An analytical study of participating policies with minimum guaranteed and surrender option
AAN ANALYTICAL STUDY OF PARTICIPATING POLICIESWITH MINIMUM GUARANTEED AND SURRENDER OPTION
MARIA B. CHIAROLLA, TIZIANO DE ANGELIS, AND GABRIELE STABILE
Abstract.
We perform a detailed theoretical study of the value of a class of partic-ipating policies with four key features: ( i ) the policyholder is guaranteed a minimuminterest rate on the policy reserve; ( ii ) the contract can be terminated by the holder atany time until maturity (surrender option); ( iii ) at the maturity (or upon surrender)a bonus can be credited to the holder if the portfolio backing the policy outperformsthe current policy reserve; ( iv ) due to solvency requirements the contract ends if thevalue of the underlying portfolio of assets falls below the policy reserve.Our analysis is probabilistic and it relies on optimal stopping and free boundarytheory. We find a peculiar structure of the optimal surrender strategy, which wasundetected by previous (mostly numerical) studies on the same topic. For that wedevelop new methods in order to study the regularity of the corresponding optimalstopping boundaries. Introduction
Participating Policies with Minimum Guaranteed are insurance contracts, appealingpredominantly to individuals during their working lives, as a form of low-risk financialinvestment. The subscriber of a participating policy (policyholder) pays a premium(either single or periodic) which is used by the insurance company to set up a so-calledpolicy reserve for the policyholder. The policy reserve is linked to a portfolio of assetsheld by the insurance company and it accrues interest tracking the performance ofsuch portfolio (the details of the contract are illustrated in Section 2). The minimumguaranteed is a minimum interest rate paid by the insurance company towards thepolicy reserve irrespective of the performance of the portfolio backing the policy (thisrate is usually lower than the risk-free rate). In the absence of any further contractspecifications, the policy terminates at a given maturity, at which the policyholderreceives an amount equal to the value of the reserve, plus a bonus , if the current valueof the portfolio is sufficiently high relative to the reserve.The contract may incur early termination. That happens if the value of the portfoliobacking the policy is not sufficient to cover the minimum guaranteed on the policyreserve. In that case we say that the insurance company fails to meet the solvencyrequirements on the participating policy and the policyholder receives the value of thepolicy reserve at that time. More interestingly, early termination of the contract maybe an embedded option in the policy specification. Indeed, along with the standardcontract, the policyholder can buy the right to an early cancellation of the policy, theso-called surrender option (SO). If the policyholder exercises the surrender option priorto the maturity of the contract, at that time she receives the value of the policy reserve,plus the above mentioned bonus . Date : April 16, 2020.2010
Mathematics Subject Classification.
JEL Classification . G22.
Key words and phrases.
Participating policies, minimum guaranteed, surrender option, solvencyrequirement, optimal stopping, free-boundary problems. a r X i v : . [ q -f i n . M F ] A p r M.B. CHIAROLLA, T. DE ANGELIS, G. STABILE
While surrender options share similarities with financial options of American type,due to their early exercise feature, they are actually rather different in nature. Indeed,the presence of a surrender option, embedded in a participating policy, changes thestructure of the whole contract. As a consequence, the price of embedded options isnormally defined as the difference between the value of a policy which includes theoption and the value of a policy which does not include the option (see (2.8) for amathematical expression).
Participating Policies with Surrender Option (PPSO) have been studied extensivelyin the academic literature. This paper contributes to that strand of the literature whichassumes that the policyholder is fully rational and the surrender option is exercisedoptimally from a financial perspective. Other papers analyse PPSO in which surren-der occurs as a randomised event (see Cheng and Li [5]) or without assuming rationalbehaviour of the policyholder (see Nolte and Schneider [20]). Several papers adopt anumerical approach to analyse PPSO without solvency requirements for the insurancecompany (see Andreatta and Corradin [1], Bacinello [2], Bacinello, Biffis and Millosso-vich [3], Grosen and Jørgensen [14] among others). Chu and Kwok [6] provide an ana-lytical approximation for the price of a participating policy without taking into accountthe surrender option. Finally, Siu [25] considers the fair valuation of a PPSO when themarket value of the portfolio backing the policy is modelled by a Markov-modulatedGeometric Brownian Motion. In [25] the author approximates the solution of a freeboundary problem by a system of second-order piecewise linear ordinary differentialequations.In this paper we develop a fully theoretical analysis of participating policies withminimum guaranteed, embedded surrender option and early termination due to solvencyrequirements. Following the example of other papers on this topic (see, e.g., Chu andKwok [6], Fard and Siu [12], Grosen and Jørgensen [14], Siu [25]), we focus purely onthe financial aspects of the policy and ignore the demographic risk, in the sense that thecontract does not account for a possible demise of the policyholder. From the point ofview of applications we may imagine that there are multiple beneficiaries of the policy,so that the demographic risk is negligible. Moreover, it is well-known (see, e.g., Chengand Li [5], Stabile [26]) that the assumption of a constant force of mortality, independentof the financial market, results in a shift in the discount rate adopted for pricing; thisdoes not affect the methods we employ and the qualitative outcomes of our work. Thestudy becomes substantially more involved if one considers a time-dependent mortalityforce (as, e.g., in De Angelis and Stabile [10]) or worse a stochastic mortality. We leavethese extensions for future work.Our main contributions are: (i) the analytical study of the pricing formula for thePPSO and (ii) a characterisation of the optimal exercise strategy for the surrenderoption, in terms of an optimal exercise boundary. The arbitrage-free price of the policyis obtained as the value function of a suitable finite-time horizon optimal stoppingproblem, on a two-dimensional degenerate diffusion which lives in an orthant of theplane and is absorbed upon leaving the orthant. After a suitable transformation, thedynamics is reduced to a one dimensional diffusion in the form of a stochastic differentialequation (SDE) with absorption upon hitting zero. We are then led to consider a stateprocess ( t, X t ) ∈ [0 , T ] × R + which is absorbed if X t reaches zero, so that our state spaceis the ( t, x )-strip with t ∈ [0 , T ] and x ≥ ARTICIPATING POLICIES WITH SURRENDER OPTION 3 problems, and relying on an explicit dependence of the process on its initial value,are not applicable (see, e.g., the American put problem in Peskir and Shiryaev [21]);the stopping payoff is independent of time but, as a function of x , it is convex withdiscontinuous first derivative.The combination of the above ingredients produces a very peculiar shape of theoptimal stopping region, which we derive from a detailed analysis of the value function.First and foremost we observe that the stopping region S , i.e., the points ( t, x ) atwhich the policyholder should instantly surrender the contract, is not connected in the x -variable, for each value of t given and fixed. Instead, S may have two connectedcomponents for each t ∈ [0 , T ]. This result was not observed in prior work on the samemodel, where a numerical approach to the problem could not detect this unusual feature(see, e.g., Siu [25]). As it turns out, the peculiar shape of the stopping set is closelyrelated to the bonus mechanism included in the PPSO and it has fine implications onthe optimal exercise of the surrender option. We will elaborate further on this point inSection 5, once the mathematical details have been laid out more clearly.The stopping set is connected in the t -variable, for each given x ≥
0. This leadsnaturally to consider an optimal stopping boundary as a function of x (rather than asa function of t , as in the vast majority of papers in the area). We obtain a wealth offine properties of the map x (cid:55)→ c ( x ) on [0 , ∞ ), which are of independent mathematicalinterest for the probabilistic theory of free boundary problems. Indeed we show that c ( · )is continuous on (0 , ∞ ) and piecewise monotonic, with two strictly increasing portionsand a strictly decreasing one. It is important to remark that questions of continuity ofthe optimal boundary x (cid:55)→ c ( x ) are much harder to address than in the more canonicalsetting of time dependent boundaries t (cid:55)→ b ( t ). Here we resolve the issue in Theorem4.16, by providing a probabilistic proof which is new in the literature and makes use ofsuitably constructed reflecting diffusions. Our proof provides a conceptually simple wayto show (in more general examples) that time dependent optimal boundaries t (cid:55)→ b ( t )cannot exhibit flat stretches, unless the smooth-fit property fails.The rest of the paper is organised as follows. In Section 2 we set up the model ina rigorous mathematical framework. In Section 3 we show Lipschitz continuity of thevalue function with respect to both t and x , and prove some other useful properties.Section 4 contains the core material concerning the analysis of the free boundary prob-lem associated with the PPSO. There we derive the geometric properties of the stoppingset, we prove that the value function is continuously differentiable in [0 , T ) × R + andit solves a suitable boundary value problem. Finally, in Section 5 we obtain numericalillustrations of the value function, the stopping set and the related sensitivity analy-sis. Results in Section 5 are complemented by a financial interpretation. The paper iscompleted by a short technical appendix.2. Actuarial model and problem formulation
In this section we provide a mathematical description of the price of a PPSO in acomplete market, under a risk-neutral probability measure. We align our setup andpart of our notations to those already used in other papers on this topic as, e.g., Chuand Kwok [6], Grosen and Jørgensen [14] and Siu [25].Given
T >
0, we consider a market with finite time horizon [0 , T ] on a completeprobability space (Ω , F , ( F t ) t ∈ [0 ,T ] , Q ) that carries a one-dimensional Brownian motion (cid:102) W := ( (cid:102) W t ) t ∈ [0 ,T ] . With no loss of generality we assume that the filtration ( F t ) t ∈ [0 ,T ] isgenerated by the Brownian motion and it is completed with Q -null sets. Our market iscomplete and Q is the risk-neutral probability measure. M.B. CHIAROLLA, T. DE ANGELIS, G. STABILE
An investor can purchase a PPSO at time zero by making a lump payment V to aninsurance company. In return, the insurer invests an amount R into a financial port-folio and commits the company to credit interests to the policyholder’s Policy Reserve according to a mechanism that will be described below. Thanks to the surrender option embedded in the contract, the policyholder has the right to withdraw her investmentat any time prior to the policy’s maturity T . In this case, she receives the so-called intrinsic value of the policy.2.1. The Policy Reserve.
First we describe the rate at which the amount R investedby the insurer accrues interest, based on the performance of the portfolio backing thepolicy (the reference portfolio ). We let A := ( A t ) t ∈ [0 ,T ] be the process denoting the valueof such portfolio and assume that A evolves as a geometric Brownian motion under Q ;that is, (cid:26) d A t = A t (cid:0) r d t + σ d (cid:102) W t (cid:1) ,A = a , (2.1)where r , σ and a are positive constants and r is the risk free rate.During the lifetime of the policy, the Policy Reserve is denoted by R := ( R t ) t ∈ [0 ,T ] and accrues interest based on a two-layer mechanism. First, the insurance companyguarantees a minimum fixed interest rate, which we denote by r G and, in line withfinancial practice, we assume r G ∈ (0 , r ) . (2.2)Second, at times when the portfolio performs particularly well, the policyholder partici-pates in the returns. In particular, we define the so-called Bonus Reserve B t := A t − R t and, as in [6], [25], [24] and [12], we consider a bonus distribution rate (BDR) of theform ln (cid:0) B t /R t (cid:1) = ln (cid:0) A t /R t (cid:1) . The BDR measures the performance of the portfolio against the performance of thepolicy reserve. The insurance company compares the BDR to a constant, long-termtarget β >
0, known as target buffer ratio . If the BDR exceeds the target buffer ratio,a proportion δ > c ( A t , R t ) = δ (cid:18) ln (cid:16) A t R t (cid:17) − β (cid:19) ∨ r G . It follows that the policy reserve R evolves under Q according to the dynamics (cid:26) d R t = c ( A t , R t ) R t d t,R = α a , (2.4)where α ∈ (0 ,
1) is fixed by the insurer. Hence, the initial reserve R covers α shares ofthe reference portfolio. Remark 2.1.
Notice that in the specification of the bonus mechanism in (2.3) we mayequivalently consider ln( αA t /R t ) instead of ln( A t /R t ) . This would emphasise that thepolicyholder only receives a bonus proportional to her share of the portfolio backing thepolicy. From the mathematical point of view, of course there is no difference since ln( αA t /R t ) = ln α + ln( A t /R t ) and the additional term, ln α , is absorbed in the specifi-cation of the target buffer ratio β > . ARTICIPATING POLICIES WITH SURRENDER OPTION 5
The policy’s intrinsic value and its arbitrage-free price.
Next we describethe so-called intrinsic value of the policy, which is the value that the policyholder receiveseither at the maturity of the policy or at an earlier time, should she decide to exercisethe surrender option.The intrinsic value is equal to the policy reserve plus a bonus component. The latter,is activated when the value of the policyholder’s shares in the portfolio A exceeds thecurrent value of the policy reserve; that is, when αA t > R t . In this case the policyholderreceives a bonus fraction γ of the surplus of her α -share.From the mathematical point of view, the intrinsic value of the policy may be writtenas g ( A t , R t ) := R t + γ [ αA t − R t ] + , (2.5)where [ x ] + := max { x, } and γ ∈ (0 ,
1) is the so-called participation coefficient .The model also takes into account that the company may fail to meet the solvencyrequirement at any time before T . In fact, denoting by τ † the stopping time ( insolvencytime ) τ † := inf { t ≥ A t ≤ R t } , (2.6)the company’s solvency requirement is satisfied for t < τ † . In the event of τ † < T thepolicy is liquidated and the policyholder receives (cf. (2.5)) g ( A τ † , R τ † ) = R τ † , i.e., the policy reserve value.Finally, we can define V , the arbitrage-free price of the PPSO at time zero. Noticethat V = V ( α ), in the sense that the contract is specified by indicating the portion α of the portfolio which backs the policy. Recalling (2.1), (2.4), (2.5) and (2.6), we have(2.7) V = sup ≤ τ ≤ T E Q (cid:104) e − r ( τ ∧ τ † ) g ( A τ ∧ τ † , R τ ∧ τ † ) (cid:105) , where E Q is the expectation under the measure Q and the supremum is taken over allstopping times τ ∈ [0 , T ] with respect to ( F t ) t ∈ [0 ,T ] . In what follows we will refer to(2.7) as the PPSO problem .The value of the surrender option embedded in the contract (usually referred to asEarly Exercise Premium in the mathematical finance literature) can be obtained as V opt := V − V E , (2.8)where V E is the arbitrage-free price of the contract without the possibility of an earlysurrender, that is V E = E Q (cid:104) e − r ( T ∧ τ † ) g ( A T ∧ τ † , R T ∧ τ † ) (cid:105) . (2.9)It is worth noticing that, in practice, the use of surrender options may be disin-centivised by insurance companies, who agree to pay out only a fraction of the policyreserve in case of early surrender. In our case that would correspond to take λg ( A τ , R τ )in (2.7) on the event { τ < τ † ∧ T } , with λ ∈ [0 , λ = 1, which is consistent with the existing literature (see, e.g., [14], [25])and provides an upper bound for the prices of contracts with λ ∈ [0 , M.B. CHIAROLLA, T. DE ANGELIS, G. STABILE Preliminary properties of the value function
As noticed in [6] and [25], the PPSO problem can be made more tractable by consid-ering the logarithm of the ratio
A/R as the observable process in the optimal stoppingformulation of (2.7). Indeed, set X := ( X t ) t ∈ [0 ,T ] with X t := ln (cid:18) A t R t (cid:19) for t ∈ [0 , T ], Q -a.s.(3.1)Then, by (2.1) and (2.4), one getsd X t = (cid:16) r − r G − σ − (cid:2) δ ( X t − β ) − r G (cid:3) + (cid:17) d t + σ d (cid:102) W t , (3.2)with initial condition X = x α := ln(1 /α ) > . (3.3)In terms of X the insolvency time becomes(3.4) τ † = inf { t ≥ X t ≤ } . Next we write the intrinsic value of the policy g ( A t , R t ) (see (2.4)) in terms of x . For x ∈ R + define the gain function h ( x ) := e − x + γ (cid:2) α − e − x (cid:3) + , (3.5)and notice that x (cid:55)→ h ( x ) is convex, striclty decreasing and | h ( x ) − h ( y ) | ≤ | x − y | , (3.6)since γ ∈ (0 , g ( A t , R t ) = A t (cid:16) e − X t + γ (cid:2) α − e − X t (cid:3) + (cid:17) = A t h ( X t ) , t ∈ [0 , T ], Q -a.s.(3.7)Now, the key to the dimension reduction is a change of measure. Define the martingaleprocess M := ( M t ) t ∈ [0 ,T ] by M t := e σ (cid:102) W t − σ t = e − rt A t /a , (3.8)and the probability measure P equivalent to Q on F T given by d P = M T d Q . ByGirsanov theorem the process W := ( W t ) t ∈ [0 ,T ] with W t := (cid:102) W t − σt, (3.9)is a P -Brownian motion. Then, under the new measure P , the dynamics of X reads(3.10) (cid:40) d X t = π ( X t )d t + σ d W t ,X = x α > , where π ( x ) := r − r G + 12 σ − (cid:2) δ ( x − β ) − r G (cid:3) + . (3.11)Using (3.7), (3.8) and the optional sampling theorem, for any stopping time τ ∈ [0 , T ]one easily obtains E Q (cid:104) e − r ( τ ∧ τ † ) g ( A τ ∧ τ † , R τ ∧ τ † ) (cid:105) = a E Q (cid:104) M τ ∧ τ † h (cid:0) X τ ∧ τ † (cid:1)(cid:105) = a E (cid:104) h (cid:0) X τ ∧ τ † (cid:1)(cid:105) , (3.12)with E [ · ] denoting the P -expectation. Hence, (2.7) may be rewritten as V = a v ,where v := sup ≤ τ ≤ T E (cid:2) h (cid:0) X τ ∧ τ † (cid:1)(cid:3) . (3.13) ARTICIPATING POLICIES WITH SURRENDER OPTION 7
Thanks to the Markovian nature of the process X the value v only depends on theinitial value of the process X = x α (see (3.3)) and on the maturity T of the contract.However, in order to be able to characterise v and the associated optimal stoppingrule, we must embed our problem in a larger state-space by considering all possibleinitial values of the time-space dynamics ( t, X ). For that we denote by X x the process X starting at time 0 from an arbitrary point x ≥ X t,x the process X starting at time t from x ≥ X is time-homogeneous, it holds Law (cid:0) ( s, X t,xs ) s ≥ t (cid:1) = Law (cid:0) ( t + s, X xs ) s ≥ (cid:1) . Then we can identify the dynamics ( s, X t,xs ) s ∈ [ t,T ] and ( t + s, X xs ) s ∈ [0 ,T − t ] , and use thelatter in the problem formulation below. Thanks to time-homogeneity, we also havethat(3.14) τ † = inf { s ≥ X xs = 0 } is independent of time. Sometimes we use τ † ( x ) to emphasise that τ † depends on X = x .For future reference we denote E x [ · ] := E [ · | X = x ] and, from now on, we will studythe finite-time horizon optimal stopping problem given by v ( t, x ) := sup ≤ τ ≤ T − t E (cid:104) h ( X xτ ∧ τ † ) (cid:105) , ( t, x ) ∈ [0 , T ] × R + . (3.15)It is clear that we can go back to our original problem formulation in two steps: first v = v (0 , x α ), and then V = a v . Remark 3.1.
It is important to emphasise that the analysis carried out in the rest ofthe paper applies to a PPSO corresponding to a participation level equal to α . Thisparameter is given and fixed at time t = 0 when the policy is purchased. Therefore, atany future time t ∈ (0 , T ] , the value v ( t, x ) should be regarded as v ( t, x ; α ) ; this is thevalue at time t (with X t = x ) of a policy whose participation level α was set at timezero. With this in mind, it should be clear that v (0 , x ) is just a useful mathematicalabstraction as the only meaningful value of the policy at time zero is v (0 , x α ) . Path properties of the underlying process.
In this section we collect somefacts about the underlying stochastic process X , defined in (3.10), which will be laterused to infer regularity of the value function (3.15). First we observe that since the driftfunction π ( · ) is Lipschitz continuous and the diffusion coefficient is constant, there existsa modification (cid:101) X of X such that the stochastic flow ( t, x ) (cid:55)→ (cid:101) X xt ( ω ) is continuous fora.e. ω ∈ Ω (see, e.g., [22, Chapter V.7]). As usual, throughout the paper we work withthe continuous modification which we still denote by X for simplicity. Lemma 3.2.
For any P -a.s. finite stopping time τ ≥ it holds (cid:12)(cid:12) X xτ − X yτ (cid:12)(cid:12) ≤ (cid:12)(cid:12) x − y (cid:12)(cid:12) e δτ , P -a.s. for x, y ∈ R + , (3.16) X yτ − X xτ ≥ ( y − x )(2 − e δτ ) , P -a.s. for y ≥ x ≥ . (3.17) Proof.
From the integral form of (3.10) (with X x = x and X y = y ), and noticing that π ( · ) is Lipschitz with constant δ >
0, it is immediate to see (cid:12)(cid:12) X xτ − X yτ (cid:12)(cid:12) ≤ (cid:12)(cid:12) x − y (cid:12)(cid:12) + (cid:90) τ (cid:12)(cid:12) π ( X xt ) − π ( X yt ) (cid:12)(cid:12) d t ≤ (cid:12)(cid:12) x − y (cid:12)(cid:12) + δ (cid:90) τ (cid:12)(cid:12) X xt − X yt (cid:12)(cid:12) d t. Then, an application of Gronwall’s inequality gives (3.16)
M.B. CHIAROLLA, T. DE ANGELIS, G. STABILE
The argument for (3.17) is similar. Using that y > x and π ( · ) Lipschitz we have X yτ − X xτ ≥ y − x − δ (cid:90) τ (cid:12)(cid:12) X yt − X xt (cid:12)(cid:12) d t ≥ y − x − ( y − x ) (cid:90) τ δe δt d t = ( y − x )(2 − e δτ ) , where the second inequality uses (3.16). (cid:3) The next estimate on the local time of the process X is particularly useful to establishthat the value function is Lipschitz in the time variable. In the rest of the paper wedenote L z := ( L zt ) t ∈ [0 ,T ] the local time of the process X at a point z ≥
0, which isdefined as (see, e.g., [21, Eq. (3.3.29), p. 68]) L zt ( X ) := lim ε ↓ ε (cid:90) t {| X s − z |≤ ε } d (cid:104) X (cid:105) s , P -a.s.(3.18) Lemma 3.3.
Let < t ≤ t ≤ T , fix N > and recall x α from (3.3) . Then, thereexists a positive constant κ := κ ( t , N ; x α ) such that sup x ∈ [0 ,N ] E x (cid:2) L x α t − L x α t (cid:3) ≤ κ ( t − t ) . (3.19) Remark 3.4.
Notice that in the lemma above t must be taken strictly positive as theconstant κ ( t , N ) might (and will) explode as t → .Proof. Thanks to (3.18) we can select a sequence ( ε n ) n ≥ such that ε n ↓ n → ∞ and L x α t − L x α t = lim n →∞ ε n (cid:90) t t {| X s − x α |≤ ε n } d (cid:104) X (cid:105) s , P x − a.s. Then, using Fatou’s lemma we get E x (cid:2) L x α t − L x α t (cid:3) ≤ lim inf n →∞ ε n (cid:90) t t P x ( | X s − x α | ≤ ε n ) σ d s. (3.20)It is well-known that X admits a continuous density with respect to its speed measure(see, e.g., [23, Thm. 50.11] or [15, Sec. 4.11]). That is P x ( | X s − x α | ≤ ε n ) = (cid:90) x α + ε n x α − ε n p ( s, x, y ) 2d yσ S (cid:48) ( y ) , where S (cid:48) is the derivative of the scale function and reads S (cid:48) ( y ) = exp (cid:18) − σ (cid:90) y π ( z )d z (cid:19) . (3.21)Moreover, the map ( s, x, y ) (cid:55)→ p ( s, x, y ) is continuous on (0 , ∞ ) × R and clearly S (cid:48) iscontinuous too. Hence, letting ε n ≤ ε , for all n ≥ ε >
0, and setting κ ( t , N ) := sup ( s,x,y ) p ( s, x, y ) S (cid:48) ( y ) , with the supremum taken over ( s, x, y ) ∈ [ t , T ] × [0 , N ] × [ x α − ε , x α + ε ], it is immediateto obtain (3.19) from (3.20). (cid:3) ARTICIPATING POLICIES WITH SURRENDER OPTION 9
Basic properties of the value function.
Some parts of the analysis in ourpaper are more conveniently performed by considering a different formulation of problem(3.15). Let L be the second order differential operator associated to the diffusion (3.10),i.e. ( L f )( x ) := σ ∂ xx f ( x ) + π ( x ) ∂ x f ( x ) , for any f ∈ C ( R + ) , with ∂ x and ∂ xx denoting the first and second order partial derivatives with respect to x , respectively. Let us also define the function(3.22) H ( x ) := e − x (cid:16) σ − π ( x ) (cid:17) , x ≤ x α (1 − γ ) e − x (cid:16) σ − π ( x ) (cid:17) , x > x α , where we recall x α from (3.3). For future reference it is worth noticing that, since r G < r , − ( r − r G ) ≤ H ( x ) ≤ δ for x ∈ R + . (3.23)Clearly H is discontinuous at x α and it is easy to check that H ( x ) = ( L h )( x ) for x (cid:54) = x α .Since x (cid:55)→ h ( x ) (see (3.5)) is a convex function and its first derivative has a single jump ∂ x h ( x α +) − ∂ x h ( x α − ) = γα, we can apply Itˆo-Tanaka’s formula to h ( X τ ∧ τ † ) in (3.15), to obtain the following equiv-alent formulation of problem (3.15) u ( t, x ) := v ( t, x ) − h ( x )(3.24) = sup ≤ τ ≤ T − t E x (cid:20) (cid:90) τ ∧ τ † H ( X s ) { X s (cid:54) = x α } d s + γα L x α τ ∧ τ † (cid:21) , where ( L zt ) t ≥ is the local time of X at a point z > u isnon-negative since v ( t, x ) ≥ h ( x ), for all ( t, x ) ∈ [0 , T ] × R + , by (3.15). Proposition 3.5.
The following properties hold for the value function of the optimalstopping problem (3.15) , i) the map t (cid:55)→ v ( t, x ) is decreasing and v ( T, x ) = h ( x ) for any fixed x ≥ ; ii) the map x (cid:55)→ v ( t, x ) is decreasing and v ( t,
0) = h (0) for any fixed t ∈ [0 , T ] .Moreover, for any ≤ x ≤ x < + ∞ and any t ∈ [0 , T ] it holds ≤ v ( t, x ) − v ( t, x ) ≤ κ ( x − x ) , (3.25) with κ := e δT . Finally, for any ≤ t ≤ t < T and any x ∈ [0 , N ] , with fixed N > ,there is a constant κ = κ ( t , N ; x α ) > such that ≤ v ( t , x ) − v ( t , x ) ≤ κ ( t − t ) . (3.26) Proof.
The monotonicity in point i ) follows from time-independence of h and τ † , whereasthe value of v at T follows from (3.15). As for ii ), v ( t,
0) = h (0) since τ † (0) = 0 P -a.s.To show monotonicity of v in x , fix x < x and note that by uniqueness of the solutionto (3.10) follows X x s ∧ τ † ( x ) ≤ X x s ∧ τ † ( x ) P -a.s. for all s ∈ [0 , T ]. Since the inequality alsoholds if we replace s by a stopping time and the gain function h is decreasing, we obtain v ( t, x ) − v ( t, x ) ≤ sup ≤ τ ≤ T − t E (cid:104) h ( X x τ ∧ τ † ( x ) ) − h ( X x τ ∧ τ † ( x ) ) (cid:105) ≤ . (3.27)Next we prove (3.25). Fix t ∈ [0 , T ], consider 0 ≤ x < x and denote by τ † := τ † ( x )and τ † := τ † ( x ) the first hitting time at zero of X x and X x , respectively. Frompathwise uniqueness of the solution of (3.10) we have τ † ≤ τ † . Then τ ∧ τ † ∧ τ † = τ ∧ τ † , P -a.s. for every admissible stopping time τ . Recalling that v ( t, · ) is decreasing and that h is strictly decreasing and 1-Lipschitz (see (3.6)) we have0 ≤ v ( t, x ) − v ( t, x ) ≤ sup ≤ τ ≤ T − t E (cid:104) h (cid:0) X x τ ∧ τ † (cid:1) − h (cid:0) X x τ ∧ τ † (cid:1)(cid:105) (3.28) ≤ E (cid:104) sup ≤ s ≤ T − t (cid:12)(cid:12)(cid:12) X x s − X x s (cid:12)(cid:12)(cid:12)(cid:105) ≤ e δT ( x − x ) , where the last inequality follows by (3.16).It remains to prove (3.26). For that it is convenient to use (3.24) and notice that for0 ≤ t ≤ t < T and x ∈ R + we have0 ≤ v ( t , x ) − v ( t , x ) = u ( t , x ) − u ( t , x ) , where the inequality is due to i ) above. For any stopping time τ ∈ [0 , T − t ] we havethat τ ∧ ( T − t ) is admissible for the problem with value u ( t , x ). Then, by directcomparison (recall that τ † only depends on x ∈ R + ) and with x ∈ [0 , N ], we have0 ≤ u ( t , x ) − u ( t , x ) ≤ sup ≤ τ ≤ T − t E x (cid:34) { τ ∧ τ † >T − t } (cid:32)(cid:90) τ ∧ τ † T − t { X s (cid:54) = x α } H ( X s )d s + αγ (cid:16) L x α τ ∧ τ † − L x α T − t (cid:17)(cid:33)(cid:35) (3.29) ≤ δ ( t − t ) + αγ E x (cid:104) L x α T − t − L x α T − t (cid:105) , where in the final inequality we used (3.23) and the fact that the local time t (cid:55)→ L x α t isnon-decreasing. Now, recalling Lemma 3.3 we obtain (3.26) by setting κ ( t , N ; x α ) := δ + αγ/ · κ ( T − t , N ; x α ). (cid:3) An immediate consequence of (3.26), (3.25), and the fact that h is bounded andnon-negative, is given in the next corollary. Corollary 3.6.
The value function v of the optimal stopping problem (3.15) is non-negative, continuous on [0 , T ] × [0 , ∞ ) and bounded by 1. As usual in optimal stopping theory, we let C = (cid:8) ( t, x ) ∈ [0 , T ] × R + : v ( t, x ) > h ( x ) (cid:9) (3.30) = (cid:8) ( t, x ) ∈ [0 , T ] × R + : u ( t, x ) > (cid:9) and S = (cid:8) ( t, x ) ∈ [0 , T ] × R + : v ( t, x ) = h ( x ) (cid:9) (3.31) = (cid:8) ( t, x ) ∈ [0 , T ] × R + : u ( t, x ) = 0 (cid:9) be respectively the so-called continuation and stopping regions. We also denote by ∂ C the boundary of the set C and introduce the first entry time of ( t + s, X s ) into S , i.e.(3.32) τ ∗ ( t, x ) := inf { s ∈ [0 , T − t ] : ( t + s, X xs ) ∈ S} . Continuity of v and h imply that C is an open set and S is closed. Moreover, standardoptimal stopping results (see [21, Cor. 2.9, Sec. 2]) guarantee that (3.32) is optimal for v ( t, x ). Finally, the process V t,x := ( V t,xs ) s ∈ [0 ,T − t ] given by V t,xs = v ( t + s, X xs ) is asupermartingale for s ∈ [0 , T − t ] and ( V t,xs ∧ τ ∗ ) s ∈ [0 ,T − t ] is a martingale for any ( t, x ) ∈ [0 , T ] × R + .Using the martingale property and continuity of the value function we obtain thenext well-known result (see, e.g. [21, Sec. 7.1, Chapter III], for a proof). ARTICIPATING POLICIES WITH SURRENDER OPTION 11
Proposition 3.7.
The value function v lies in C , ( C ) and it solves the boundary valueproblem ∂ t v + L v = 0 , in C (3.33) with v = h on ∂ C . We close the section with a simple technical lemma that is a consequence of themaximum principle.
Lemma 3.8.
For all ( t, x ) ∈ C it holds ∂ t v ( t, x ) < .Proof. By contradiction we assume there is ( t , x ) ∈ C such that ∂ t v ( t , x ) = 0.Since v ( t , x ) > h ( x ) and v ( T, x ) = h ( x ) there must exists t ∈ ( t , T ) such that( t , x ) ∈ C and ∂ t v ( t , x ) < − ε , for some ε >
0. By continuity of ∂ t v inside C , and thefact that C is open, there exists δ > ∂ t v ( t , x ) < − ε/ x ∈ ( x − δ, x + δ ).Now, letting O := ( t , t ) × ( x − δ, x + δ ) we have that ∂ t v ∈ C , ( O ), thanks tointernal regularity results for solutions of partial differential equations applied to (3.33)(see, e.g., [13, Thm. 10, Ch. 3, Sec. 5]). Moreover, differentiating (3.33) with respect totime and using (i)-Proposition 3.5 with the observations above, we obtain that ˆ v := ∂ t v solves ( ∂ t ˆ v + L ˆ v )( t, x ) = 0 , for ( t, x ) ∈ O (3.34) ˆ v ( t, x ± δ ) ≤ , for t ∈ [ t , t )(3.35) ˆ v ( t , x ) < − ε/ , for x ∈ ( x − δ, x + δ ) . (3.36)Setting τ O := inf { s ≥ t + s, X x s ) / ∈ O} , an application of Dynkin’s formula gives0 = ˆ v ( t , x ) = E (cid:2) ˆ v ( t + τ O , X x τ O ) (cid:3) ≤ − ε P (cid:0) τ O = t − t (cid:1) , (3.37)which leads to a contradiction as the process ( t + s, X x s ) exits O by crossing thesegment { t } × ( x − δ, x + δ ) with positive probability. (cid:3) It is clear by (3.24) that u inherits the same continuity and boundedness propertiesof v (see (3.26), (3.25) and Corollary 3.6). Moreover, ∂ t u < C with u ∈ C , in C \ ([0 , T ] × { x α } ) due to (3.5) and (3.22). Finally, in C \ ([0 , T ] × { x α } ) the function u solves ∂ t u + L u = − H, (3.38)with u = 0 on ∂ C .3.3. A comment on the perpetual problem.
For completeness we notice that theperpetual version of our problem, i.e., with T = + ∞ in (3.15), admits a trivial solution.Let us define v ∞ ( x ) := sup τ ≥ E x [ h ( X τ ∧ τ † )] , for x ≥ , and τ n := inf { t ≥ X t ≥ n } . Recall the scale function S ( x ) := (cid:82) x S (cid:48) ( y )d y , with S (cid:48) asin (3.21), and notice that P x ( τ † < + ∞ ) = lim n →∞ P x ( τ † < τ n ) = lim n →∞ S ( n ) − S ( x ) S ( n ) = 1 , by straightforward calculations for all x ≥ S ( n ) → ∞ as n → ∞ , since π ( z ) ∼ − δz as z → ∞ ). Then we have the following proposition. Proposition 3.9.
The value of the perpetual problem is v ∞ ( x ) = 1 for all x ≥ and τ † is optimal. Proof.
Since 0 ≤ h ( x ) ≤
1, it is immediate that 0 ≤ v ∞ ( x ) ≤ x ≥
0. Thestopping time τ n is admissible and provides the lower bound v ∞ ( x ) ≥ E x (cid:2) h ( X τ n ∧ τ † ) (cid:3) = h (0) P x ( τ † < τ n ) + h ( n ) P x ( τ † > τ n ) . Letting n → ∞ we obtain v ∞ ( x ) ≥ v ∞ ( x ) = 1. It then follows that thecontinuation set reads { x ≥ v ∞ ( x ) > h ( x ) } = (0 , + ∞ ) , and therefore τ † is optimal, since it is also finite P x -a.s. (cid:3) A free boundary problem
In this section we study the free boundary problem associated with the stoppingproblem (3.24). We derive geometric properties of the continuation region C and regu-larity of its boundary ∂ C . These have a close interplay with the smoothness of the valuefunction v in the whole space.4.1. Analysis of the stopping region.
We can now start the study of the stoppingregion. The next statement is an immediate consequence of the fact that t (cid:55)→ u ( t, x ) isnon-increasing (see (i) in Proposition 3.5). Proposition 4.1.
For any ( t, x ) ∈ [0 , T ] × R + it holds (4.1) ( t, x ) ∈ S ⇒ [ t, T ] × { x } ∈ S . Some of the arguments that we need in order to characterise the stopping regionrequire the next lemma. Its proof is somewhat standard but we provide it in theappendix for completeness.
Lemma 4.2.
For ε > define ρ ε := inf { s ≥ X x α s / ∈ ( x α − ε, x α + ε ) } . Then, for any c > there exists t ε,c > such that E x α (cid:2) L x α t ∧ ρ ε (cid:3) > cγα E x α [ t ∧ ρ ε ] for all t ∈ (0 , t ε,c ) . Now we can use the lemma to show that it is never optimal to stop at x α . Proposition 4.3.
It holds [0 , T ) × { x α } ⊂ C .Proof. Fix ε > ρ ε be as in Lemma 4.2. Take t ∈ [0 , T ) and s ∈ [0 , T − t ). Sincestopping at s ∧ ρ ε is admissible for the problem with value function u ( t, x α ), andinf | ζ |≤ ε H ( x α + ζ ) ≥ − c ε , for some c ε > ε , one obtains u ( t, x α ) ≥ E x α (cid:20) (cid:90) s ∧ ρ ε H ( X u ) { X u (cid:54) = x α } du + γα L x α s ∧ ρ ε ( X . ) (cid:21) ≥ γα E x α (cid:2) L x α s ∧ ρ ε ( X ) (cid:3) − c ε E x α (cid:2) s ∧ ρ ε (cid:3) . Now, applying Lemma 4.2 and picking s > u ( t, x α ) >
0. Hence( t, x α ) ∈ C . Since t ∈ [0 , T ) was arbitrary, the claim follows. (cid:3) ARTICIPATING POLICIES WITH SURRENDER OPTION 13
For any initial point ( t, x ) with t ∈ [0 , T ) and x ∈ R + \ { x α } such that H ( x ) > x . Since H > P -a.s., this well-knownargument gives u ( t, x ) >
0. Then, it follows that
R ⊆ C , where R := { ( t, x ) ∈ [0 , T ) × ( R + \{ x α } ) : H ( x ) > } . (4.2)Combining this observation with Proposition (4.3) we get R ∪ ([0 , T ) × { x α } ) ⊆ C . (4.3)It is clear that the shape of the set R varies depending on the parameters of theproblem. Interestingly, this gives rise to two possible shapes of the stopping region, aswe will see in the rest of the paper. Let us start by noticing that H ( x ) > ⇐⇒ σ − π ( x ) > ⇐⇒ x > ¯ x := β + rδ , (4.4)where we used (3.11) and r > r G . Then, based on the fact that S ⊆ (cid:0) R c ∩ { x (cid:54) = x α } (cid:1) ∪ (cid:0) { T } × [0 , ∞ ) (cid:1) , where R c is the complement of R , we distinguish two cases: Case 1 : x α < ¯ x , then we have S ⊆ (cid:110) [0 , T ) × (cid:16) [0 , x α ) ∪ ( x α , ¯ x ] (cid:17)(cid:111) ∪ (cid:16) { T } × [0 , ∞ ) (cid:17) . (4.5) Case 2 : x α ≥ ¯ x , then we have S ⊆ (cid:110) [0 , T ) × [0 , ¯ x ) (cid:111) ∪ (cid:16) { T } × [0 , ∞ ) (cid:17) . (4.6)Notice that in Case 1, we might (and will) find portions of the stopping region bothabove and below the line [0 , T ) × { x α } . Hence we shall find a disconnected stoppingregion. This is an interesting feature from the point of view of the analysis of thestopping boundary and it was never observed before in models of participating policieswith surrender option.We now focus on the study of the optimal stopping region in Case 1. Case 2 iseasier and can be handled with simpler methods. Properties of the boundary ∂ C willbe analysed separately in [0 , T ) × [0 , x α ) and [0 , T ) × ( x α , ¯ x ]. We will find that ∂ C isfully characterised in terms of a continuous, piece-wise monotonic curve, x (cid:55)→ c ( x ), on[0 , ∞ ).Thanks to Proposition 4.1 we may define c ( x ) := inf { t ∈ [0 , T ] : u ( t, x ) = 0 } , for x ∈ R + .(4.7)Since u ( T, x ) = 0 for x ≥ i )), the set in (4.7) always contains T . Moreover, c ( x α ) = T and c ( x ) = T, for x > ¯ x , (4.8)due to Proposition 4.3 and (4.3), respectively.The next proposition (combined with Proposition 4.1) shows that each of the portionsof the stopping region S lying in the box [0 , T ] × [0 , x α ] and in the box [0 , T ] × [ x α , ¯ x ]is connected. Proposition 4.4.
Assume x α < ¯ x . Then, (i) for z < x α and t ∈ (0 , T ] it holds ( t, z ) ∈ S = ⇒ [ t, T ] × [0 , z ] ⊆ S ;(ii) for z , z ∈ ( x α , ¯ x ) , with z < z , and t ∈ (0 , T ] it holds ( t, z ) , ( t, z ) ∈ S = ⇒ [ t, T ] × [ z , z ] ⊆ S . Proof.
Recall that ( t, ∈ S for all t ∈ [0 , T ] (see Proposition 3.5, ( ii )) and notice thatthe two claims are similar. We prove point ( ii ) as the proof of point ( i ) is analogous upto obvious changes.Let ( t, z ) and ( t, z ) belong to S and x α < z < z ≤ ¯ x . If t = T the result is trivial.Then let t < T and recall that H ( x ) < x ∈ ( x α , ¯ x ). By Proposition 4.1 we knowthat [ t, T ] × { z i } ⊆ S for i = 1 ,
2. Then it suffices to show that also { t } × ( z , z ) ⊆ S .Arguing by contradiction assume there exists z ∈ ( z , z ) such that ( t, z ) ∈ C . Let τ ∗ = τ ∗ ( t, z ) be optimal for the problem with value u ( t, z ). Then u ( t, z ) = E z (cid:20) (cid:90) τ ∗ ∧ τ † H ( X s ) { X s (cid:54) = x α } d s + γα L x α τ ∗ ∧ τ † ( X ) (cid:21) . Since [ t, T ] × { z i } ⊆ S for i = 1 ,
2, we have that τ ∗ ≤ ζ := inf { s ≥ t + s, X z s ) / ∈ [ t, T ) × ( z , z ) } . Hence, u ( t, z ) ≤ s ≥ L x α s ∧ ζ = 0 and H ( X s ∧ ζ ) < P z -a.s. and P z ( τ ∗ >
0) = 1 by assumption. Thus we have a contradiction. (cid:3)
We now provide several properties of the boundary x (cid:55)→ c ( x ), which we present in aseries of propositions and corollaries below. Proposition 4.5.
Assume x α < ¯ x . The map x (cid:55)→ c ( x ) is non-decreasing and left-continuous on the interval [0 , x α ) .Proof. The monotonicity follows from Proposition 4.1 and Proposition 4.4, ( i ).As for the left-continuity we use a standard argument. Fix x ∈ [0 , x α ) and take anincreasing sequence ( x n ) n ≥ that converges to x . Thenlim n →∞ ( x n , c ( x n )) = ( x, c ( x − )) , where c ( x − ) denotes the left limit of c at x , which exists by monotonicity of c . Since( x n , c ( x n )) n ≥ ⊆ S and S is a closed set, it must be ( x, c ( x − )) ∈ S . Hence c ( x − ) ≥ c ( x )and, by monotonicity of c , this implies c ( x − ) = c ( x ). (cid:3) Proposition 4.6.
Assume x α < ¯ x . Then the map x (cid:55)→ c ( x ) is never strictly posi-tive and constant (simultaneously) on intervals ( z , z ) contained in [0 , x α ) ∪ ( x α , ¯ x ) .Moreover, c ( x ) < T on [0 , x α ) . (4.9) Proof.
The proof borrows some ideas from [7]. Arguing by contradiction, assume thatthere exists an interval ( z , z ) ⊂ [0 , x α ) ∪ ( x α , ¯ x ) where c ( x ) takes the constant value¯ c >
0. Then the open set O := (0 , ¯ c ) × ( z , z ) is contained in C and u ∈ C , ( O ) sinceso are v , by Proposition 3.7, and h (away from x α ). It follows that u satisfies (cid:26) ∂ t u + L u = − H, in O , u (¯ c, x ) = 0 , x ∈ ( z , z ) .(4.10) ARTICIPATING POLICIES WITH SURRENDER OPTION 15
Pick ϕ ∈ C ∞ c ( z , z ), with ϕ ≥
0. Thanks to (4.10), for s ∈ [0 , ¯ c ) we have (cid:90) z z ∂ t u ( s, y ) ϕ ( y )d y = − (cid:90) z z ( L u )( s, y ) ϕ ( y )d y − (cid:90) z z H ( y ) ϕ ( y )d y = (cid:90) z z u ( s, y )( L ∗ ϕ )( y )d y − (cid:90) z z H ( y ) ϕ ( y )d y where we used integration by parts and L ∗ is the adjoint operator of L . Recalling that u t ≤
0, we use dominated convergence to obtain0 ≥ lim s ↑ ¯ c (cid:90) z z u ( s, y )( L ∗ ϕ )( y )d y − (cid:90) z z H ( y ) ϕ ( y )d y = (cid:90) z z lim s ↑ ¯ c u ( s, y )( L ∗ ϕ )( y )d y − (cid:90) z z H ( y ) ϕ ( y )d y (4.11) = − (cid:90) z z H ( y ) ϕ ( y )d y > , where the last equality is due to u (¯ c, y ) = 0 and the final inequality follows from thefacts that H < , ¯ x ) and ϕ is arbitrary. Hence a contradiction.Now, Proposition 4.3 implies c ( x α ) = T ; then monotonicity and left-continuity of c ( · )on [0 , x α ), together with the above result, give c ( x ) < T on [0 , x α ). (cid:3) The next corollary gives us some information about the shape of S in the box [0 , T ] × ( x α , ¯ x ). Corollary 4.7.
Assume x α < ¯ x . For every interval ( z , z ) contained in ( x α , ¯ x ) itholds S ∩ (cid:0) (0 , T ) × ( z , z ) (cid:1) (cid:54) = ∅ . Proof.
The claim must hold, otherwise O := (0 , T ) × ( z , z ) ⊆ C for some interval ( z , z ) ⊂ ( x α , ¯ x ). That would imply c ( x ) = T on ( z , z ), contradicting Proposition 4.6. (cid:3) Corollary 4.8.
Assume x α < ¯ x . The map x (cid:55)→ c ( x ) can be constant with zero constantvalue at most on intervals (0 , x ) , ( x , x ) for some x < x α and ( x , x ) ⊂ ( x α , ¯ x ) .Proof. It follows from the connected property of S in [0 , x α ) and in ( x α , ¯ x ) (see Propo-sition 4.4) and from Proposition 4.6. (cid:3) Proposition 4.9.
Assume x α < ¯ x . The map x (cid:55)→ c ( x ) is lower semi-continuous on [0 , ∞ ) , with c ( x ) = T for x ∈ (¯ x , ∞ ) , and lim sup x → x α c ( x ) = lim inf x → x α c ( x ) = T. Proof.
The fact that c ( x ) = T for x ∈ (¯ x , ∞ ) follows from (4.5). Hence c ( · ) is continu-ous on (¯ x , ∞ ).Fix z ∈ (0 , ¯ x ] and take a sequence ( z n ) n ≥ ⊆ (0 , ∞ ) with z n → z as n → ∞ . Thenlim inf n →∞ (cid:0) c ( z n ) , z n (cid:1) = (cid:0) lim inf n →∞ c ( z n ) , z (cid:1) , and since ( c ( z n ) , z n ) n ≥ ⊆ S and S is closed, it must be (cid:0) lim inf n →∞ c ( z n ) , z (cid:1) ∈ S . The latter implies lim inf n →∞ c ( z n ) ≥ c ( z ), by definition of c ( z ), and lower semi-continuity follows.If z = 0, then c (0) = 0 by Proposition 3.5, point ( ii ). Obviously lim inf n →∞ c ( z n ) ≥ z n → Finally, if z = x α , then c ( x α ) = T by Proposition 4.3 and lim inf n →∞ c ( z n ) ≥ c ( x α ) = T , together with c ( · ) ≤ T , implylim inf n →∞ c ( z n ) = T = lim sup n →∞ c ( z n )for any z n → x α . (cid:3) In the next proposition we prove that c ( · ) is piecewise monotonic. Proposition 4.10.
Assume x α < ¯ x . The map x (cid:55)→ c ( x ) attains a global minimum ≤ ˆ c ≤ T on [ x α , ¯ x ] . Moreover, there exist x ∈ [0 , x α ) , x ∈ ( x α , ¯ x ) and x ∈ [ x , ¯ x ) such that c ( · ) is (i) equal to zero on [0 , x ] , strictly increasing on ( x , x α ] and left-continuous on [0 , x α ) (notice that it might be x = 0 ); (ii) strictly decreasing on [ x α , x ) and right-continuous on [ x α , x ) ; (iii) strictly increasing on [ x , ¯ x ) and left-continuous on [ x , ¯ x ) .(Notice that in ( ii ) and ( iii ) it might be x = x .)In all cases ˆ c := c ( x ) = c ( x ) and, if x < x , then ˆ c = 0 . Finally, T = lim x ↓ ¯ x c ( x ) ≥ c (¯ x ) = lim x ↑ ¯ x c ( x ) . (4.12) Proof.
Recall that c ( · ) is lower semi-continuous on [0 , ∞ ) (see Proposition 4.9), with c ( x ) = T for x ∈ (¯ x , ∞ ). Moreover c ( x ) < T on [0 , x α ) and at least at one point in[ x α , ¯ x ], by Proposition 4.6 and Corollary 4.7 respectively. Then x as in (i) exists andthere must be a minimum of c ( · ) on [ x α , ¯ x ], denoted ˆ c ≥
0. Notice that it might be x = 0, in which case c ( · ) > , x α ).For the minimum on [ x α , ¯ x ] we have two possible cases. Either ˆ c = 0 or ˆ c >
0. Itfollows from Corollary 4.8 and Propositions 4.4 and 4.6 that(a) If ˆ c = 0, then it may occur at most in the interval [ x , x ] by Corollary 4.8.However the interval ( x , x ) may collapse into a single point x = x (in whichcase c ( · ) > x α , x ) ∪ ( x , ¯ x ]);(b) If ˆ c >
0, then it may only occur at a single point x (= x ) ∈ ( x α , ¯ x ].The monotonicity properties now follow by Corollary 4.8. Left-continuity and right-continuity are obtained by arguments as in Proposition 4.5. (cid:3) Repeating the same arguments as above we obtain analogous results for the case of x α ≥ ¯ x . Therefore we omit the proof of the next proposition. Proposition 4.11.
Assume x α ≥ ¯ x . Then, on the interval [0 , ¯ x ) the map x (cid:55)→ c ( x ) is non-decreasing, left-continuous, with c ( x ) < T . On the interval [¯ x , + ∞ ) it holds c ( x ) = T and lim x ↑ ¯ x c ( x ) = c (¯ x ) ≤ T. Moreover, there exists at most a point x ≤ ¯ x such that c ( x ) = 0 for x ∈ [0 , x ] and c ( · ) is strictly increasing on ( x , ¯ x ] . Higher regularity of value function and optimal boundary.
Thanks to thegeometry of the optimal boundary we obtain a lemma that will be useful to establishglobal C -regularity of the value function.As shown in [9] the key to C -regularity of the value function is the probabilisticregularity of the stopping boundary. Since the 2-dimensional process ( t, X t ) t ≥ is notof strong Feller type, we will actually use probabilistic regularity for the interior S ◦ of the stopping region. For completeness we recall that a process Z ∈ R d is said to ARTICIPATING POLICIES WITH SURRENDER OPTION 17 be of strong Feller type if z (cid:55)→ E z [ f ( Z t )] is continuous for any t > boundedmeasurable function f : R d → R .More precisely, letting σ ∗ ( t, x ) := inf { s ∈ (0 , T − t ] : ( t + s, X xs ) ∈ S} , and σ ◦∗ ( t, x ) := inf { s ∈ (0 , T − t ] : ( t + s, X xs ) ∈ S ◦ } . we say that a boundary point ( t, x ) ∈ ∂ C is (probabilistically) regular for S (or S ◦ ) if P ( σ ∗ ( t, x ) = 0) = 1 (or P ( σ ◦∗ ( t, x ) = 0) = 1) . (4.13)Clearly, probabilistic regularity for S ◦ implies the one for S . However, regularity for S ◦ is meaningless at points ( t , z ) ∈ ∂ C such that S has empty interior in a neighbourhoodof ( t , z ). Therefore in what follows we need both. Lemma 4.12.
The boundary ∂ C is probabilistically regular for S . Moreover, for any ( t , z ) ∈ ∂ C and any sequence ( t n , x n ) → ( t , z ) as n → ∞ , it holds lim n →∞ τ ∗ ( t n , x n ) = 0 , P -a.s. (4.14) where τ ∗ is defined in (3.32) .Proof. By the law of iterated logarithm and the geometry of the stopping region, it isclear that σ ∗ ( t, z ) = σ ◦∗ ( t, z ) = τ ∗ ( t, z ) , P -a.s.(4.15)for all ( t, z ) ∈ ∂ C except at most along vertical stretches of the boundary correspondingto x = 0 and x = x , as defined in Proposition 4.10. Indeed, at such points a spike may occur so that S ◦ may be (locally) empty. For simplicity let us denote E := (cid:16)(cid:0) , c ( x +) (cid:1) × { x } (cid:17) ∪ (cid:16)(cid:0) ˆ c, c ( x − ) ∧ c ( x +) (cid:1) × { x } (cid:17) . By definition τ ∗ ( t, z ) = 0, P -a.s., for all ( t, z ) ∈ ∂ C . Then, by (4.15) we have regularityof ∂ C \ E for S ◦ in the sense of (4.13). Hence (4.14) holds for any ( t, z ) ∈ ∂ C \ E (see,e.g., Corollary 6 in [9]).Thanks to lower semi-continuity of c , it only remains to consider regularity at E inthe cases: (a) x = x but ˆ c < c ( x ± ), and (b) x = 0 but c ( x +) >
0. We give a fullargument for case (a), then case (b) may be handled analogously.Let us assume x = x but ˆ c < c ( x ± ). Then σ ∗ ( t, x ) = τ ∗ ( t, x ), P -a.s., continuesto hold for all t ∈ [0 , T ) such that ( t, x ) ∈ ∂ C , by the law of iterated logarithm.Hence the first in (4.13) holds. Since the hitting time σ ◦∗ ( t, x ) is no longer zero forˆ c ≤ t < c ( x +) ∧ c ( x − ), because there is no interior part to the stopping region in aneighbourhood of ( t, x ), the argument provided in [9] needs a small tweak.Fix ( t , x ) ∈ ∂ C with ˆ c ≤ t < c ( x +) ∧ c ( x − ) and a sequence ( t n , x n ) n ≥ ⊂ C thatconverges to ( t , x ) as n → ∞ . Recall that we work with a continuous modification ofthe stochastic flow and let us pick ω ∈ Ω outside a null set such that ( t, x ) (cid:55)→ X xt ( ω ) iscontinuous. Then for any δ >
0, there exist 0 < s ,ω < s ,ω < δ such that X x s ,ω ( ω )
For all ( t, x ) ∈ C and < s < ( δ − ln(2)) ∧ ( T − t ) it holds e δs (cid:0) E (cid:2) { τ ∗ ≤ s } ∂ x h ( X xτ ∗ ) (cid:3) − κ P ( τ ∗ > s ) (cid:1) (4.16) ≤ ∂ x v ( t, x ) ≤ (cid:0) − e δs (cid:1) E (cid:104) { τ ∗ t, x + ε ) ∈ C and ( t, x − ε ) ∈ C . Notice that τ † ( x − ε ) ≤ τ † ( x ) ≤ τ † ( x + ε ) , P -a.s.and, by (ii) in Proposition 3.5, that τ † ( x ) ≥ τ ∗ ( t, x ) a.s., because [0 , T ] × { } ⊆ S . Set τ ∗ := τ ∗ ( t, x ) to simplify notation. Then for all s < T − t , using the (super)martingaleproperty, we have v ( t, x + ε ) ≥ E (cid:2) v (cid:0) t + ( s ∧ τ ∗ ) , X x + εs ∧ τ ∗ (cid:1)(cid:3) v ( t, x ) = E (cid:2) v (cid:0) t + ( s ∧ τ ∗ ) , X xs ∧ τ ∗ (cid:1)(cid:3) . Thus v ( t, x + ε ) − v ( t, x ) ≥ E (cid:2) v (cid:0) t + ( s ∧ τ ∗ ) , X x + εs ∧ τ ∗ (cid:1) − v (cid:0) t + ( s ∧ τ ∗ ) , X xs ∧ τ ∗ (cid:1)(cid:3) = E (cid:104) { τ ∗ ≤ s } (cid:16) v (cid:0) t + τ ∗ , X x + ετ ∗ (cid:1) − v (cid:0) t + τ ∗ , X xτ ∗ (cid:1)(cid:17)(cid:105) (4.17) + E (cid:104) { τ ∗ >s } (cid:16) v (cid:0) t + s, X x + εs (cid:1) − v ( t + s, X xs ) (cid:17)(cid:105) ≥ E (cid:104) { τ ∗ ≤ s } (cid:16) h (cid:0) X x + ετ ∗ (cid:1) − h ( X xτ ∗ ) (cid:17)(cid:105) − κ E (cid:104) { τ ∗ >s } (cid:12)(cid:12) X x + εs − X xs (cid:12)(cid:12)(cid:105) , where κ > { τ ∗ ≤ s } , the decreasing property of h and (cid:12)(cid:12) X x + εs − X xs (cid:12)(cid:12) ≤ εe δs (see (3.16)) give h (cid:0) X x + ετ ∗ (cid:1) ≥ h (cid:0) X xτ ∗ + εe δs (cid:1) . Hence v ( t, x + ε ) − v ( t, x ) ≥ E (cid:104) { τ ∗ ≤ s } (cid:16) h (cid:0) X xτ ∗ + εe δs (cid:1) − h (cid:0) X xτ ∗ (cid:1)(cid:17)(cid:105) − κ ε e δs P ( τ ∗ > s )= E (cid:34) { τ ∗ ≤ s } (cid:90) εe δs ∂ x h ( X xτ ∗ + z ) d z (cid:35) − κ ε e δs P ( τ ∗ > s ) , ARTICIPATING POLICIES WITH SURRENDER OPTION 19 since h ∈ AC( R + ). Then ∂ x v ( t, x ) = lim ε ↓ ε (cid:16) v ( t, x + ε ) − v ( t, x ) (cid:17) ≥ lim ε ↓ E (cid:34) { τ ∗ ≤ s } ε (cid:90) εe δs ∂ x h ( X xτ ∗ + z ) d z (cid:35) − κ e δs P ( τ ∗ > s )= E (cid:34) { τ ∗ ≤ s } lim ε ↓ ε (cid:90) εe δs ∂ x h ( X xτ ∗ + z ) d z (cid:35) − κ e δs P ( τ ∗ > s ) , where the final equality follows by dominated convergence since (cid:12)(cid:12) ∂ x h (cid:12)(cid:12) ≤ ω ∈ { τ ∗ ≤ s } we have X xτ ∗ ( ω ) (cid:54) = x α by Proposition 4.3 since s < T − t .Hence, there exists ¯ ε ω > z (cid:55)→ ∂ x h ( X xτ ∗ ( ω ) + z ) is continuouson (cid:2) , ¯ ε ω e δs (cid:3) and an application of the fundamental theorem of calculus giveslim ε ↓ ε (cid:90) εe δs ∂ x h ( X xτ ∗ ( ω ) + z ) d z = e δs ∂ x h ( X xτ ∗ ( ω )) . (4.18)Hence ∂ x v ( t, x ) ≥ e δs (cid:0) E (cid:2) { τ ∗ ≤ s } ∂ x h ( X xτ ∗ ) (cid:3) − κ P ( τ ∗ > s ) (cid:1) . Next we want to bound from above the difference v ( t, x ) − v ( t, x − ε ). This requiresa slight modification of the previous argument in order to account for the fact that τ † ( x − ε ) ≤ τ † ( x ), a.s. In particular, with no loss of generality we assume that ε ∈ (0 , ε ]for some ε > τ † := τ † ( x − ε ) for simplicity, we have τ † ≤ τ † ( x − ε ).Then, arguing as in (4.17) gives v ( t, x ) − v ( t, x − ε ) ≤ E (cid:20) v (cid:16) t + ( s ∧ τ ∗ ∧ τ † ) , X xs ∧ τ ∗ ∧ τ † (cid:17) − v (cid:18) t + ( s ∧ τ ∗ ∧ τ † ) , X x − εs ∧ τ ∗ ∧ τ † (cid:19)(cid:21) ≤ E (cid:104) { τ ∗ ≤ s ∧ τ † } (cid:0) h ( X xτ ∗ ) − h (cid:0) X x − ετ ∗ (cid:1)(cid:1)(cid:105) + E (cid:20) { τ ∗ >s ∧ τ † } (cid:18) v (cid:16) t + ( s ∧ τ † ) , X xs ∧ τ † (cid:17) − v (cid:18) t + ( s ∧ τ † ) , X x − εs ∧ τ † (cid:19)(cid:19)(cid:21) . Notice that the second term in the last expression is negative thanks to (ii)-Proposition3.5. Moreover, on the event { τ ∗ ≤ s ∧ τ † } we have h ( X xτ ∗ ) − h (cid:0) X x − ετ ∗ (cid:1) = (cid:90) X xτ ∗ − X x − ετ ∗ ∂ x h (cid:0) X x − ετ ∗ + z (cid:1) d z ≤ (cid:90) X xτ ∗ − X x − ετ ∗ ∂ x h ( X xτ ∗ + z ) d z where the last step follows from the convexity of h ( · ). Also, on the event { τ ∗ ≤ s ∧ τ † } ,using (3.17) we have X xτ ∗ − X x − ετ ∗ ≥ ε (cid:16) − e δτ ∗ (cid:17) ≥ ε (cid:16) − e δs (cid:17) > , by assuming s < δ − ln(2) with no loss of generality. It follows that, since ∂ x h ≤
0, wehave h ( X xτ ∗ ) − h (cid:0) X x − ετ ∗ (cid:1) ≤ (cid:90) ε ( − e δs ) ∂ x h ( X xτ ∗ + z ) d z on { τ ∗ ≤ s ∧ τ † } . Thus v ( t, x ) − v ( t, x − ε ) ≤ E (cid:34) { τ ∗ ≤ s ∧ τ † } (cid:90) ε ( − e δs ) ∂ x h ( X xτ ∗ + z ) d z (cid:35) , and, by arguments as in (4.18), we obtain ∂ x v ( t, x ) ≤ (cid:16) − e δs (cid:17) E (cid:104) { τ ∗ ≤ s ∧ τ † } ∂ x h ( X xτ ∗ ) (cid:105) , for s < ( δ − ln(2)) ∧ ( T − t ) . To conclude we let ε ↓ τ † = τ † ( x − ε ) ↑ τ † ( x ) and the upper bound in (4.16)holds by monotone convergence. (cid:3) Proposition 4.14.
Fix any ( t , z ) ∈ ∂ C with t < T and z > . Then, for anysequence ( t n , x n ) n ≥ ⊂ C such that ( t n , x n ) → ( t , z ) as n ↑ ∞ , we have lim n →∞ ∂ x v ( t n , x n ) = ∂ x h ( z )(4.19) and lim n →∞ ∂ t v ( t n , x n ) = 0 . (4.20) Proof.
First we prove (4.19). Notice, that (4.16) holds for any point ( t n , x n ) ∈ C from asequence that converges to ( t , z ) as n ↑ ∞ , where ( t , z ) ∈ ∂ C with t < T and z > τ ∗ n := τ ∗ ( t n , x n ) → n ↑ ∞ , by Lemma 4.12, and ∂ x h (cid:16) X x n τ ∗ n (cid:17) → ∂ x h ( z ) (recallthat z (cid:54) = x α ), then for 0 < s < ( δ − ln(2)) ∧ ( T − t ) dominated convergence and (4.16)give e δs ∂ x h ( z ) ≤ lim inf n →∞ ∂ x v ( t n , x n ) ≤ lim sup n →∞ ∂ x v ( t n , x n ) ≤ (2 − e δs ) ∂ x h ( z ) . Letting s → t, x ) ∈ C with t < T and ε > t + ε, x ) ∈ C . Let τ N := inf { u ≥ X xu ≥ N } , and pick s < T − ( t + ε ). Then, proceeding as in the proof of Lemma 4.13 we have v ( t + ε, x ) ≥ E (cid:2) v (cid:0) t + ε + ( s ∧ τ ∗ ∧ τ N ) , X xs ∧ τ ∗ ∧ τ N (cid:1)(cid:3) ,v ( t, x ) = E (cid:2) v (cid:0) t + ( s ∧ τ ∗ ∧ τ N ) , X xs ∧ τ ∗ ∧ τ N (cid:1)(cid:3) . Combining the above with (3.26) gives v ( t + ε, x ) − v ( t, x ) ≥ E (cid:104) { τ ∗ ≤ s ∧ τ N } (cid:16) h ( X xτ ∗ ) − h ( X xτ ∗ ) (cid:17)(cid:105) + E (cid:104) { τ ∗ >s ∧ τ N } (cid:16) v (cid:0) t + ε + ( s ∧ τ N ) , X xs ∧ τ N (cid:1) − v (cid:0) t + ( s ∧ τ N ) , X xs ∧ τ N (cid:1)(cid:17)(cid:105) ≥ − κ ( t + ε + s, N ) ε P x ( τ ∗ > s ∧ τ N ) . With no loss of generality we may assume that ε ∈ (0 , ε ] for some ε > s < T − ( t + ε ) and κ ( t + ε + s, N ) ≤ ˆ κ ( ε , N ) for some constant ˆ κ ( ε , N ) > ≥ ∂ t v ( t, x ) ≥ − ˆ κ ( ε , N ) P x ( τ ∗ > s ∧ τ N ) . The result holds for any ( t n , x n ) ∈ C from a sequence converging to ( t , x ). Moreover,with no loss of generality we can assume that x n ≤ x + 1 < N for all n ≥ τ N ( x n ) ≥ τ N ( x + 1) > P -a.s., and P (cid:16) τ ∗ ( t n , x n ) > s ∧ τ N ( x n ) (cid:17) ≤ P (cid:16) τ ∗ ( t n , x n ) > s ∧ τ N ( x + 1) (cid:17) . Then, thanks to Lemma 4.12 we get0 ≥ lim sup n →∞ ∂ t v ( t n , x n ) ≥ lim inf n →∞ ∂ t v ( t n , x n ) ≥ . Hence (4.20) holds. (cid:3)
ARTICIPATING POLICIES WITH SURRENDER OPTION 21
Proposition 4.14 and Proposition 3.7 imply continuous differentiability of v . Corollary 4.15.
The value function v is continuously differentiable on the set [0 , T ) × (0 , + ∞ ) . Moreover, v ∈ C , (cid:0) C ∩ (cid:0) [0 , T ) × (0 , + ∞ ) (cid:1)(cid:1) with lim C(cid:51) ( t,x ) → ( t ,z ) ∈ ∂ C ∂ xx v ( t, x ) = − σ π ( z ) ∂ x h ( z )(4.21) with t < T and z > .Proof. We only need to prove (4.21). In order to do that it is sufficient to take limitsin (3.33) and use Proposition 4.14. (cid:3)
The next theorem shows that the optimal boundary is continuous as a function of x . Notice that this type of continuity is not a standard result in optimal stoppingproblems for time-space processes ( t, X ). Indeed, in the probabilistic literature, onenormally proves continuity of the boundary as a function of time .If x α < ¯ x our boundary c is strictly monotonic on the intervals [ x , x α ), ( x α , x )and ( x , ¯ x ) (with x , x and x as in Proposition 4.10). Hence, the map x (cid:55)→ c ( x ) canbe inverted on [ x , x α ) ∪ ( x α , x ) ∪ ( x , ¯ x ], to obtain three time-dependent continuousfunctions that describe the boundary ∂ C . Recalling ˆ c from Proposition 4.10, we candefine the so-called generalised inverse functions b ( t ) := inf { x ∈ [0 , x α ) : c ( x ) > t } , t ∈ [0 , T ) , (4.22) b ( t ) := sup { x ∈ ( x α , x ) : c ( x ) > t } , t ∈ [ˆ c, T ) , (4.23) b ( t ) := inf { x ∈ ( x , ¯ x ) : c ( x ) > t } , t ∈ [ˆ c, T ) . (4.24)Clearly 0 ≤ b ( t ) ≤ b ( t ) ≤ b ( t ) for t ∈ [ˆ c, T ). Moreover, b ( T − ) = b ( T − ) = x α and b ( T − ) = ¯ x . The optimal stopping time can be expressed in terms of these functionsas τ ∗ = inf { t ∈ [0 , T ) : X t ≤ b ( t ) or X t ∈ [ b ( t ) , b ( t )] } ∧ T. Similarly, if x α ≥ ¯ x we can define a single time dependent boundary b as in (4.22) butwith ¯ x instead of x α . It is now clear that proving continuity of x (cid:55)→ c ( x ) is equivalentto proving that b , b and b have no flat portions. This is an interesting result in itsown right and, to the best of knowledge, it was never proven before with probabilisticmethods.We can now give the theorem. Its proof relies on the use of a suitably constructedreflecting diffusion. Theorem 4.16.
The mapping x (cid:55)→ c ( x ) is continuous on (0 , ∞ ) . If c (0+) = 0 thencontinuity holds on [0 , ∞ ) .Proof. We give a full proof in the case x α < ¯ x and consider the interval [ x , ∞ ), with x as in Proposition 4.10, where the boundary is increasing. It will be clear that theintervals [ x , x α ] and [ x α , x ] and the case x α ≥ ¯ x can be treated analogously.Arguing by contradiction let us assume that there exists z ∈ [ x , ∞ ) such that c ( z ) < c ( z +) and let I := ( c ( z ) , c ( z +)). Then I × { z } ⊂ ∂ C and there exists z > z such that ∂ t u ( t, z ) < − ε for some ε > t ∈ I := ( t , t ) ⊂ I (see Lemma 3.8).Since u ∈ C ([0 , T ) × R + ) by Corollary 4.15 and I × { z } ⊂ ∂ C , we have ∂ t u ( t, z ) = ∂ x u ( t, z ) = 0 for t ∈ I . Then for any ε > δ ε > z + δ ε < z and 0 ≥ ∂ t u ≥ − ε and | ∂ x u | ≤ ε on I × [ z , z + δ ε ],(4.25)by uniform continuity on any compact. Now we consider a process that equals ( X t ) t ≥ away from z + δ ε and is reflected(upwards) at z + δ ε . It is well-known (see, e.g., [18] or [4, Sec. 12, Chapter I]) thatthere exists a unique strong solution of the stochastic differential equationd (cid:101) X εt = π ( (cid:101) X εt )d t + σ d W t + d A δ ε t , (cid:101) X ε = z + δ ε , where A δ ε is a continuous, non-decreasing process that guarantees, P -a.s., (cid:101) X εt ≥ z + δ ε , for all t ≥ (cid:90) T { (cid:101) X εt >z + δ } d A δ ε t = 0 . (4.26)As in Lemma 3.8 we appeal to classical results on interior regularity for solutions ofPDEs that guarantee ∂ t u ∈ C , (cid:0) I × ( z , z ) (cid:1) and ( ∂ t + L ) ∂ t u = 0 on I × ( z , z ).Then, setting τ ε := inf { s ≥ (cid:101) X εs = z } and ˆ u := ∂ t u , an application of Itˆo’s formulafor semi-martingales gives, for any t ∈ I E (cid:104) ˆ u ( t + τ ε ∧ ( t − t ) , (cid:101) X ετ ε ∧ ( t − t ) ) (cid:105) =ˆ u ( t, z + δ ε ) + E (cid:34)(cid:90) τ ε ∧ ( t − t )0 ∂ x ˆ u ( t + s, (cid:101) X εs ) d A δ ε s (cid:35) (4.27) ≥ − ε + E (cid:34)(cid:90) τ ε ∧ ( t − t )0 ∂ tx u ( t + s, z + δ ε ) d A δ ε s (cid:35) where the inequality follows from (4.25) and the second condition in (4.26) impliesd A δ ε s = { (cid:101) X εs = z + δ ε } d A δ ε s . For the expression on the left-hand side of (4.27) we have E (cid:104) ˆ u ( t + τ ε ∧ ( t − t ) , (cid:101) X ετ ε ∧ ( t − t ) ) (cid:105) ≤ E (cid:104) { τ ε 0. However, the regularity of ∂ tx u as δ ε ↓ ϕ ∈ C ∞ c ( I ), ϕ ≥ (cid:82) I ϕ ( t ) dt = 1. Then, multiplying both sides of ARTICIPATING POLICIES WITH SURRENDER OPTION 23 (4.28) by ϕ , integrating over I and using Fubini’s theorem we obtain − ε (cid:90) I P ( τ ε < t − t ) ϕ ( t ) dt ≥ − ε + E (cid:20)(cid:90) τ ε (cid:18)(cid:90) I { t Law ( (cid:101) X ε | (cid:98) P ε ) = Law ( Y ε | (cid:98) P ) . Then, setting ρ ε := inf { t ≥ Y εt = z } , denoting (cid:98) E ε the expectation under (cid:98) P ε andletting ξ εT be defined as the D´oleans-Dade exponential in (4.31) but with ( Y ε , B ) insteadof ( (cid:101) X ε , (cid:99) W ε ), we obtain P ( τ ε < t − t ) = (cid:98) E ε (cid:104) Z εT { τ ε Corollary 4.17. It holds lim x → ¯ x c ( x ) = T. Numerical results, financial interpretation and final remarks Here we present a numerical study of the PPSO problem that complements the the-oretical results obtained in the previous sections. Since we have studied the optimalsurrender boundary as a function of x , all the figures presented below are drawn in the( x, t ) plane. A sample path of the dynamics (2.1) with the corresponding sample pathsfor (2.4) (left plot) and (3.1) (right plot) are shown in Figure 1. The insolvency time τ † is indicated in both plots. 𝑥 𝑥 Figure 1. A sample path for A and R (left plot) and X (right plot) for T = 1, r = 0 . σ = 1, r G = 0 . δ = γ = β = 0 . α = 0 . , T ] with N +1 equally spaced time points. At each nodein the tree, we associate a value of the underlying process X and of the correspondingtime: that is, in the ( n, j )-node we have the couple ( n, x jn ). At the subsequent time-stepthe process can move to one of the two nodes ( n + 1 , x jn ± σ √ ∆) with ∆ := T /N , sothat the tree is recombining. If x jn > p jn of moving upwards from the n -th node is calculated as in [19] as p jn = 0 ∨ [1 ∧ ( + √ ∆ · π ( x jn ) / σ )]. If instead x jn ≤ n + 1 , 0) with probability one. We compute the numer-ical approximation of the value function ˜ v n ( x jn ) of the PPSO with the usual backwardrecursion, starting from ˜ v N ( x jN ) = h ( x jN ) for x jN ≥ 0. For any n < N , if x jn ≤ 0, then˜ v n ( x jn ) = h (0) = 1; if instead x jn > 0, then ˜ v n ( x jn ) = max { h ( x jn ) , E [˜ v n +1 ( X n +1 ) | X n = x jn ] } . Since the binomial-tree has recombining nodes, the evaluation of the continuationvalue E [˜ v n +1 ( X n +1 ) | X n = x jn ] reduces to the average of the payoff at the next two nodes.Notice that the regularity we have obtained for the value function v allows us, inprinciple, to obtain an integral equation for the optimal boundary (see [21] for someexamples). However, as the explicit form of the transition density of the process X is not known, solving such integral equation numerically would not be possible. Thismotivates our use of binomial-trees.Unless otherwise specified, in what follows we set T = 1, r = 0 . σ = 1, r G = 0 . δ = 0 . α = 0 . γ = 0 . β = 1 . 6. For the sake of realistic applications, T = 1 01 5 Figure 2. The optimal surrender regions and boundaries in the case x > x = x with ˆ c = c ( x ) > one decade and the values r , σ and r G are scaled accordingly. It is usefulto define x G := β + r G δ , since for X larger than x G the bonus on the policy reserveis active (cf. (2.3)). For such parameter values x α = 2 . x = 4 . 11 (see (3.3) and(4.4) respectively), therefore we are in the setting of Case 1 (see (4.5)), and x G = 3 . S and the boundary c ( · ) (see (3.31)and (4.7) respectively) on the ( x, t ) plane. As expected, the numerics confirm thebehaviour of c ( · ) described in Proposition 4.10 and Theorem 4.16. Notice in particularthat [0 , T ) × { x α } ⊂ C as proved in Proposition 4.3. In a real-world application thecontract is determined by specifying the parameter α that links the value at time zero ofthe reference portfolio A and the initial reserve R (see (2.1) and (2.4)). Therefore, froman actuarial/financial perspective the relevant initial condition is X = x α . Startingat time 0 from x α , the policyholder exercises the SO as soon as X hits the boundary c ( · ). It is worth noticing that in Figure 2 the solvency requirement is always fulfilled ifthe policyholder exercises the SO optimally (i.e., τ ∗ < τ † ). This is due to the fact that x > x α = 0 . 2. There we have x = 0 with c (0+) > X hits zero prior to time c (0+).There is a natural interpretation of the results. On the one hand, when X is close to 0, c ( A, R ) = r G and g ( A, R ) = R (see (2.3) and (2.5) respectively); thus, the policy reservegrows at a rate r G which is lower than the discount rate r used in (2.7). So, a persistentlylow value of X will induce the policyholder to surrender, in order to avoid a furthererosion of the present value of the reserve. On the other hand, for values of X largerthan ¯ x , the policy reserve grows at a rate greater than r , due to the bonus mechanismin (2.3). In this case the policyholder has no incentive to surrender the contract. Thepeculiar shape of the continuation region near x α can be explained as follows. Thevalue x α is the critical value at which the participation bonus in the intrinsic value ofthe policy becomes active (see (2.5)). Then, if X = x α , the investor delays the surrenderwith a view to possibly receiving the bonus. Moreover, in a neighbourhood of x α the ARTICIPATING POLICIES WITH SURRENDER OPTION 27 drift in the dynamic (3.10) is positive (hence pulling towards the bonus), so that thepolicyholder has an incentive to wait also if X < x α but not too small. When X > x α the participation bonus in the policy’s intrinsic value is active and can be collected bythe policyholder upon immediate surrender. However, when X > x G the bonus on thepolicy reserve’s growth rate is also active (see (2.3)) and creates an incentive to wait.The effect of discounting will eventually push the policyholder to exercise the SO (unless X exceeds also ¯ x as discussed above). This trade-off between continuing to stay in thecontract and exercising the surrender option when X ∈ ( x α , ¯ x ) produces the peculiarhump-shaped stopping region.Figure 3 shows the optimal surrender boundary c ( · ) for several values of the parameter α , or equivalently x α (see (3.3)). The possible presence of a portion of continuationregion below the cusp (as in Figure 2) and the depth of the cusp depend on severalfactors, including the value of α . Large values of α push X = x α towards zero sothat the chances of benefiting from the bonus on the policy reserve’s growth rate areslim and the policyholder will prioritise the participation bonus in the intrinsic valueof the policy. This widens the area above the cusp until the continuation region (3.30)becomes completely disconnected (see the curves in Figure 3 for x α ∈ { . , . } ). As α increases this mechanism is reversed: the activation threshold x α of the participatingbonus and x G of the bonus on the reserve’s growth rate become closer. Then the areaabove the cusp shrinks as x α approaches ¯ x from the left (see the curves in Figure 3for x α ∈ { . , , . } ). In the limit we arrive at the case of x α ≥ ¯ x (cf. (4.6)), whichis also illustrated in Figure 3 (see the curves for x α = ¯ x = 4 . x α = 4 . X has already exceeded ¯ x , so that the reserve is already growing at a rate higher thanthe discount rate. In that case the policyholder’s waiting strategy is aimed at collectingboth a large reserve and the participating bonus. The exercise of the SO in this settingis only optimal when X is sufficiently small and it is purely triggered by the discounting.The shape of the stopping set in our problem is remarkable and almost unique in theexisting literature on optimal stopping (besides, it was never observed in the context ofparticipating policies with surrender options). To the best of our knowledge the onlyother example of a similar geometry in a finite horizon (parabolic) setting is containedin [11]. However, the problem studied in [11] concerns the optimal prediction of themaximum of a Brownian motion with drift. Hence, the similarities in the stopping ruleappear to be a mere coincidence.In Figure 4 we study the sensitivity of the optimal surrender boundary c ( · ) withrespect to γ (left plot) and to r G (right plot). We remark that γ only affects the intrinsicvalue of the policy (see (2.5)) but it does not affect either the reference portfolio A orthe policy reserve R . As γ increases, the participating bonus increases and countersthe effect of discounting. So the policyholder is inclined to stay in the contract abit longer to see if the the bonus mechanism on the reserve will also be activated.Likewise, as r G increases the policyholder has progressively more benefits from stayingin the contract and the SO becomes less appealing. In both situations the incentive tosurrender the contract decreases and the continuation region expands. As a result theoptimal boundary c ( · ) is pushed upwards in our plots.We conclude the section by analysing the impact of the surplus distribution mech-anism and of the minimum guaranteed on the value of the policy and on the value ofthe embedded SO. This is done by comparing the value of the PPSO to the value of itsEuropean counterpart (i.e., with no SO). We collect in table 1 the value V of the PPSO(see (2.7)), the value V E of the contract without SO (see (2.9)) and the value V opt of the SO (see (2.8)). As in [14], we consider the following three scenarios depending 01 5.5 Figure 3. The optimal surrender boundary as a function of x α . Here¯ x = 4 . x G = 3 . 1. For x α ∈ { . , . , . } we omit the plot of therightmost portion of the boundary. This is done for clarity of expositionand it is in line with the financial application: since X = x α stoppingshould occur when X leaves the bell-shaped continuation set around x α . 𝑥𝑥 Figure 4. Sensitivity of the optimal surrender boundary c ( · ) with re-spect to γ (left plot) and to r G (right plot). For γ = 0 . r G = 0 . X = x α stopping should occur when X leaves the bell-shaped continuation set around x α .on the level of participation in the returns generated by the reference portfolio: low ( δ = γ = 0 . β = 3 . medium ( δ = γ = 0 . 25 and β = 1 . high ( δ = γ = 0 . β = 0 . δ and γ , the less the policyholder participates in the reserve and terminalbonuses. Moreover, the higher the value of the target buffer ratio β , the smaller is thesurplus that the policyholder receives. The value of V E is evaluated by using the same ARTICIPATING POLICIES WITH SURRENDER OPTION 29 binomial-tree method described above, without the complication of the optimisationwhich is necessary at each node in the PPSO. As expected, V is always greater than V E and their difference is equal to V opt .Spread Scenario V V E V opt r − r G = 0 . r − r G = 0 . r − r G = 0 . Table 1. The values V of the PPSO, V E of the contract without theoption of an early surrender and the value V opt of the SO.On the one hand, the value of the european contract V E is below par in the low scenario and, if the spread is high, also in the medium scenario. In those cases, theminimum guaranteed rate r G is much smaller than the market risk-free rate and the levelof participation is also relatively small. So the policy would not be financially appealingto an investor when compared, for example, to bond investments. The appeal howevermay stem from the fact that the initial reserve R will have a higher value than the fairprice V E of the contract (see Table 1). On the other hand, the value V of the PPSOis always at or above par due to the American-type option embedded in the contract(at par the SO is immediately exercised). Contract values V and V E increase movingfrom low towards high scenario while, at the same time, the value of V opt decreases.This shows that the incentive to exercise the SO is reduced by higher participation ofthe investor in the bonus mechanisms. On the contrary, as r − r G increases the contractvalues decrease, whereas V opt increases. This is in line with the intuition that the higherthe spread, the less the contract is profitable for the policyholder, hence creating a bigincentive to exercise the SO. Appendix Proof of Lemma 4.2. Here we borrow arguments from the proof of [8, Thm. 1]. Tokeep a simple notation, in what follows we set X t = X x α t everywhere. From the equality | X t − x α | = (cid:90) t sign( X s − x α )d X s + L x α t ( X ) , we deduce that L x α s ∧ ρ ε ( X ) = (cid:18) | X s ∧ ρ ε − x α | − (cid:90) s ∧ ρ ε sign( X u − x α )d X u (cid:19) = (cid:18) | X s ∧ ρ ε − x α | − (cid:90) s ∧ ρ ε sign( X u − x α ) σ d W u − (cid:90) s ∧ ρ ε sign( X u − x α ) π ( X u )d u (cid:19) . Notice that π ( · ) is bounded on [ x α − ε, x α + ε ] by a constant c π,ε > 0. Moreover, since | X s ∧ ρ ε − x α | ≤ ε , then 1 ≥ ε − p | X s ∧ ρ ε − x α | p , with p > 0. Taking expectation and usingthese two observations we get E x α (cid:2) L x α s ∧ ρ ε ( X ) (cid:3) = E x α [ | X s ∧ ρ ε − x α | ] − E x α (cid:20)(cid:90) s ∧ ρ ε sign( X u − x α ) π ( X u )d u (cid:21) ≥ E x α [ | X s ∧ ρ ε − x α | ] − c π,ε E x α [ s ∧ ρ ε ] ≥ ε − p E x α (cid:2) | X s ∧ ρ ε − x α | p (cid:3) − c π,ε s. The expectation of the absolute value above can be estimated using the integral formof the SDE for X and the inequality | a + b | p ≥ p | a | p − | b | p , for all a, b ∈ R (see, [17, Ch. 8, Sec. 50, p. 83]). That is, E x α (cid:2) L x α s ∧ ρ ε ( X ) (cid:3) ≥ ε − p E x α (cid:34)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s ∧ ρ ε π ( X z )d u + σW s ∧ ρ ε (cid:12)(cid:12)(cid:12)(cid:12) p (cid:35) − c π,ε s ≥ ε − p (cid:40)(cid:16) σ (cid:17) p E x α (cid:104) | W s ∧ ρ ε | p (cid:105) − E x α (cid:34)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s ∧ ρ ε π ( X u )d u (cid:12)(cid:12)(cid:12)(cid:12) p (cid:35)(cid:41) − c π,ε s (5.1) ≥ ε − p (cid:26)(cid:16) σ (cid:17) p E x α (cid:104) | W s ∧ ρ ε | p (cid:105) − c pπ,ε s p (cid:27) − c π,ε s. Now, Burkholder-Davis-Gundy inequality and Doob’s inequality, imply that there existsa positive constant c p such that E x α (cid:104) | W s ∧ ρ ε | p (cid:105) ≥ c p E x α (cid:104) ( s ∧ ρ ε ) p (cid:105) ≥ c p E x α (cid:2) { s<ρ ε } (cid:3) s p = c p s p (cid:0) − P x α ( ρ ε ≤ s ) (cid:1) . Inserting the last inequality in (5.1) we get E x α (cid:2) L x α s ∧ ρ ε ( X ) (cid:3) ≥ c s p − c (cid:16) s + s p + P x α ( ρ ε ≤ s ) s p (cid:17) , for some suitable positive constants c = c ( ε, p ) and c = c ( ε, p ). Since p ∈ (0 , p ∈ ( , s ↓ E x α (cid:2) L x α s ∧ ρ ε ( X ) (cid:3) ≥ (cid:96) ( s ) with (cid:96) ( s ) ∼ s p (1 − P x α ( ρ ε ≤ s )) . Hence, the claim follows from P x α ( ρ ε ≤ s ) ↓ s → (cid:3) Acknowledgment : The authors were financially supported by Sapienza Universityof Rome, research project “ Life market: a renewal boost for quantitative managementof longevity and lapse risks ”, grant no. RM11916B8953F292. T. 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Stabile: Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Fi-nanza, Sapienza-Universit`a di Roma, Roma, Italy E-mail address ::