Dynamic programming and efficient hedging for unit-linked insurance contracts

Transcription

1 Dynamic programming and efficient hedging for unit-inked insurance contracts Johannes Morsing Johannesen Thomas Møer PFA Pension PFA Pension Sundkrogsgade 4 Sundkrogsgade 4 DK-2100 Copenhagen Ø DK-2100 Copenhagen Ø Denmark Denmark e-mai: jmj@pfa.dk e-mai: thm@pfa.dk Abstract. This paper considers hedging of the combined insurance and financia risk in a basic unit-inked insurance contract by using dynamic programming. The contracts studied specify a payment to the poicy-hoders at a given time conditiona on surviva of the poicy-hoders. The payment is inked to the vaue of a traded stock. The insurance company can invest in this stock and in a savings account. Due to the mortaity risk invoved, the financia market is incompete such that a perfect hedge is not possibe. We determine optima sef-financing investment strategies which minimize the probabiity of shortfa, i.e. the probabiity that the capita avaiabe at the time of payment is ess than the integrated caim. We compare our resuts with resuts obtained in the iterature via quadratic hedging approaches. In genera, ony existence resuts are avaiabe for the case of incompete markets using the shortfa probabiity criterion. Hence, the dynamic programming approach has been appied. One contribution of the paper is an understanding of soutions for the shortfa probabiity in the one-period case. In addition, computationa procedures (dynamic programming agorithms) for the genera discrete time muti-period and muti-poicy-hoder case are deveoped. The procedures expoit discrete properties of the probem. Thus, discretization with respect to capita and investment is avoided. Key words: shortfa probabiity, expected shortfa, sef-financing strategy, unit-inked insurance, incompete market, discrete time dynamic programming JEL Cassification: G10. Mathematics Subject Cassification (2000): 62P05, 91B28. This version: May 6, 2009

2 1 Introduction Unit-inked ife insurance contracts differ from traditiona insurance contracts in that benefits (and possiby premiums) are inked directy to the vaue of a unit of some investment portfoio. We study unit-inked contracts in their most basic form, which specify a ump sum payment (to the poicy-hoders) determined as a function of the vaue of a singe stock at a given future time conditiona on surviva. It is assumed that the insurance company can invest in the same stock and in a savings account with a risk free interest. The probem is then to determine optima sef-financing investment strategies for the company. The integrated financia and insurance caim cannot be hedged perfecty since we are deaing with an incompete market, see Møer and Steffensen (2007) and references therein. We study the shortfa probabiity criterion (and simiar criteria) for the optimization probem and compare these resuts with previous resuts on quadratic hedging that have been obtained in the iterature. Quadratic hedging has been appied in ife insurance, see Møer (2001a), where the above probem with unit-inked insurance contracts is considered in a discrete time setting, and Møer (2001b), where more genera payment processes in a continuous time setting are treated. For further deveopments and appications aowing for systematic mortaity and ongevity effects, see Dah and Møer (2006) and Dah, Mechior and Møer (2008). When studying risk-minimizing strategies, a key quantity is the so-caed cost process, defined as the current vaue of the strategy reduced by trading gains. A strategy is said to be risk-minimizing if it, at any time during the term of the poicy, minimizes the expected squared vaue of a future (discounted) costs. One advantage of the quadratic hedging approach is that it is anayticay tractabe and provides soutions that agree we with intuition, e.g. for the above basic probem with unit-inked insurance. Moreover, as the above references show, it has proven appicabe for advanced appications. However, one possibe drawback of quadratic hedging is that the hedging error is symmetric, i.e. redundant capita is punished in the same manner as ack of capita. A different approach is to appy so-caed efficient hedging methods, see Fömer and Schied (2002) for a detaied account in the discrete time case; continuous time probems have been studied in Fömer and Leukert (1999, 2000). Within the efficient hedging approach, quantie hedging adopts the shortfa probabiity as the object for the minimization. This amounts to minimizing an expected hedging error, where the hedging error is 1 if the capita is smaer than the caim and 0 otherwise. This criterion may be modified to the case of expected shortfa, where the hedging error is defined as the deficit in case of ack of capita and 0 otherwise. The methodoogy used in the iterature is typicay based on the Neyman-Pearson Lemma from statistica test theory, combined with martingae measures and appropriatey chosen knock-out options. Ceary, efficient hedging differentiates between redundant capita and ack of capita as opposed to quadratic hedging. However, the mathematica tractabiity is typicay ost. Methods for obtaining optima strategies by efficient hedging in compete markets have been derived, whereas typica resuts ony provide existence proofs in case of incompete markets such that the approach is not readiy appicabe for our use. Instead we appy dynamic programming methods. Dynamic programming can be appied in virtuay any probem with a sequence of decisions to be taken. Numerous appications appear in economics, see e.g. Stokey and Lucas (1989). Life insurance aso benefits from dynamic programming, in particuar in a continuous time setting through the we-known HJBequation, see e.g. Björk (2004). A more advanced mathematica treatment can be found in e.g. Yong and Zhou (1998). The HJB-equation re-appears in many stochastic contro 1

3 probems in insurance, e.g. for optima consumption and investment combined with ife insurance, see Schmidi (2008) and references therein. The paper is organized as foows. In Section 2, the one-period probem for hedging basic unit-inked contracts is stated. Soey from an agebraic point of view, anaytica soutions for strategies and shortfa probabiities are presented for the case with a singe poicy-hoder and a singe period. These soutions do not reay contribute to the overa understanding of the structure of the soutions. However, reaizing that hedging the caim for a any number of (surviving) poicy-hoders imposes a inear reation between initia capita and the number of stocks to be purchased, we can expain the one-period resuts and a basis for muti-period considerations is estabished. Section 3 contains a brief introduction to the finite time horizon dynamic programming agorithm providing a backward recursion procedure for a probem with sequentia decisions and costs in each step, speciaized to costs at the termina time ony. The presentation draws upon Hernández-Lerma and Lasserre (1995). For a comprehensive review on discrete time dynamic programming, see aso Bertsekas and Shreve (1978). In Section 4, computationa procedures for the genera muti-period and muti-poicy-hoder case of the basic unit-inked contract are deveoped using dynamic programming. First, the basis for a naïve discretization approach is outined. We refer to this method as the brute force approach, due to its computationa requirements. The brute force approach is mainy reevant for verification of aternatives. Then, with the observations from Section 2 in mind, we expoit that in the one-period case the minimum expected shortfa probabiity is a piecewise constant function of the initia capita. Moreover, this property propagates in the backward recursion in the dynamic programming agorithm. These observations ead to an agorithm with the shortfa probabiity criterion which is soey based on discrete properties of the probem and not a discretization of a continuous probem. We refer to the procedure as the discrete properties approach. Section 5 contains a number of numerica exampes. 2 The mode and main probem We consider a portfoio of unit-inked ife insurance contracts in a discrete time setting, where the sum insured, which may depend on the vaue of some stock, is payabe at a fixed time conditiona on surviva of the poicy-hoders. We first introduce the financia market and define the basic hedging criterion. In addition, we sove the probem in the one-period case. 2.1 The financia market and trading strategies Consider a financia market consisting of a stock S and a savings account B. We denote by S t the vaue of the stock at time t = 0,1,...,T, where T is a fixed finite time horizon. Formay, the stock price process and a other processes introduced in the foowing are defined on some probabiity space (Ω, F,P) equipped with a fitration IF = (F t ) t {0,1,...,T }. We assume that the underying price process S = (S t ) t {0,1,...,T } can be traded in the financia market in addition to a savings account with price process B given by B t = (1 + r) t. Thus, the savings account pays a fixed interest r during each time period. We mainy work with the discounted price processes S = S/B and B = B/B, where we have used the savings account as numeraire. A trading strategy is a two-dimensiona process ĥ = (h0,h 1 ), where h 1 t is the number of 2

4 stocks hed at time t and where h 0 tb t is the amount deposited in the savings account. More precisey, h 1 t is the number of stocks chosen at time t 1 and hed unti time t. This means that h 1 t needs to be determined based on the information avaiabe at time t 1. The (undiscounted) vaue at time t of the portfoio ĥt = (h 0 t,h1 t ) is given by V t (ĥ) = h0 t B t + h 1 t S1 t, (2.1) and the discounted vaue is Vt (ĥ) = Vt(ĥ)/B t. We restrict to sef-financing strategies, i.e. strategies ĥ, where the vaue process V (ĥ) has dynamics given by V t (ĥ) = V t 1 (ĥ) + h1 t S t, (2.2) and where St = S t S t 1. Thus, the vaue process at time t depends on the strategy via the initia vaue V 0 (ĥ) and the number of stocks hed h1 1,...,h1 t. We take S t = (1+ρ t )S t 1, such that S t = S t 1(1 + ρ t )/(1 + r). (2.3) In the present paper, we work with the so-caed binomia mode, where the random variabes ρ 1,...,ρ T are i.i.d., and where ρ 1 {a,b} and 0 < P(ρ 1 = b) = p < 1. In addition, we assume that a < r < b. It is convenient to introduce the quantities ρ t, defined by 1 + ρ t = (1 + ρ t )/(1 + r), which attain the vaues ã = a r 1+r and b = b r 1+r. Thus, St = S t 1 (1+ ρ t). For a thorough treatment of the binomia mode, see e.g. Piska (1997). It is often reevant to introduce some additiona constraints on the strategies. In the foowing sections, we typicay require that the sef-financing strategies are chosen such that the vaue process remains non-negative. In particuar, this condition ensures that one cannot use so-caed doubing strategies. In Section 2.4, we consider the one-period case without this non-negativity constraint. 2.2 The iabiity and the choice of criterion We study a portfoio of n poicy-hoders and denote by Y t the number of survivors at time t. The poicy-hoders remaining ife-times are modeed via some random variabes T 1,...,T n, which are assumed to be independent of the traded price process S. The discounted iabiity payabe at time T is given by H = Y T f(s T )B 1 T = Y T f(s T ), (2.4) where f is some measurabe function. Thus, the sum payabe upon surviva to T is assumed to be a function of the termina vaue of the stock ony. We assume that the insurance company receives some premium at time 0, which is invested in the financia market via a dynamic trading strategy ĥ in order to hedge the risk associated with the iabiity H. A iabiity H payabe at time T is said to be attainabe if there exists a sef-financing strategy ĥ such that the termina vaue of the investments VT(ĥ) coincides with the iabiity H amost surey, i.e. if V T (ĥ) = H, P-a.s. In this case, we say that the iabiity is hedged perfecty. A sef-financing strategy ĥ is said to be a super-hedging strategy for H if V T (ĥ) H, P-a.s., i.e. if the vaue process at time T exceeds the iabiity H with probabiity 1. Since the iabiity (2.4) is assumed to depend on the number of survivors at time T, which is considered to be a non-traded risk, it is in genera not possibe to hedge the 3

5 iabiity perfecty, see Møer (2001a). We therefore study the criterion of minimizing the probabiity of a shortfa, i.e. the probabiity of having insufficient capita at time T, where the iabiity is payabe. For a sef-financing strategy ĥ and a iabiity H, the shortfa probabiity is given by [ ] P sf (ĥ) = P(VT(ĥ) < H) = E 1 {VT (ĥ)<h}. (2.5) In addition, we study the so-caed expected shortfa given by E sf (ĥ) = E [ ( H V T (ĥ) ) + ], (2.6) which measures the expected deficit associated with the iabiity H and the trading strategy ĥ. We determine dynamic sef-financing strategies ĥ that minimize Psf(ĥ) and Esf(ĥ) by use of dynamic programming methods in a muti-period setting. First, however, we study the probem in the one-period case, where dynamic programming is not needed. The study of the one-period probem aready aows for some important observations. 2.3 The roe of the information avaiabe It is of reevance to study the impact of the amount of information that is avaiabe to the insurer. We denote by IG the natura fitration associated with the traded price processes and et IH be the natura fitration associated with the process for the number of survivors Y. One exampe is the natura situation, where the insurance company observes the current number of survivors at each time t {0,1,...,T }. This is the case where the process Y is adapted to the fitration IF. For exampe, we coud define IF = (F t ) t {0,1,...,T } by F t = G t H t = σ(g t H t ), such that we have access to the fu information from G t and H t at time t. Thus, the insurance company can base investments at each time on exact information about the current number of survivors. Another exampe is the case, where the insurance company receives information about the financia market but is restricted to information about the number of poicy-hoders at time 0 and the fina number of survivors at time T. At the intermediate times t = 1,2,...,T 1, the insurance company does not observe Y t. This can for exampe be modeed by working with a fitration IF defined by F t = G t for t < T and F T = G T H T. We compare the minimum obtainabe shortfa probabiity in these two cases and show in an exampe in Section 5.2 that the optima strategies based on the fitration IF may indeed ead to ower shortfa probabiities than the ones based on IF in the genera case. This underines the importance of the choice of the fitration and shows that the insurance company in genera wi benefit from adapting their investment strategies to the current number of survivors. 2.4 Minimizing the shortfa probabiity in the one-period case We study the probem of minimizing (2.5) in the case where T = 1 via direct cacuations. It foows from (2.2) that V1 (ĥ) = V 0 + h1 ρ 1S0. Thus, the termina vaue V 1 of the strategy depends on the initia vaue of the portfoio v = V0, the initia vaue of the stock s = S0, the number h1 1 of stocks purchased at time 0 and the reative change ρ 1 in the discounted vaue of the stock. Since we are working with the binomia mode, where ρ 1 {ã, b}, we see that P(V 1 = v + h 1 1 bs ) = p = 1 P(V 1 = v + h 1 1ãs ). 4

6 We denote by F Y1 (y) = P(Y 1 y) the distribution function for Y 1, and we et Y = {0,1,...,n}. Then, we may write the expected shortfa probabiity on the form ] [ ] E [1 = E P(V {V1(ĥ)<H} 1 (ĥ) < y f(s 1))F Y1 (dy) Y ( ) = p1 {v +h 1 1 bs <y f(s + bs )} + (1 p)1 {v +h 1 1ãs <y f(s +ãs )} F Y1 (dy). Y Here, the second equaity foows by conditioning on the two possibe outcomes at time 1 for the stock price process. If we in addition assume that the remaining ife-times are i.i.d. with one-period surviva probabiity 1 p x = e µx, the number of survivors at time 1 is binomiay distributed with parameters (n, 1 p x ). Thus, the minimum shortfa probabiity P sf,min is given by P sf,min = min h 1 1 [ n y=0 ( p1 {v +h 1 1 bs <y f(s + bs )} +(1 p)1 {v +h 1 1ãs <y f(s +ãs )} ) n! (n y)!y! 1 p x y (1 1 p x ) n y]. (2.7) Finding P sf,min amounts to minimizing a sum of 2(n+1) terms, where each term invoves the indicator function of the event v +h ρs < f(s + ρs )y, ρ {ã, b}, and y = 0,1,...,n. One poicy-hoder In the case of a singe poicy-hoder with y = 1 and with P Y1 (1) = 1 p x = 1 P Y1 (0), the equation (2.7) reduces to: P sf,min = min h 1 1 [ (p1 {v +h 1 1 bs <0} + (1 p)1 {v +h 1 1ãs <0} )(1 1p x ) (2.8) + (p1 {v +h 1 1 bs < f(s + bs )} + (1 p)1 {v +h 1 1ãs < f(s +ãs )} ) 1p x ]. If we further impose the natura assumptions ã < 0, b > 0, and s > 0, we see from (2.8) that P sf,min = 0, if the foowing four inequaities are met by the optima vaue h opt of h 1 1 : h opt bs v h opt v ãs h opt f(s + bs ) v bs h opt f(s +ãs ) v ãs (b0), (a0), (b1), (a1). (2.9) Each of these criteria represents a possibe combined outcome of the stock price process and the number of survivors. One can interpret the 4 criteria (2.9) as foows. The first one, referred to as criterion b0, is reated to the first term in (2.8) and represents the situation where the poicy-hoder does not survive and the stock jumps upwards. In this case, the inequaity impies that the capita requirement 0 is hedged, i.e. the capita is non-negative at time 1, and this is sufficient to super-hedge the iabiity in this scenario. Simiary, the second criterion (criterion a0) is the case where the stock jumps downwards and the poicy-hoder does not survive. The ast two criteria (criterion b1 and a1, respectivey) represent the case where the poicy-hoder survives and the stock jumps either upwards or downwards. In these outcomes, the conditions on h 1 1 are sufficient to require that the 5

7 vaue process at time 1 exceeds the iabiity. Further, we take f(s ) = f(s)/((1 + r) T ) > 0, where s = s(1 + r) T. One exampe is the case f(s) = max(k,s), K > 0. If f is stricty positive, we see that and f(s + ãs ) v ãs f(s + bs ) v bs < v ãs, > v bs. This shows that the probabiity of insufficient capita is zero for [ f(s h opt + bs ) v ; f(s ] + ãs ) v bs ãs, (2.10). If the condition f(s + bs ) v < f(s +ãs ) v bs ãs is not satisfied, there are severa cases, depending on the ordering of the r.h.s. in the four inequaities above, to be considered. It is noted that if the initia capita v is such that the condition if f(s + bs ) v < f(s +ãs ) v bs ãs f(s + bs ) v bs probabiity. < f(s +ãs ) v ãs is not satisfied, it is not possibe to obtain a zero shortfa h III II a0 a1 b1 b0 I v Figure 1: Optima amount of stock h versus start capita v shown as grey areas for a = 0.10, b = 0.15, r = 0, p = 0.7, f(s) = max(s,k), K = 100, S 0 = 100, and µ = 1. The approach outined above provides compete resuts regarding shortfa probabiities and the corresponding (non-unique) optima strategies. However, the considerations do not reay contribute much to the genera understanding of the overa structure of soutions. Moreover, they do not appear to be constructive when considering muti-period probems. The four inequaities representing criteria a0, a1, b0, and b1 in (2.9) can be transated into ines which characterize the optima soution. Indeed, the first inequaity eads to combinations of h and v such that h v, which gives a ower bound on the optima bs 6

8 number of stocks h. Anaogousy, the 2nd, 3rd, and 4th inequaities give upper, ower, and upper bounds, respectivey, represented by ines. The above ower and upper bounds on the number of stocks h as a function of the initia capita v given by ines and corresponding optima combinations of v and h (grey areas) are shown in Figure 1. The three areas, which represent optima combinations of h and v, are abeed I, II and III, respectivey. In addition, we have shown the 4 optimaity ines derived from the criteria (2.9). Area I represents combinations of initia capita v and number of stocks h, where, irrespective of the outcome of the stock and the number of survivors, the integrated caim is super-hedged. In area II, the case of 1 (and 0) survivors is hedged, provided that the stock jumps upwards. If the stock jumps downwards, no survivor is hedged (criterion a0). Finay, in area III, 1 (and 0) survivors are hedged (criteria b1 and b0) if the stock jumps upwards. If the stock jumps downwards, the capita becomes negative. 3 Dynamic programming with finite time horizon In this section, we reformuate the main probem in a dynamic programming framework with a finite time horizon. For a genera treatment of discrete time dynamic programming, see Hernández-Lerma and Lasserre (1995). Since we restrict to sef-financing strategies, we may focus on the component h 1 1 of the strategy (h0,h 1 ) and refer to this component as h = (h 1,...,h T 1 ). Further, denote by X t ( h) = (Vt ( h),s t,y t ) tr the vector-vaued process consisting of the current (discounted) vaue Vt ( h) of the investment strategy, the (discounted) vaue of the stock St and the current number of survivors Y t. In the genera case, this process can be observed by the insurance company. Moreover, the company is abe to contro the process Vt ( h) via the initia vaue V0 and the number of risky assets hed. We assume that h is a Markov contro, i.e. h t+1 = h t+1 (Vt ( h),s t,y t ). Thus, the process X t ( h) is a Markov process. The goa is to minimize the finite horizon performance criterion given by J h(t,x) [ )] ( ] = E t,x c T (X T ( h) = E [c T X T ( h)) Xt ( h) = x, (3.1) for a t = 0,1,...,T, for x = (v,s,y) tr. As above, we study shortfa probabiity minimization with c T (v,s,y) = 1 {v<y f(s)} and the expected shortfa, where c T (v,s,y) = (y f(s) v) +. We denote by J opt the optima vaue function J opt (x) = inf J h(0,x), (3.2) h H where H is the set of a admissibe strategies. This probem can be soved by a simpified version of the genera dynamic programming theorem, see Hernández-Lerma and Lasserre (1995, Theorem 3.2.1), which provides an agorithm for finding both the vaue function J opt and an optima strategy h opt. In order to formuate this resut we introduce some additiona notation. We denote by Q(dy x,h) = P(X t+1 ( h) dy X t ( h) = x,h t+1 = h), the distribution of X t+1 ( h) given that X t ( h) = x and given the contro h t+1 = h. In addition, we introduce the sets H(x) of possibe vaues for the contro h t+1 given the 7

9 present state X t ( h) = x. For exampe, if we are working with the condition that the process V t ( h) may not attain negative vaues, this eads to a condition on the admissibe vaues for the number of stocks hed h. Finay, we denote by X the space of possibe vaues for the process X t ( h). We can now formuate the dynamic programming theorem. Theorem 3.1 Let J 0,J 1,...,J T be functions defined by and J T (x) = c T (x), (3.3) [ ] J t (x) = min J t+1 (y)q(dy x,h), (3.4) h H(x) X for t = T 1,T 2,...,0. Suppose that these functions are measurabe and that there exists a strategy h = {h 1,...,h T } such that h t+1 = h t+1 (x) H(x), t = 1,...,T, and h t+1 (x) attains the minimum in (3.4) for a x X and t = 1,...,T, i.e., J t (x) = J t+1 (y)q(dy x,h t+1 ). (3.5) X Then the strategy h = {h 1,...,h T } is optima and the vaue function J opt equas J 0, i.e., J opt (x) = J 0 (x) x X. (3.6) Proof. See Hernández-Lerma and Lasserre (1995). The main resut in the theorem is that the vaue function J opt (x) which is defined in (3.2) as the infimum with respect to a strategies of the performance criterion in (3.1), can be cacuated as J 0 (x) from the backward recursion (3.4). A further assumption, the socaed measurabe seection condition, ensures that we obtain a minimum, and not just an infimum in (3.4). The measurabe seection condition, see Hernández-Lerma and Lasserre (1995), adapted to the current presentation, can be stated as in the foowing assumption. Assumption 3.2 The mode and a given measurabe function w : X IR are such that [ ] ŵ(x) = w(y)q(dy x, h), x X, (3.7) inf h H(x) X is measurabe, and there exists a measurabe function g : X H satisfying g(x) H(x) for a x X, such that the function within brackets attains its minimum at g(x) for a x, i.e., ŵ(x) = w(y)q(dy x,g) x X, (3.8) X It is noted that we have confined ourseves to stationary Markov contro modes in that X, H, H(x), Q, and c t, t < T, are time-invariant (c t = 0 for t < T). 8

10 3.1 Reduced information on the number of survivors In order to investigate the roe of the amount of information avaiabe, we aso study the situation where the insurance company is restricted to information concerning the number of survivors in the insurance portfoio. This situation may be described by the process Xt ( h) = (Vt ( h),s t,yt )tr, where Yt := Y 0 for t < T and YT = Y T. This means that the company does not observe the current number of survivors during t = 1,...,T 1. An aternative approach is discussed beow. We consider the case from Section 2.3, where the insurance company ony receives information about the number of poicy-hoders at time 0 and the fina number of survivors Y T at time T. Such a state component for which the vaue is unknown at intermediate time points cannot be deat with directy by the dynamic programming agorithm, since it assumes fu information on a state components entering the contro. Generay, methods for imperfect state information in dynamic programming may be empoyed, see e.g. Bertsekas and Shreve (1978). However, if the number Y of survivors is not observed, it may be eiminated from the dynamic programming agorithm, if it is aso uncontroabe and independent of the other components. Thus, Y enters the dynamic programming agorithm through the cost function c T ony. Hence, assuming a fu state vector X = (Y,Z tr ) tr, where Z represents a state components except for Y, we adopt the modified costs with Y averaged out, c T (z) = c T (z,y)p T (dy). Y Here, P T is the probabiity distribution for Y T. Finay, we have the dynamic programming agorithm [ ] J t (z) = min J t+1 ( z)q(d z z,h), h H(z) Z where H(z) = {h H(z, y) y Y}, i.e. feasibe contros expressed through the observabe state component Z t, t = 0,1,...,T, ony. The restriction to information on the number of survivors is aso studied in case of stochastic surviva probabiities. For this appication, a standard Bayesian updating procedure is adopted with the surviva probabiities being Beta distributed with a uniform prior. 4 Computationa procedures 4.1 Introduction In this section, a computationa procedure for shortfa probabiity hedging is proposed for handing the genera case with an arbitrary number of periods and an arbitrary number of poicy-hoders. A simiar procedure for expected shortfa hedging is commented. We start out in Section 4.2 with a straightforward discretization with respect to capita and strategy (number of stocks). This approach is extremey computationay intensive and is thus unsuitabe for practica appications. However, due to its simpicity, it is usefu for verification of more advanced aternatives. We refer to the approach as the brute force approach. We proceed in Section 4.3 by observing that the vaue function for shortfa probabiity hedging is piecewise constant considered as a function of capita for given stock vaue (in each step). Aternativey formuated, the vaue function as a function of the capita can attain a finite number of different vaues ony in each step. We obtain a dramatic reduction 9

11 in the computation effort as compared to the above brute force approach by utiizing this property systematicay for propagating the point seection in the backward recursion in the dynamic programming agorithm and in the minimization within each time step. We refer to the new procedure as the discrete properties approach. It is important to note that the discrete properties approach does not provide a discretization of a continuous probem but rather utiizes discrete properties of the probem and thus eads to cacuations that are exact within the imitations of foating point arithmetics. Further optimization of the agorithms for arge scae probems is considered to be outside the scope of this paper. Naturay, it woud then be necessary to give up the property that cacuations be exact in the above sense by introducing resoution imits on capita and strategies. Moreover, one coud consider the vast iterature on sub-optima methodoogies, see e.g. Bertsekas (2005), possiby combined with the discrete properties approach. A detaied derivation of the discrete properties approach for shortfa probabiity hedging, see Section 4.3 and Appendix A, eads to a dynamic programming agorithm, see Proposition 4.1. For expected shortfa hedging, see Remark Brute force approach The straightforward way for soving the muti-period probem by dynamic programming is by introducing a discretization of capita and strategy. Irrespective of the criterion, e.g. shortfa probabiity or expected shortfa hedging, the vaue function as a function of v is treated as constant between discretization points for v. The basis for the brute force approach is outined beow. The method is mainy interesting for verification of and as a first step to more advanced approaches. Since the vaue of the stock ony takes a finite number of vaues, discretization is irreevant for the stock. We introduce: v = v min + v, = 0,1,...,L v, h = h min + h, = 0,1,...,L h, where (discounted) capita v is imited to the interva [vmin,v max ] and is discretized with resoution v = (vmax v min )/L v. Simiary, the strategy is imited to the interva [h min,h max ] with resoution h = (h max h min )/L h. For notationa convenience, we introduce s t,u := s 0 (1 + b) u (1 + ã) t u, t = 0,1,...,T, u = 0,1,...,t, (4.1) which is the (discounted) stock price at time t, given that the number of jumps upwards is u within time steps 1,2,...,t, and s 0 is the vaue of stock at time 0. Hence, for m = 0,1,...,L v, we have for t = 0,1,...,T 1, the dynamic programming equation corresponding to (3.4), [ y J t (vm,u,y) = min p k y (4.2) h {h 0,h 1,...,h L } h k=0 ( ) ] pj t+1 (vm + h bs t,u,u + 1,k) + (1 p)j t+1 (vm + hãs t,u,u,k), where s t,u is represented by u in the vaue function J t and p k y is the probabiity of k surviving poicy-hoders at the end of a time step, given that the number is y at the start of the step. It is noted that the J-vaues entering the sum in (4.2) are found by ocating the capita vaue intervas at the prior time t+1 (in the recursion) that contain v m+h bs t,u and v m+hãs t,u corresponding to the case where the vaue of the stock jumps up and down, respectivey. 10

12 4.3 Discrete properties approach for shortfa probabiity hedging Singe-period case We cacuate the shortfa probabiity after one period starting with y poicy-hoders, see aso Section 2.4. We have the shortfa probabiity P sf (h) = = y (p1 {v +h bs <k f(s + bs )} + (1 p)1 {v +hãs <k f(s +ãs )} )p k y (4.3) k=0 y (p1 {h<hb,k } + (1 p)1 {h>ha,k })p k y, k=0 where p k y is the probabiity of k survivors after one period starting with y survivors and h b,k = k f(s + bs ) v, k = 0,1,...,y, (4.4) bs h a,k = k f(s + ãs ) v ãs, k = 0,1,...,y. (4.5) We see from (4.3) that min h [P sf (h)] is determined by the ordering of the joint sequence consisting of both h a,k and h b,k, k = 0,1,...,y. Hence, we may change v without affecting min h [P sf (h)] as ong as the ordering of h a,k and h b,k, k = 0,1,...,y is not changed. The idea is essentiay to identify vaues v at (time T 1) at which the ordering of the h a,k s and h b,k s is changed. The ordering is changed for v -vaues for which h a,j = h b,k, j = 0,1,...,y and k = 0,1,...,y, i.e. vaues where j f(s + ãs ) v ãs = k f(s + bs ) v, bs considered separatey for different pairs j,k. Thus, we need to focus on vaues for v on the form j b f(s + ãs ) kã f(s + bs ) b ã,t 1 =: vjk,j = 0,1,...,y, k = 0,1...,y. (4.6) It is noted that the above outined procedure for seection of v -vaues according to (4.6) can aternativey be obtained as a specia case of the procedure for an arbitrary step in the muti-period probem, see Appendix A.1.2.,T 1,T 1 We introduce the stricty increasing sequence v, = 0,1,...,L T 1 where v0 <,T 1,T 1,T 1 v1 < < vl T 1 are obtained by rearrangement of vjk, j = 0,1,...,y,k = 0,1...,y and skipping mutipes. Hence, L T (y + 1) 2 (equaity corresponding to no mutipes). In order to dea propery with intervas and not ony points between intervas, we add a point corresponding to a capita vaue greater than any of the above,t 1 vaues. We choose to add vl T 1 +1 =. Now we are abe to imit the number of capita vaues to be considered in vaue functions, i.e., J T 1 (v,s,y) =,T 1,T 1 L T 1 +1 =1 1 {v,t 1 [v 1,T 1 ;v )} J T 1(ζ,T 1,s,y), (4.7) where ζ,t 1 [v 1 ;v ) can be chosen freey. The singe step (or first step) determination of capita vaues is a specia case of the genera procedure depicted for an 11

13 arbitrary step in Figure 2. The input points appearing on the vertica axis for the first step are: v,t,u+1 = f(s + bs ), = 0,...,y, and v,t,u = f(s + ãs ), = 0,...,y. Muti-period case We present a proposition for the muti-period case mainy based on the observation that the imitation of capita into a finite number of vaues in the one-period case propagates to the subsequent time steps in the backward recursion (not necessariy the same capita vaues and same number of capita vaues). The resuts are formuated as Proposition 4.1, and Remarks beow. Proposition 4.1 The minimum shortfa probabiity P sf,min = min Prob( f(s T )Y T > VT ), given that the initia capita is vm,0,0, is J 0 (vm,0,0,n), which is determined from the subsequent dynamic programming agorithm. The corresponding optima number of stocks to be purchased at time 0 is the h-vaue by which J 0 (vm,0,0,n) is attained. The dynamic programming agorithm: J T (vm,t,u,y) = 1 {y>m}, (4.8) where vm,t,u = m f(s T,u ), y = 0,1,...,n, m = 0,1,...,n, and u = 0,1,...,T. For t = 0,1,...,T 1, we have: [ y J t (v,u,t m +,y) = min h H (t+1,u) a (vm,u,t ) L t+1,u+1 1 (t+1,u+1) {h=h b, (vm,u,t )} =0 L t+1,u =0 1 (t+1,u) {h=h a, (vm,u,t )} (t+1,u) H b (vm,u,t ) ( ( k=0 pj t+1 (v,t+1,u+1 pj t+1 (v,t+1,u ( (t+1,u+1) b,m p k y [1 {h Hb (v,u,t m )} (4.9) ),k) + (1 p)j t+1 (v,t+1,u+1 ( a,m (t+1,u) ),k) ),k) + (1 p)j t+1 (v,t+1,u,k)) ], for m = 0,1,...,L t,u with v,t,u m the stricty increasing sequence corresponding to,t+1,u bv m, ˆm = m ãv,t+1,u+1 ˆm, m = 0,1,...,L t+1,u, ˆm = 0,1,...,L t+1,u+1, b ã v,t,u and a, (v ) = v,t+1,u v, = 0,1,...,L t+1,u, h (t+1,u) h (t+1,u+1) ãs t,u b, (v ) = v,t+1,u+1 v, = 0,1,...,L t+1,u+1, bs t,u (t+1,u) a,m (t+1,u+1) b,m = max{ v,t+1,u = max{ v,t+1,u+1 v,t,u () = v,t,u, v,t,u m v,t,u m + h (t+1,u+1) b, + h (t+1,u) a, (vm,t,u )ãs t,u}, (vm,t,u ) bs t,u}, H a (t+1,u) (v ) = {h (t+1,u) a,0 (v ),h (t+1,u) a,1 (v ),...,h (t+1,u) a,l t+1,u (v )}, H (t+1,u) b (v ) = {h (t+1,u+1) b,0 (v ),h (t+1,u+1) b,1 (v ),...,h (t+1,u+1) b,l t+1,u+1 (v )}, H (t+1,u) b (v ) = H (t+1,u) b (v ) \ H a (t+1,u) (v ). 12

14 Proof: see Appendix A. Remark 4.2 An anaogous dynamic programming agorithm can be derived for the expected shortfa criterion. The capita vaue point seection is simiar, whereas the use of constant J-functions between seected points is repaced by ineary varying J-functions. Remark 4.3 According to the proof, see Appendix A.1.2, we have L t,u + 1 (L t+1,u + 1)(L t+1,u+1 + 1) with equaity in case of no mutipes. Note that the number of points increases exponentiay. Hence, it is is highy advantageous with regard to performance to carry out a screening after cacuation of J(v,t,u m ), m = 0,1,...,L t,u, ony keeping v,t,u m where the J-function as a function of v is discontinuous. After the screening (and renumbering), a reduced vector is kept for the backward recursion for the foowing dynamic programming step. Remark 4.4 If the non-negativity constraint on capita, see Section 2.1, is imposed, this is refected in corresponding constraints on the strategy h. For each h-vaue, it is checked that the impied capita in the previous step is non-negative, both for jumps up and down for the stock vaue. If this condition is not satisfied, the h-vaue is discarded. Accounting for the number of survivors Y, too, in the capita vaue points seection procedure might seem reevant since fewer survivors impy fewer points to be incuded. However, we do not know Y (ooking ahead from time 0) and hence must account for the case that a poicy-hoders survive. Moreover, introducing Y into the point seection procedure compicates matters severey and has not been used. The procedure according to Proposition 4.1 is described in the foowing. For detais, see Appendix A. Points seection The procedure for deriving a set of capita vaues to be considered at time t based on the corresponding vaues for time t + 1 is iustrated in Figure 2. Here, one sees the capita v,t+1,u 0,...,v,t+1,u 2 and v,t+1,u+1 0,...,v,t+1,u 3 for u and u + 1 jumps upwards reaized at time t + 1. In this agorithm, v,t+1,u 0 = v,t+1,u+1 0 = 0. The vaues are determined by the points seection procedure in the dynamic programming agorithm. The three ines with sope 1/(ãs t,u) through one of the points with v = 0 and h = v,t+1,u 0,...,v,t+1,u 2 represent h t+1,u a, (v ), = 0,...,2, in Proposition 4.1. Simiary, the four ines with sope 1/( bs t,u) through one of the points with v = 0 and h = v,t+1,u+1 0,...,v,t+1,u+1 3, represent h (,t+1,u+1) (v ), = 0,...,3. The functions h (,t+1,u) and h (,t+1,u+1) determine b, the number of stocks h to be purchased at time t such that, given that the stock price jumps down and up, respectivey, we obtain the th capita vaue according to the points seection procedure for time t + 1 in the recursion. Next, we find the intersections between the ines corresponding to h (,t+1,u) a, intersection points, i.e. (v,t,u 0,...,v,t,u a, (v ) and h (,t+1,u+1) b, b, (v ). The v -vaues in the 8 ) = (v 0,...,v 8 ) in the current exampe, constitute the set of v -vaues to be considered in the dynamic programming agorithm for time t assuming u upward jumps of the stock up to this time. It is noted that since the v -vaue from the intersection generated by v,t+1,u+1 2 and v,t+1,u 2 coincides with the one from the intersection generated by v,t+1,u+1 1 and v,t+1,u 1, it is discarded (both intersections resut in v,t,u 2 = v 2 ony). Minimization within each time step We expoit that we ony need to consider the strategies h (,t+1,u+1) b, (v ), in case the stock 13

15 vaue jumps up, see Appendix A.1.3. The reason is that we then obtain capita vaues in the previous step that are seected according to the points seection agorithm. Simiary, we need ony consider h (,t+1,u) a, (v ), if the stock jumps downs. Hence, the strategies h (,t+1,u+1) b, (v ) and h (,t+1,u) a, (v ) cover a reevant strategies, and we may pick separatey from each group (inner sums in (4.9)). However, since it is unknown in advance whether the stock jumps up or down, capita vaues have to be ocated if we are considering h (,t+1,u+1) b, (v ), in case the stock jumps down (anaogousy for h (,t+1,u+1) b, (v )). Located capita vaues appear e.g. in the second J-function in the first inner sum in (4.9). h v,t+1,u+1 3 1/(ãs t,u) v,t+1,u v,t+1,u v,t+1,u 2 1/( bs t,u ) v,t+1,u 1 v,t+1,u+1 0 = v,t+1,u v 0 = v 0 v 1 0 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v = v,t,u, = 0,...,8 Exampe: (v,t+1,u+1 2,v,t+1,u 1 ) v,t,u 2,1 = v,t,u 3 (= v 3 ) Figure 2: Iustration of backward recursion for seection of capita vaues to be considered in the dynamic programming agorithm, Proposition Numerica exampes In this section, a number of numerica exampes are presented. Throughout, we impose the non-negativity constraint on capita. In Section 5.1, an exampe is presented considering a times and time steps with one poicy-hoder and four periods. It is demonstrated that the strategies for the first time step are equa for the shortfa probabiity and the expected shortfa hedging criteria and that the strategies corresponding to the same outcomes of the financia market (number of jumps upwards of stock) in subsequent time steps are quite simiar. Moreover, comparisons with the quadratic hedging criterion indicate that the above criteria impy that it is optima to buy more stocks than according to quadratic hedging. In Section 5.2, a number of exampes are presented focusing on the first time (time 0) 14

16 and the first time step (time step 1). Firsty, expected shortfa hedging with 8 or 9 poicy-hoders and one period is treated. It seems that adding a singe poicy-hoder may affect the optima strategy significanty. Secondy, the focus is on observing the number of survivors or not. An exampe cacuation (not reported herein) for shortfa probabiity hedging with two poicy-hoders and two periods indicates that the effect from observations is moderate. However, taking the surviva probabiity to be stochastic with a uniform prior, see Section 3.1, in an exampe with three poicy-hoders and four periods shows that the importance of observing the number of poicy-hoders can be high. Finay, an exampe with three poicy-hoders and three periods, covering both shortfa probabiity and expected shortfa hedging, is reported. It is demonstrated that the vaue for p is of great significance. Moreover, the resuts from shortfa probabiity and from expected shortfa hedging are quaitativey not very different but there are differences with regard to compexity of strategy versus start capita and the effect from optima investment reative to no action. Uness otherwise stated, the foowing parameter vaues are appied: reative (signed) stock jump down a = 0.10, reative stock jump up b = 0.15, probabiity of stock jumping up p = 0.7, risk-free interest rate r = 0, purey financia caim at time T, f(s T ) = max(s T,K) with K = 100, initia vaue of stock s 0 = 100, mortaity-intensity µ = 0.25, and initia capita v 0 = Exampe considering a times and steps We study the case with one poicy-hoder and four periods. The aim is to iustrate optima strategies and the corresponding shortfa probabiities. Depending on how the financia market deveops, the strategies and minimization resuts for a time steps are shown. The strategy aso refects the deveopment of the number of survivors. However, since we impose the non-negativity constraint on capita, zero survivors triviay ead to zero shortfa probabiity. Hence, we ony show resuts conditiona on surviva of the (one) poicy-hoder. Even though strategies are non-unique, we may set up rues for seection of a singe strategy in each time step, thus obtaining a unique strategy. Without much consideration on the choice of such a rue, we adopt the foowing principe: Out of a optima strategies, we seect the one which is numericay smaest. Vaue functions are found for a singe capita vaue in each time point, where this capita vaue foows from a forward cacuation as opposed to the backward cacuation in the dynamic programming agorithm. In Figure 3, optima strategies and minimum shortfa probabiities from shortfa probabiity hedging are shown. Optima strategies and corresponding shortfa probabiities are shown above and beow the tree nodes. It is noted that whereas the vaue of the stock can be represented by a recombining tree, the present trees are not recombining. At time points 0, 1, and 2, super-hedging does not occur. At time point 3, it occurs ony if the stock vaue has jumped up in a three of the preceding time steps (jump sequence 111). At time point 4, 7 jump sequences impy super-hedging. Out of 6 sequences with 2 jumps up, ony the 0011 and 0101 sequences give super-hedging, i.e. the two sequences with two jumps up that are ocated as ate as possibe. Moreover, a of the 4 sequences with 3 jumps up and the one sequence with 4 jumps up ead to super-hedging. Cacuations as in Figure 3 based on expected shortfa have been carried out but are not incuded here. However, it is mentioned that the two hedging criteria give the same 15

17 (0.00) (0.00) (0.00) (0.055) (0.00) (0.234) (1.00) (0.102) (0.00) (0.234) (1.00) (0.309) (1.00) (0.779) (1.00) (0.096) (0.00) (0.234) (1.00) (0.182) (0.00) (0.234) (1.00) (0.171) (0.00) (0.234) (1.00) (0.309) 0.00 (1.00) (0.779) (1.00) Figure 3: Tree showing the optima strategy (the upper numbers) and the minimum shortfa probabiity (ower numbers) in case of shortfa probabiity hedging in exampe with one poicy-hoder and four periods. The resuts are conditiona on surviva of the poicy-hoder (up to the time in question). strategy at time 0. Furthermore, the strategies according to the two hedging criteria are not very different at ater time points, and the outcomes of the financia market that ead to super-hedging are common for the two criteria. In Figure 4, strategies corresponding to shortfa probabiity hedging and quadratic hedging, as described in Møer (2001a) (interest rate set to 0), are compared. It appears that the initia number of stocks is more than three times bigger for shortfa probabiity hedging than quadratic hedging. At ater time points, this factor is in most cases even bigger. However, ater time points are not directy comparabe, since the quadratic hedging strategy depicted is not sef-financing, i.e. it is based on different capita at ater time points. Intuitivey, the observation that shortfa probabiity hedging resuts in a arger strategy than quadratic hedging may be expained by the fact that with quadratic hedging, a arge strategy may ead to a arge amount of redundant capita which is punished just as much as ack of capita. With the shortfa probabiity strategy, such considerations on redundant capita are irreevant, since redundant capita is not punished according to this strategy. 5.2 Iustrations for the first time point and time step Eight or nine poicy-hoders and one period In this section, we study two cases with a singe period, one with 8 and one other with 9 poicy-hoders. The purpose is to demonstrate that significant differences in optima strategies can occur in cases that appear to be very simiar. In Figure 5, the optima strategy h versus start capita v is shown for 9 poicy-hoders. In the foowing, we focus on arge initia vaues for the initia capita such that the nonnegativity constraint on capita is not active. Hence, we consider initia capita vaues of 16

18 Figure 4: Tree showing optima strategies according to shortfa probabiity hedging (the upper numbers) and quadratic hedging (ower numbers) in exampe with one poicy-hoder and four periods. The resuts are conditiona on surviva of the poicy-hoder (up to the time in question). 400 or arger. It appears that for n = 9 poicy-hoders, it is aways optima to choose a strategy which hedges surviva of a n poicy-hoders when the stock vaue jumps up, except for two intervas of capita, where the trianges in the figure are ocated beow the ine defining the bottom sides the other trianges. Beow, we refer to the two v-intervas by 1 and 2 (1 for the interva with the arges v-vaues). In these intervas, the surviva of ony n 1 = 8 poicy-hoders, is hedged in case the stock vaue jumps up. In case of n = 8 poicy-hoders (not shown), no such exceptions occur, i.e. in the considered capita eves, it is aways optima to hedge a n poicy-hoders. The above mentioned differences can be justified by ooking at the probabiities more cosey. Beow we show that for n = 9, referring to the above v-intervas, 1 and 2: 1 It is more optima to 1-i) hedge n 1 survivors, irrespective of the outcome for the stock than to 1-ii) hedge n and n 2 survivors provided that the stock vaue jumps up and down, respectivey. 2 It is more optima to 2-i) hedge n 1 and n 2 survivors provided that the stock vaue jumps up and down, respectivey, than to 2-ii) hedge n and n 3 survivors provided that the stock vaue jumps up and down, respectivey. With p y n, the probabiity of y survivors, given that the number of poicy-hoders is n, we find the shortfa probabiity using the strategies 1-i) and 1-ii) introduced above: P sf,1i = pp n n + (1 p)p n n = p n n, P sf,1ii = p 0 + (1 p)(p n 1 n + p n n ) = (1 p)(p n 1 n + p n n ). Strategy i) is more optima than ii) if P sf,1 =: P sf,1ii P sf,1i = (1 p)p n 1 n pp n n > 0. 17

19 In our case with p n y being the Binomia probabiity distribution function with parameters (n,q) (and argument y), we find P sf,1 = q n 1 [(1 p)(1 q)n pq], i.e. P sf,1 > 0, i.e. strategy 1-i) is more optima than 1-ii), for n n 1 where n 1 = min{n IN n pq (1 p)(1 q) }. With q = exp( 0.25) and p = 0.7, we find n 1 = 9. Hence, for n = 8 poicyhoders, strategy 1-ii) is optima, whereas strategy 1-i) is optima for n = 9. Simiary, one finds that strategy 2-i) is more optima than 2-ii) for n n 2 where n 2 = min{n IN n 2pq 2 (1 q) 2 }. Again, with q = exp( 0.25) and p = 0.7, n 2 = 9. h v Figure 5: Optima strategy h versus start capita v in exampe with 9 poicy-hoders and 1 period using shortfa probabiity hedging. Three poicy-hoders and four periods - importance of restrictions to observations We adopt a stochastic surviva probabiity as discussed in Section 3.1. We consider 4 periods and 3 poicy-hoders. Shortfa probabiities in case of observations and no observations are shown in Figure 6. A considerabe reduction of the shortfa probabiity obtained through observations is seen. For start capita 200 and above, the probabiity is roughy reduced by 50 %. Naturay, choosing a uniform prior as in this exampe woud be too conservative in most appications. The interpretation is then that for the case considered, we have roughy an upper bound for the effect from observations. Three poicy-hoders and three periods Here we study an exampe with minimum shortfa probabiities and expected shortfa and corresponding strategies for the case with three poicy-hoders and three periods. Optima strategies for shortfa probabiity hedging are given in Figure 7 for p = 0.7. The corresponding minimum shortfa probabiities are shown in Figure 8 for p = 0.7 and p = 0.5, incuding shortfa probabiities with no action for comparison. It appears that the reationship between non-unique strategies and start capita eve is rather compicated. Even though the shortfa probabiity without investment is higher for p = 0.7 than for p = 0.5, the minimum shortfa probabiity from optima investments is much ower with p = 0.7 than p = 0.5, see Figure 8, as expected due to the more efficient contro possibiity. Anaogous resuts based on expected shortfa hedging are reported in Figure 9 and 10. The 18