Valuation of EquityLinked Insurance Products and Practical Issues in Equity Modeling. March 12, 2013


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1 Valuation of EquityLinked Insurance Products and Practical Issues in Equity Modeling March 12, 2013 The University of Hong Kong (A SOA Center of Actuarial Excellence) Session 2 Valuation of EquityLinked Insurance Products Professor, ASA
2 Department of Statistics and Actuarial Science The University of Hong Kong Hong Kong Based on a paper with Hans Gerber and Elias Shiu
3 Introduction Equitylinked products are very popular in the market nowadays. Example: Guaranteed Minimum Death Benefits Payoff: max(s(t x ), K) = S(T x ) + [K S(T x )] + = K + [S(T x ) K] +, where T x is the timeuntildeath random variable for a life age x, S(t) is the price of equityindex at time t, and K is the guaranteed amount.
4 What is a contingent option? Option s payoff, e.g. European call: (S(T ) K) + Contingent option s payoff: (S(T x ) K) +, where T x is a random variable, independent of S(t).
5 Notation S(t), t 0, value of one unit of a fund at time t T x time of death of a life age x Equitylinked death benefit defined by b(s): Payment b(s(t x )) at time T x Problem: Calculate E[e δtx b(s(t x ))] δ > 0 valuation force of interest
6 Distribution of T x Any continuous distribution on (0, ) can be approximated by a linear combination of exponential distributions f Tx (t) = n A i λ i e λ i t = i=1 n A i f i (t), t > 0, i=1
7 Distribution of T x = = = = E[e δtx b(s(t x ))] 0 0 e δt E[b(S(t))]f Tx (t)dt [ n e δt E[b(S(t))] i=1 ] A i f i (t) dt n A i e δt E[b(S(t))]f i (t)dt i=1 0 n A i E[e δτ i b(s(τ i ))]. i=1
8 The distribution of T x can be approximated by a linear combination of exponential distributions. Thus we limit ourself to the following problem: Calculate E[e δτ b(s(τ))] with τ an exponential random variable notation: E[τ] = 1/λ.
9 Model for S(t): S(t) = S(0)e X (t), t 0 with (1) X (t) = µt + σw (t) W (t) is a standard Brownian motion E[X (t)] = µt, Var[X (t)] = σ 2 t or (2) jump diffusion upward jumps: exponential mean 1/v, frequency ν downward jumps: exponential mean 1/w, frequency ω
10 In this talk, we focus on (1). Thus {X (t)} Brownian motion parameters µ, σ 2, D = 1 2 σ2 E[e zx (t) ] = e tψ(z) with Ψ(z) = Dz 2 + µz We need the distribution of X (τ) (well known, exponential stopping of a Wiener process )
11 E[e zx (τ) ] = E[E[e zx (τ) τ]] = E[e τψ(z) λ ] = λ Ψ(z) = λ Dz 2 µz + λ. Now we rewrite this with partial fractions. We need α < 0, β > 0 solutions of Dz 2 + µz λ = 0
12 Then E[e zx (τ) ] = with κ = αβ β α κ z α κ z β Thus the pdf of X (τ) is { κe f X (τ) (x) = αx, if x 0, κe βx, if x > 0,
13 Discounted density functions: f δ X (τ) (x) = E[e δτ f (x, τ)] = λ = λ λ + δ 0 e (λ+δ)t f (x, t)dt (λ + δ) e (λ+δ)t f (x, t)dt 0 } {{ } = f X (τ) (x) with λ replaced by λ + δ
14 Thus we find the following Recipe: Discounted density of X (τ) = λ λ+δ pdf of X (τ) with λ replaced by λ + δ
15 f δ X (τ) (x) = { κe αx, if x 0, κe βx, if x 0, where α < 0 and β > 0 are the solutions of Dξ 2 + µξ (λ + δ) = 0 and κ = λ αβ λ+δ β α.
16 Value of a death benefit defined by b(s): E[e δτ b(s(τ))] = λ = = κ 0 0 e (λ+δ)t b(s(0)e x )f (x, t)dxdt b(s(0)e x )f δ X (τ) (x)dx b(s(0)e x )e αx dx + κ 0 b(s(0)e x )e βx dx
17 Factorization formula If τ is exponential with mean 1/λ, then the following factorization formula holds, E[e δτ g τ (X )] = E[e δτ ] E[g τ (X )], where τ is an exponential random variable with mean 1/(λ + δ) and independent of X. Remarks (i) E[e δτ ] = λ λ+δ. (ii) The condition δ > 0 can be replaced by the condition δ > λ.
18 Examples (1) b(s) = (K s) +, K < S(0) outofthemoney put option E[e δτ (K S(τ)) + ] = (K S(0) e x ) + fx δ (τ)(x) dx = κk [ K ] α α(1 α) S(0) (2) b(s) = (s K) +, K > S(0) outofthemoney call option E[e δτ (S(τ) K) + ] = κk [ S(0) β(β 1) K ] β
19 The Greek letters (1) of the outofthemoney put option κ [ ] K (α 1) 1 α S(0) (2) of the outofthemoney call option [ κ S(0) β 1 K ] β 1
20 The Greek letters (1) Γ of the outofthemoney put option [ ] κ K (α 2) K S(0) (2) Γ of the outofthemoney call option κ K [ S(0) K ] β 2
21 inthemoney options (3) b(s) = (K s) +, K > S(0) put (4) b(s) = (s K) +, K < S(0) call Use the putcall parity and (1) and (2) [K S(τ)] + [S(τ) K] + = K S(τ) yields E[e δτ [K S(τ)] + ] E[e δτ [S(τ) K] + ] = λ λ + δ K E[e δτ S(τ)]
22 Inthemoney formulas Call E[e δτ [S(τ) K] + ] if S(0) > K [ ] κk K α = λ α(1 α) S(0) λ + δ K + E[e δτ S(τ)] Put E[e δτ [K S(τ)] + ] if S(0) < K [ ] κk S(0) β = + λ β(β 1) K λ + δ K E[e δτ S(τ)]
23 The Greek letters (under risk neutral probability) (1) of the inthemoney call option 1 κ [ ] K (α 1) 1 α S(0) (2) of the inthemoney put option [ ] κ S(0) β 1 1 β 1 K
24 The Greek letters (under risk neutral probability) (1) Γ of the inthemoney call option [ ] κ K (α 2) K S(0) (2) Γ of the inthemoney put option κ K [ S(0) K ] β 2
25 Illustration We consider 90strike lifecontingent call and put options on a stock with initial price S(0) = 100. We assume δ = 8% and µ = δ D. We assume that the distribution of T x is exponential with mean 125/6. Table 1: Contingent call and put values call put σ = σ = σ = σ =
26 rollup GMDB options Payoff: [Ke pτ S(τ)] + at time of death p: rollup rate Discounted payoff e δτ [Ke pτ S(τ)] + = e (δ p)τ [K e pτ S(τ)] + Valuation of a put option, with the substitutions δ δ p, µ µ p
27 T year contingent options Finite expiry date T > 0 Payoff [K S(τ)] + I (τ T ) Can be written as [K S(τ)] + [K S(τ)] + I (τ>t ) The time0 cost of the T year deferred contingent put option is E[e δτ [K S(τ)] + I (τ>t ) ] = Pr(τ > T )E[e δτ [K S(τ)] + τ > T ] = e (λ+δ)t E[e δ(τ T ) [K S(T )e X (τ) X (T ) ] + τ > T ].
28 The conditional expectation given S(T ) is [ ] κk K α I α(1 α) S(T ) (S(T )>K) { [ ] κk S(T ) β + + λ } β(β 1) K λ + δ K E[e δτ+x (τ) ]S(T ) I (S(T )<K). Its expectation can be evaluated by the factorization formula in the method of Esscher transforms.
29 Result: [ ] κk K α Φ( z α ) + κk [ ] S(0) β Φ(z β ) α(1 α) S(0) β(β 1) K +e (λ+δ)t λ λ + δ KΦ(z 0) e (λ+δ ϑ)t E[e δτ S(τ)]Φ(z 1 ). where z h = k (µ + hσ2 )T σ T,
30 Value of the T year Kstrike contingent put option S(0) > K S(0) < K [ ] κk K α Φ(z α ) κk [ ] S(0) β Φ(z β ) α(1 α) S(0) β(β 1) K e (λ+δ)t λ λ + δ KΦ(z 0) + e (λ+δ ϑ)t E[e δτ S(τ)]Φ(z 1 ) [ ] κk K α Φ( z α ) + κk [ ] S(0) β Φ( z β ) α(1 α) S(0) β(β 1) K + λ λ + δ K[1 e (λ+δ)t Φ(z 0 )] E[e δτ S(τ)][1 e (λ+δ ϑ)t Φ(z 1 )]
31 Illustration We consider T year 90strike lifecontingent put options on a stock with initial price S(0) = 100. We assume δ = 8% and µ = δ D. We assume that the distribution of T x is exponential with mean 125/6. Table 2: Contingent T =year put values T σ = σ = σ = σ =
32 M(t) = max 0 s t X (s) running maximum m(t) = min 0 s t X (s) running minimum For certain options we need f M(τ) (x) pdf of M(τ) f m(τ) (x) pdf of m(τ) It is well known that f M(τ) (x) = βe βx, x > 0, f m(τ) (x) = αe αx, x < 0.
33 Proof: Stop the martingale {e βx (t) I (τ>t) } at the time min(τ, first passage time at x) and use the optional sampling theorem: 1 = 0 + e βx Pr(M(τ) > x) Thus Pr(M(τ) > x) = e βx and f M(τ) (x) = βe βx, x > 0.
34 Discounted density functions: fm(τ) δ (x) = λ e (λ+δ)t f M(t) (x)dt f δ M(τ) (x) = = 0 λ λ + δ λ λ+δ βe βx, x > 0, (λ + δ) e (λ+δ)t f M(t) (x)dt 0 } {{ } = f M(τ) (x) with λ replaced by λ + δ where β > 0 solution of Dz 2 + µz (λ + δ) = 0.
35 Similarly, f δ m(τ) (x) = λ λ+δ ( α)e αx, x < 0, where α < 0 solution of Dz 2 + µz (λ + δ) = 0.
36 Bivariate distributions: We know that for a Lévy process M(τ) and M(τ) X (τ) are independent M(τ) X (τ) has the same distribution as m(τ) = m(τ) f M(τ),M(τ) X (τ) (y, z) = f M(τ) (y)f M(τ) X (τ) (z) = f M(τ) (y)f m(τ) ( z) = αβe βy e αz for y 0, z 0.
37 Because of X (τ) = M(τ) [M(τ) X (τ)], we find that f X (τ),m(τ) (x, y) = f M(τ),M(τ) X (τ) (y, y x) = f M(τ) (y)f M(τ) X (τ) (y x) = αβe αx (β α)y for y max(x, 0)
38 Then the discounted density is f δ X (τ),m(τ) (x, y) = αβ λ λ + δ e αx (β α)y for y max(x, 0) where α < 0 and β > 0 are the solutions of Dz 2 + µz (λ + δ) = 0
39 Lookback options Many equityindexed annuities credit interest using a high water mark method or a low water mark method
40 Outofthemoney fixed strike lookback call option Payoff: Time0 value k [S(0)e M(τ) K] + [S(0)e y K]fM(τ) δ (y)dy = λ [S(0)βe (β 1)k λ + δ β 1 [ ] S(0) β =. λ λ + δ Another expression for the option value [ ] λ K S(0) β. D αβ(β 1) K K β 1 K Ke βk ]
41 Inthemoney fixed strike lookback call option Payoff Rewriting as max(h, S(0)e M(τ) ) K. H K + [S(0)e M(τ) H] + Time0 value { λ H K + H λ + δ β 1 [ S(0) H ] β }.
42 Floating strike lookback put option Payoff where H S(0). Time0 value { λ H + H λ + δ β 1 max(h, max S(t)) S(τ), (1) 0 t τ [ S(0) H ] β } E[e δτ S(τ)].
43 Floating strike lookback put option Special case: H = S(0), the time0 value λ β λ + δ β 1 S(0) E[e δτ S(τ)] = 1 α α E[e δτ S(τ)] E[e δτ S(τ)] = 1 α E[e δτ S(τ)]. (2) This result can be reformulated as E[e δτ max S(t)] = 0 t τ ( 1 α + 1 ) E[e δτ S(τ)].
44 Floating strike lookback put option Milevsky and Posner (2001) have evaluated (1) with a riskneutral stock price process and H = S(0). They also assume that the stock pays dividends continuously at a rate proportional to its price. With l denoting the dividend yield rate, δ = r, and µ = r D l, the RHS of (2) is 2D (r D l) + (r D l) 2 + 4D(λ + r) S(0) λ λ + l. Although it seems rather different from formula (38) on page 117 of Milevsky and Posner (2001), they are the same.
45 Fractional floating strike lookback put option Payoff Notice [γ max 0 t τ S(t) S(τ)] + = S(0)[γe M(τ) e X (τ) ] +. [γe M(τ) e X (τ) ] + = e M(τ) [γ e X (τ) M(τ) ] +
46 Fractional floating strike lookback put option Hence E[e δτ e M(τ) [γ e X (τ) M(τ) ] + ] = e y [γ e z ] + fm(τ),m(τ) X δ (τ)(y, z)dydz 0 0 = λ [ ][ ] e y e βy dy [γ e z ] + e αz dz D 0 = λ 1 D β 1 α(1 α) = γ 1 α λ β λ + δ (1 α)(β 1) = γ 1 α 1 α E[e δτ e X (τ) ]. γ 1 α 0
47 Fractional floating strike lookback put option This can be rewritten as E[e δτ [γe M(τ) e X (τ) ] + ] = γ 1 α E[e δτ (e M(τ) e X (τ) )]. Time0 value E[e δτ [γ max 0 t τ S(t) S(τ)] +] = γ1 α α E[e δτ S(τ)],
48 Outofthemoney fixed strike lookback put option Payoff [K S(0)e m(τ) ] +, Time0 value k [K S(0)e y ]fm(τ) δ (y)dy = λ [ ] K K α. λ + δ 1 α S(0)
49 Inthemoney fixed strike lookback put option Payoff K min(h, S(0)e m(τ) ) = K H + [H S(0)e m(τ) ] +, Time0 value { λ K H + H [ ] H α }. λ + δ 1 α S(0)
50 Floating strike lookback call option Payoff where 0 < H S(0). Time0 value E[e δτ S(τ)] + S(τ) min(h, min 0 t τ S(t)), λ { H + λ + δ H [ ] H α }. 1 α S(0) In the special case where H = S(0), the time0 value E[e δτ S(τ)] λ λ + δ α 1 α S(0) = E[e δτ S(τ)] β 1 β E[e δτ S(τ)] = 1 β E[e δτ S(τ)]. This result can be reformulated as ( )
51 Fractional floating strike lookback call option Payoff [S(τ) γ min 0 t τ S(t)] + = S(0)[e X (τ) γe m(τ) ] +. = S(0)e m(τ) [e X (τ) m(τ) γ] +
52 Fractional floating strike lookback call option Its expected discounted value is S(0) times the following expectation E[e δτ e m(τ) [e X (τ) m(τ) γ] + ] = λ [ 0 ][ ] e y e αy dy [e z γ] + e βz dz D 0 = λ 1 γ 1 β D 1 α β(β 1) 1 λ α = γ β 1 λ + δ (1 α)(β 1) 1 1 = γ β 1 β E[e δτ e X (τ) ].
53 Fractional floating strike lookback call option This can be rewritten as E[e δτ [e X (τ) γe m(τ) ] + ] = γ (β 1) E[e δτ (e X (τ) e m(τ) )]. We have E[e δτ [S(τ) γ min 0 t τ S(t)] +] = 1 βγ β 1 E[e δτ S(τ)].
54 Highlow option Payoff max(h, max S(t)) min(h, min S(t)), 0 t τ 0 t τ where 0 < H S(0) H. Time0 value { λ H + H [ ] S(0) β H + H [ ] H α }. λ + δ β 1 H 1 α S(0) In the special case where H = S(0) = H, time0 value λ β α S(0) λ + δ (β 1)(1 α) = β α αβ E[e δτ S(τ)]. This can be rewritten as ( 1 α + 1 ) E[e δτ S(τ)], β
55 Illustration We consider 90strike lifecontingent lookback call and put options on a stock with initial price S(0) = 100. We assume H = 100, δ = 8% and µ = δ D. We assume that the distribution of T x is exponential with mean 125/6. Table 1: Contingent call and put values call put σ = σ = σ = σ =
56 Barrier options A barrier option is an option whose payoff depends on whether or not the price of the underlying asset has reached a predetermined level or barrier. Knockout options are those which go out of existence if the asset price reaches the barrier, and knockin options are those which come into existence if the barrier is reached.
57 Parity relation Knockout option + Knockin option = Ordinary option. Notation: L denotes the barrier and l = ln[l/s(0)]
58 Upandout and upandin options (L > S(0) (l > 0)) Payoffs I ([max0 t τ S(t)]<L)b(S(τ)) = I (M(τ)<l) b(s(0)e X (τ) ) I ([max0 t τ S(t)] L)b(S(τ)) = I (M(τ) l) b(s(0)e X (τ) )
59 The expected discounted values Upandout Upandin λ D 0 = λ D l [ y I (y<l) b(s(0)e x )fx δ (τ),m(τ) ]dy (x, y)dx [ y l 0 b(s(0)e x )e αx dx ] e (β α)y dy [ y ] b(s(0)e x )e αx dx e (β α)y dy;
60 Downandout and downandin options (0 < L < S(0) (l < 0)) Payoffs I ([min0 t τ S(t)]>L)b(S(τ)) = I (m(τ)>l) b(s(0)e X (τ) ) I ([min0 t τ S(t)] L)b(S(τ)) = I (m(τ) l) b(s(0)e X (τ) )
61 The expected discounted values λ D 0 l [ y ] b(s(0)e x )e βx dx e (β α)y dy λ l [ ] b(s(0)e x )e βx dx e (β α)y dy, D y
62 Notation A 1 (n) = λ S(0) n D (n α)(β n), A 2 (n) = λ L n [ ] S(0) β, D (n α)(β n) L A 3 (n) = λ L n [ ] L α, D (n α)(β n) S(0) A 4 = λ K n [ ] K α = κk n [ ] K α, D (n α)(β α) S(0) n α S(0)
63 Notation A 5 = λ K n α L α [ ] S(0) β = κk n α L α [ ] S(0) β, D (n α)(β α) L n α L A 6 = λ K n [ ] S(0) β = κk n [ ] S(0) β, D (β n)(β α) K β n K A 7 = λ K (β n) L β [ ] L α = κk (β n) L β [ ] L α, D (β n)(β α) S(0) β n S(0) A 8 = λ [ ] K K α [ ] κk K α =, D α(1 α)(β α) S(0) α(1 α) S(0)
64 Notation A 9 = λ K 1 α L α [ ] S(0) β D α(1 α)(β α) L = κk 1 α L α [ ] S(0) β, α(1 α) L A 10 = λ [ ] K S(0) β = D β(β 1)(β α) K A 11 = λ K (β 1) L β [ ] L α D β(β 1)(β α) S(0) = κk (β 1) L β [ ] L α. β(β 1) S(0) κk β(β 1) [ ] S(0) β, K
65 Upandout allornothing call option The option value is 0, if L < K, λ l D 0 [ y k S(0)n e nx e αx dx]e (β α)y dy, if L K and S(0) > K, λ l D k [ y k S(0)n e nx e αx dx]e (β α)y dy, if L K and S(0) K 0, if L < K, = A 1 (n) A 2 (n) A 4 + A 5, if L K and S(0) > K, A 6 A 2 (n) + A 5, if L K and S(0) K.
66 Upandout allornothing put option The option value is = λ l D 0 [ y λ D S(0)n e nx e αx dx]e (β α)y dy, if L < K, l 0 [ k S(0)n e nx e αx dx]e (β α)y dy, if L K&S(0) > K λ D { k 0 [ y S(0)n e nx e αx dx]e (β α)y dy + l k [ k S(0)n e nx e αx dx]e (β α)y dy}, A 1 (n) A 2 (n), if L < K, A 4 A 5, if L K and S(0) > K, A 1 (n) A 5 A 6, if L K and S(0) K. if L K&S(0) K
67 upandout option with payoff S(τ) n λ D l 0 [ y ] S(0) n e nx e αx dx e (β α)y dy = A 1 (n) A 2 (n). This is the sum of the value of the upandout allornothing put option and the value of the upandout allornothing call option.
68 Upandout call option The value is 0, if L < K, A 1 (1) A 2 (1) A 1 (0)K +A 2 (0)K + A 8 A 9, if L K and S(0) > K, A 2 (0)K + A 10 A 2 (1) A 9, if L K and S(0) K.
69 Upandout put option The value is A 1 (0)K A 2 (0)K A 1 (1) + A 2 (1), if L < K, A 8 A 9, if L K and S(0) > K, A 1 (0)K A 1 (1) + A 10 A 9, if L K and S(0) K.
70 Double barrier option Payoff: π(s(τ))i {a < m(τ), M(τ) < b}
71 Several stocks µ X(t) = (X 1 (t), X 2 (t),, X n (t)) ndimensional Brownian motion. the mean vector C the covariance matrix of X(1) h g t (X) a realvalued functional of the process up to time t. an ndimensional vector of real numbers
72 E[e δτ e h X(τ) g τ (X)] = E[e δ(h)τ g τ (X); h], (3) where δ(h) = δ ln[m X(1) (h)] = δ h µ 1 2 h Ch.
73 Proof of (3) Conditioning on τ = t, the LHS (3) is 0 e δt E[e h X(t) g t (X)]f τ (t)dt. By the factorization formula in the method of Esscher transforms, the expectation inside the integrand can be written as the product of two expectations, Hence 0 E[e h X(t) ] E[g t (X); h] = [M X(1) (h)] t E[g t (X); h]. e δt E[e h X(t) g t (X)]f τ (t)dt = 0 e δ(h)t E[g t (X); h]f τ (t)dt.
74 Application of (3) k q t (k X) ndimensional vector of real numbers realvalued functional of the process up to time t E[e δτ e h X(τ) q τ (k X)] = E[e δ(h)τ q τ (k X); h]. The quadratic equation becomes 1 2 Var[k X(1); h]ξ 2 + E[k X(1); h]ξ [λ + δ(h)] = 1 2 k Ckξ 2 + k (µ + Ch)ξ (λ + δ h µ 1 2 h Ch)
75 Special case: n = 2 S 1 (t) = S 1 (0)e X 1(t) and S 2 (t) = S 2 (0)e X 2(t) µ = (µ 1, µ 2 ) ( ) σ 2 C = 1 ρσ 1 σ 2 ρσ 1 σ 2 σ2 2
76 Margrabe option Payoff: [S 1 (τ) S 2 (τ)] +. (4) If we rewrite (4) as e X 2(τ) [S 1 (0)e X 1(τ) X 2 (τ) S 2 (0)] +,
77 [ ] E[e δτ [S 1 (τ) S 2 (τ)] + S 1 (0) < S 2 (0)] = κ S 2 (0) S1 (0) β β (β. 1) S 2 (0) Here, κ = λ D (β α ), D = 1 2 Var[X 1(1) X 2 (1)] = 1 2 (σ2 1 + σ 2 2 2ρσ 1 σ 2 ), and α < 0 and β > 0 are the zeros of D ξ 2 + (µ 1 µ 2 + ρσ 1 σ 2 σ 2 2)ξ (λ + δ µ σ2 2) = ln[m X(1) ((ξ, 1 ξ) )] (λ + δ).
78 If we write (4) as Here, e X 1(τ) [S 1 (0) S 2 (0)e X 2(τ) X 1 (τ) ] +, E[e δτ [S 1 (τ) S 2 (τ)] + S 1 (0) < S 2 (0)] κ [ ] S 1 (0) S1 (0) α = α (1 α. ) S 2 (0) κ = λ D (β α ), D = 1 2 Var[X 2(1) X 1 (1)] = D, and α < 0 and β > 0 are the zeros of ln[m X(1) ((1 ξ, ξ) )] (λ + δ).
79 Hence α = 1 β and β = 1 α. Thus, κ = κ
80 What is different for jump diffusions? Ψ(z) = Dz 2 z + µz + ν v z ω z w + z E[e zx (τ) λ ] = λ Ψ(z) partial fraction expansion makes inversion possible α 2 < w < α 1 < 0 < β 1 < v < β 2 the solution of Ψ(z) λ = 0 { a1 e f X (τ) (x) = α1x + a 2 e α2x, if x 0, b 1 e β1x + b 2 e β2x, if x > 0.
81 f M(τ) (x) = β 2(v β 1 ) v(β 2 β 1 ) β 1e β 1x + β 1(β 2 v) v(β 2 β 1 ) β 2e β 2x for x 0 f m(τ) (x) = α 2(w + α 1 ) w(α 1 α 2 ) ( α 1)e α 1x + α 1(w + α 2 ) w(α 1 α 2 ) ( α 2)e α 2x for x 0
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