# On Valuing Equity-Linked Insurance and Reinsurance Contracts

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4 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 4 rigger sop-loss payoffs. The double rigger producs are riggered by he asse level rising above or dropping below a criical level. The sum of he wo double rigger payoffs resuls in a pure sop-loss reinsurance conrac; however, he sum of heir values do no equal o he value of he sop-loss conrac. This is indicaive of he non-linear pricing rules implied by uiliy indifference. 2. The modeling and pricing framework To model he problem for insurers exposed o equiy-linked losses, we assume ha here is a risky asse whose price process follows a Geomeric Brownian moion, and ha losses arrive a Poisson imes wih claim sizes depending on he price of he risky asse a he claim arrival ime. More specifically, le S()} 0 T denoe he price process for a risky asse; le L()} 0 T denoe he loss process for he insurer; le F S F S } 0 T denoe he naural filraion generaed by S(); le F L F L } 0 T denoe he naural filraion generaed by L(); le F F S F L denoe he produc filraion generaed by he pair S(), L()}; and le (Ω, P, F) represen he corresponding filered probabiliy space wih saisical probabiliy (or real-world) measure P. The insurer is assumed o inves coninuously in he risky asse S() and a risk-free money marke accoun wih consan yield of r 0. Furhermore, he risky asse s price process saisfies he SDE: ds() = S() µ d + σ dx()}, (1) where X()} 0 T is a sandard P-Brownian process, and µ > r. Equivalenly, S() = S(0)e (µ 1 2 σ2 ) +σ X(). (2) The claims arrive a he arrival imes i } of an inhomogenous Poisson process wih deerminisic hazard rae λ(), and each claim size is a funcion of he prevailing index level and possibly ime: g(s( i ), i ). Noice ha he loss size depends on he price of he risky asse prevailing a he ime he loss arrives. This is a defining feaure of equiy-linked insurance producs and inroduces a new dimension o he opimal sochasic conrol problem associaed wih pricing he premium sream. The loss process may be wrien in erms of an underlying Poisson couning process N()} 0 T as follows L() = N() n=1 g(s( i ), i ), We implicily assume ha he claims are posiive g(s, ) 0 and are bounded for every finie pair (S, ) [0, ) [0, T ]. Since our assumpions on he dynamics of he risky asse and he loss process have been addressed, we urn aenion o he dynamics of he wealh process for he insurer. There are wo separae siuaions of ineres: (i) he insurer does no ake on he insurance risk, however, he insurer does inves in he risky asse and he riskless money-marke accoun; and (ii) he insurer akes on he insurance risk in exchange for receiving a coninuous premium of q and simulaneously invess in he risky asse and he riskless money-marke accoun. Le W ()} 0 T and W L ()} 0 T denoe, respecively, he wealh process of he insurer who does no ake on he insurance risk (3)

7 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 7 The indifference price is once again defined hrough he balancing equaion U RC (w P (L, S, ), L, S, ; q) = U(w, L, S, ; q). (12) We find ha couner-pary risk serves only o add one addiional non-linear, bu local, erm o he pricing PDE which vanishes as he defaul inensiy reduces o zero. 3. The indifference premium rae problem Now ha he sochasic model for he insurer has been described, and he pricing principle has been specified, we can focus on he deails of he pricing problem iself. In he nex subsecion, he value funcion wihou he insurance risk is reviewed. The resuls of ha secion are essenially hose of Meron (1969). Those resuls are hen used in 3.2 o solve he HJB equaion for he insurer exposed o he insurance risk. In 3.3, we deermine he indifference premium for a general loss funcion and provide specific examples. In 3.4, we address he issue of hedging he risk associaed wih his premium choice The value funcion wihou he insurance risk The value funcion of he insurer who does no ake on he insurance risk is defined in (6), and we now use he dynamic programming principle o deermine he opimal invesmen sraegy and he value funcion iself. Given a paricular invesmen sraegy π, we deermine ha V saisfies he following SDE: dv (W, s) = [ V + (r W + (µ r) π) V w σ2 π 2 V ww ] ds + πσ Vw dx. (13) The subscrips denoe he usual parial derivaives of V, and he ime dependence of he various processes are suppressed for breviy. Through he usual dynamic programming principle, V solves he HJB equaion: V + r w V w + max π [ (µ r) π Vw σ2 π 2 V ww ] = 0, V (w, T ) = u(w). We may assume ha he opimal invesmen is provided by he firs order condiion, and he Verificaion Theorem confirms he resul. To his end, he opimal invesmen sraegy is π () = µ r σ 2 V w V ww. On subsiuing π ino (14), V is found o saisfy he PDE V 1 ( ) µ r 2 Vw 2 + r w V w = 0. (16) 2 σ V ww Assuming ha (14) (15) V (w, ) = 1ˆα e α() w+β(), (17) wih β(t ) = 0 and α(t ) = ˆα, he HJB equaion reduces o ( ) µ r 2 = 0, (18) (α + rα) w + β 1 2 σ

8 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 8 which mus hold for all w and. Therefore, α() = ˆα e r(t ) and (19) ( ) µ r 2 (T ), (20) β() = 1 2 σ resuling in he sandard opimal (Meron, 1969) invesmen of π () = µ r ˆα σ 2 e r(t ). Since he soluion saisfies he requiremens of he Verificaion Theorem, π corresponds o he opimal invesmen sraegy for (6), and V, given in (17), is he soluion of he original opimal sochasic problem The value funcion wih insurance risk While assuming he insurance company akes on he insurance risk and receives a premium rae of q, we mus solve for he opimal invesmen and value funcion U, given in (7). Through sraighforward mehods, we esablish he following HJB equaion for he value funcion U: 0 = U + (r w + q)u w + µ S U S σ2 S 2 U SS +λ() (U(w g(s, ), S, ) U(w, S, )) U(w, S, T ; q) = u(w). + max π 1 2 σ2 U ww π 2 + π [ (µ r)u w + σ 2 S()U ws ]}, The nonlocal erm appears due o he presence of he Poisson claims, and can be explained by observing ha a claim arrives in (, + d] wih probabiliy λ() d, causing he wealh o drop by g(s(), ). A firs sigh, he presence of his nonlocal erm appears o render he problem inracable. However, on closer inspecion, we find ha he HJB equaion can be solved explicily for arbirary claims. Theorem. 3.1 The soluion o he HJB sysem (22) is } U(w, S, ; q) = V (w, ) exp ˆα q er(t ) 1 + γ(s, ), (23) r (21) (22) where γ(s(), ) = E Q [ T λ(u) ( ) ] e α(u) g(s(u),u) 1 du, (24) and he process S() saisfies he following SDE in erms of he Q-Wiener process X()} 0 T, ds() = S() r d + S() σ dx(). (25) Furhermore, he opimal invesmen sraegy is independen of wealh and equals } π e r(t ) µ r (S, ) = ˆα σ 2 + S γ S. (26)

10 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 10 Lemma 3.3 The indifference premium rae q is monoone in he risk-aversion level ˆα if he claim size g(s(), ) is sricly posiive. Proof. I is easy o check ha p(y) := (e y x 1) /y is monoonically increasing in y for every x > 0. Then, under he posiiviy assumpion of g, for every even ω Ω, we have, T λ(u) eα(u) g(s(u),u; ω) 1 ˆα is increasing in ˆα and he resul follows. In analyzing he value funcion U, we assumed ha q was consan; however, on glancing a (32) i can be inferred ha q is no a consan, and herefore, our assumpions are false, discrediing he analysis. This iniial reacion is premaure. The siuaion is bes explained by appealing o he familiar case of a forward conrac. On signing of a forward conrac, he delivery price is se such ha he conrac has zero value. This delivery price is a funcion of he prevailing spo price of he asse and bond prices a he ime of signing. Alhough he conrac value on signing is zero, he forward price, a any fuure dae, will no equal o he delivery price, and he conrac s value is no longer zero. In he presen conex, he insurer is looking forward o a fuure ime horizon, and is deciding on a rae o charge so ha she is indifferen o aking he risk. Our analysis shows ha he rae (32), which depends on he prevailing price of he risky asse, should be charged. This rae is fixed unil he end of he ime horizon, and does indeed render he insurer indifferen o he insurance risk a he curren ime. (33) However, as ime evolves, he prevailing indifference premium a ha fuure poin in ime may be higher or lower han he rae he insurer iniially se. Consequenly, if he insurer ook on he insurance risk a ime in exchange for q(s(), ) unil he horizon end, hen a some fuure ime she may develop a preference eiher owards releasing he insurance risk or for holding ono i. Wih he forward conrac analogy, i is no surprise hen ha he premium rae depends on he risky asse s spo price. The indifference premium (32) has some addiional noeworhy properies. The risk-neural measure Q appearing in he premium calculaion is independen of he risk-aversion level of he insurer. Wihin his risk-neural measure, he disribuion of claim sizes has no been disored from is real world disribuion. Indeed, he Radon-Nikodym derivaive process which performs he measure change is ( ) dq η() = exp 1 ( ) } µ r 2 + µ r X(). (34) dp 2 σ σ This is he same measure change ha Meron (1976) uses in his jump-diffusion model and corresponds o risk-adjusing only he diffusion componen. Since he equiy risk and he losses are inerrelaed only hrough he loss size, and no hrough any insananeous correlaion, his risk-neural measure corresponds o he minimal maringale one. This is similar o he case of indifference pricing for financial opions, where only he radable asse s risk process are disored such ha heir drif is he risk-free one under he pricing measure, while orhogonal sochasic degrees of freedom are lef undisored. Even hough he risk-aversion level does no feed ino he

11 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 11 probabiliy measure used for compuing expecaions, i does manifes iself in he disorion of he claim sizes hrough he exponenial erm. This exponenial disorion is, no surprisingly, inheried from he choice of uiliy funcion. A paricularly ineresing inerpreaion of he premium is supplied by spliing he expecaion erm appearing in (32) ino wo and wriing each erm as he expeced number of claims under wo differen measures. Specifically, [ T ( ) ] [ T ] [ T λ(u) e α(u) g(s(u),u) 1 du = E Q λ(u) e α(u) g(s(u),u) du E Q E Q = E e Q [N(T )] EQ ] λ(u) du [N(T )]. (35) Under he measure Q, he process N() is a doubly sochasic Poisson process wih aciviy rae λ() λ() e α() g(s(),). When ineres raes are zero and he losses hemselves are consan, i is easy o check ha he measure Q is he minimizer of he penalized enropy: min E e [ ( ) ] Q d Q ln ˆα g N(T ). (36) eq<<p dp T As a final poin of ineres, alhough he premium is a non-linear funcional of he claim sizes i is linear in he arrival rae of he claims λ(). This observaion suggess ha he generalizaion o muliple claims disribuions is sraighforward. In he heorem below, we provide he resuls for muliple claims disribuions. The proof is omied for breviy as i follows along he same lines as hose in he previous wo secions. Theorem. 3.4 Suppose ha he insurer is exposed o losses from m differen sources of risk. Explicily, he loss process is modeled as follows: L() = m j=1 N j () n=1 g j (S( n j ), n j ), (37) where N j () : j = 1,..., m} are independen inhomogenous Poisson processes wih arrival raes λ j () : j = 1,..., m}, g j (S, ) denoes he loss funcions for he j-h source of risk, and n j denoes he n-h arrival ime for he j-h process. Then, he value funcion of he insurer who akes on he insurance risk and receives a premium of q(w, S, ) is } U(w, S, ; q) = V (w, ) exp ˆα q er(t ) 1 + γ(s, ), (38) r where γ(s(), ) = m j=1 E Q [ T λ j (u) ( ) ] e α(u) gj(s(u),u) 1 du, (39) and he process S() saisfies he following sochasic differenial equaion in erms of he Q-Wiener process X()} 0 T : ds() = S() r d + S() σ dx(). (40)

12 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 12 Furhermore, he insurer s indifference premium is independen of wealh and is explicily [ r m ] T q(w, S, ) = e r(t ) E Q λ j (u) eα(u) gj(s(u),u) 1 du. (41) 1 ˆα j=1 This lineariy is an ineresing consequence of he coninual arrival of claims a Poisson imes. If insead, we assumed ha here was a maximum finie number of claims, hen he problem would no be linear. We are currenly invesigaing how a large, bu finie, number of claims alers he resuls in his aricle Consan losses An ineresing esing ground for our resuls is he case when he losses hemselves are consan, i.e. g(s, ) = l. This case is a paricular example of he model in Young and Zariphopoulou (2003) where he claims disribuion is a Dirac dela disribuion; however, hey did no repor he resul for his simple consan claims case. We deermine he indifference premium rae as λ ( ) q = ˆα ( e r(t ) 1 ) Ei(ˆα l e r(t ) ) Ei(ˆα l) (T )r, (42) where Ei(x) denoes he so called exponenial inegral, defined as he following Cauchy principle value inegral: Ei(x) x e d. The exponenial inegral has he following asympoic expansion:ei(x) = γ + ln(x) + n=1 xn n! n ; where γ is Euler s consan. As expeced, he indifference premium is non-linear in he claim size l. There is good evidence ha insurers who have well diversified porfolios and large reserves exhibi near risk-neural behavior. I is herefore ineresing o invesigae he impac his has on he valuaion of he insurance sream. If he insurer is near risk-neural, hen an expansion in ˆα l can be carried ou, and we find he indifference premium rae o linear order is ( q = λ l ( ) ) e r(t ) + 1 ˆα l + o(ˆα l). (44) 4 As such, a risk-neural insurer exposed o fixed losses, will charge a rae equal o he expeced loss per uni ime λ l an inuiively sound resul. As expeced, he sign of he firs order correcion is posiive Near risk-neural insurer Exending he near risk-neural insurer analysis o general claims funcion is no difficul. For losses ha grow a mos power like, i.e. here exiss c > 0, b() > 0 and S () > 0 such ha for each and S > S (), g(s, ) b() S c, he rae has he following perurbaive expansion in erms of he risk-aversion parameer ˆα: λ r ˆα n 1 T q = e r(t ) e n r(t u) E Q 1 n! [ gn (S(u), u)] du. (45) n=1 (43)

13 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 13 This series converges by appealing o he Lebesgue dominaed convergence heorem and noing ha E Q [Sn (u)] = S()e n(r+ 1 2 σ2 (n 1))(u ). This growh condiion can be weakened considerably; however, a his poin we are concerned wih aiding inuiion and as a resul, omi such deails from he analysis. he opions conex. Expansions similar o he one above have been explored in Davis (1998) in He demonsraes ha he zeroh order erm is equivalen o price of an infiniesimal posiion in he opion he so called marginal price. Based on (45), a risk-neural insurer would hen charge a premium rae of q = r e r(t ) 1 T λ(u) e r(t u) E Q [ g(s(u), u)] du. (46) Observe ha he facor in fron of he expecaion can be represened as (a) 1 wih a = T e r(t u) du. The expression a is precisely he accumulaed value of \$1 per annum received coninuously over he ime span (, T ]. Furhermore, he expecaion can be inerpreed as he risk-neural expeced claims accumulaed o he mauriy dae T. Wih hese poins in mind, he risk-neural indifference premium (46) rae balances, in expecaion, beween coninually receiving he premium and paying ou he claims Floor, capped, and marke paricipaion claims In his secion we provide an explici example of he premium when he losses are funcions of he logarihm of he sock index. While sill mainaining he essenial properies of linear claim sizes, we use he logarihm of he sock price because i allows for parially closed form soluions. To his end, define A(u) as he expecaion appearing under he inegral in he indifference premium (32), i.e. A(u) E Q [ e α(u) g(s(u),u)]. Then, he indifference premium q can be wrien in erms of A(u) explicily as q = r ˆα ( e r(t ) 1 ) T (47) λ(u) A(u) 1} du. (48) Consider insurance claims which have a cap and a floor proecion in addiion o a paricipaion in he risky asse s reurn independen of ime ypical feaures found in equiy-linked insurance payoffs. In his case, he claim sizes are θ, S() < c 1, g(s(), ) = θ + β (log (S()) log(c 1 )), c 1 S() < c 2, θ + β (log(c 2 ) log(c 1 )), S() c 2. To mainain posiiviy of he claim sizes in all oucomes, we resric θ > 0, β > 0 and c 1 < c 2. Afer some edious calculaions, he inegrand A(u) reduces o ( ) β α(u) A(u) = e Φ(d α(u)θ 1 (c 1 )) + c2 c Φ( d1 1 (c 2 )) ( ) } β α(u) (50) + S() c 1 e β α(u)(r 1 2 σ2 (1 β α(u)))(u ) (Φ(d 1 (c 3 )) Φ(d 2 (c 1 ))). (49)

14 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 14 Indifference Premium [ q ] \$130 \$110 α = 0 α = 0.1 α = 0.2 \$90 \$70 \$80 \$90 \$100 \$110 \$120 \$130 Indifference Premium [ q ] \$130 \$110 α = 0 α = 0.1 α = 0.2 \$90 \$70 \$80 \$90 \$100 \$110 \$120 \$130 Spo Price [ S() ] Spo Price [ S() ] Figure 1. The dependence of he indifference premia on he underlying equiy spo price for losses given in equaion (49). The model parameers are θ = 1, β = 1, c 1 = 90, c 2 = 110, r = 4%, σ = 15%, and λ = 100. The erms in he lef/righ panels are one and five years respecively. In Figure 1, he dependence of he premium on he underlying spo price is illusraed for hree choices of he risk-aversion parameer ˆα and for erms of one and five years respecively. The boxed line shows he pure loss funcion (49) scaled by he aciviy rae for comparison purposes. As he risk-aversion parameer increases, he premia increases illusraing he monooniciy propery. Noice ha when he erm increases he premium decreases for large spo prices, while i increases for small spo prices. This is analogous o he pricing for a sandard bull-spread opion in he Black-Scholes model Hedging he insurance risk Now ha we have deermined he indifference premium ha he insurer charges, i is ineresing o explore he hedging sraegy ha she would follow. In his incomplee marke seing, i is impossible o replicae he insurance claims; noneheless, he insurer holds differen unis of he risky asse when she is exposed o he insurance risk and when she is no exposed o he insurance risk. As a resul, we can define an analog of he Black-Scholes Dela hedging parameer. To his end, he Dela is defined as he excess unis of he risky asse ha he insurer holds when aking on he risk and receiving he premiums, and when here is an absence of insurance risk. Corollary 3.5 The Dela of he insurer s posiion is (S, ) 1 S (π U π V ) = e r(t ) ˆα γ S (S, ). (51) Proof. The opimal invesmen in he risky asse wihou he insurance risk appears in (15), and wih he insurance risk appears in (26). The resul is quie similar o he Black-Scholes Dela for an opion. Alhough i is possible o rewrie he Dela in erms of he indifference premium rae q, i is mos naurally represened in erms of he auxiliary funcion γ. Moreover, as T, he Dela vanishes; his is quie differen from he behavior of he Black-Scholes Dela of an opion wih payoff g(s(t ), T ). In he case of

15 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 15 Dela α = 0 α = 0.1 α = 0.2 Dela α = 0 α = 0.1 α = \$70 \$80 \$90 \$100 \$110 \$120 \$130 Spo Price 0.0 \$70 \$80 \$90 \$100 \$110 \$120 \$130 Spo Price Figure 2. The dependence of he Dela on he underlying equiy spo price for losses given in equaion (49). The model parameers are θ = 1, β = 1, c 1 = 90, c 2 = 110, r = 4%, σ = 15%, and λ = 100. The erms in he lef/righ panels are one and five years respecively. a European opion, he Dela becomes equal o he derivaive of he payoff funcion wih respec o he spo price, and is zero only where he opion s payoff becomes fla. In he presen conex, he Dela vanishes as mauriy approaches because he probabiliy of a loss arriving in he nex small ime inerval close o mauriy is λ T. Therefore, probabilisically, here is no need o hold addiional shares of he risky-asse near mauriy. In Figure 2, we show how he Dela behaves as a funcion of he spo-level, risk-aversion parameer, and ime o mauriy for he example in The general shape of hese curves is expeced. The payoff is asympoically fla ouside of he paricipaion region (see Figure 1); implying a decaying Dela in he ails. The Dela is wider when mauriy is furher away and, conrary o an opion s dela, i is increases wih mauriy due o he larger number of poenial losses. 4. The indifference price for reinsurance Now ha we have deermined he indifference premium ha he insurers charges, we can address he dual problem of pricing a reinsurance conrac which makes paymens a he end of he ime horizon. In secion 2, we describe he value funcion associaed wih he insurer who akes on he insurance risk, receives he premium rae q, and receives a reinsurance paymen of h(l(t ), S(T )). The value funcion of such an insurer was denoed U R as defined in equaion (9). The associaed HJB equaion for his value funcion is essenially he same as he one for U (see equaion (22)); however, he boundary condiion is now alered o accoun for he presence of he reinsurance, and we mus also keep rack of he loss process explicily. Through he usual dynamic programming principle, we deermine ha U R saisfies he following HJB equaion: 0 = U R + (rw + q)uw R + µ S US R σ2 S 2 USS R +λ() (U R (w g(s, ), L + g(s, ), S, ) U R (w, L, S, )) U R (w, L, S, ; q) = u(w + h(l, S)). + max π 1 2 σ2 U R ww π 2 + π [ (µ r)u R w + σ 2 S()U R ws]}, (52)

16 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 16 The nonlocal erm now conains wo ypes of shifing: he firs, due o he decrease in he wealh of he insurer; and he second, due o he increase in he loss process. However, boh shifs come from he same risk source. We can immediaely address he issue of couner-pary risk by modifying he payoff o include an indicaor of he even ha he reinsurer survives o he mauriy dae. As described a he end of secion 2, he reinsurer s defaul ime is modeled by a separae inhomogeneous Poisson process M() wih aciviy rae κ(). I is easy o see ha he value funcion in he presence of couner-pary risk, which we denoed by U RC, saisfies he HJB: 0 = U RC + (rw + q)uw RC + µ S US RC σ2 S 2 USS RC +λ() (U RC (w g(s, ), L + g(s, ), S, ) U RC (w, L, S, )) U R (w, L, S, ; q) = u(w + h(l, S)). +κ() (U(w, L, S, ) U RC (w, L, S, )) + max 1 π 2 σ2 Uww RC π 2 + π [ (µ r)uw RC + σ 2 S()Uws RC ]}, The only difference beween (52) and (53) is he presence of he nonlocal erm proporional o he defaul inensiy κ(). Is appearance can be undersood as follows: if a defaul occurs over he nex infiniesimal ime, he value funcion U RC revers o U since he insurer will no longer receive he reinsurance paymen a mauriy; however, she is sill exposed o he insurance risks and sill receives he premium rae q. Since we can recover he defaul-free case by seing κ() = 0, he remaining analysis focus only on solving equaion (53). Once again, exponenial uiliy allows us o obain a soluion of he HJB equaion in a semi-explici form. Theorem. 4.1 The soluion o he HJB sysem (53) can be wrien as U RC (w, L, S, ) = U(w, S, )φ(l, S, ), (54) (53) where φ saisfies he non-linear PDE ( ) 0 = φ + r Sφ S σ2 S 2 φ SS φ2 S φ + κ() (1 φ(l, S, )) +λ() e α()g(s,) (φ(l + g(s, ), S, ) φ(l, S, )), φ(l, S, T ) = e ˆα h(l,s). (55) Furhermore, he opimal invesmen in he risky-asse is [ π e r(t ) µ r (S, ) = ˆα σ 2 + S γ S + φ ]} S. (56) φ Proof. Assuming ha U RC ww < 0, he firs order condiions allow he opimal invesmen sraegy o be wrien, π () = (µ r)u RC w + σ 2 S()U RC ws σ 2 U RC ww. (57) On subsiuing he ansäz (54) and he opimal invesmen (57) ino he HJB equaion (52), we

17 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 17 esablish 0 = φ U + (rw + q)u w + µsu S σ2 S 2 U SS 1 2 +U φ + } ((µ r)u w+σ 2 SU ws) 2 σ 2 U ww ( ) µ (µ r) U w 2 U U ww S φ S σ2 S (φ 2 SS U w 2 φ 2 S U U ww φ 2 +κ() U(w, S, ) [1 φ(l, S, )] +λ() [U(w g(s, ), S, ) φ(l + g(s, ), S, ) U(w, S, ) φ(l, S, )], ( UwS U w U U ww ) )} U S U φ S subjec o he boundary condiion U(w, S, T )φ(l, S, T ) = u(w + h(l, S)). From (22), he erms inside } in he firs line of he above expression equals λ() [U(w g(s, ), S, ) U(w, S, )]; collecing his wih he las line and making use of he ideniies U(w g(s, ), S, ) = U(w, S, ) e α()g(s,), (58) U 2 w U ww U = 1, and U wsu w U ww U = U S U = γ S, (59) we find, hen, equaion (58) disills o (55). I can hen be proven ha U R ww < 0. Using he ansäz (54), he opimal invesmen π can be rewrien as (56). For smooh g, he Verificaion Theorem allows us o confirm ha he consruced soluion is he value funcion for he original problem and ha he described sraegy is clearly opimal. Corollary 4.2 The insurer s indifference price P (L(), S(), ) for he reinsurance conrac saisfies he nonlinear nonlocal PDE: r P = P + r S P S σ2 S 2 P SS κ() α() P (L, S, T ) = h(l, S). ( 1 e α() P (L,S,) ) + λ() α() eα()g(s,) ( 1 e α()[p (L+g(S,),S,) P (L,S,)]), Proof. The indifference price P saisfies U RC (w P, L, S, ) = U(w, S, ). (60) The facorizaion (54) ogeher wih he ideniy U(w g(s, ), S, ) = U(w, S, ) e α()g(s,) implies ha P (L, S, ) = 1 α() ln φ(l, S, ). On subsiuing φ in erms of P in (55), we obain (60). Noice ha if he payoff funcion h(l, S) is independen of he loss level, i.e. h(l, S) = h(s), hen (60) reduces o r P = P + r S P S σ2 S 2 P SS κ() α() P (L, S, T ) = h(s). ( 1 e α() P ), When here is no couner-pary risk, he above pricing equaion is precisely ha of a European opion wih payoff h(s) in he Black and Scholes (1973) model. (61) This resul is expeced since he reinsurance conrac is hen exposed only o he hedgable risk he risky asse and no o he non-hedgable claims risk or couner-pary risk. Therefore, our resul should reduce o he no arbirage Black-Scholes price for an insurer of any degree of risk-aversion. When couner-pary risk is he only non-hedgable risk, i.e. κ() 0, bu h is a funcion only of S, hen he pricing equaion is idenical o he indifference pricing equaion for an equiy-linked pure endowmen paying h(s) condiional on survival o mauriy as sudied in Moore and Young (2003). Alernaively, his price may be viewed as he value of a defaulable opion wih payoff h(s).

18 On Valuing Equiy-Linked Insurance and Reinsurance Conracs Near risk-neural insurer Le he price of a risk-neural insurer, aken as he limi of a risk-averse insurer, be denoed P 0 (L, S, ) = limˆα 0 + P (L, S, ). Then, he pricing PDE for P 0 following from (60) is (r + κ()) P 0 = P 0 + r S PS σ2 S 2 PSS 0 + λ() P 0 P 0 (L, S, T ) = h(l, S), where P 0 denoes he increase in he price due o a loss arrival: P 0 (L, S, ) P 0 (L + g(s, ), S, ) P 0 (L, S, ). (63) Consequenly, hrough he Feynman-Kac Formula, a risk-neural insurer would be willing o pay P 0 (L, S, ) = E Q [e R ] T (r+κ(s))ds h(l(t ), S(T )) = E Q [e R ] T r ds h(l(t ), S(T )) I(τ > T ) (64) for he reinsurance conrac, where he Q-dynamics of S() appears in (40), while he loss arrival rae and couner-pary defaul rae are boh unalered from heir real world values. (62) Alhough his marke is incomplee, and herefore here exiss many risk-neural measures equivalen o he real world measure (Harrison and Pliska, 1981), he indifference pricing mehodology selecs a unique measure. This measure is he minimal maringale measure under which sochasic degrees of freedom ha are orhogonal o he driving diffusion of he radable asse s price process remain undisored. I is ineresing o invesigae he firs order correcion in he risk-aversion parameer ˆα o gain some undersanding of he perurbaions around he risk-neural price. This is similar o he work of Soikov (2005) where he invesigaes he linear correcions of he price of volailiy derivaives when he invesor has already aken posiions in a porfolio of derivaives. In our conex we are pricing he reinsurance opion in he presence of he insurance risk. The main difference here is ha our reinsurance payoff is no considered small relaive o he background porfolio as in Soikov (2005). If we assume ha he payoff funcion is bounded from above, and hence he price is also bounded, hen he price can be expanded in a power series in ˆα. Specifically, wrie P (L, S, ) = P 0 (L, S, ) + ˆαP 1 (L, S, ) + o(ˆα), (65) subjec o P 0 (L, S, T ) = h(l, S) and P 1 (L, S, T ) = 0. The linear correcion vanishes a mauriy since we have fully accouned for he payoff in he zeroh order erm. When insering his ansäz ino (60) and using (62), we deermine P 1 (L, S, ) saisfies he following PDE: (r + κ()) P 1 = P 1 + r S PS σ2 S 2 PSS 1 + λ() P 1 ( +e r(t ) 1 2 κ() ( P 0) 2 + λ() g 2 (S, ) [ P 0 (L, S, ) g(s, ) ] }) 2 + o(ˆα), P 1 (L, S, T ) = 0. Through Feynman-Kac, he firs order correcion can be represened as a risk-neural expecaion as well, and we find he following resul: [ T P 1 (L, S, ) = E Q e R [ u 1 κ(s) ds 2 κ(u) ( P 0 (L(u), S(u), u)) 2) (66)

19 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 19 +λ(u) g 2 (S(u), u) [ P 0 (L(u), S(u), u) g(s(u), u) ] } ] ] 2 du.(67) Ineresingly, he payoff funcion h(l, S) does no explicily appear in P 1 ; raher, i feeds from he risk-neural price funcion P 0 which does explicily depend on he payoff. The sign of his correcion erm is difficul o discern on firs observaion due o he erm proporional o λ(u). However, we may deduce ha if (i) h is increasing in L, (ii) g is non-negaive, and (iii) h is Lipschiz-coninuous wih Lipschiz consan 2, hen he correcion erm is non-negaive Probabilisic inerpreaion of he indifference price Alhough explici soluions o he general pricing PDE (60) were no consruced, we follow Musiela and Zariphopoulou (2003) and show ha he price funcion solves a paricular sochasic opimal conrol problem. By using he convex dual of he non-linear erm, he PDE is linearized and resuls in a pricing resul similar o he American opion problem. However, in he curren conex, he opimizaion is no over sopping imes. Insead, we find ha he opimizaion is over he hazard raes of he driving Poisson processes. Theorem. 4.3 The soluion of he sysem (60) is given by he value funcion [ P (S, L, ) = sup inf EˆQ z Y y Y e R T (r+ˆκ(s)) ds h(l(t ), S(T )) + 1ˆα T R T e u ( ) }] ˆλ(u) ˆκ(s) ds y(u) ˆβ(y(u)) ˆκ(u) ˆβ(z(u)) du (68) where Y is he se of non-negaive F -adaped processes, he loss process L() = ˆN() n=1 g(s( i ), i ) (69) and i are he arrival imes of he doubly-sochasic Poisson process ˆN(). In he measure ˆQ, he F -adaped hazard rae process for ˆN() is ˆλ() = y() λ() e α()g(s(),), (70) and ˆκ() = z()κ(). Finally, S() saisfies he SDE: ds() = r S() d + σ S() d ˆX(), (71) where ˆX()} 0 T is a ˆQ-Wiener process. Proof. Le β(x) denoe he non-linear erm in (60), i.e. β(x) = 1 e x, (72) and le ˆβ(y) denoe is convex-dual so ha ˆβ(y) = max (β(x) x y) = 1 y + y ln y. (73) x

20 On Valuing Equiy-Linked Insurance and Reinsurance Conracs 20 Clearly, ˆβ(y) is defined on (0, ) and is non-negaive on is domain of definiion. Furhermore, ( ) β(x) = min ˆβ(y) + y x. (74) y 0 Rewriing he exponenial erms in (60) in erms of heir convex dual, we find ha he PDE becomes linear in P : r P = P + r S P S σ2 ( S 2 P SS + κ() α() max z() 0 ˆβ(z()) ) z() α() P (L, S, ))] ( ) + λ() α() eα()g(s,) min y() 0 ˆβ(y()) + y() α() P (L, S, ) P (L, S, T ) = h(l, S). Through he usual dynamic programming principle, we find ha he value funcion (68) saisfies he above HJB equaion. We have shown ha he pricing problem reduces o simulaneously finding he aciviy rae which minimizes and he ineres rae ha maximizes he Black-Scholes price of he reinsurance conrac, subjec o a penaly erm which is iself a funcion of he aciviy rae and ineres raes. I is useful o illusrae how he risk-neural resul of he previous subsecion is recovered. In he limi in which ˆα 0 +, he penaly erm increases o infiniy and he process y which minimizes (68) is clearly he one in which ˆβ(y(u)) = 0 for all u [, T ]. Similarly, he process z which minimizes (68) saisfies ˆβ(z(u)) = 0. This is achieved when y(u) = z(u) = 1. The opimal hazard rae is hen equal o is real world value ˆλ() = λ() and he rae ˆκ() = κ() implying an ineres rae of r + κ(). The price herefore reduces o (64) Numerical examples In he absence of explici soluions, we now demonsrae how he pricing PDE can be used, noneheless, o obain he value of reinsurance conracs hrough a simple implici-explici finiedifference scheme. Since we are no concerned wih proving ha he scheme converges in a wide class of scenarios, we ake a praciioner s viewpoin and apply he scheme o siuaions in which he loss funcion and reinsurance conrac iself are boh bounded and asympoically consan. To his end, i is convenien o rewrie he problem using he log of he forward-price process z() ln S() + r(t ). Also, i is appropriae o scale he price funcion by he risk-aversion parameer and he discoun facor by inroducing he funcion P (L, z, ) α() P (L, e z r(t ), ). (76), (75) Wih hese subsiuions, he pricing PDE (60) becomes 0 = P 1 2 σ2 P z σ2 P zz ) ) κ() (1 e P (L,z,) + λ(z, ) (1 e (P (L+g(z,),z,) P (L,z,)) P (L, z, T ) = ˆα h(l, e z ),, (77) where g(z, ) = g(e z r(t ), ) and λ(z, ) = λ() e α()g(z). Now, we inroduce a M L M z N grid for he (L, z, ) plane wih sep sizes of ( L, z, ) so ha L j = j L, z k = z min + k L, n = n. (78)

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