where it is only C . But u ( x ) is strictly convex on (0 , b ) (i.e., u ′′ ( x ) > )and linear on ( b, ∞ ) , and we can define the action Lu ( x ) at x = b by using theone-sided second derivative, say, u ′′ ( b − ) = P q ∗ b − . (3.18)In this situation, a generalization of the Itˆo formula holds, known as the Itˆo–Meyerformula (see [39, Ch. VIII, § Lu ( x ) − ˜ ru ( x ) = 0 , x ∈ (0 , b ) . (3.19)Moreover, it is easy to check using (3.18) that the equality (3.19) also extends to x = b . On the other hand, on account of the condition (2.11) and the definition of b in (3.13), for x > b we get Lu ( x ) − ˜ r u ( x ) = µβ x − ˜ r ( β x − P )= β x ( µ − ˜ r ) + ˜ rP< β b ( µ − ˜ r ) + ˜ rP = P ( µq ∗ − ˜ r ) q ∗ − < , (3.20)because, due to the equation (3.11) and the inequality q ∗ > , µq ∗ − ˜ r = − σ q ∗ ( q ∗ − < . Thus, combining (3.19) and (3.20) we obtain Lu ( x ) − ˜ ru ( x ) ≤ x > . (3.21)Substituting the inequality (3.21) into formula (3.16), we conclude that, for any x > and all t ≥ , u ( x ) + M t ≥ e − ˜ rt u ( X t ) ( P x -a.s. ) . (3.22)According to formula (3.17), ( M t ) is a continuous local martingale (see, e.g., [39,Ch. II, § ( τ n ) be a localizing sequence of bounded stopping times, sothat τ n ↑ ∞ (P x -a.s.) and the stopped process ( M τ n ∧ t ) is a martingale, for each n ∈ N .17ow, let τ be an arbitrary stopping time of ( X t ) . From (3.22) we get u ( x ) + M τ n ∧ τ ≥ e − ˜ r ( τ n ∧ τ ) u ( X τ n ∧ τ ) ≥ e − ˜ r ( τ n ∧ τ ) g ( X τ n ∧ τ ) ( P x -a.s. ) , (3.23)using that u ( x ) ≥ g ( x ) for all x ≥ (see (3.15)). Taking expectation on both sidesof the inequality (3.23) gives u ( x ) ≥ E x (cid:2) e − ˜ r ( τ n ∧ τ ) g ( X τ n ∧ τ ) (cid:3) , (3.24)since by Doob’s optional sampling theorem (see, e.g., [44, Theorem 8.10, p. 131])E x [ M τ n ∧ τ ] = E x [ M ] = 0 . By Fatou’s lemma (see, e.g., [38, § II.6, Theorem 2(a), p. 187]), from (3.24) it follows u ( x ) ≥ E x (cid:2) lim inf n →∞ e − ˜ r ( τ n ∧ τ ) g ( X τ n ∧ τ ) (cid:3) = E x (cid:2) e − ˜ rτ g ( X τ ) (cid:3) . (3.25)Finally, taking in (3.25) the supremum over all stopping times τ , we obtain u ( x ) ≥ sup τ E x (cid:2) e − ˜ rτ g ( X τ ) (cid:3) = v ( x ) ( x > , as claimed.(ii) Let us now prove the opposite inequality, u ( x ) ≤ v ( x ) ( x > ). According to (3.1)and (3.15), we readily have u ( x ) = g ( x ) ≤ v ( x ) for x ∈ [ b, + ∞ ) . Next, fix x ∈ (0 , b ) and consider the representation (3.16) with t replaced by τ n ∧ τ b , where ( τ n ) is thelocalizing sequence of stopping times for ( M t ) as before. Then, by virtue of theidentity (3.19) (which, as has been explained, is also true for x = b ), it follows that u ( x ) + M τ n ∧ τ = e − ˜ r ( τ n ∧ τ b ) u ( X τ n ∧ τ b ) ( P x -a.s. ) . (3.26)Similarly as above, taking expectation on both sides of the equality (3.26) and againapplying Doob’s optional sampling theorem to the martingale ( M τ n ∧ t ) , we obtain u ( x ) = E x (cid:2) e − ˜ r ( τ n ∧ τ b ) u ( X τ n ∧ τ b ) (cid:3) . (3.27)Note that, for < x < b , we have ≤ u ( x ) ≤ u ( b ) and ≤ X τ n ∧ τ b ≤ b (P x -a.s.),hence ≤ e − ˜ r ( τ n ∧ τ b ) u ( X τ n ∧ τ b ) ≤ u ( b ) ( P x -a.s. ) . Using that τ n ↑ ∞ , observe that, P x -a.s., lim n →∞ e − ˜ r ( τ n ∧ τ b ) u ( X τ n ∧ τ b ) = e − ˜ rτ b u ( X τ b ) { τ b < ∞} + lim n →∞ e − ˜ rτ n u ( X τ n ) { τ b = ∞} = e − ˜ rτ b u ( b ) { τ b < ∞} , (3.28)because X τ b = b on the event { τ < ∞} , while ≤ u ( X τ n ) ≤ u ( b ) on the event { τ = ∞} . Hence, letting n → ∞ in (3.27) and using the dominated convergencetheorem (see, e.g., [38, § II.6, Theorem 3, p. 187]), we get, on account of (3.28), u ( x ) = E x (cid:2) e − ˜ rτ b u ( b ) { τ b < ∞} (cid:3) = E x (cid:2) e − ˜ rτ b g ( b ) { τ b < ∞} (cid:3) = E x (cid:2) e − ˜ rτ b g ( X τ b ) { τ b < ∞} (cid:3) ≤ v ( x ) , according to (2.17). That is, we have proved that u ( x ) ≤ v ( x ) for all < x < b , asrequired.Thus, the proof of the verification theorem is complete.18 . Elementary solution of the reduced problem In view of the formula (2.2), the hitting problem for the process X t is reduced to that forthe Brownian motion with drift, τ b := inf { t ≥ X t = b } ≡ inf { t ≥ B t + ˜ µt = ˜ b } , (4.1)where ˜ µ = µ − σ σ , ˜ b = 1 σ ln bx . (4.2)Suppose that x ≤ b , so that ˜ b ≥ . The explicit expression for the Laplace transform ofthe hitting time (4.1) is well known (see, e.g., [9, Exercises 6.29 and 6.31, p. 268] or [12,Proposition 3.3.5, p. 61]). Proposition 4.1.
For x ≤ b and any θ > , set Φ x,b ( θ ) := E x (e − θτ b ) ≡ E x (cid:0) e − θτ b { τ b < ∞} (cid:1) . (4.3) Then Φ x,b ( θ ) = exp n − ˜ b (cid:16)p ˜ µ + 2 θ − ˜ µ (cid:17)o , θ > , (4.4) where ˜ µ and ˜ b are defined in (4.2) . Substituting the expressions (4.2), the formula (4.4) is rewritten as Φ x,b ( θ ) = (cid:16) xb (cid:17) q ( θ ) , θ > , (4.5)where q ( θ ) is given by (cf. (2.24)) q ( θ ) = 1 σ (cid:18) − (cid:0) µ − σ (cid:1) + q(cid:0) µ − σ (cid:1) + 2 θσ (cid:19) . (4.6)As usual, it is straightforward to extract from the Laplace transform (4.3) some ex-plicit information about the distribution of the hitting time τ b . First, by the monotoneconvergence theorem (see, e.g., [38, § II.6, Theorem 1(a), p. 186] we have lim θ ↓ Φ x,b ( θ ) = E x ( { τ b < ∞} ) = P x ( τ b < ∞ ) . Hence, noting from (4.6) that q (0) = if µ − σ ≥ , − µσ if µ − σ < , (4.7)we obtain P x ( τ b < ∞ ) = (cid:16) xb (cid:17) q (0) = , µ − σ ≥ , (cid:16) xb (cid:17) − µ/σ , µ − σ < . (4.8)19 emark . The result (4.8) shows that hitting the critical threshold b = b ∗ , as requiredby the stopping rule, is only certain when the wage growth rate is large enough, µ ≥ σ .Thus, the “dangerous” case is when µ < σ , whereby relying only on the optimal stoppingrecipe may not be practical. This observation may serve as a germ of the idea to connectthe optimality problem in the insurance context with the notion of utility (cf. the discussionin Section 7.1 below).Via the Laplace transform Φ x,b ( θ ) , we can also obtain the mean hitting time E x ( τ b ) inthe case µ ≥ σ , where τ b < ∞ (P x -a.s.). Namely, again using the monotone convergencetheorem we have lim θ ↓ ∂ Φ x,b ( θ ) ∂θ = − lim θ ↓ E x (cid:0) τ b e − θτ b (cid:1) = − E x ( τ b ) . Hence, differentiating formula (4.5) at θ = 0 and noting from (4.6) that q (0) = 0 (cf. (4.7)) and q ′ (0) = ∞ , µ = σ , µ − σ , µ > σ , we get E x ( τ b ) = − ln (cid:16) xb (cid:17) (cid:16) xb (cid:17) q (0) q ′ (0) = ∞ , µ = σ , ln( b/x ) µ − σ , µ > σ . (4.9) An alternative (and more direct) method to derive the formulas (4.8) and (4.9) is based ongeneral theory of Markov processes by solving the suitable boundary value problems (see,e.g., [34, § π ( x ) := P x ( τ b < ∞ ) as a function of x > satisfies the Dirichlet problem [34, § ( Lπ ( x ) = 0 (0 < x < b ) ,π ( b ) = 1 . (4.10)The differential equation in (4.10) reads σ x π ′′ ( x ) + µxπ ′ ( x ) = 0 (0 < x < b ) , which is easily solved to give π ( x ) = c x − µ/σ + c . If − µ/σ < (i.e., µ − σ > ) then c = 0 (since π ( x ) is bounded), and due tothe boundary condition π ( b ) = 1 it follows that c = 1 and π ( x ) ≡ . A similar argumentshows that π ( x ) ≡ in the case − µ/σ = 0 . Finally, if − µ/σ > then, noting that π (0) = 0 , we conclude that c = 0 and, due to the boundary condition, c = b − µ/σ .Thus, formula (4.8) is proved.Similarly, the mean hitting time m ( x ) := E x ( τ b ) (with µ − σ > ) satisfies the Poissonproblem [34, § ( Lm ( x ) = − < x < b ) ,m ( b ) = 0 . (4.11)20s usual, to solve the problem (4.11), it is convenient to approximate it with a two-sidedboundary problem by adding an auxiliary Neumann (reflection) condition at ε > , Lm ε ( x ) = − ε < x < b ) ,m ε ( b ) = 0 ,m ′ ε ( ε ) = 0 , (4.12)and then taking the limit of m ε ( x ) as ε ↓ . This procedure will produce the correctsolution m ( x ) since lim ε ↓ P x ( τ ε < ∞ ) = P x ( τ < ∞ ) = 0 (for any x > ).A particular solution to the inhomogeneous differential equation σ x m ′′ ε ( x ) + µxm ′ ε ( x ) = − ε < x < b ) can be sought in the form m ( x ) = c ln x , which gives c = − / ( µ − σ ) . Thus, thegeneral solution of (4.12) can be expressed as m ε ( x ) = − ln xµ − σ + c x − µ/σ + c . (4.13)Now, using the boundary conditions in (4.12) it is straightforward to check that lim ε ↓ c = 0 , lim ε ↓ c = ln bµ − σ . Hence, from (4.13) we get m ( x ) = lim ε ↓ m ε ( x ) = ln ( b/x ) µ − σ , which retrieves the result (4.9). Remark . The same method applied to the killed process e X t with generator ˜ L = L − ˜ rI (see (3.8)) provides a neat interpretation of the value function u ( x ) as given by (3.14).Namely, rewrite the expectation in (3.4) (i.e., eNPV ( x ; τ b ) ) in the form ˜ E x (cid:2) g ( e X τ b ) (cid:3) , where ˜ E x denotes expectation with respect to the killed process ( e X t ) , and note that, for b ≥ , ˜ E x (cid:2) g ( e X τ b ) (cid:3) = ( g ( b ) ˜ P x ( τ b < ∞ ) , x ∈ [0 , b ] ,g ( x ) , x ∈ [ b, ∞ ) . In turn, the hitting probability ˜ π ( x ) := ˜ P x ( τ b < ∞ ) can be easily found by solving thecorresponding Dirichlet problem (cf. (4.10)), ( ˜ L ˜ π ( x ) = 0 (0 < x < b ) , ˜ π ( b ) = 1 . Indeed, repeating the calculations in Section 3.2, it is straightforward to get ˜ π ( x ) = ( x/b ) q ∗ .21 .3. Direct maximization Using the results of the previous sections, we can easily solve the optimal stopping problem(2.17), at least in the subclass of hitting times τ = τ b (see (3.4)), u ( x ) = sup b ≥ eNPV ( x ; τ b ) = sup b ≥ E x (cid:2) e − ˜ rτ b ( β X τ b − P ) (cid:3) . (4.14)Observe that if x ≥ b then τ b = 0 and X τ b = x (P x -a.s.), so that eNPV ( x ; τ b ) ≡ β x − P for all b ∈ [0 , x ] . Let now b ∈ [ x, ∞ ) . As already noted, on the event { τ b < ∞} we have X τ b = b (P x -a.s.), hence, according to (2.17) and (4.5),eNPV ( x ; τ b ) = ( β b − P ) E x (cid:0) e − ˜ rτ b (cid:1) = ( β b − P ) (cid:16) xb (cid:17) q ∗ ( b ≥ x ) , (4.15)where q ∗ = q ( θ ) | θ =˜ r (cf. (2.24) and (4.6)). It is straightforward to find the maximizer forthe function (4.15). Indeed, the condition ( ∂/∂ b ) eNPV ( x ; τ b ) ≥ , equivalent to β b − q ∗ − q ∗ ( β b − P ) b − q ∗ − ≥ , holds for all b ∈ [0 , b ∗ ] , where b ∗ = P q ∗ β ( q ∗ − , (4.16)which is the same optimal threshold as before (cf. (2.23)). Thus, the supremum ofeNPV ( x ; τ b ) over b ≥ x is attained at b = b ∗ if x ≤ b ∗ or at b = x if x ≥ b ∗ .The corresponding value function u ( x ) is then calculated as (cf. (2.25)) u ( x ) = ( ( β b ∗ − P ) (cid:16) xb ∗ (cid:17) q ∗ , x ∈ [0 , b ∗ ] ,β x − P, x ∈ [ b ∗ , ∞ ) . (4.17)Finally, substituting (4.16) into (4.17), we obtain explicitly (cf. (2.26)) u ( x ) = Pq ∗ − (cid:18) β ( q ∗ − xP q ∗ (cid:19) q ∗ , ≤ x ≤ P q ∗ β ( q ∗ − ,β x − P, x ≥ P q ∗ β ( q ∗ − . (4.18)
5. Statistical issues and numerical illustration
From the practical point of view, in order to exercise the stopping rule (2.22) the individualconcerned needs to be able to compute the critical threshold b ∗ expressed in (2.23), for whichthe knowledge is required about β (defined in (2.9)) and therefore about the parameters r , λ , µ and β (see (2.5)); furthermore, to evaluate the quantity q ∗ defined in (2.24), oneneeds to estimate µ − σ and σ itself. Specifically: • The loss-of-job rate λ can be extracted from the publicly available data about themean length at work, which is theoretically given by E ( τ ) = 1 /λ . • Likewise, the inflation rate r is also in the public domain.22 To specify the wage growth rate µ , a simple approach is just to set µ = r as a crudeversion of a “tracking” rule. However, it may be possible that the individual’s wagegrowth rate µ is, to some extent, stipulated by the job contract — for example, that itmust not exceed the inflation rate r by more than 1% per annum (applicable, e.g., tocivil servants) or, by contrast, that it must be no less than r minus 0.5% per annum(more realistic in the private sector). In practical terms, this would often mean thatthe actual growth rate µ is kept on the lowest predefined level. • More generally, the wage growth rate µ can be estimated by observing the wageprocess X t . This can be implemented by first using regression analysis on Y t = ln X t and estimating the regression line slope µ − σ (see (2.2)). In addition, the volatility σ can be estimated by using a suitable quadratic functional of the sample paths Y t . • Finally, knowing the benefit schedule (which should be available through the insurancepolicy’s terms and conditions), it is in principle possible to calculate, or at leastestimate the value β .To summarize, certain estimation procedures need to be carried out along with the on-lineobservation of the sample path ( X t ) . More details (most of which are quite standard) areprovided in the next two subsections. Denote for short a := µ − σ . According to the geometric Brownian motion model (2.2),we have Y t := ln X t = ln x + σB t + at, Y = ln x. Suppose the process X t is observed over the time interval t ∈ [0 , T ] on a discrete-time grid t i = iT /n ( i = 0 , . . . , n ), and consider the consecutive increments Z i := Y t i − Y t i − = σ ( B t i − B t i − ) + a ( t i − t i − ) ( i = 1 , . . . , n ) . (5.1)Note that the increments of the Brownian motion in (5.1) are mutually independent andhave normal distribution with zero mean and variance t i − t i − = T /n , respectively. There-fore, ( Z i ) is an independent random sample with normal marginal distributions, Z i ∼ N (cid:18) aTn , σ Tn (cid:19) ( i = 1 , . . . , n ) . Then, it is standard to estimate the parameters via the sample mean and sample variance, ˆ a n := nT · ¯ Z = Z + · · · + Z n T = Y T − Y T , (5.2) ˆ σ n := nT · n − n X i =1 ( Z i − ¯ Z ) . (5.3)These estimators are unbiased,E (ˆ a n ) = a = µ − σ , E ( ˆ σ n ) = σ , with mean square errors Var (ˆ a n ) = σ T ,
Var (ˆ σ n ) = 2 σ n − .
23n turn, the parameter µ is estimated by ˆ µ n = ˆ a n + ˆ σ n , with mean E (ˆ µ n ) = E (ˆ a n ) + E (ˆ σ n ) = a + σ = µ and mean square errorVar (ˆ µ n ) = Var (ˆ a n ) + Var (ˆ σ n ) = σ T + σ n − (due to independence of the estimators ˆ a n and ˆ σ n ).Note that the estimator ˆ a n in (5.2) only employs the last observed value, Y T ; in partic-ular, its mean square error is not sensitive to the grid size ∆ t i = T /n , and only tends tozero with increasing observational horizon, T → ∞ . This makes the estimation of the driftparameter a difficult in the sense that very long observations over Y t are required to achievean acceptable precision (see, e.g., [10, Example 2.1, p. 3]). For instance, let µ = 0 . and σ = 0 . (per week), then a = 0 . ; if T = 25 (weeks) then the 95%-confidence boundsfor a are given by ˆ a ± . σ/ √ T = ˆ a ± . , so the margin of error is about twice asbig as the value of a itself. To reduce it, say to . a , one needs T ≈ (weeks), whichexemplifies slow convergence.In contrast, the mean square error of the estimator ˆ σ n in (5.3) tends to zero as n → ∞ ,with T fixed. Thus, estimation of the parameter σ can be made asymptotically precise.A numerical example illustrating the estimation of µ and σ using simulated data willbe given at the end of Section 5.4. A brief discussion of practical choices of µ , based onsensitivity analysis, is provided at the end of Section 6.3. In view of the drawback in the general solution of the optimal stopping problem in thatthe stopping time τ b ∗ may be infinite, that is, P x ( τ b ∗ = ∞ ) > (which occurs when a = µ − σ < , see Section 4.1), a reasonable pragmatic approach to decision makingin our model may be based on testing the null hypothesis H : a ≥ versus the alternative H : a < (at some intuitively acceptable significance level, e.g. α = 0 . ). Namely, aslong as H remains tenable, one keeps waiting for the hitting time τ b ∗ to occur, but once H has been rejected, it is reasonable to terminate waiting and buy the policy immediately.The corresponding test is specified as follows. Again, suppose that the process Y t isobserved on a discrete time grid t i = iT /n , and set Z i = Y t i − Y t i − ( i = 1 , . . . , n ).Let z ( α ) be the upper α -quantile of the standard normal distribution N (0 , , that is, − Φ( z ( α )) = α , where Φ( x ) = √ π R x −∞ e − u / d u . Then the null hypothesis H : a ≥ is to be rejected at significance level α whenever Z + · · · + Z n ≤ inf a ≥ n aT − z ( α ) σ √ T o , that is, Y T − Y ≤ − z ( α ) σ √ T . (5.4)This test is uniformly most powerful among all tests with probability of error of type I notexceeding α , that is, P ( reject H | H true ) ≤ α .The normal test (5.4) assumes that the variance σ is known. As mentioned before,this presents no real restriction if the process Y t is observable continuously (i.e., if the grid24 t i ) can be refined indefinitely). If this is not the case (e.g., because the wage process canonly be observed on the weekly basis) then the test (5.4) is replaced by the t -test, Y T − Y ≤ − t n − ( α ) ˆ σ √ T , where ˆ σ is the sample variance (see (5.3)) and t n − ( α ) is the upper α -quantile of the t -distribution with n − degrees of freedom.In practice, the hypothesis testing is carried out sequentially (e.g., weekly) as the obser-vational horizon T increases. The advantage of this approach is that the resulting stoppingtime is finite with probability one (i.e., P x -a.s.); indeed, it is the minimum between theoptimal stopping time τ b ∗ (which is finite P x -a.s. under the null hypothesis H : a ≥ ) andthe first time of rejecting H (which is finite P x -a.s. if H is false). To be specific, we use euro as the monetary unit. First of all, the value of the constant β ,which encapsulates information about the benefit schedule as well as the rate λ of findingnew job (see (2.5)), is chosen to be β = 30 . Thus, the overall expected benefit payable over the lifetime of the policy (and projected tothe beginning of unemployment) is taken to be equal to 30 weekly wages; that is, if thefinal wage is 400 (euro per week) then the total to be received is . ×
30 = 12 000 . (euro) . Further, we set λ = 0 . , r = 0 . . This means that the expected time until loss of job is /λ = 100 (weeks), that is, about1 year and 11 months, whereas the annual inflation rate is e (365 / · . − . ≈ . , which is quite realistic.Next, we need to specify the premium P and the parameters of the wage process X t ,First, choose the initial value x = X as x = 346 . (euro) . This is motivated by the French labour legislation, whereby the current minimum pay rateis set as 9.88 euro per hour [42], with a 35-hour workweek [11, 16], giving . ×
35 = 345 . (euro per week) . As for the premium, it is set at the value P = 9 000 . (euro) , which equates to about 26 minimum weekly wages (i.e., income over about half a year).For simplicity, we also choose µ = r = 0 . , (5.5)25o that the wage growth rate is the same as inflation r (in reality, it could be slightly less).Then from (2.9), using (5.5), we get β = λ β ˜ r − µ = β = 30 . For the volatility σ , we will illustrate two opposite cases, µ < σ and µ > σ . Example . Set σ = 0 . , then µ − σ = − . < . From (2.24) we calculate q ∗ = 3 . , then (3.13) yields b ∗ = 404 . . (euro) . Using (4.8), the hitting probability is calculated asP x ( τ b ∗ < ∞ ) = 0 . . Finally, using (3.14), we obtain the value of this contract, v (346) = 1 714 . . (euro) . Example . Now, set σ = 0 . , then µ − σ = 0 . > . Furthermore, using (2.24)we calculate q ∗ = 6 . , and from (3.13) b ∗ = 352 . . (euro) . Hence, using (4.9), the expected hitting time is found to beE ( τ b ∗ ) = 91 . . (weeks) . Finally, according to formula (2.25), the value of this contract is calculated as v (346) = 1 389 . . (euro) . In the simulation of the process X t shown in Fig. 3, the drift a = µ − σ is estimatedusing formula (5.2) as ˆ a . = 0 . . Estimation of the variance σ according to formula(5.3) (on a weekly time grid) gives ˆ σ . = 0 . , while the true value is σ = 0 . .Hence, the parameter µ is estimated by ˆ µ . = 0 . ; recall that the true value is µ = 0 . .
6. Parametric dependencies
In this section, we aim to explore the parametric dependencies of the solution of ourinsurance problem, that is, of the optimal threshold b ∗ given by (2.23) and the value function v = v ( x ) given by (2.25). In particular, it is helpful to analyse different asymptotic regimesas well as (the sign of) appropriate partial derivatives, so as to ascertain the direction ofchanges under small perturbations and to understand their economic meaning. This is a keyingredient of sensitivity analysis and of the so-called comparative statics [30, Section VII].In what follows, we confine ourselves to a discussion of the two most important exoge-nous parameters — the wage drift µ and the unemployment rate λ . The constraint (2.11)implies that the range of the parameters µ and λ is specified as follows, −∞ < µ < ˜ r = r + λ , ∨ ( µ − r ) < λ < ∞ . Remark . The next two technical subsections are elementary but rather tedious, and thereader wishing to grasp the results quickly may just inspect the plots in Figs. 4 and 5.26 .1. Monotonicity
By virtue of the quadratic equation (3.11), the formula (2.23) can be conveniently rewrittenas b ∗ = P ( σ q ∗ + ˜ r ) β λ . (6.1)First, fix λ and consider the function µ b ∗ . Differentiating the equation (3.11) and thenagain using (3.11) to eliminate µ , we obtain ∂q ∗ ∂µ = − q ∗ σ (2 q ∗ −
1) + µ = − q ∗ σ q ∗ + ˜ r < . (6.2)Hence, using (6.1) and (6.2), d b ∗ d µ = ∂b ∗ ∂µ + ∂b ∗ ∂q ∗ · ∂q ∗ ∂µ = − P ( σ q ∗ ) β λ ( σ q ∗ + ˜ r ) < , (6.3)and, therefore, b ∗ is a decreasing function of µ (see Fig. 4(a)).Similarly, the equation (3.11) yields ∂q ∗ ∂λ = 1 σ (2 q ∗ −
1) + µ = q ∗ σ q ∗ + r + λ > . (6.4)From (6.1) and (6.4), after some rearrangements we obtain d b ∗ d λ = ∂b ∗ ∂λ + ∂b ∗ ∂q ∗ · ∂q ∗ ∂λ = − P ( σ q ∗ + r ) β λ + P ( σ q ∗ ) β λ ( σ q ∗ + r + λ )= − P (cid:2) ( σ q ∗ + r )( σ q ∗ + r ) + λ r (cid:3) β λ ( σ q ∗ + r + λ ) < , (6.5)and it follows that the function λ b ∗ is decreasing (see Fig. 4(b)).Let us now turn to the value function v = v ( x ) . First, consider v as a function of µ , thus keeping λ fixed. Using the expression (2.23), we can rewrite the first line of theformula (2.25) (i.e., for x ≤ b ∗ ) as v = Pq ∗ − (cid:16) xb ∗ (cid:17) q ∗ . (6.6)Differentiating (6.6), we get ∂v∂q ∗ = − P ( q ∗ − (cid:16) xb ∗ (cid:17) q ∗ (cid:18) q ∗ −
1) ln (cid:18) b ∗ x (cid:19)(cid:19) < , (6.7) ∂v∂b ∗ = − P q ∗ ( q ∗ − b ∗ (cid:16) xb ∗ (cid:17) q ∗ < . (6.8)Hence, on account of the inequalities (6.2), (6.4), (6.7) and (6.8), d v d µ = ∂v∂µ + ∂v∂q ∗ · ∂q ∗ ∂µ + ∂v∂b ∗ · d b ∗ d µ > . (6.9)27 Wage drift m O p t i m a l t h r e s ho l d b * ( eu r o ) rx IIIIIIIVV I: λ = 0 . II: λ = 0 . III: λ = 0 . IV: λ = 0 . V: λ = 0 . (a) µ b ∗ Unemployment rate l O p t i m a l t h r e s ho l d b * ( eu r o ) x P/β VIVIIIII I I: µ = − . II: µ = 0 . r ) III: µ = 0 . IV: µ = 0 . V: µ = 0 . (b) λ b ∗ Fig. 4: Graphs illustrating parametric dependencies of the optimal threshold (2.23): (a) onthe wage drift µ < ˜ r and (b) on the unemployment rate λ > ∨ ( µ − r ) , for selectedvalues of λ and µ , respectively. The values of other model parameters used throughout areas in Example 5.2: r = 0 . , P = 9 000 , β = 30 , and σ = 0 . . The dashed horizontalline in both plots indicates the initial wage x = 346 . The dashed vertical line in (a)indicates µ = r . The lower dashed horizontal line in (b) shows the asymptote P/β = 300 (see (6.19)).If x ≥ b ∗ , then from the second line of (2.25) we readily obtain d v d µ = β λ x (˜ r − µ ) > . (6.10)Thus, in all cases d v/ d µ > , which implies that the function µ v is increasing (seeFig. 5(a)).Finally, fix µ and consider the function λ v . If x ≥ b ∗ then v is given by the secondline of (2.25), that is, v = β λ xλ + r − µ − P. (6.11)In particular, if µ = r then (6.11) is reduced to v ≡ v ∗ := β x − P . From (6.11) it followsthat d v d λ = β x ( r − µ )( λ + r − µ ) < , µ > r, = 0 , µ = r,> , µ < r. Due to monotonicity of the function λ b ∗ (see (6.5)), v is given by (6.11) as long as λ ≥ λ ∗ , for some critical value λ ∗ ≡ λ ∗ ( µ ) ≤ ∞ . It will be shown below (see (6.19)) that lim λ →∞ b ∗ = P/β , so λ ∗ < ∞ if and only x > P/β . Clearly, λ ∗ is determined by thecondition b ∗ = x (see (2.23)) together with the equation (3.11). In the special case µ = r (assuming that x > P/β ), these equations can be solved to yield λ ∗ = Pβ x (cid:18) σ β xβ x − P + r (cid:19) . (6.12)28 Wage drift m V a l ue f un c t i on v ( x ) ( eu r o ) (a) µ v ( x ) v ∗ r I II III IVV I: λ = 0 . II: λ = 0 . III: λ = 0 . IV: λ = 0 . V: λ = 0 . Unemployment rate l V a l ue f un c t i on v ( x ) ( eu r o ) (b) λ v ( x ) v ∗ λ ∗ IIIIIIIVVVI I: µ = 0 . II: µ = 0 . r ) III: µ = 0 . IV: µ = 0 . V: µ = 0 . VI: µ = − . Fig. 5: Graphs illustrating parametric dependencies of the value function (2.25): (a) on thewage drift µ < ˜ r and (b) on the unemployment rate λ > ∨ ( µ − r ) , for selected valuesof λ and µ , respectively. The values of other model parameters used throughout are asin Example 5.2: r = 0 . , P = 9 000 , β = 30 , σ = 0 . , and x = 346 . The dashedhorizontal lines in both plots correspond to the value v ∗ := β x − P = 1380 . The dashedvertical line in (a) indicates µ = r ; in this case, shown as curve ii in plot (b), v ( x ) ≡ v ∗ for all λ ≥ λ ∗ . = 0 . (see (6.12)). That is why curves iii , iv and v in plot (a) allintersect at µ = r .In particular, in Example 5.2 this gives λ ∗ . = 0 . . From the consideration above, italso follows that if x > P/β then (see (6.11)) lim λ →∞ v = v ∗ = β x − P. (6.13)In the case x ≤ b ∗ , we use formula (6.6). Similarly to (6.9), d v d λ = ∂v∂λ + ∂v∂q ∗ · ∂q ∗ ∂λ + ∂v∂b ∗ · d b ∗ d λ . (6.14)Substituting the expressions (6.2), (6.4), (6.7) and (6.8) into (6.14), cancelling immaterialfactors and recalling formula (6.1), the condition d v/ d λ < is reduced to (cid:18) σ q ∗ + r (cid:19) (cid:18) σ q ∗ + r (cid:19) + λ r < (cid:18) q ∗ − (cid:18) b ∗ x (cid:19)(cid:19) (cid:18) σ q ∗ + r + λ (cid:19) . (6.15)It can be proved that if µ ≥ r then the inequality (6.15) holds for all λ < λ ∗ , butthe analysis becomes difficult for µ < r . Numerical plots (see Fig. 5(b)) suggest that inthe latter case the function λ v may be non-monotonic, with the derivative d v/ d λ possibly vanishing in up to two points, provided that r − ε < µ < r with ε > smallenough. To be more specific, the plots in Fig. 5(b) illustrate the case x > P/β , with thecommon asymptote (6.13). For x ≤ P/β , the plots look similar (not shown here) but with lim λ →∞ v = 0 (see (6.22) below), so the derivative d v/ d λ may vanish in at most onepoint. 29 .2. Limiting values Let us investigate the functions b ∗ and v in the limits (i) µ → −∞ or µ ↑ ˜ r , and (ii) λ → ∞ or λ ↓ ( µ < r ), λ ↓ µ − r ( µ ≥ r ). Start by observing, using equation (3.11),that lim µ →−∞ q ∗ = ∞ , lim µ ↑ ˜ r q ∗ = 1 , (6.16)and moreover, q ∗ − ∼ ˜ r − µ σ + ˜ r ( µ ↑ ˜ r ) . (6.17)Similarly, lim λ →∞ q ∗ = ∞ ; on the other hand, if µ < r then lim λ ↓ q ∗ = q ∗ | λ =0 > ,while if µ ≥ r then q ∗ − ∼ λ − ( µ − r ) σ + µ ( λ ↓ µ − r ) . (6.18)Hence, from (6.1) and (6.16) it readily follows that b ∗ → ∞ ( µ → −∞ ) and b ∗ → P ( σ + ˜ r ) β λ ( µ ↑ ˜ r ) . Also, using that q ∗ → ∞ ( λ → ∞ ), from (2.23) we get b ∗ → Pβ ( λ → ∞ ) . (6.19)In the opposite limit, if µ > r then, according to (6.1) and (6.18), b ∗ → P ( σ + µ ) β ( µ − r ) ( λ ↓ µ − r ) , (6.20)while if µ ≤ r then lim λ ↓ b ∗ = ∞ ; in particular, for µ = rb ∗ ∼ P ( σ + r ) β λ ( λ ↓ . (6.21)For the value function v = v ( x ) , from formula (6.6) we get, using (6.16) and (6.17), lim µ →−∞ v = 0 , lim µ ↑ ˜ r v = ∞ . Furthermore, according to (6.13), if x > P/β then v → v ∗ = β x − P as λ → ∞ .In the opposite case, due to monotonicity of b ∗ (see (6.5)) and the limit (6.19) we have b ∗ > P/β ≥ x , so using formula (6.6) and recalling that q ∗ → ∞ , we get v ≤ Pq ∗ − → λ → ∞ ) . (6.22)Now, consider the limit of v as λ approaches the lower edge of its range. If µ < r then(6.6) implies that lim λ ↓ v = 0 , since b ∗ → ∞ and q ∗ → q ∗ | λ =0 > . If µ = r then, using(6.18) and (6.21) (with µ = r ), we obtain v ∼ β xλ q ∗ − = β x exp (cid:8) ( q ∗ −
1) ln λ (cid:9) → β x ( λ ↓ . (6.23)Finally, if µ > r then from (6.6) it readily follows, according to (6.18) and (6.20), v ∼ β x ( µ − r ) λ − ( µ − r ) → ∞ ( λ ↓ µ − r ) . (6.24)30 .000 0.010 0.020 0.030 − . . . . . Unemployment rate l W age d r i ft m (a) Isolines of b ∗ IIIIIIIV I: b ∗ = 330 II: b ∗ = 340 III: b ∗ = 355 IV: b ∗ = 385 − . . . Unemployment rate l W age d r i ft m (b) Isolines of v = v ( x ) r λ ∗ V IV IIIIII I: v = 1510 II: v = 1380 III: v = 1250 IV: v = 1100 V: v = 900 Fig. 6: Isolines (level curves) of the optimal stopping problem solution on the ( λ , µ ) -plane: (a) b ∗ ( λ , µ ) = const (optimal threshold (2.23)); (b) v ( λ , µ ) = const (valuefunction (2.25)). The values of other parameters used throughout are as in Example 5.2: r = 0 . , P = 9 000 , β = 30 , σ = 0 . , and x = 346 . The slanted dashed lines in bothplots show the boundary µ = λ + r (see (2.11)). In plot (b), the horizontal dashed lineindicates µ = r and the vertical dashed line shows λ ∗ . = 0 . (cf. Fig. 5(b)). The goal of comparative statics is to understand how varying values of exogenous parametersaffect a target function of interest. For instance, consider the optimal threshold b ∗ as afunction of both unemployment rate λ and wage drift µ . Rather then fixing one of theseparameters and then plotting b ∗ against the remaining parameter (as was done in Figs. 4(a)and 4(b)), it is useful to plot a family of comparative statics plots showing the isolines (or level curves ) for different values (levels) of the function, that is, b ∗ ( λ , µ ) = const (see Fig. 6(a)). As may be expected from Figs. 4(a) and 4(b), the plots of the function λ = λ ( µ ) (determined implicitly by the level condition) behave as monotone decreasinggraphs. Analogous plots for the value function are presented in Fig. 6(b); the plots becomenon-monotonic for v large enough. If λ is fixed then the value v grows with µ , in agreementwith (6.9) and (6.10). Similarly, if µ > r is fixed then v decreases with λ , convergingto the limit v ∗ = β x − P as λ → ∞ (see (6.13)), represented by curve II in Fig. 5(b).If v > v ∗ then there are up to two different values of λ (and common µ ) producing thesame value v , while for v smaller than but close enough to v ∗ , the number of such rootsmay increase to three (see the discussion in Section 6.4).Let us also comment on the sensitivity of our numerical examples presented in Sec-tion 5.4. The question here is, how much the output values (say, the optimal threshold b ∗ and the value v ) would change under a small variation of one of the background pa-rameters. In the linear approximation, the change factor is given by the correspondingpartial derivative. As in the previous sections, we address the sensitivity with regard to thewage drift µ (around the set value µ = 0 . ) and the unemployment rate λ (around λ = 0 . ). Other model parameters are fixed as in Section 5.4, that is, r = 0 . , P = 9 000 , β = 30 , and x = 346 . As for the volatility parameter σ , it is set to be σ = 0 .
31s in Example 5.1 or σ = 0 . as in Example 5.2. The required partial derivatives of b ∗ and v can be computed using the formulas derived in Section 6.1; the results are presented inTable 1(a).Table 1: Sensitivity check of numerical results for the functions b ∗ and v in Examples 5.1and 5.2: (a) parametric derivatives; (b) increments in response to a -change in thebackground parameters. (a) Derivatives Derivative Example 5.1 Example 5.2 d b ∗ / d µ −
16 037 . −
13 962 . v/ d µ
842 062 .
30 993 991 . b ∗ / d λ − . − . v/ d λ −
46 485 . − . (b) Increments (euro) Increment Example 5.1 Example 5.2 ∆ b ∗ ( µ ) − . − . v ( µ ) . . b ∗ ( λ ) − . − . v ( λ ) − . − . Numerical values in Table 1(a) may seem quite big, but they should be offset by smallbackground values of the parameters, µ = 0 . and λ = 0 . . If we increase them bya small amount, say by , then the absolute increments would be ∆ µ = 0 . /
100 = 4 · − , ∆ λ = 0 . /
100 = 10 − . Hence, using Table 1(a), we obtain the corresponding approximate increments of the targetfunctions b ∗ and v (see Table 1(b)), which look more palatable. One interesting observationis that the value v reacts about times stronger to the change of the unemployment rate λ when the volatility σ gets times bigger (from σ = 0 . in Example 5.2 to σ = 0 . inExample 5.1); in contrast, the change of v in response to an increase of the wage drift ismuch less pronounced. This highlights the primary significance of the unemployment rate,which is of course only natural.Sensitivity analysis with regard to the wage drift µ is also useful in the light of thedifficulty in estimation of µ from the data, mentioned in Section 5.2. The results inTable 1(b) suggest that a reasonably small error in selecting µ has only a minor effect onthe identification of the optimal threshold b ∗ and the value v ; for instance, overestimatingit by will decrease b ∗ by just . euro, while the value v will be up by about . euro. Thus, an individual using a moderately inflated value of their wage rate would takea slightly over-optimistic view about the timing of joining the insurance scheme and itsexpected benefit. On the other hand, a risk-averse individual may take a more conservativeview and prefer to underestimate their wage drift µ , which will raise the threshold b ∗ resultingin a longer waiting time. For the insurance company though, it may be reasonable to tryand avoid underestimation of the wage drift of potential customers, so as to reduce therisk of overpaying the benefits. 32 .4. Economic interpretation Monotonic decay of the optimal threshold b ∗ with an increase of the unemployment rate λ (see (6.5) and Fig. 4(b)) has a clear economic appeal: a bigger unemployment rate λ means a higher risk of losing the job, which demands a lower target threshold b ∗ in orderto expedite joining the insurance scheme. The economic rationale for the monotonicity of b ∗ as a function of µ (see (6.3) and Fig. 4(a)) is different — a bigger wage drift µ makesit more likely to reach a higher final wage X τ by the time of loss of job, so lowering thethreshold b ∗ adds incentive to an earlier entry.Monotonic growth of the value v as a function of the wage drift µ (see (6.9), (6.10),and Fig. 5(a)) is also meaningful — indeed, when the wage drift µ gets bigger, there is morepotential to reach a higher final wage X τ by the time of loss of job, which increases theexpected benefit β (see (2.9)) and, therefore, the value v = v ( x ) of the insurance policy.The behaviour of the value function v = v ( x ) in response to a varying unemploymentrate λ is more interesting, as indicated by the plots in Fig. 5(b). In the case µ < r , it issatisfactory to see that the value v , vanishing in the limit as λ ↓ , starts growing with λ , thus reflecting a good efficiency of the insurance policy against an increasing risk ofunemployment. On the other side of the policy, this may present a growing risk for theinsurance company which will have to finance an increasing number of claims. But withthe unemployment rate λ getting ever bigger, the value v should stay bounded, so mustconverge to a limit as λ → ∞ , given by v ∗ = β x − P if x > P/β (see (6.13)) or v ∗ = 0 otherwise (see (6.22)). In particular, Fig. 5(b) shows that, for a certain range of µ , thevalue v achieves its maximum at some λ . However, the graphs also reveal that if µ keepsincreasing then the value plots may have a more complicated non-monotonic behaviour,which is harder to interpret economically.On the other hand, as is evident from Fig. 5(b), in the case µ ≥ r our model producesa counter-intuitive increase of the value v as λ approaches the left edge of its range — itis hard to believe that the value may grow as the risk of unemployment falls. Moreover,as was computed in (6.24), for µ > r the corresponding limit of v is infinite! But perhapsthe most striking example emerges in the borderline case µ = r , whereby formally setting λ = 0 we would get, according to (6.21), that the threshold b ∗ is infinite (unlike the case µ > r , see (6.21)), so that the wage process ( X t ) never reaches it; therefore, we neverbuy the insurance policy (understandably so, as there is no risk of losing the job), andnonetheless its value is positive in this limit (see (6.13)). The explanation of this paradoxlies in the way how the optimal stopping is exercised for small λ > : here, the threshold b ∗ is high and there is only a very small probability that it is ever reached; before this happens,we stay idle, but if and when the threshold is hit then the expected payoff is rather big,which contributes enough to the expected net present value to keep it positive in the limit λ ↓ (see (6.23)).Thus, the artefacts in our model as indicated above are caused by not putting anyconstraint on the waiting times. This can be rectified, for example, by introducing mortality ,as was sketched in Section 2.3; in particular, such a regularization should restore a zerolimit of v at the lower edge of λ . 33 . Including utility considerations Our model (and its solution) resembles that of the optimal stopping problem for the (perpet-ual) American call option (see a detailed discussion in [39, Ch. VIII, § K , where the decision is based on observationsover the random process of stock prices ( S t ) , assumed to follow a geometric Brownianmotion model. The term perpetual is used to indicate that there is no expiration date, sothe right to buy extends indefinitely.The optimal time instant τ = τ ∗ to buy, bearing in mind a purely financial target ofmaximizing the profit S τ − K , is the solution of the following optimal stopping problem, V ( x ) = sup τ E x (cid:0) e − rτ ( S τ − K ) + (cid:1) , (7.1)where S t is a geometric Brownian motion with parameters µ < r and σ > , the supremumis taken over all stopping times τ adapted to the filtration associated with ( S t ) . The positivetruncation ( · ) + corresponds to the constraint that the option holder is not in a positionto buy at the price K higher than the current spot price S t . The solution to (7.1) is wellknown (see, e.g., [39, Ch. VIII, § τ ∗ = τ b ∗ , with theoptimal threshold b ∗ = Kq ∗ q ∗ − , where q ∗ is given by formula (2.24) but with ˜ r = r + λ replaced by r . The correspondingvalue function is given by V ( x ) = ( b ∗ − K ) (cid:16) xb ∗ (cid:17) q ∗ , x ∈ [0 , b ∗ ] ,x − K, x ∈ [ b ∗ , ∞ ) . Observe that our optimal stopping problem (2.17) can be rewritten as v ( x ) = β sup τ E x (cid:2) e − ˜ rτ ( X τ − ˜ K ) (cid:3) , ˜ K := P/β , (7.2)which makes it look very similar to the perpetual American call option problem (7.1).However, there are several important differences. Firstly, unlike the gain function in theAmerican call option problem (7.1), no truncation is applied in (7.2), because the financialgain is not the sole priority in this context and therefore the individual is prepared to toleratenegative values of β X τ − P (despite the fact that, under the optimal strategy, the valuefunction v ( x ) is always non-negative, see Lemma 2.2(i) and formula (2.25)). In addition,as was mentioned in Remark 4.1 and in Section 5.3, the hitting time τ b ∗ may be infinitewith a positive probability (i.e., when µ < σ ), which may be deemed impractical in theinsurance context, but is considered to be acceptable for exercising the American call option.This simple observation helps to realize the fundamental conceptual difference between the The equivalence of the problems (7.1) and (7.2), which we have established directly, is not a coinci-dence: it is known [41, Proposition 3.1, p. 185] that, under mild assumptions, the solution of the generaloptimal stopping problem v ( x ) = sup τ E x (e − rτ g ( X τ )) does not change with the positive truncation of g ( · ) . per se . Hence, amore realistic formulation of the optimal stopping problem in the UI model should involvea certain utility , which specifies the individual’s weighted preferences for satisfaction — forexample, impatience against waiting for too long before joining the UI scheme. Here, we present a few informal thoughts about the possible inclusion of utility in theoptimality analysis. As already mentioned, in the case µ < σ the probability of hittingthe critical threshold b ∗ is less than 1, so there is a probability that the individual will neverjoin the insurance scheme if the optimal stopping rule is strictly followed. This is of coursenot desirable, as the individual puts high priority on getting insured at some point in time(hopefully, prior to loss of job).One simple way to take these additional requirements into account is to extend theoptimal stopping problem (2.17) as follows: v † ( x ) = sup τ (cid:2) κ P x ( τ < ∞ ) + eNPV ( x ; τ ) (cid:3) = sup τ E x (cid:2) κ { τ< ∞} + e − ˜ rτ g ( X τ ) (cid:3) , (7.3)where the supremum is again taken over all stopping times τ adapted to the process ( X t ) ,and the coefficient κ ≥ is a predefined weight representing the individual’s personalattitude (preference) towards the two contributing terms. If P x ( τ < ∞ ) = 1 then the firstterm in (7.3) is reduced to a constant ( κ ), leading to a pure optimal stopping problemas before; however, if P x ( τ < ∞ ) < then the first term enhances the role of candidatestopping times τ that are less likely to be infinite.The problem (7.3) can be rewritten in a more standard form by pulling out the commondiscounting factor under expectation, v † ( x ) = sup τ E x (cid:2) e − ˜ rτ G ( τ, X τ ) (cid:3) , (7.4)with G ( t, x ) := κ e ˜ rt + g ( x ) , ( t, x ) ∈ [0 , ∞ ] × [0 , ∞ ) . (7.5)Unfortunately, the optimal stopping problem (7.4) is not amenable to an exact solution asbefore, because the gain function (7.5) depends also on the time variable (see [35, Ch. IV]).In this case, the problem (7.4) may again be reduced to a suitable (but more complex)free-boundary problem, but the hitting boundary (of a certain set on the ( t, x ) -plane) is nolonger a straight line.More generally, our optimal stopping problem can be modified by replacing the indicatorin (7.3) with the expression e − ρτ ( ρ > ), v † ( x ) = sup τ E x (cid:2) κ e − ρτ + e − ˜ rτ g ( X τ ) (cid:3) , (7.6)which retains the flavour of progressively penalizing larger values of τ , including τ = ∞ .Here, the gain function (7.5) takes the form G ( t, x ) = κ e (˜ r − ρ ) t + g ( x ) .
35n particular, by choosing ρ = ˜ r the problem (7.6) is transformed into v † = sup τ E x (cid:2) e − ˜ rτ ( β X τ + κ − P ) (cid:3) , which is the same problem as (2.17) but with the premium P replaced by P − κ .Another, more drastic approach to amending the standard optimal stopping problem(2.17) stems from the observation that even if τ < ∞ (P x -a.s.), it may take long to waitfor τ to happen — for instance, if E x ( τ ) = ∞ . In other words, it is reasonable to take intoaccount the expected value of τ , leading to the combined optimal stopping problem v † ( x ) = sup τ (cid:2) κ P x ( τ < ∞ ) + κ exp {− E x ( τ ) } + eNPV ( x ; τ ) (cid:3) . (7.7)If P x ( τ < ∞ ) < then E x ( τ ) = ∞ and the problem (7.7) is reduced to (7.3), whereasif P x ( τ < ∞ ) = 1 then, effectively, only the term with the expectation remains in (7.7).However, a disadvantage of the formulation (7.7) is that it cannot be expressed in theform (7.4). Trying to amend this would take us back to the version (7.6).It is interesting to look at how the value function depends on the preference parameter κ .The next property is intuitively obvious. Proposition 7.1.
For each x > , the value function v † ( x ) of the optimal stopping problem (7.6) is a strictly increasing function of κ . The same is true for the problem (7.7) .Proof. We use the notation v † ( x ; κ ) to indicate the dependence of the value function onthe parameter κ . For κ < κ and any stopping time τ
6≡ ∞ , we haveE x (cid:2) κ e − ρτ + e − ˜ rτ g ( X τ ) (cid:3) < E x (cid:2) κ e − ρτ + e − ˜ rτ g ( X τ ) (cid:3) ≤ v † ( x ; κ ) . (7.8)Suppose that τ ∗ is a maximizer for the optimal stopping problem (7.6) with κ = κ . Then,according to (7.8), v † ( x ; κ ) = E x (cid:2) κ e − ρτ ∗ + e − ˜ rτ ∗ g ( X τ ∗ ) (cid:3) < v † ( x ; κ ) , that is, v † ( x ; κ ) < v † ( x ; κ ) as claimed. Similar arguments apply to the problem (7.7). As already mentioned, the optimal stopping problems outlined in Section 6.2 are difficultto solve in full generality. To gain some insight about the qualitative effects of the addedutility-type terms, it may be reasonable to restrict our attention to solutions in the subclassof hitting times τ b . Despite such solutions will only be suboptimal, the advantage is thatthe reduced problems can be solved using that all the ingredients are available explicitly(see Section 4.1).For example, the original problem (7.3) is replaced by u † ( x ) = sup b ≥ (cid:2) κ P x ( τ b < ∞ ) + eNPV ( x ; τ b ) (cid:3) . (7.9)Similarly as in Section 4.3, we only need to maximize the functional in (7.9) over b ≥ x .Indeed, if b ≤ x then τ b = 0 (P x -a.s.) and, according to (2.7) and (2.16), sup b ≤ x (cid:2) κ P x ( τ b < ∞ ) + eNPV ( x ; τ b ) (cid:3) = κ + eNPV ( x ; 0) = κ + β x − P, sup b ≥ x (cid:2) κ P x ( τ b < ∞ ) + eNPV ( x ; τ b ) (cid:3) ≥ (cid:2) κ P x ( τ b < ∞ ) + eNPV ( x ; τ b ) (cid:3)(cid:12)(cid:12) b = x = κ + β x − P. Assume that µ − σ < (for otherwise P x ( τ b < ∞ ) = 1 , thus leading to the sameoptimal stopping problem as before). Then, according to (4.8), the probability P x ( τ b < ∞ ) becomes a strictly decreasing function of b ∈ [ x, ∞ ) , and so the maximum in (7.9) isachieved by a different stopping strategy, with a lower optimal threshold b † . More precisely,by virtue of formulas (4.8) and (4.15), the problem (7.9) is explicitly rewritten as u † ( x ) = sup b ≥ x (cid:20) κ (cid:16) xb (cid:17) − µ/σ + ( β b − P ) (cid:16) xb (cid:17) q ∗ (cid:21) , (7.10)where q ∗ > is defined in (2.24). Differentiating with respect to b , it is easy to check thatthe maximizer for the problem (7.10) is given by b † = min ( b ≥ x : aκ (cid:18) bx (cid:19) q ∗ − a + ( q ∗ − β b ≥ P q ∗ ) , where a := 1 − µ/σ < < q ∗ .The following (slightly artificial) version of the utility keeps the spirit of (7.9) but isamenable to the exact analysis: u † ( x ) = sup b ≥ (cid:20) κ (cid:8) P x ( τ b < ∞ ) (cid:9) q ∗ / (1 − µ/σ ) + eNPV ( x ; τ b ) (cid:21) . (7.11)Indeed, using the same substitutions (4.8) and (4.15) as before, (7.11) is reduced to(cf. (7.10)) u † ( x ) = sup b ≥ x h ( β b + κ − P ) (cid:16) xb (cid:17) q ∗ i , (7.12)which is the same problem as (4.14) but with P replaced by P − κ (cf. (4.15)). Therefore,from (4.16) we immediately obtain the maximizer b † = ( P − κ ) q ∗ β ( q ∗ −
1) = b ∗ − κq ∗ β ( q ∗ − ≤ b ∗ . (7.13)This is a strictly decreasing (linear) function of κ ; in particular, b † = b ∗ if κ = 0 and b † = 0 if κ = P . The corresponding value function is given by (cf. (4.17)) u † ( x ) = ( ( β b † + κ − P ) (cid:16) xb † (cid:17) q ∗ , x ∈ [0 , b † ] ,β x + κ − P, x ∈ [ b † , ∞ ) , (7.14)or more explicitly (cf. (4.18)) u † ( x ) = P − κq ∗ − (cid:18) β ( q ∗ − x ( P − κ ) q ∗ (cid:19) q ∗ , ≤ x ≤ ( P − κ ) q ∗ β ( q ∗ − ,β x + κ − P, x ≥ ( P − κ ) q ∗ β ( q ∗ − . (7.15)37
100 200 300 400
Weight k (euro) T h r e s ho l d b ( eu r o ) † κ † x (a) κ b † Weight k (euro) V a l ue f un c t i on u ( x ) ( eu r o ) † κ † u † ( x ) | κ = κ † (b) κ u † ( x ) Fig. 7: Functional dependence on the preference weight κ in the reduced optimal stoppingproblem (7.11): (a) the optimal threshold b † (see (7.13)); (b) the value function u † ( x ) (see (7.15)). Numerical values of the parameters used are as in Example 5.2: r = µ =0 . , P = 9 000 , β = 30 , σ = 0 . , and x = 346 . In particular, if κ = 0 then b † coincides with b ∗ . = 352 . and u † ( x ) coincides with v ( x ) . = 1389 . . The dashedvertical lines on both plots indicate the value κ † . = 162 . (see (7.16)) separating differentregimes for u † ( x ) according to (7.15). When κ = κ † , we have b † = x = 346 , shown asa dashed horizontal line in plot (a); the corresponding value function is given by u † ( x ) = β x + κ † − P . = 1542 . (see (7.14)), shown as a dashed horizontal line in plot (b).Note that the graph of u † ( x ) in plot (b) looks almost linear for κ ∈ [0 , κ † ] , because theratio κ/P is quite small, ≤ κ/P ≤ κ † /P . = 0 . ; the slope here is approximately v ( x )( q ∗ − /P . = 0 . , as compared to slope of the linear graph for κ ≥ κ † .If x is fixed then the problem value u † , as a function of κ , is given by the first or the secondline in (7.15) according as κ ∈ [0 , κ † ] or κ ∈ [ κ † , ∞ ) , respectively, where κ † := P − β ( q ∗ − xq ∗ . (7.16)The dependence of b † and u † ( x ) upon the utility parameter κ ∈ [0 , P ] is illustratedin Fig. 7, while Fig. 8 demonstrates the functional dependence of the hitting probabilityP x ( τ b < ∞ ) and the mean hitting time E x ( τ b ) upon the variable threshold b ≥ , alongwith the corresponding plots of the expected net present value eNPV ( x ; τ b ) . Remark . Note that u † ( x ) is a strictly increasing function of κ ∈ [0 , P ] , in accord withProposition 7.1. In particular, u † ( x ) coincides with the original value function u ( x ) givenby (4.18), but with the premium P replaced by P − κ . This can be interpreted as theindividual’s consent to convert additional satisfaction, gained by virtue of pursuing theoptimal stopping problem (7.11) instead of (2.17), into a higher premium, P † = P + κ .Such an effect is characteristic of the use of risk-averse utility functions under the ExpectedUtility Theory [23] (see also a discussion below in Section 6.4).In the case µ > σ , instead of (7.7) we may consider the simplified problem u † ( x ) = sup b ≥ (cid:2) κ exp {− E x ( τ b ) } + eNPV ( x ; τ b ) (cid:3) . (7.17)38
200 400 600 800 1000 1200 . . . . . . . Threshold b (euro) P x ( t b < ¥ ) x b ∗ (a) b P x ( τ b < ∞ ) (cid:0) µ < σ (cid:1)
300 350 400 450 500
Threshold b (euro) E x ( t b ) ( eu r o ) x b ∗ (b) b E x ( τ b ) (cid:0) µ > σ (cid:1) Threshold b (euro) e N PV ( x ; t b ) ( eu r o ) x b ∗ (c) b eNPV ( x ; τ b ) (cid:0) µ < σ (cid:1)
300 350 400 450 500
Threshold b (euro) e N PV ( x ; t b ) ( eu r o ) x b ∗ (d) b eNPV ( x ; τ b ) (cid:0) µ > σ (cid:1) Fig. 8: Theoretical graphs for functionals of the hitting time τ b versus threshold b ≥ . Upper row: (a) the hitting probability P x ( τ b < ∞ ) (see (4.8)); (b) the mean hitting timeE x ( τ b ) (see (4.9)). Bottom row: the expected net present value eNPV ( x ; τ b ) (see (4.15))with µ < σ (c) or µ > σ (d). The values of parameters used throughout are as inSection 5.4: x = 346 , P = 9 000 , β = 30 , µ = 0 . , and σ = 0 . (left) or σ = 0 . (right). The dashed vertical lines on each plot indicate x and the optimal threshold b ∗ ,respectively; specifically, b ∗ . = 404 . on the left (see Example 5.1) and b ∗ . = 352 . on the right (see Example 5.2).Upon the substitution of formulas (4.9) and (4.15), it is rewritten in the form (cf. (7.10)) u † ( x ) = sup b ≥ x (cid:20) κ ln( b/x ) µ − σ + ( β b − P ) (cid:16) xb (cid:17) q ∗ (cid:21) . (7.18)Again, the maximization problem (7.18) can be solved (at least, numerically). For ananalytic solution, it is convenient to modify the problem (7.17) as follows, u † ( x ) = sup b ≥ (cid:20) κ exp (cid:18) − q ∗ µ − σ E x ( τ b ) (cid:19) + eNPV ( x ; τ b ) (cid:21) . The considerations above can be linked to the standard Expected Utility Theory [23]. Inthe usual setting, it is assumed that an individual uses (perhaps, subconsciously) a certainutility U ( w ) , as a function of financial wealth w , to assess losses, gains and the resultingsatisfaction. Generically, given the current wealth w and some random future loss Y , theexpected loss (measured via utility U ( · ) ) may be expressed as E (cid:2) U ( w − Y ) (cid:3) . The individualis inclined to pay a premium P and buy the insurance policy as long as the expected utilitywithout insurance is no more than U ( w − P ) ,E (cid:2) U ( w − Y ) (cid:3) ≤ U ( w − P ) . (7.19)The balance condition E (cid:2) U ( w − Y ) (cid:3) = U ( w − P ) (7.20)determines the maximum premium P max the customer is prepared to pay (in fact, at thispoint it makes no difference whether to buy the insurance or not).In the baseline case with U ( w ) ≡ w , the conditions (7.19) and (7.20) are reduced to P ≤ P max = E ( Y ) . (7.21)However, choosing a different utility function may well change this threshold. For instance,if the random loss Y has exponential distribution with parameter θ = 0 . , then accordingto (7.21) we have P max = E ( Y ) = 1 /θ = 1 000 . In contrast, let the utility function bechosen as U ( w ) = 1 − exp (cid:0) − θw (cid:1) . Here, the utility is between and if the wealth w ispositive, but it becomes increasingly negative for a negative wealth; that is, strong weightis placed against negative wealth, which may be characteristic of a risk-averse individual.In this case, it is easy to check that P max = 2 ln 2 θ = 1 386 . > . Thus, the individual is happy to pay more than before to protect themselves from theperceived risk of significant losses. That is to say, an additional amount of satisfaction isconvertible into an extra premium.In our case, if the UI was to be entered immediately, at time t = 0 , then the value ofthis decision would be eNPV ( x ; 0) = β x − P (see (2.8) and (2.16)). Clearly, in order forthis to be non-negative, the premium P must satisfy the condition P ≤ P max = β x. For instance, in the setting of the numerical example in Section 5.4, we get P max =30 ×
346 = 10 380 , while the set premium is P = 9 000 .Similarly, if the decision was taken at a stopping time τ , then, conditional on the wage X τ , the maximum premium payable would be given by P max = β X τ . Thus, the value of P max goes up or down together with the current wage. However, in our setting the entrytime is not decided in advance, being subject to the stopping rule based on observationsover ( X t ) . As a result, the value function v ( x ) ( x > ) of the optimal stopping problem is40lways positive for any premium P , no matter how high (see formula (2.26)). Apparently,this is manufactured by selecting the threshold b ∗ high enough, which guarantees that, inthe (rare) event of hitting it, the mean value of this strategy will be positive.This may not be satisfactory from the standpoint of the Expected Utility Theory; how-ever, there is no contradiction, because in its standard version this theory does not allowfor an optional stopping. Adding utility terms to the gain function in the spirit of Sections6.2 and 6.3 helps to amend the situation (see Remark 7.1), but the maximum premiumpayable still remains indeterminate.The explanation of this paradox lies in the simple fact that the gain function in theoptimal stopping problems considered so far does not include any losses. A simple wayto account for such losses is to include consumption in the model. Namely, suppose forsimplicity that the consumption rate c is constant; for instance, the net present value ofconsumption over time interval [0 , t ] is given by Z t e − rs c d s = c (1 − e − rt ) r . It is natural to assume that the wage X t is sufficient to finance the consumption, so thatE x ( X t ) = x e µt ≥ c for all t ≥ (see (2.3)). In turn, for this to hold it suffices to assumethat X = x ≥ c and µ ≥ . Hence, we need to take into account consumption only overthe unemployment spell [ τ , τ + τ ] , where the wage is replaced by the UI benefit. Theexpected net present value of this consumption is given by γ := E (cid:18) e − rτ Z τ e − rs c d s (cid:19) = E (cid:0) e − rτ (cid:1) · E (cid:18) c (1 − e − rτ ) r (cid:19) = λ c ( r + λ )( r + ∗ λ ) , using independence of τ and τ and their exponential distributions (with parameters λ and λ , respectively). Thus, our basic optimal stopping problem (2.17) is modified to v ‡ ( x ) = sup τ E x (cid:2) e − ˜ rτ g ( X τ ) − γ (cid:3) , which has the same solution as before (see Section 2.5) but with the new value function v ‡ ( x ) = v ( x ) − γ , that is (cf. (2.25)), v ‡ ( x ) = ( ( β b ∗ − P ) (cid:16) xb ∗ (cid:17) q ∗ − γ, x ∈ [0 , b ∗ ] ,β x − P − γ, x ∈ [ b ∗ , ∞ ) . Now, the inequality v ‡ ( x ) ≥ can be easily solved for P to yield P ≤ P ‡ max := β b ∗ − γ (cid:18) b ∗ x (cid:19) q ∗ , x ∈ [0 , b ∗ ] ,β x − γ, x ∈ [ b ∗ , ∞ ) . (7.22)Note that P ‡ max in (7.22) is a decreasing function of γ , but an increasing function of x .Thus, as could be expected, the maximum affordable premium gets lower with the increaseof consumption, but becomes higher with the increase of the wage. Remark . Of course, consumption can also be incorporated into the optimal stoppingmodels involving utility (see Sections 6.2 and 6.3), but we omit technical details.41 . Concluding remarks
In this paper, we have set up and solved an optimal stopping problem in a stylized UI model.The model and its solution are useful by illustrating approaches to optimal strategy of anindividual seeking to get insured. By including consumption in the model, we have alsodemonstrated how a fair premium can be calculated, which makes our UI model usable alsofrom the insurer’s perspective.An explicit closed-form solution of the corresponding optimal stopping problem waspossible due to some simplifying assumptions — in particular, exponential distribution oftime τ to loss of job and constant inflation rate r . The analysis also strongly relied onthe simplest model for the wage process ( X t ) , that is, geometric Brownian motion withconstant drift µ and volatility σ .Let us indicate a few directions of making our UI model more realistic. Firstly, indefiniteterm of UI insurance could be replaced by a finite expiration term for the benefit schedule(akin to American call option with finite horizon), which would lead to a harder (time-dependent) optimal stopping problem (cf. [35, § τ needs to be tested on the basis of real unemployment data. Note, however,that fitting a different distribution for τ will invalidate the expression (2.13) for the expectednet present value eNPV ( x ; τ ) and, therefore, will change the gain function in the optimalstopping problem (2.17), making it more difficult to solve.The parameters of the model may also need to be made time-dependent, causing obviouscomplications to the model. On the other hand, the implicit assumption of passive waitingfor a new job during the unemployment spell may not be realistic, or at least not desirableas individuals would rather be expected to seek jobs more pro-actively. Thus, it may beinteresting to combine our UI model with job-seeking models such as in [4].The inclusion of utility terms in the optimal setting is novel in this context, and illumi-nates significant changes in the individual’s behaviour when driven by utility considerations.In particular, the value of the optimal stopping problem (7.6) is an increasing function ofthe preference coefficient κ (see Proposition 7.1). This result is intuitively appealing, asit conforms with the usual impact of utility function (under the Expected Utility Theory),allowing one to convert extra satisfaction into extra premium. This is confirmed by ouranalysis of suboptimal solutions in Section 6.3 (see Fig. 7). Finally, it would be interestingto study the optimal stopping problem (7.6) in more detail. Acknowledgements
J.S.A. was supported by a Leeds Anniversary Research Scholarship (LARS) from the Uni-versity of Leeds. Both authors have greatly benefited from many useful discussions withTiziano De Angelis, who has also contributed to the design of this study. J.S.A. is gratefulto Elena Issoglio for helpful comments. We thank three anonymous reviewers for their use-ful feedback. In particular, Reviewer eferences [1] Acemoglu, D. and Shimer, R. Productivity gains from unemploy-ment insurance.
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