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

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1 Valuation of Equity-Linked 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 Equity-Linked 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 Equity-linked 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 time-until-death random variable for a life age x, S(t) is the price of equity-index 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 Equity-linked 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) out-of-the-money 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) out-of-the-money call option E[e δτ (S(τ) K) + ] = κk [ S(0) β(β 1) K ] β

19 The Greek letters (1) of the out-of-the-money put option κ [ ] K (α 1) 1 α S(0) (2) of the out-of-the-money call option [ κ S(0) β 1 K ] β 1

20 The Greek letters (1) Γ of the out-of-the-money put option [ ] κ K (α 2) K S(0) (2) Γ of the out-of-the-money call option κ K [ S(0) K ] β 2

21 in-the-money options (3) b(s) = (K s) +, K > S(0) put (4) b(s) = (s K) +, K < S(0) call Use the put-call 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 In-the-money 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 in-the-money call option 1 κ [ ] K (α 1) 1 α S(0) (2) of the in-the-money put option [ ] κ S(0) β 1 1 β 1 K

24 The Greek letters (under risk neutral probability) (1) Γ of the in-the-money call option [ ] κ K (α 2) K S(0) (2) Γ of the in-the-money put option κ K [ S(0) K ] β 2

25 Illustration We consider 90-strike life-contingent 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 time-0 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 K-strike 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 90-strike life-contingent 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 equity-indexed annuities credit interest using a high water mark method or a low water mark method

40 Out-of-the-money fixed strike lookback call option Payoff: Time-0 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 In-the-money fixed strike lookback call option Payoff Rewriting as max(h, S(0)e M(τ) ) K. H K + [S(0)e M(τ) H] + Time-0 value { λ H K + H λ + δ β 1 [ S(0) H ] β }.

42 Floating strike lookback put option Payoff where H S(0). Time-0 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 time-0 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 risk-neutral 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) )]. Time-0 value E[e δτ [γ max 0 t τ S(t) S(τ)] +] = γ1 α α E[e δτ S(τ)],

48 Out-of-the-money fixed strike lookback put option Payoff [K S(0)e m(τ) ] +, Time-0 value k [K S(0)e y ]fm(τ) δ (y)dy = λ [ ] K K α. λ + δ 1 α S(0)

49 In-the-money fixed strike lookback put option Payoff K min(h, S(0)e m(τ) ) = K H + [H S(0)e m(τ) ] +, Time-0 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). Time-0 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 time-0 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 High-low option Payoff max(h, max S(t)) min(h, min S(t)), 0 t τ 0 t τ where 0 < H S(0) H. Time-0 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, time-0 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 90-strike life-contingent 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. Knock-out options are those which go out of existence if the asset price reaches the barrier, and knock-in options are those which come into existence if the barrier is reached.

57 Parity relation Knock-out option + Knock-in option = Ordinary option. Notation: L denotes the barrier and l = ln[l/s(0)]

58 Up-and-out and up-and-in 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 Up-and-out Up-and-in 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 Down-and-out and down-and-in 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 Up-and-out all-or-nothing 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 Up-and-out all-or-nothing 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 up-and-out 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 up-and-out all-or-nothing put option and the value of the up-and-out all-or-nothing call option.

68 Up-and-out 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 Up-and-out 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)) n-dimensional Brownian motion. the mean vector C the covariance matrix of X(1) h g t (X) a real-valued functional of the process up to time t. an n-dimensional 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) n-dimensional vector of real numbers real-valued 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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