THIS study focuses on the valuation problem of an
|
|
- Richard Cross
- 5 years ago
- Views:
Transcription
1 IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ The Valuaion of an Equiy-inked ife Insurance Using he Theory of Indifference ricing Jungmin Choi Absrac This sudy addresses he valuaion problem of an equiy-linked erm life insurance in wo moraliy models - a deerminisic moraliy and a sochasic moraliy. For each case, he Hamilon-Jacobi-Bellman (HJB) arial Differenial Equaion (DE) for he corresponding uiliy funcion is derived wih a coninuous ime model, and he principle of equivalen uiliy is applied o obain a DE for he indifferen price of he premium when an exponenial uiliy funcion is employed. Numerical examples are performed wih Gomperz s law of moraliy for he deerminisic model and wih a mean-revering Brownian Gomperz (MRBG) process for he sochasic model. Index Terms Equiy-inked ife Insurance, indifference pricing, sochasic moraliy. I. INTRODUCTION THIS sudy focuses on he valuaion problem of an equiy-linked erm life insurance using he heory of indifference pricing. In addiion o a moraliy risk like any oher life insurance produc, an equiy-linked erm life insurance has a marke risk from he underlying asse. In recen years, insurers have offered more flexible life insurance producs ha combine he deah benefi coverage wih an invesmen componen, o compee wih oher forms of he policy holder s savings, for example, muual funds or banks. An equiy-linked life insurance produc can offer a benefi from he performance of an underlying asse by defining he deah benefi o depend on he accoun value of he underlying asse. The pricing and hedging problem of he equiy-linked life insurance has been invesigaed exensively, and i is well summarized by Melnikov and Romanyuk [2]. As menioned in heir aricle, insurance firms do no consider he moraliy risk in valuaion of he policies nor adop adequae moraliy raes. This leads o an overpricing or underpricing of he premiums and he burden falls on he cusomers or he firm iself. Young [9] considered he same problem using he heory of indifference pricing when he moraliy rae is compued using Gomperz s law of moraliy. The purpose of his sudy is o exend heir idea o include a sochasic moraliy rae. Indifference pricing, also known as reservaion pricing or privae valuaion, is a mehod of pricing financial derivaives wih regard o a uiliy funcion. I is one of he pricing ools in incomplee financial markes, and i uses he principle of equivalen uiliy. Uiliy funcions are widely used for problems in pricing and hedging of financial derivaives, see [8] and [5], for example. Uiliy indifference pricing was firs inroduced by Hodges and Neuberger [6] when hey considered ransacion coss in replicaing coningen claims. The main idea of indifference pricing is ha by comparing he maximal expeced uiliies wih and wihou a Jungmin Choi is wih he deparmen of Mahemaics a Eas Carolina Universiy, Greenville, Norh Carolina, USA. ( choiju@ecu.edu) coningen claim, one can find a value of he price funcion which is indifferen o he exisence of he coningen claim. This idea was used o price insurance risks in a dynamic financial marke seing by Young and Zariphopoulou [2] using an exponenial uiliy. The same idea was exended by Young [9] o sudy an equiy-indexed life insurance, and he derived DE for premiums and reserves generalized he Black-Scholes equaion by including a nonlinear erm reflecing he nonhedgeable moraliy risk. We will adop heir model o sudy he indifference pricing of an equiyindexed life insurance wih wo differen moraliy risk models: a deerminisic and a sochasic. Indifference pricing is also used in pricing problems in an incomplee marke. The valuaion of opions in a sochasic volailiy model for sock price using indifference pricing was sudied by Sircar and Surm [8] and Kumar [7]. Sochasic moraliy became imporan especially for he moraliy coningen claim. In [4], Milevsky and romislow sudied he pricing problem of an opion o annuiize when considering sochasic moraliy raes and sochasic ineres raes. They also sudied how o hedge an opion o annuiize using pure endowmens, defaul free bonds, and life insurance conracs. Variable annuiies under sochasic moraliy were also considered by Balloa and Haberman []. ricing, reserving and hedging of a guaraneed annuiy opion (GAO) valuaion problem was sudied when moraliy risk was incorporaed via a sochasic model of he underlying hazard raes. Assuming a sochasic moraliy ha is independen of he financial risk, a general pricing model was proposed, and he Mone Carlo mehod was used for he esimaion of he value of GAO. Their sochasic moraliy model was also used by iscopo and Haberman [6], who considered a Guaraneed ifelong Wihdrawal Benefis (GWB) conrac under he hypohesis of a predeermined wihdrawal amoun. The valuaion approach was based on he decomposiion of he produc ino living and deah benefis, and a no arbirage model was used o derive he valuaion formula, wih a fixed moraliy and a sochasic moraliy. Indifference pricing of moraliy coningen claims was invesigaed by udkovsky and Young [] wih boh sochasic hazard raes in he populaion moraliy and he sochasic ineres raes. The resuling DEs were linear for pure endowmens and emporary life annuiies in a coninuous ime model, and i was found ha he price-per-risk increases as more conracs are sold. A sudy of he indifference pricing of a radiional life insurance and pension producs porfolio wih sochasic moraliy was presened by Delong [5], when a financial marke consiss of a risk-free asse wih a consan rae of reurn and a risky asse whose price is driven by a evy process. He applied echniques from sochasic conrol heory o solve he opimizaion problems. In his paper, we consider he pricing problem of an equiy- (Advance online publicaion: 26 November 26)
2 IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ linked erm life insurance using he heory of indifference pricing, when he moraliy of he insured is described by wo differen moraliy models: a deerminisic moraliy and a sochasic moraliy. The remainder of his paper is organized as follows. In Secion II, we presen he pricing DE of an equiy-indexed erm life insurance using an exponenial uiliy in he coninuous-ime model wih a deerminisic moraliy funcion, as described in [9]. The numerical soluions of he DE wih Gomperz s law of moraliy and he sensiiviies of he indifference prices wih respec o various parameers are also provided. The pricing DE and numerical examples wih a sochasic moraliy model are given in Secion III. We adop he MRBG process o model he moraliy risk. Finally, Secion IV summarizes our findings and oulines fuure works o exend our model. II. INDIFFERENCE RICING OF AN EQUITY-INDEXED IFE INSURANCE WITH A DETERMINISTIC MORTAITY A. The Financial Marke: Meron s Model We presen he classical model by Meron [3] which invesigaes he opimal invesmen sraegies of an individual wih an iniial wealh, who seeks o maximize he expeced uiliy of he erminal wealh. The invesor has he opporuniy o rade beween a risky asse (sock) and a risk-free asse (U.S. reasury bond). The price of he risky asse S s for some ime s >, wih a fixed ime, follows { dss = µs s ds + σs s db s, S = S >, where B s is a sandard Brownian moion on a probabiliy space (Ω, F, ) wih a filraion F and a probabiliy measure. The rae of reurn µ and he volailiy σ are posiive consans. The price of he risk-free bond s for some ime s > follows d s = r s ds, where r is a consan rae of reurn (or force of reurn) of he risk-free bond, and we assume µ > r >. e w be he iniial wealh of he insurer a ime, and W s be he wealh of he insurer a ime s in [, T ], where T is he erminal ime. Suppose he insurer rades dynamically beween he sock and he bond. e π s be he amoun of wealh invesed in he sock a ime s. Then he amoun invesed in he bond is πs b = W s π s, and he dynamics of he wealh process becomes ( ) ( ) d dw s = πs b s dss + π s s S s = (W s π s )rds + π s (µds + σdb s ). Hence we have { dws = (rw s + (µ r)π s )ds + σπ s db s, s T, W = w. B. Expeced Uiliy Wihou he insurance risk Suppose he invesor wans o maximize he expeced uiliy of he erminal wealh, and define he value funcion V (, w) as V (, w) = sup E[u(W T ) W = w], π A where A is he se of admissible policies, and u : R R is a uiliy funcion, which is increasing, concave, and smooh. We will use an exponenial uiliy funcion o derive a DE for he indifference price. I has been shown in [2] ha V saisfies he following HJB equaion: { V + max π [(µ r)π V w + 2 σ2 π 2 V ww ] + rwv w =, V (T, w) = u(w). Since he maximum funcion is quadraic in π and he concaviy of he uiliy funcion u is inheried o he value funcion, he maximum exiss and we have he opimal invesmen process π = µ r V w (w, ) σ 2 V ww (w, ). This gives a closed form DE for V : V + rwv w (µ r)2 Vw 2 2σ 2 =, V ww V (T, w) = u(w). One of he advanages o considering an exponenial uiliy funcion is ha we can find he closed form soluion o (). Suppose u(w) = e w, for some >, hen we obain he soluion V (, w) o be V (, w) = [ ] exp we r(t ) (µ r)2 2σ 2 (T ). We also can find he corresponding opimal sraegy π (, w) = µ r σ 2 e r(t ), which is no sochasic and independen of w. I is generally observed when considering exponenial uiliy. Since he absolue risk aversion for he exponenial uiliy funcion is measured by a consan, ( u (w)/u (w) = ), one can observe ha as he invesor s risk aversion () increases, he amoun of money invesed in he risky asse (π ) decreases [2]. C. Expeced Uiliy wih he insurance risk The insurer has an opporuniy o insure a person whose age is x + a ime. The deah benefi of his life insurance is defined o be G = max(a, A τ ), where τ < T is he ime of deah of he policy holder, A is he iniial accoun value of he underlying muual fund a he ime when he conrac is made, and A s is he accoun value a ime s. This insurance policy is an equiy-indexed produc since i is ied o an accoun value hrough he funcion G. Suppose he insurer charges an insurance fee o hedge he marke risk, and we assume ha i is deduced from he accoun value as an ongoing fracion, α. The dynamics of A s follow da s = (µ α)a s ds + σa s db s, where B s is a sandard Brownian moion on (Ω, F, ), and µ (rae of reurn) and σ (volailiy) are consans. Suppose For simpliciy, we assume i is a fixed consan. () (2) (Advance online publicaion: 26 November 26)
3 IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ TABE I ARAMETER VAUES he insurer wans o maximize he expeced uiliy of he erminal wealh, and define he value funcion Age a incepion x Risk free ineres rae r Volailiy of he risky asse σ Insurance fee α Term of policy T U (, w, A) = sup E{u(WT ) W = w, A = A}, π A where A is he se of admissible policies, and u : R R is a uiliy funcion, which is increasing, concave, and smooh. The wealh process should follow dws = [rws + (µ r)πs ]ds + σπs dbs, W = w, Wτ + = Wτ Gτ, if τ < T. The HJB equaion for U can be obained as follows (see Appendix) U + rwuw + (µ α)aua + 2 σ 2 A2 UAA +λx ()[V (w G, ) U ] + maxπ [(µ r)π Uw + σ 2 π AUwA (3) σ π U ] =, ww 2 U (T, w, A) = u(w), Using his relaionship in (5), we can derive a DE for (, A) r + + (r α)aa + 2 σ 2 A2 AA r(t ) ( G) = e r(t ) λx ()[ e e ], (6) (T, A) =. D. Numerical Example and Sensiiviy Analysis where λx () is he force of moraliy of a person aged x a ime. The corresponding opimal sraegy π is π = µ r Uw UwA A. σ2 Uww Uww.8 Suppose we use he exponenial uiliy u(w) = e w, for some >. Because of he naure of he exponenial uiliy, we propose he soluion of (3) o be in he form of U (, w, A) = V (, w) ϕ(, A) [9]. Then he opimal sraegy becomes µ r Vw Vw ϕa π = 2 A. σ Vww Vww ϕ e U = V ϕ in (3). From (2), we also have Vw2 = V, Vww and years A Fig.. rice wih respec o ime and accoun value V (, w G) = V (, w) exp[ger(t ) ]. Then we obain he DE for ϕ from (3):.2 ϕ + (r α)aϕa + 2 σ 2 A2 (ϕaa Ger(T ) +λx ()(e ϕ(t, A) =. ϕ2a ϕ ) ϕ) =, By inroducing η(, A) as ϕ(, A) = eη(,a), we have 2 2 η + (r α)aηa + 2 σ A ηaa r(t ) η +λx ()(ege ) =, η(t, A) =..8 (4) (5) Now le (, A) be he indifference price, ha is, he minimum premium he insurer should have in exchange for insuring he person whose age is x + a ime for a erm life insurance which expires a ime T. Then (, A) should solve V (, w) = U (, w +, A) = V (, w + )ϕ(, A), and using he closed form of V in (2), we have a formula for (, A) wih respec o η as (, A) = e r(t ) η(, A) Fig. 2. The relaionship beween and (, A ) In his secion, we solve (6) assuming he moraliy λx () follows Gomperz s law of moraliy λx () = B C x+, wih B =.64 5 and C =.96. These parameer esimaions were obained in [] using moraliy daa for Sweden. To solve (6) numerically, we use he finie (Advance online publicaion: 26 November 26)
4 IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ α x 3 Fig. 3. The relaionship beween insurance fee and (, A ) σ=. σ=.5 σ=.2 σ= Age a incepion Fig. 4. The relaionship beween volailiy and (, A ) difference mehod, paricularly, backward difference scheme for he ime and he cenral difference scheme for he accoun value A. We se he minimum accoun value o be zero and he maximum accoun value o be 2, wih he iniial accoun value (A ) of. We assume he DE holds on he boundary since he domain is runcaed. For he boundary condiion when he accoun value is maximum (A = 2), we assume he lineariy of he premium in erms of he accoun value, namely, we se AA =, and solve r + +(r α)a A = e r(t ) λ[ e er(t ) ( G) ]. When he accoun value is minimum (A = ), we solve r + = e r(t ) λ[ e er(t ) ( G) ]. A ypical soluion for he premium funcion is ploed in Figure wih respec o he ime and he accoun value, using he parameers in Table I for he base case. From he plo, we can observe ha increases as he accoun value A increases, which is expeced since he higher he accoun value is, he higher he deah benefi G = max(a, A τ ) will be. The insurer should receive more premium for a higher deah benefi. The premium decreases as he ime increases,which is a common rend for an equiy linked financial derivaive (for example, he hea, he rae of change of he derivaive wih respec o he ime, is negaive for a European call opion). Figure 2 shows he relaionship beween he risk aversion and he premium. As observed in oher sudies of moraliy coningen claims ( [], [9]), when he risk aversion increases, he indifference price of he premium also increases. Figure 3 shows he impac of he insurance fee α on he premium. I is clear ha hey should have a negaive relaionship, since if one has o pay more insurance fees, he price of he produc a ime zero should decrease. A posiive relaionship beween he volailiy σ of he risky asse and he premium is refleced in Figure 4, which is consisen wih a general financial heory ha he financial produc is more expensive when he volailiy is high. The premium is ploed agains he age a incepion x for various volailiies in he same figure, and we can observe ha if a person oped o purchase he life insurance produc a laer daes (as x increases), he should pay a higher price for he benefi. This can be jusified as follows: when x increases, he moraliy of he person increases, and, hence, he price of he life insurance produc should also increase. III. INDIFFERENCE RICING OF AN EQUITY-INDEXED IFE INSURANCE WITH A STOCHASTIC MORTAITY A. ricing DE wih sochasic moraliy Now we consider a sochasic model for he force of moraliy for an individual or a se of individuals of he same age. We adop he model proposed by udkovsky and Young [] and assume he force of moraliy λ follows a diffusion process as dλ s = µ(s, λ s )ds + σ(s)λ s db λ s, (7) where Bs λ is a Brownian moion on a probabiliy space (Ω, F, ) which is independen of B s in he previous secion. The volailiy σ is a nonzero coninuous funcion of ime s bounded below by a posiive consan κ on [, T ]. The drif µ(s, λ) is a coninuous funcion of s and λ which is posiive for all s in [, T ]. We will use he mean-revering Brownian Gomperz model in [4] for he numerical examples. Suppose he accoun value A s and he wealh process W s are defined as in Secion II. The insurer agrees o pay G τ = max(a, A τ ) upon deah a τ < T given a person aged x a = purchased he life insurance produc. Suppose he insurer wans o maximize he expeced uiliy of he erminal wealh, and define he value funcion U(, w, A, λ) = sup {u(w T ) W = w, A = A, λ = λ}, π A where A is he se of admissible policies, and u : R R is a uiliy funcion, which is increasing, concave, and smooh. The HJB equaion for U can be obained as follows (see Appendix) U + rwu w + (µ α)au A + 2 σ2 A 2 U AA +λ[v (, w G) U] + µλu λ + 2 σ2 λ 2 Uλλ + max π [(µ r)πu w + σ 2 πau wa + 2 σ2 π 2 U ww ] =, U(T, w, A) = u(w). Since he opimized erms are he same as in (3), i will assume he same opimal sraegy π = µ r U w σ 2 A U wa. U ww U ww (8) (Advance online publicaion: 26 November 26)
5 IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ Following he idea in Secion II and using he same exponenial uiliy u(w) = e w, we assume he soluion of (8) o be in he form of U(, w, A, λ) = V (, w) ϕ(, A, λ). Then he opimal sraegy becomes π = µ r V w σ 2 A V w ϕa V ww V ww ϕ. e U = V ϕ in (8). Afer collecing V erms and ϕ erms, (8) becomes V [ϕ s + (r α)aϕ A + 2 σ2 A 2 ϕ AA +λ[exp(ge r(t ) ) ϕ] +µλϕ λ + 2 σ2 λ 2 ϕ λλ 2 σ2 A 2 ϕ 2 A ϕ ] +ϕ[v s + rwv w (µ r)2 2σ 2 V ] =. The muliple o ϕ in he las erm is zero because of (), hence we obain he DE for ϕ ϕ + (r α)aϕ A + 2 σ2 A 2 (ϕ AA ϕ2 A ϕ ) +µλϕ λ + 2 σ2 λ 2 ϕ λλ + λ(e Ger(T ) ϕ) =, ϕ(t, A, λ) =. To eliminae he nonlinear erm ϕ2 A in (4), le us define ϕ η(, A, λ) by ϕ(, A, λ) = e η(,a,λ). Then we have a DE for η(, A, λ): (9) η + (r α)aη A + 2 σ2 A 2 η AA + µλη λ + 2 σ2 λ 2 (η 2 λ + η λλ) + λ(e Ger(T ) η ) =, η(t, A, λ) =. () Now le (, A, λ) be he indifference price, ha is, he minimum premium he insurer should have in exchange for a erm life insurance which expires a ime T. Then (, A, λ) should solve V (, w) = U(, w +, A, λ) = V (, w + )ϕ(, A, λ), and using he closed form of V in (2), we have a formula for (, A, λ) wih respec o η as (, A, λ) = e r(t ) η(, A, λ). Using his relaionship in (), we can derive a DE for (, A, λ) r + + (r α)a A + 2 σ2 A 2 AA +µλ λ + 2 σ2 λ 2 (e r(t r) 2 λ + λλ ) = e r(t ) λ[ e ) ( er(t G) () ], (T, A, λ) =. B. Numerical Example For numerical examples ha solves (), we use he following MRBG process proposed in [4]: ( dλ s = g + ) 2 σ2 + κ(gs + ln λ ln λ s ) λ s ds+σλ s dbs λ, wih κ =.5. The process ln λ s follows an Ornsein- Uhlenbeck model wih a linear drif g. The parameer values TABE II ARAMETER VAUES Risk free ineres rae r.8 Volailiy of he risky asse σ.2 Insurance fee α. Term of policy T years Volailiy of he force of moraliy σ.2 Force of moraliy a incepion λ.3 Gomperz parameer g Fig. 5. 3D plo of he soluion (, A, λ) λ.25 in Table II are used o obain he soluions unless noed oherwise. The DE for he premium (, A, λ) is solved in he domain [, T ] [, 2] [,.25] wih an iniial accoun value (A ), using he backward in ime finie difference mehod. As in he previous secion, we assume he DE holds on he boundary, and assuming lineariy when A = A max and λ = λ max. The boundary condiions imposed are ) A = : r + + +µλ λ + 2 σ2 λ 2 (e r(t r) 2 λ + λλ ) = e r(t ) λ[ e er(t ) ( G) ] 2) A = A max : 3) λ = : 4) λ = λ max : r + + (r α)a A +µλ λ + 2 σ2 λ 2 (e r(t r) 2 λ + λλ ) = e r(t ) λ[ e er(t ) ( G) ] r + + (r α)a A + 2 σ2 A 2 AA = r + + (r α)a A + 2 σ2 A 2 AA +µλ λ + 2 σ2 λ 2 (e r(t r) 2 λ) = e r(t ) λ[ e er(t ) ( G) ] A ypical soluion for he premium funcion (, A, λ) is given in Figure 5 a he ime of incepion ( = ). I.5. A.5 2. (Advance online publicaion: 26 November 26)
6 IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ λ=.25 λ=.5 λ=. λ=.5 λ= λ=.25 λ=.5 λ=. λ= A Fig. 6. The relaionship beween A and (, A, λ) for various values of λ Time o Expiraion Fig. 8. The relaionship beween ime o expiraion T and (T,., λ) for various values of λ Fig. 7. The relaionship beween λ and (, A, λ) for various values of A λ x Fig. 9. The plo of (,,.) wih respec o σ σ shows ha is an increasing funcion in A and λ, which is expeced since he price of a life insurance produc should increase when he accoun value increases or he moraliy rae increases. We can observe his more clearly in Figures 6 and 7. The premium funcion is ploed agains A for various λ in Figure 6. The premium is an increasing convex funcion in he accoun value A for various values of λ; he rae of change is more significan wih a higher value of A. I is clearer wih a higher force of moraliy λ. The premium funcion is ploed agains λ for various values of A in Figure 7. The premium is an increasing concave funcion in λ; he rae of change is more significan wih a smaller value of λ. I is clearer wih a higher value of A. As in he deerminisic moraliy model, he premium funcion has a posiive relaionship wih he ime o expiraion (T ) in Figure 8. Similar rends can be observed in oher lieraure, for example, in he descripion of he deah benefi in he sudy by iscopo and Haberman [6]. To see he effec of he volailiy of he force of moraliy in (7), he value of (,,.) is ploed for differen values of σ. We observe ha as he volailiy increases, he premium also increases in Figure 9, which reflecs he common phenomenon in he financial markes ha he price of a moraliy coningen produc increases when here is more risk in moraliy. The effecs of he risk aversion rae and he insurance fee α on he premium is similar as in he case wih a deerminisic moraliy model in Secion II. The premium (,,.) wih respec o he risk aversion raes is ploed in Figure, and we observe ha he premium increases as he risk aversion rae increases. The premium (,,.) wih respec o he insurance fee α is ploed in Figure, which shows ha he premium of a life insurance produc should decrease when he policy holder pays a higher insurance fee α. IV. CONCUSION We have considered he valuaion problem of an equiyindexed erm life insurance wih wo differen moraliy models, a deerminisic moraliy and a sochasic moraliy. For he financial marke, we employ Meron s model and use an exponenial uiliy o obain HJB equaions for he uiliy funcions wih and wihou he life insurance risks. By using he heory of equivalen uiliy, we derive he DEs for he indifference price of he premium for boh moraliy models. The DE wih a deerminisic moraliy is solved numerically using Gomperz law of moraliy, while he MRBG process is adoped for he sochasic moraliy case. The derived DEs are no simple and closed form soluions canno be found, bu sraighforward applicaions of he finie difference mehod wih proper boundary condiions produce soluions ha are reasonable for a life insurance conrac. The sensiiviy analysis shows ha he models are appropriae o explain he premiums of he equiy-indexed erm life insurance. Fuure research should consider he effec of sochasic ineres raes, since i is unreasonable o assume he risk-free rae is consan for a long period of ime. We can also apply he indifference pricing heory o variable annuiy producs exposed o similar risks, for example, wih a Guaraneed Minimum Deah Benefi opion or a Guaraneed ifelong Wihdrawal Benefi opion. (Advance online publicaion: 26 November 26)
7 IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ Fig.. The plo of (,,.) wih respec o [4] Milevsky, M. A., and S. D. romislow, Moraliy Derivaives and he Opion o Annuiise. Insurance: Mahemaics and Economics, 29:299-38, 2. [5] Mukupa G. M., and Offen E. R., The Impac of Uiliy Funcions on The Equilibrium Equiy remium In A roducion Economy Wih Jump Diffusion, IAENG Inernaional Journal of Applied Mahemaics, vol. 45, no. 2, pp2-27, 25 [6] iscopo, G., and S. Haberman, The Valuaion of Guaraneed ifelong Wihdrawal Benefi Opions in Variable Annuiy Conracs and he Impac of Moraliy Riak. Norh American Acuarial Journal, 5(): 59-76, 2. [7] Sihole, T. Z., S. Haberman, and R. J. Verrall, An Invesigaion ino arameric Models for Moraliy rojecions, wih Applicaions o Immediae Annuians and ife Office ensioners daa. Insurance: Mahemaics and Economics, 27: , 2. [8] Sircar, S. and S. Surm, From Smile Asympoics o Marke Risk Measures. Mahemaical Finance, : , 22. doi:./mafi.25. [9] Young, V. R., Equiy-Indexed ife Insurance: ricing and Reserving Using he rinciple of Equivalen Uiliy. Norh American Acuarial Journal, 7(): 68-86, 23. [2] Young, V. R., and T. Zariphopoulou, ricing Dynamic Insurance Risks Using he rinciple of Equivalen Uiliy. Scandinavian Acuarial Journal, 4: , α x 3 Fig.. The plo of (,,.) wih respec o α AENDIX A HJB EQUATION FOR U WITH THE INSURANCE RISK Here we derive he HJB equaion for U. Assume ha he insurer follows an arbirary invesmen policy {π s } beween and + h, and afer + h, he insurer follows he opimal invesmen policy {π s}. If he insured aged x + survives unil ime + h, he conrac goes on. If he insured aged x+ dies before + h, he insurer pays G +h, and coninues under V, he value funcion wihou he claim. Thus REFERENCES [] Balloa,., and S. Haberman, The Fair Valuaion roblem of Guaraneed Annuiy Opions: The Sochasic Moraliy Environmen Case. Insurance: Mahemaics and Economics, 38:95-24, 26. [2] Bjork, T. Arbirage Theory in Coninuous Time. Oxford Universiy ress, 998. [3] Carmona, R. Indifference ricing. rinceon Universiy ress, 29. [4] Choi, J., and M. Gunzburger, Opion ricing in he resence of Random Arbirage Reurn. Inernaional Journal of Compuer Mahemaics, 86(6): [5] Delong,., Indifference ricing of a ife Insurance orfolio wih Sysemaic Moraliy Risk in a Marke wih an Asse Driven by a evy rocess. Scadinavian Acuarial Journal, :-26, 29. [6] Hodges, S. D., and A. Neuberger, Opimal Replicaion of Coningen Claims under Transacion Coss. Review of Fuures Markes, 8: , 989. [7] Kumar, R., Effec of Volailiy Clusering on Indifference ricing of Opions by Convex Risk Measures. Applied Mahemaical Finance 22():63-82, 25, doi:.8/35486x [8] Qiang i, and ap Keung Chu, A Muli-sage Financial Hedging Sraegy for a Risk-averse Firm wih Coningen aymen, IAENG Inernaional Journal of Applied Mahemaics, vol. 45, no., pp7-76, 25 [9] oeve, M. robabiliiy Theory. New York: Springer, 977. [] udkovski, M. and V.R. Young, Indifference ricing of ure Endowmens and ife Annuiies under Sochasic Hazard and Ineres Raes. Insurance: Mahemaics and Economics, 42:4-3, 28. [] Melnikov, A. and Y. Romaniuk, Evaluaing he erformance of Gomperz, Makeham and ee-carer moraliy models for risk managemen wih uni-linked conracs. Insurance: Mahemaics and Economics, 39:3-329, 26. [2] Melnikov, A. and Y. Romaniuk, Efficien Hedging and ricing of Eruiy-inked ife Insurance Conracs on Several Risky Asses. Inernaional Journal of Theoreical and Applied Finance, (3): , 28. [3] Meron, R. C., ifeime orfolio Selecion under uncerainy: he Coninuous-ime Case. The Review of Economics and Saisics, 5(3): , 969. U(, w, A) E[ + h, U(W +h, A +h ) W = w, A = A] h p x+ +E[V ( + h, W +h G +h ) W = w] h q x+, (2) where h p x+ is he probabiliy of a person aged x+ survives unil x + + h, and h q x+ = h p x+. This will have an equaliy if and only if he invesmen policy is opimal beween and + h. Assuming U and V are smooh enough o have all he derivaives, we have U( + h, W +h, A +h ) = U(, w, A) + +h du, (3) where du is he differenial of U. Using Iô s formula, du = [U s + U w (rw + (µ r)π) + (µ α)au A + 2 σ2 π 2 U ww + 2 σ2 A 2 U AA + σ 2 πau wa ]ds +σπu w db + σau A db. The righ hand side of (3) becomes U(, w, A) + +h Uds + +h σπu w db + σau A db, where +h U = U s + U w (rw + (µ r)π) + (µ α)au A + 2 σ2 π 2 U ww + 2 σ2 A 2 U AA + σ 2 πau wa, and aking expecaions yields he las wo inegrals zero. Similarly, for he value funcion wihou he claim V, we have V (+h, w +h ) = V (, w)+ +h 2 V + +h σπv w dbds, where 2 V = V s + (rw + (µ r)π)v w + 2 σ2 π 2 V ww. (Advance online publicaion: 26 November 26)
8 IAENG Inernaional Journal of Applied Mahemaics, 46:4, IJAM_46_4_ or Now (2) becomes U(, w, A) E,w,A [U(, w, A) + +h Uds] h p x+ +E w, [V (, w G) + +h 2 V ds] h q x+, U(, w, A) h q x+ E,w,A [ +h Uds] h p x+ +E w, [V (, w G) + +h 2 V ds] h q x+h. If we divide boh sides by h and ake limi h, hen λ x () U U + V (w G, ) λ x (), since h q x+ λ x (), he force of moraliy of a person h aged x a ime, and h q x+ as h. If he invesmen policy is opimal, we have an equaliy, which gives he following HJB equaion of U; U + rwu w + (µ α)au A + 2 σ2 A 2 U AA +λ x ()[V (w G, ) U] + max π [(µ r)πu w + σ 2 πau wa + 2 σ2 π 2 U ww ] =, U(T, w, A) = u(w). (4) or U(, w, A, λ) h q x+ E,w,A,λ [ +h 3 Uds] h p x+ +E,w [V (, w G) + +h 2 V ds] h q x+h. If we divide boh sides by h and ake limi h, hen λ U 3 U + V (, w G) λ. If he invesmen policy is opimal, we have an equaliy, which gives he following HJB equaion of U; U + rwu w + (µ α)au A + 2 σ2 A 2 U AA +λ[v (w G, ) U] + µλu λ + 2 σ2 λ 2 Uλλ + max π [(µ r)πu w + σ 2 πau wa (7) + 2 σ2 π 2 U ww ] =, U(w, A, T ) = u(w). AENDIX B HJB EQUATION FOR U WITH THE INSURANCE RISK The derivaion of HJB equaion for U is similar o he process for U. Wih he same assumpion in he previous secion, we have U(, w, A, λ) E[U( + h, W +h, A +h, λ +h ) W = w, A = A, λ = λ] h p x+ +E[V ( + h, W +h G +h ) W = w] h q x+. (5) This will have an equaliy if and only if he invesmen policy is opimal beween and + h. Assuming U and V are smooh enough o have all he derivaives, we have U( + h, W +h, A +h, λ +h ) = U(, w, A, λ) + +h du, (6) where du is he differenial of U. Using Iô s formula, du = [U s + U w (rw + (µ r)π) + (µ α)au A +µλu λ + 2 σ2 π 2 U ww + 2 σ2 A 2 U AA + σ 2 πau wa + 2 σ2 λ 2 U λλ ]ds +(σπu w + σau A )db + σλu λ db λ. The righ hand side of (6) becomes where 3 U = U(, w, A, λ) + +h + +h 3 Uds + +h σπu w db σau A db + +h σλu λ db λ, U s + U w (rw + (µ r)π) + (µ α)au A +µλu λ + 2 σ2 π 2 U ww + 2 σ2 A 2 U AA + σ 2 πau wa + 2 σ2 λ 2 U λλ, and aking expecaions yields he las hree inegrals zero. Now (5) becomes U(, w, A, λ) E,w,A,λ [U(, w, A, λ) + +h 3 Uds] h p x+ +E,w [V (, w G) + +h 2 V ds] h q x+, (Advance online publicaion: 26 November 26)
A pricing model for the Guaranteed Lifelong Withdrawal Benefit Option
A pricing model for he Guaraneed Lifelong Wihdrawal Benefi Opion Gabriella Piscopo Universià degli sudi di Napoli Federico II Diparimeno di Maemaica e Saisica Index Main References Survey of he Variable
More informationPricing FX Target Redemption Forward under. Regime Switching Model
In. J. Conemp. Mah. Sciences, Vol. 8, 2013, no. 20, 987-991 HIKARI Ld, www.m-hikari.com hp://dx.doi.org/10.12988/ijcms.2013.311123 Pricing FX Targe Redempion Forward under Regime Swiching Model Ho-Seok
More informationThe Mathematics Of Stock Option Valuation - Part Four Deriving The Black-Scholes Model Via Partial Differential Equations
The Mahemaics Of Sock Opion Valuaion - Par Four Deriving The Black-Scholes Model Via Parial Differenial Equaions Gary Schurman, MBE, CFA Ocober 1 In Par One we explained why valuing a call opion as a sand-alone
More informationPolicyholder Exercise Behavior for Variable Annuities including Guaranteed Minimum Withdrawal Benefits 1
Policyholder Exercise Behavior for Variable Annuiies including Guaraneed Minimum Wihdrawal Benefis 1 2 Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, 11h Floor; Alana, GA
More informationIntroduction to Black-Scholes Model
4 azuhisa Masuda All righs reserved. Inroducion o Black-choles Model Absrac azuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York, NY 6-439 Email:
More informationEquivalent Martingale Measure in Asian Geometric Average Option Pricing
Journal of Mahemaical Finance, 4, 4, 34-38 ublished Online Augus 4 in SciRes hp://wwwscirporg/journal/jmf hp://dxdoiorg/436/jmf4447 Equivalen Maringale Measure in Asian Geomeric Average Opion ricing Yonggang
More informationMAFS Quantitative Modeling of Derivative Securities
MAFS 5030 - Quaniaive Modeling of Derivaive Securiies Soluion o Homework Three 1 a For > s, consider E[W W s F s = E [ W W s + W s W W s Fs We hen have = E [ W W s F s + Ws E [W W s F s = s, E[W F s =
More informationIJRSS Volume 2, Issue 2 ISSN:
A LOGITIC BROWNIAN MOTION WITH A PRICE OF DIVIDEND YIELDING AET D. B. ODUOR ilas N. Onyango _ Absrac: In his paper, we have used he idea of Onyango (2003) he used o develop a logisic equaion used in naural
More informationLIDSTONE IN THE CONTINUOUS CASE by. Ragnar Norberg
LIDSTONE IN THE CONTINUOUS CASE by Ragnar Norberg Absrac A generalized version of he classical Lidsone heorem, which deals wih he dependency of reserves on echnical basis and conrac erms, is proved in
More informationINSTITUTE OF ACTUARIES OF INDIA
INSIUE OF ACUARIES OF INDIA EAMINAIONS 23 rd May 2011 Subjec S6 Finance and Invesmen B ime allowed: hree hours (9.45* 13.00 Hrs) oal Marks: 100 INSRUCIONS O HE CANDIDAES 1. Please read he insrucions on
More informationAvailable online at ScienceDirect
Available online a www.sciencedirec.com ScienceDirec Procedia Economics and Finance 8 ( 04 658 663 s Inernaional Conference 'Economic Scienific Research - Theoreical, Empirical and Pracical Approaches',
More informationFAIR VALUATION OF INSURANCE LIABILITIES. Pierre DEVOLDER Université Catholique de Louvain 03/ 09/2004
FAIR VALUATION OF INSURANCE LIABILITIES Pierre DEVOLDER Universié Caholique de Louvain 03/ 09/004 Fair value of insurance liabiliies. INTRODUCTION TO FAIR VALUE. RISK NEUTRAL PRICING AND DEFLATORS 3. EXAMPLES
More informationOption pricing and hedging in jump diffusion models
U.U.D.M. Projec Repor 21:7 Opion pricing and hedging in jump diffusion models Yu Zhou Examensarbee i maemaik, 3 hp Handledare och examinaor: Johan ysk Maj 21 Deparmen of Mahemaics Uppsala Universiy Maser
More informationUCLA Department of Economics Fall PhD. Qualifying Exam in Macroeconomic Theory
UCLA Deparmen of Economics Fall 2016 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and you are o complee each par. Answer each par in a separae bluebook. All
More informationHEDGING SYSTEMATIC MORTALITY RISK WITH MORTALITY DERIVATIVES
HEDGING SYSTEMATIC MORTALITY RISK WITH MORTALITY DERIVATIVES Workshop on moraliy and longeviy, Hannover, April 20, 2012 Thomas Møller, Chief Analys, Acuarial Innovaion OUTLINE Inroducion Moraliy risk managemen
More informationOption Valuation of Oil & Gas E&P Projects by Futures Term Structure Approach. Hidetaka (Hugh) Nakaoka
Opion Valuaion of Oil & Gas E&P Projecs by Fuures Term Srucure Approach March 9, 2007 Hideaka (Hugh) Nakaoka Former CIO & CCO of Iochu Oil Exploraion Co., Ld. Universiy of Tsukuba 1 Overview 1. Inroducion
More informationMay 2007 Exam MFE Solutions 1. Answer = (B)
May 007 Exam MFE Soluions. Answer = (B) Le D = he quarerly dividend. Using formula (9.), pu-call pariy adjused for deerminisic dividends, we have 0.0 0.05 0.03 4.50 =.45 + 5.00 D e D e 50 e = 54.45 D (
More informationBlack-Scholes Model and Risk Neutral Pricing
Inroducion echniques Exercises in Financial Mahemaics Lis 3 UiO-SK45 Soluions Hins Auumn 5 eacher: S Oriz-Laorre Black-Scholes Model Risk Neural Pricing See Benh s book: Exercise 44, page 37 See Benh s
More informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 05 h November 007 Subjec CT8 Financial Economics Time allowed: Three Hours (14.30 17.30 Hrs) Toal Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1) Do no wrie your
More informationMatematisk statistik Tentamen: kl FMS170/MASM19 Prissättning av Derivattillgångar, 9 hp Lunds tekniska högskola. Solution.
Maemaisk saisik Tenamen: 8 5 8 kl 8 13 Maemaikcenrum FMS17/MASM19 Prissäning av Derivaillgångar, 9 hp Lunds ekniska högskola Soluion. 1. In he firs soluion we look a he dynamics of X using Iôs formula.
More informationIncorporating Risk Preferences into Real Options Models. Murat Isik
Incorporaing Risk Preferences ino Real Opions Models Mura Isik Assisan Professor Agriculural Economics and Rural Sociology Universiy of Idaho 8B Ag Science Building Moscow, ID 83844 Phone: 08-885-714 E-mail:
More informationBrownian motion. Since σ is not random, we can conclude from Example sheet 3, Problem 1, that
Advanced Financial Models Example shee 4 - Michaelmas 8 Michael Tehranchi Problem. (Hull Whie exension of Black Scholes) Consider a marke wih consan ineres rae r and wih a sock price modelled as d = (µ
More informationMEAN-VARIANCE ASSET ALLOCATION FOR LONG HORIZONS. Isabelle Bajeux-Besnainou* James V. Jordan** January 2001
MEAN-VARIANCE ASSE ALLOCAION FOR LONG HORIZONS Isabelle Bajeux-Besnainou* James V. Jordan** January 1 *Deparmen of Finance he George Washingon Universiy 3 G S., NW Washingon DC 5-994-559 (fax 514) bajeux@gwu.edu
More informationAn Analytical Implementation of the Hull and White Model
Dwigh Gran * and Gauam Vora ** Revised: February 8, & November, Do no quoe. Commens welcome. * Douglas M. Brown Professor of Finance, Anderson School of Managemen, Universiy of New Mexico, Albuquerque,
More informationSingle Premium of Equity-Linked with CRR and CIR Binomial Tree
The 7h SEAMS-UGM Conference 2015 Single Premium of Equiy-Linked wih CRR and CIR Binomial Tree Yunia Wulan Sari 1,a) and Gunardi 2,b) 1,2 Deparmen of Mahemaics, Faculy of Mahemaics and Naural Sciences,
More informationTentamen i 5B1575 Finansiella Derivat. Måndag 27 augusti 2007 kl Answers and suggestions for solutions.
Tenamen i 5B1575 Finansiella Deriva. Måndag 27 augusi 2007 kl. 14.00 19.00. Answers and suggesions for soluions. 1. (a) For he maringale probabiliies we have q 1 + r d u d 0.5 Using hem we obain he following
More informationJarrow-Lando-Turnbull model
Jarrow-Lando-urnbull model Characerisics Credi raing dynamics is represened by a Markov chain. Defaul is modelled as he firs ime a coninuous ime Markov chain wih K saes hiing he absorbing sae K defaul
More informationSystemic Risk Illustrated
Sysemic Risk Illusraed Jean-Pierre Fouque Li-Hsien Sun March 2, 22 Absrac We sudy he behavior of diffusions coupled hrough heir drifs in a way ha each componen mean-revers o he mean of he ensemble. In
More informationOn the Impact of Inflation and Exchange Rate on Conditional Stock Market Volatility: A Re-Assessment
MPRA Munich Personal RePEc Archive On he Impac of Inflaion and Exchange Rae on Condiional Sock Marke Volailiy: A Re-Assessmen OlaOluwa S Yaya and Olanrewaju I Shiu Deparmen of Saisics, Universiy of Ibadan,
More informationPricing formula for power quanto options with each type of payoffs at maturity
Global Journal of Pure and Applied Mahemaics. ISSN 0973-1768 Volume 13, Number 9 (017, pp. 6695 670 Research India Publicaions hp://www.ripublicaion.com/gjpam.hm Pricing formula for power uano opions wih
More informationAnalyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective
Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive AFIR Colloquium Madrid, Spain June 22, 2 Alexander Bohner and Nadine Gazer Universiy
More informationProceedings of the 48th European Study Group Mathematics with Industry 1
Proceedings of he 48h European Sudy Group Mahemaics wih Indusry 1 ADR Opion Trading Jasper Anderluh and Hans van der Weide TU Delf, EWI (DIAM), Mekelweg 4, 2628 CD Delf jhmanderluh@ewiudelfnl, JAMvanderWeide@ewiudelfnl
More informationResearch Article On Option Pricing in Illiquid Markets with Jumps
ISRN Mahemaical Analysis Volume 213, Aricle ID 56771, 5 pages hp://dx.doi.org/1.1155/213/56771 Research Aricle On Opion Pricing in Illiquid Markes wih Jumps Youssef El-Khaib 1 and Abdulnasser Haemi-J 2
More informationOptimal Consumption and Investment with Habit Formation and Hyperbolic discounting. Mihail Zervos Department of Mathematics London School of Economics
Oimal Consumion and Invesmen wih Habi Formaion and Hyerbolic discouning Mihail Zervos Dearmen of Mahemaics London School of Economics Join work wih Alonso Pérez-Kakabadse and Dimiris Melas 1 The Sandard
More informationVALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION
Aca Universiais Mahiae Belii ser. Mahemaics, 16 21, 17 23. Received: 15 June 29, Acceped: 2 February 21. VALUATION OF THE AMERICAN-STYLE OF ASIAN OPTION BY A SOLUTION TO AN INTEGRAL EQUATION TOMÁŠ BOKES
More informationOptimal Early Exercise of Vulnerable American Options
Opimal Early Exercise of Vulnerable American Opions March 15, 2008 This paper is preliminary and incomplee. Opimal Early Exercise of Vulnerable American Opions Absrac We analyze he effec of credi risk
More informationErratic Price, Smooth Dividend. Variance Bounds. Present Value. Ex Post Rational Price. Standard and Poor s Composite Stock-Price Index
Erraic Price, Smooh Dividend Shiller [1] argues ha he sock marke is inefficien: sock prices flucuae oo much. According o economic heory, he sock price should equal he presen value of expeced dividends.
More informationFINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004
FINAL EXAM EC26102: MONEY, BANKING AND FINANCIAL MARKETS MAY 11, 2004 This exam has 50 quesions on 14 pages. Before you begin, please check o make sure ha your copy has all 50 quesions and all 14 pages.
More informationPARAMETER ESTIMATION IN A BLACK SCHOLES
PARAMETER ESTIMATIO I A BLACK SCHOLES Musafa BAYRAM *, Gulsen ORUCOVA BUYUKOZ, Tugcem PARTAL * Gelisim Universiy Deparmen of Compuer Engineering, 3435 Isanbul, Turkey Yildiz Technical Universiy Deparmen
More informationAgenda. What is an ESG? GIRO Convention September 2008 Hilton Sorrento Palace
GIRO Convenion 23-26 Sepember 2008 Hilon Sorreno Palace A Pracical Sudy of Economic Scenario Generaors For General Insurers Gareh Haslip Benfield Group Agenda Inroducion o economic scenario generaors Building
More informationValuing Real Options on Oil & Gas Exploration & Production Projects
Valuing Real Opions on Oil & Gas Exploraion & Producion Projecs March 2, 2006 Hideaka (Hugh) Nakaoka Former CIO & CCO of Iochu Oil Exploraion Co., Ld. Universiy of Tsukuba 1 Overview 1. Inroducion 2. Wha
More informationModels of Default Risk
Models of Defaul Risk Models of Defaul Risk 1/29 Inroducion We consider wo general approaches o modelling defaul risk, a risk characerizing almos all xed-income securiies. The srucural approach was developed
More informationDYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Krzysztof Jajuga Wrocław University of Economics
DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 2006 Krzyszof Jajuga Wrocław Universiy of Economics Ineres Rae Modeling and Tools of Financial Economerics 1. Financial Economerics
More informationVolatility and Hedging Errors
Volailiy and Hedging Errors Jim Gaheral Sepember, 5 1999 Background Derivaive porfolio bookrunners ofen complain ha hedging a marke-implied volailiies is sub-opimal relaive o hedging a heir bes guess of
More informationA UNIFIED PDE MODELLING FOR CVA AND FVA
AWALEE A UNIFIED PDE MODELLING FOR CVA AND FVA By Dongli W JUNE 2016 EDITION AWALEE PRESENTATION Chaper 0 INTRODUCTION The recen finance crisis has released he counerpary risk in he valorizaion of he derivaives
More informationOptimal Portfolios when Volatility can Jump
Opimal Porfolios when Volailiy can Jump Nicole Branger Chrisian Schlag Eva Schneider Finance Deparmen, Goehe Universiy, Meronsr. 7/Uni-Pf 77, D-60054 Frankfur am Main, Germany. Fax: +49-(0)69-798-22788.
More informationLi Gan Guan Gong Michael Hurd. April, 2006
Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis Li Gan Guan Gong Michael Hurd April, 2006 ABSTRACT When he age of deah is uncerain, individuals will leave bequess even if hey have
More informationChange of measure and Girsanov theorem
and Girsanov heorem 80-646-08 Sochasic calculus I Geneviève Gauhier HEC Monréal Example 1 An example I Le (Ω, F, ff : 0 T g, P) be a lered probabiliy space on which a sandard Brownian moion W P = W P :
More informationOptimal Portfolio Choices and the Determination of. Housing Rents in the Context of Housing Price Uncertainty
Opimal Porfolio Choices and he Deerminaion of Housing Rens in he Conex of Housing Price Uncerainy Gang-Zhi FAN * Ming PU Xiaoying DENG Seow Eng ONG Nov 015 Absrac This paper develops a uiliy indifference-based
More informationAsymmetry and Leverage in Stochastic Volatility Models: An Exposition
Asymmery and Leverage in Sochasic Volailiy Models: An xposiion Asai, M. a and M. McAleer b a Faculy of conomics, Soka Universiy, Japan b School of conomics and Commerce, Universiy of Wesern Ausralia Keywords:
More informationEffect of Probabilistic Backorder on an Inventory System with Selling Price Demand Under Volume Flexible Strategy
Inernaional Transacions in Mahemaical Sciences and compuers July-December 0, Volume 5, No., pp. 97-04 ISSN-(Prining) 0974-5068, (Online) 0975-75 AACS. (www.aacsjournals.com) All righ reserved. Effec of
More informationMA Advanced Macro, 2016 (Karl Whelan) 1
MA Advanced Macro, 2016 (Karl Whelan) 1 The Calvo Model of Price Rigidiy The form of price rigidiy faced by he Calvo firm is as follows. Each period, only a random fracion (1 ) of firms are able o rese
More informationThe Investigation of the Mean Reversion Model Containing the G-Brownian Motion
Applied Mahemaical Sciences, Vol. 13, 219, no. 3, 125-133 HIKARI Ld, www.m-hikari.com hps://doi.org/1.12988/ams.219.918 he Invesigaion of he Mean Reversion Model Conaining he G-Brownian Moion Zixin Yuan
More informationNumerical probabalistic methods for high-dimensional problems in finance
Numerical probabalisic mehods for high-dimensional problems in finance The American Insiue of Mahemaics This is a hard copy version of a web page available hrough hp://www.aimah.org Inpu on his maerial
More informationAn Incentive-Based, Multi-Period Decision Model for Hierarchical Systems
Wernz C. and Deshmukh A. An Incenive-Based Muli-Period Decision Model for Hierarchical Sysems Proceedings of he 3 rd Inernaional Conference on Global Inerdependence and Decision Sciences (ICGIDS) pp. 84-88
More informationThe Binomial Model and Risk Neutrality: Some Important Details
The Binomial Model and Risk Neuraliy: Some Imporan Deails Sanjay K. Nawalkha* Donald R. Chambers** Absrac This paper reexamines he relaionship beween invesors preferences and he binomial opion pricing
More informationOPTIMALITY OF MOMENTUM AND REVERSAL
OPTIMALITY OF MOMENTUM AND REVERSAL XUE-ZHONG HE, KAI LI AND YOUWEI LI *Finance Discipline Group, UTS Business School Universiy of Technology, Sydney PO Box 13, Broadway, NSW 7, Ausralia **School of Managemen
More informationt=1 C t e δt, and the tc t v t i t=1 C t (1 + i) t = n tc t (1 + i) t C t (1 + i) t = C t vi
Exam 4 is Th. April 24. You are allowed 13 shees of noes and a calculaor. ch. 7: 137) Unless old oherwise, duraion refers o Macaulay duraion. The duraion of a single cashflow is he ime remaining unil mauriy,
More informationPrinciples of Finance CONTENTS
Principles of Finance CONENS Value of Bonds and Equiy... 3 Feaures of bonds... 3 Characerisics... 3 Socks and he sock marke... 4 Definiions:... 4 Valuing equiies... 4 Ne reurn... 4 idend discoun model...
More informationPricing Vulnerable American Options. April 16, Peter Klein. and. Jun (James) Yang. Simon Fraser University. Burnaby, B.C. V5A 1S6.
Pricing ulnerable American Opions April 16, 2007 Peer Klein and Jun (James) Yang imon Fraser Universiy Burnaby, B.C. 5A 16 pklein@sfu.ca (604) 268-7922 Pricing ulnerable American Opions Absrac We exend
More informationSupplement to Models for Quantifying Risk, 5 th Edition Cunningham, Herzog, and London
Supplemen o Models for Quanifying Risk, 5 h Ediion Cunningham, Herzog, and London We have received inpu ha our ex is no always clear abou he disincion beween a full gross premium and an expense augmened
More informationYou should turn in (at least) FOUR bluebooks, one (or more, if needed) bluebook(s) for each question.
UCLA Deparmen of Economics Spring 05 PhD. Qualifying Exam in Macroeconomic Theory Insrucions: This exam consiss of hree pars, and each par is worh 0 poins. Pars and have one quesion each, and Par 3 has
More informationStochastic Interest Rate Approach of Pricing Participating Life Insurance Policies with Embedded Surrender Option
American Journal of Mahemaical and Compuer Modelling 28; 3(): -2 hp://www.sciencepublishinggroup.com/j/ajmcm doi:.648/j.ajmcm.283.2 Sochasic Ineres Rae Approach of ricing aricipaing Life Insurance olicies
More informationEFFICIENT POST-RETIREMENT ASSET ALLOCATION
EFFICIENT POST-RETIREMENT ASSET ALLOCATION Barry Freedman* ABSTRACT To examine pos-reiremen asse allocaion, an exension o he classic Markowiz risk-reurn framework is suggesed. Assuming ha reirees make
More informationA Decision Model for Investment Timing Using Real Options Approach
A Decision Model for Invesmen Timing Using Real Opions Approach Jae-Han Lee, Jae-Hyeon Ahn Graduae School of Managemen, KAIST 207-43, Cheongrangri-Dong, Dongdaemun-Ku, Seoul, Korea ABSTRACT Real opions
More informationEstimating Earnings Trend Using Unobserved Components Framework
Esimaing Earnings Trend Using Unobserved Componens Framework Arabinda Basisha and Alexander Kurov College of Business and Economics, Wes Virginia Universiy December 008 Absrac Regressions using valuaion
More informationFinancial Markets And Empirical Regularities An Introduction to Financial Econometrics
Financial Markes And Empirical Regulariies An Inroducion o Financial Economerics SAMSI Workshop 11/18/05 Mike Aguilar UNC a Chapel Hill www.unc.edu/~maguilar 1 Ouline I. Hisorical Perspecive on Asse Prices
More informationABSTRACT JEL: G11, G20, G23. KEYWORDS: Hedge Fund Replication, Dynamic Portfolio Optimization, Martingale Method, Malliavin Calculus INTRODUCTION
GLOBAL JOURNAL OF BUSINESS RESEARC VOLUME 4 NUMBER 4 A NEW EDGE FUND REPLICAION MEOD WI E DYNAMIC OPIMAL PORFOLIO Akihiko akahashi, he Universiy of okyo Kyo Yamamoo, he Universiy of okyo ABSRAC his paper
More information(1 + Nominal Yield) = (1 + Real Yield) (1 + Expected Inflation Rate) (1 + Inflation Risk Premium)
5. Inflaion-linked bonds Inflaion is an economic erm ha describes he general rise in prices of goods and services. As prices rise, a uni of money can buy less goods and services. Hence, inflaion is an
More informationUtility portfolio optimization with liability and multiple risky assets under the extended CIR model
Uiliy porfolio opimizaion wih liabiliy and muliple risky asses under he exended CIR model HAO CHANG Tianjin Polyechnic Universiy Deparmen of Mahemaics Bin-shui Wes Road 399, 3387 Tianjin CHINA ch8683897@126.com
More informationLeveraged Stock Portfolios over Long Holding Periods: A Continuous Time Model. Dale L. Domian, Marie D. Racine, and Craig A.
Leveraged Sock Porfolios over Long Holding Periods: A Coninuous Time Model Dale L. Domian, Marie D. Racine, and Craig A. Wilson Deparmen of Finance and Managemen Science College of Commerce Universiy of
More informationOn Valuing Equity-Linked Insurance and Reinsurance Contracts
On Valuing Equiy-Linked Insurance and Reinsurance Conracs Sebasian Jaimungal a and Suhas Nayak b a Deparmen of Saisics, Universiy of Torono, 100 S. George Sree, Torono, Canada M5S 3G3 b Deparmen of Mahemaics,
More informationOptimal investment problem for an insurer with dependent risks under the constant elasticity of variance (CEV) model
Opimal invesmen problem for an insurer wih dependen riss under he consan elasiciy of variance (CEV) model YAJIE WANG Tianjin Universiy School of Science 92, Weijin Road, 300072 Tianjin CHINA wangyajie92@126.com
More informationDynamic Asset Allocation with Commodities and Stochastic Interest Rates
World Review of Business Research Vol.. No. 4. July 0. Pp. 5 9 Dynamic Asse Allocaion wih Commodiies and Sochasic Ineres Raes Sakkakom Maneenop* his research aims a finding an explici invesmen policy wih
More informationTentamen i 5B1575 Finansiella Derivat. Torsdag 25 augusti 2005 kl
Tenamen i 5B1575 Finansiella Deriva. Torsdag 25 augusi 2005 kl. 14.00 19.00. Examinaor: Camilla Landén, el 790 8466. Tillåna hjälpmedel: Av insiuionen ulånad miniräknare. Allmänna anvisningar: Lösningarna
More informationCENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T. J. KEHOE MACROECONOMICS I WINTER 2011 PROBLEM SET #6
CENTRO DE ESTUDIOS MONETARIOS Y FINANCIEROS T J KEHOE MACROECONOMICS I WINTER PROBLEM SET #6 This quesion requires you o apply he Hodrick-Presco filer o he ime series for macroeconomic variables for he
More informationOptimal Tax-Timing and Asset Allocation when Tax Rebates on Capital Losses are Limited
Opimal Tax-Timing and Asse Allocaion when Tax Rebaes on Capial Losses are Limied Marcel Marekwica This version: January 15, 2007 Absrac Since Consaninides (1983) i is well known ha in a marke where capial
More informationINFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES.
INFORMATION ASYMMETRY IN PRICING OF CREDIT DERIVATIVES. Join work wih Ying JIAO, LPMA, Universié Paris VII 6h World Congress of he Bachelier Finance Sociey, June 24, 2010. This research is par of he Chair
More informationData-Driven Demand Learning and Dynamic Pricing Strategies in Competitive Markets
Daa-Driven Demand Learning and Dynamic Pricing Sraegies in Compeiive Markes Pricing Sraegies & Dynamic Programming Rainer Schlosser, Marin Boissier, Mahias Uflacker Hasso Planer Insiue (EPIC) April 30,
More informationFORECASTING WITH A LINEX LOSS: A MONTE CARLO STUDY
Proceedings of he 9h WSEAS Inernaional Conference on Applied Mahemaics, Isanbul, Turkey, May 7-9, 006 (pp63-67) FORECASTING WITH A LINEX LOSS: A MONTE CARLO STUDY Yasemin Ulu Deparmen of Economics American
More informationFinal Exam Answers Exchange Rate Economics
Kiel Insiu für Welwirhschaf Advanced Sudies in Inernaional Economic Policy Research Spring 2005 Menzie D. Chinn Final Exam Answers Exchange Rae Economics This exam is 1 ½ hours long. Answer all quesions.
More informationRisk-Neutral Probabilities Explained
Risk-Neural Probabiliies Explained Nicolas Gisiger MAS Finance UZH ETHZ, CEMS MIM, M.A. HSG E-Mail: nicolas.s.gisiger @ alumni.ehz.ch Absrac All oo ofen, he concep of risk-neural probabiliies in mahemaical
More informationAlexander L. Baranovski, Carsten von Lieres and André Wilch 18. May 2009/Eurobanking 2009
lexander L. Baranovski, Carsen von Lieres and ndré Wilch 8. May 2009/ Defaul inensiy model Pricing equaion for CDS conracs Defaul inensiy as soluion of a Volerra equaion of 2nd kind Comparison o common
More informationAN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM MODEL WITH STOCHASTIC INTEREST RATES
Inernaional Journal of Pure and Applied Mahemaics Volume 76 No. 4 212, 549-557 ISSN: 1311-88 (prined version url: hp://www.ijpam.eu PA ijpam.eu AN EASY METHOD TO PRICE QUANTO FORWARD CONTRACTS IN THE HJM
More informationA Note on Forward Price and Forward Measure
C Review of Quaniaive Finance and Accouning, 9: 26 272, 2002 2002 Kluwer Academic Publishers. Manufacured in The Neherlands. A Noe on Forward Price and Forward Measure REN-RAW CHEN FOM/SOB-NB, Rugers Universiy,
More informationHEDGING VOLATILITY RISK
HEDGING VOLAILIY RISK Menachem Brenner Sern School of Business New York Universiy New York, NY 00, U.S.A. Email: mbrenner@sern.nyu.edu Ernes Y. Ou ABN AMRO, Inc. Chicago, IL 60604, U.S.A. Email: Yi.Ou@abnamro.com
More informationChapter 8 Consumption and Portfolio Choice under Uncertainty
George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 8 Consumpion and Porfolio Choice under Uncerainy In his chaper we examine dynamic models of consumer choice under uncerainy. We coninue, as
More informationEquilibrium in Securities Markets with Heterogeneous Investors and Unspanned Income Risk
Equilibrium in Securiies Markes wih Heerogeneous Invesors and Unspanned Income Risk July 9, 29 Peer Ove Chrisensen School of Economics and Managemen, Aarhus Universiy, DK-8 Aarhus C, Denmark email: pochrisensen@econ.au.dk
More informationA Note on Missing Data Effects on the Hausman (1978) Simultaneity Test:
A Noe on Missing Daa Effecs on he Hausman (978) Simulaneiy Tes: Some Mone Carlo Resuls. Dikaios Tserkezos and Konsaninos P. Tsagarakis Deparmen of Economics, Universiy of Cree, Universiy Campus, 7400,
More informationForecasting with Judgment
Forecasing wih Judgmen Simone Manganelli DG-Research European Cenral Bank Frankfur am Main, German) Disclaimer: he views expressed in his paper are our own and do no necessaril reflec he views of he ECB
More informationMATURITY GUARANTEES EMBEDDED IN UNIT-LINKED CONTRACTS VALUATION & RISK MANAGEMENT *
ABSRAC MAURIY GUARANEES EMBEDDED IN UNI-LINKED CONRACS VALUAION & RISK MANAGEMEN * Floren PERNOUD hierry FAVRE-BONVIN A key feaure of mauriy guaranees aached o uni-linked life insurance conracs is he uncerainy
More informationFee Structure and Surrender Incentives in Variable Annuities
Fee Srucure and Surrender Incenives in Variable Annuiies by Anne MacKay A hesis presened o he Universiy of Waerloo in fulfillmen of he hesis requiremen for he degree of Docor of Philosophy in Acuarial
More informationPricing options on defaultable stocks
U.U.D.M. Projec Repor 2012:9 Pricing opions on defaulable socks Khayyam Tayibov Examensarbee i maemaik, 30 hp Handledare och examinaor: Johan Tysk Juni 2012 Deparmen of Mahemaics Uppsala Universiy Pricing
More informationAdvanced Tools for Risk Management and Asset Pricing
MSc. Finance/CLEFIN 214/215 Ediion Advanced Tools for Risk Managemen and Asse Pricing May 215 Exam for Non-Aending Sudens Soluions Time Allowed: 13 minues Family Name (Surname) Firs Name Suden Number (Mar.)
More informationOn the Edge of Completeness
On he Edge of Compleeness May 2000 Jean-Paul LAURENT Professor, ISFA Acuarial School, Universiy of Lyon, Scienific Advisor, BNP Paribas Correspondence lauren.jeanpaul@online.fr On he Edge of Compleeness:
More informationThe Impact of Stochastic Volatility on Pricing, Hedging, and Hedge Efficiency of Variable Annuity Guarantees
The Impac of Sochasic Volailiy on Pricing, Hedging, and Hedge Efficiency of Variable Annuiy Guaranees Alexander Kling *, Frederik Ruez and Jochen Ruß This Version: Augus 14, 2009 Absrac We analyze differen
More informationLecture Notes to Finansiella Derivat (5B1575) VT Note 1: No Arbitrage Pricing
Lecure Noes o Finansiella Deriva (5B1575) VT 22 Harald Lang, KTH Maemaik Noe 1: No Arbirage Pricing Le us consider a wo period marke model. A conrac is defined by a sochasic payoff X a bounded sochasic
More informationDynamic Programming Applications. Capacity Expansion
Dynamic Programming Applicaions Capaciy Expansion Objecives To discuss he Capaciy Expansion Problem To explain and develop recursive equaions for boh backward approach and forward approach To demonsrae
More informationParameter Uncertainty: The Missing Piece of the Liquidity Premium Puzzle?
Parameer Uncerainy: The Missing Piece of he Liquidiy Premium Puzzle? Ferenc Horvah Tilburg Universiy November 14, 2016 Absrac I analyze a dynamic invesmen problem wih sochasic ransacion cos and parameer
More informationASSIGNMENT BOOKLET. M.Sc. (Mathematics with Applications in Computer Science) Mathematical Modelling (January 2014 November 2014)
ASSIGNMENT BOOKLET MMT-009 M.Sc. (Mahemaics wih Applicaions in Compuer Science) Mahemaical Modelling (January 014 November 014) School of Sciences Indira Gandhi Naional Open Universiy Maidan Garhi New
More information