Article from: ARCH Proceedings

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1 Artcle from: ARCH 204. Proceedngs July 3-August 3, 203

2 Opton Prcng Wthout Tears: Valung Equty-Lnked Death Benefts Elas S. W. Shu Department of Statstcs & Actuaral Scence The Unversty of Iowa U.S.A. Jont work wth Hans U. Gerber & Halang Yang

3 Let T denote the tme-untl-death random varable for a lfe aged. Let S(t) be the tme-t prce of a stock or mutual fund. Consder death benefts that depend on the value of S(T ),.e., consder b(s(t )) for some functon b(.). Eamples: b(s) = Ma(s, K) b(s) = (s K) + Problem: Evaluate δ T E[e b(s(t ))] where the epectaton s taken wth respect to an approprate probablty dstrbuton and δ s a contnuously compounded nterest rate.

4 δt E[e b(s(t ))] = δt E[ E[ e b(s(t )) T ]] = 0 0 δt f T s ndependent of {S(t)}. E[ e b(s(t)) T =t] f (t) dt δt = E[ e b(s(t)) ] f T (t) dt T

5 So we want to calculate If then 0 f T (t) = c f τ (t), 0 δt E[e b(s(t))]f (t)dt. δt c 0 E[e b(s(t))]f (t)dt δt = E[e b(s(t))]f (t)dt = E[e b ) δτ T T c (S( τ ) ]. τ

6 The tme-untl-death densty functon can be appromated by lnear combnatons of eponental densty functons λ f T (t) c f τ (t) = c e. Thus, our valuaton problem becomes fndng E[e δτ b(s(τ))], where τ s an eponental random varable ndependent of {S(t)}. It turns out to be an elementary calculus eercse for geometrc Brownan moton {S(t)}. λ t

7 Let S(t) = S(0)e µt + σz(t), t 0, where {Z(t)} s a standard Brownan moton. Let τ be an ndependent eponental random varable wth mean /λ. Then, E[e δτ b(s(τ), Ma{S(t); 0 t τ})] m 2λ m α ( β α)m = 2 b( S(0)e, S(0)e ) e d e dm σ 0 where α < 0 and β > 0 are the solutons of ½σ µ (λ + δ) = 0.

8 E[e δτ b(s(τ), Ma{S(t); 0 t τ})] = m 2λ m α ( β α)m b( S(0)e, S(0)e ) e d e dm. 2 σ 0 Eamples: b(s, u) = (s K) + b(s, u) = (K s) + b(s, u) = u call opton put opton hgh water mark payoff

9 Barrer Optons Assume S(0) < B, a barrer. b(s, u) = I(u < B) π(s) Up-and-out opton b(s, u) = I(u B) π(s) Up-and-n opton Useful for ncorporatng lapses or surrenders.

10 Assume S(t) = S(0)e X(t), t 0, where N (t) N (t) X(t) =µ t +σ Z(t) + J K m ν ω = k = v f () = A v e, > 0 J = n f () B w e, 0 K m = w = > A =, B = n = = k

11 S(t) = S(0)e X(t), t 0 Runnng mamum M(t) := Ma{X(u); 0 u t} Runnng mnmum m(t) := Mn{X(u); 0 u t} Because {X(u)} s a Levy process, () M(τ) and [X(τ) M(τ)] are ndependent random varables, () [X(τ) Μ(τ)] has the same dstrbuton as m(τ). () s hard to prove; () s easy.

12 In fact, () s true for each fed t. X(t) M(t) = X(t) Ma{X(s); 0 s t} = X(t) + Mn{ X(s); 0 s t} = Mn{X(t) X(s); 0 s t} = Mn{X(t s); 0 s t} n dstrbuton = Mn{X(s); 0 s t} = m(t)

13 Runnng mamum M(t) := Ma{X(u); 0 u t} Runnng mnmum m(t) := Mn{X(u); 0 u t} () M(τ) and [X(τ) M(τ)] are ndependent r.v. s. () [X(τ) M(τ)] and m(τ) have the same dstrbuton. Then, E[e zx(τ) ] = E[e z[x(τ) Μ(τ)+M(τ)] ] = E[e z[x(τ) M(τ)] ] E[e zm(τ) ] = E[e zm(τ) ] E[e zm(τ) ], whch s a verson of Wener-Hopf factorzaton.

14 Assume where N (t) N (t) X(t) =µ t +σ Z(t) + J K m = k = f () = A v e, > 0 J = f () = B w e, > 0 Then, E[e zx(t) ] = e tψ(z) for each t 0, wth n K = m ν v w 2 2 z z (z) =µ z + ½σ z +ν A ω B v z w + z ω = = n k

15 can be etended by analytc contnuaton. The moment-generatng functon of X(τ) s E[e zx(τ) ] = E[E[e zx(τ) τ]] m 2 2 z z (z) =µ z + ½σ z +ν A ω B v z w + z = = n = E[e Ψ(z)τ ] λ = λ Ψ (z). The zeros of the RHS are the poles of Ψ(z). The poles of the RHS are the zeros of λ Ψ(z).

16 Label the parameters (the poles of Ψ(z)) such that v < v 2 < < v m w < w 2 < < w n If the weghts A s and B s are postve, then <α n+ < w n <...< w <α <0<β < v <...<v m <β m+ <

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19 Label the parameters (the poles of Ψ(z)) such that v < v 2 < < v m w < w 2 < < w n If the weghts A s and B s are postve, then <α n+ < w n <...< w <α <0<β < v <...<v m <β m+ < Wener-Hopf: E[e zx(τ) ] = E[e zm(τ) ] E[e zm(τ) ]. For z > 0, 0 < E[e zm(τ) ]. No postve zeros or poles. For z < 0, 0 < E[e zm(τ) ]. No negatve zeros or poles.

20 <α n+ < w n <...< w <α <0<β < v <...<v m <β m+ < n n+ zm( τ) E[e ] (z + w ) = = z α m m+ zm( τ) E[e ] (z v ) = = z β n n zm( ) z w + τ + α E[e ] = = w = z α zm( τ) E[e ] m v m z + β = = v = z β

21 zm( τ) E[e ] m v m z + β = v β z = = m+ m v m β + k β βk = v β β β k= =, = k k k z. m+ Thus, f ( ) = b e k, > 0, where M( τ) b k k= k β mv m β + k β = β v = β, = k βk k

22 zm( τ) E[e ] n z + w n+ α = w z α = = n+ n α n k + w + α αk = w α α z α k= = =, k k k n+ Thus, f ( ) e k, 0, m( τ) k= k α n α n k + w + α where ak = ( αk) w α α = a < =, = k k

23 For y ma(, 0), f X(τ), M(τ) (, y) = f M(τ), X(τ) Μ(τ) (y, y) = f M(τ) (y) f X(τ) Μ(τ) ( y) = f M(τ) (y) f m(τ) ( y) m+ n+ β y α ( y) k k = = k = be ae m+ n+ = ab e e k = = k α ( β α )y k

24 Further Work. Use SOA CAE research grant to hre graduate students to estmate the parameters and to program the formulas. 2. Bnomal tree verson.

25 References Asmussen, S., Avram, F., Pstorus, M.R., Russan and Amercan put optons under eponental phase-type Lévy models. Stochastc Processes and Ther Applcatons 09, 79-. Bowers, N., Gerber, H.U., Hckman, J., Jones, D., Nesbtt, C., 997. Actuaral Mathematcs, 2nd ed., Schaumburg, IL: Socety of Actuares. Ca, N., Kou, S.G., 20. Opton prcng under a med-eponental ump dffuson model. Management Scence, 57, Dufresne, D., Fttng combnatons of eponentals to probablty dstrbutons. Appled Stochastc Models n Busness and Industry 23, Dufresne, D., Stochastc lfe annutes. North Amercan Actuaral Journal (),

26 Gerber, H.U., Shu, E.S.W., Yang, H., 202. Valung equty-lnked death benefts and other contngent optons: a dscounted densty approach. Insurance: Mathematcs and Economcs 5, Gerber, H.U., Shu, E.S.W., Yang, H., 203. Valung equty-lnked death benefts n ump dffuson models. Insurance: Mathematcs and Economcs, 53, Kou, S.G., Jump-dffuson models for asset prcng n fnancal engneerng. In Brge, J. R., Lnetsky, V. (Eds.). Handbooks n Operatons Research and Management Scence, Vol. 5. Elsever, Mordeck, E., The dstrbuton of the mamum of a Lévy process wth postve umps of phase-type. Theory of Stochastc Processes, 8 (24),