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


 Julia Cobb
 1 years ago
 Views:
Transcription
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
Loaded Participation Rates for EquityIndexed Annuities
Loaded Participation Rates for EquityIndexed Annuities Patrice Gaillardetz and Joe Youssef Lakhmiri Abstract In this paper, we will introduce premium principles for equityindexed annuities (EIAs). We
More informationA linear algebraic method for pricing temporary life annuities
A linear algebraic method for pricing temporary life annuities P. Date (joint work with R. Mamon, L. Jalen and I.C. Wang) Department of Mathematical Sciences, Brunel University, London Outline Introduction
More informationWhich Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal Portfolios
Which Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal Portfolios Riaz Ahmad Course Director for CQF, 7city, London Paul Wilmott Wilmott Associates, London Abstract:
More informationA SIMPLE OPTION FORMULA FOR GENERAL JUMPDIFFUSION AND OTHER EXPONENTIAL LÉVY PROCESSES
A SIMPLE OPTION FORMULA FOR GENERAL JUMPDIFFUSION AND OTHER EXPONENTIAL LÉVY PROCESSES ALAN L. LEWIS Envision Financial Systems and OptionCity.net August, 00 Revised: September, 00 Abstract Option values
More informationFair valuation of pathdependent participating life insurance contracts
Insurance: Mathematics and Economics 33 (2003) 595 609 Fair valuation of pathdependent participating life insurance contracts Antti Juho Tanskanen a,, Jani Lukkarinen b a Varma Mutual Pension Insurance
More informationManaging the Risk of Variable Annuities: a Decomposition Methodology. Thomas S. Y. Ho Ph.D. President. Thomas Ho Company Ltd. 55 Liberty Street, 4 B
Managing the Risk of Variable Annuities: a Decomposition Methodology By Thomas S. Y. Ho Ph.D. President Thomas Ho Company Ltd 55 Liberty Street, 4 B New York NY 10005 tom.ho@thomasho.com And Blessing Mudavanhu
More informationOn Some Aspects of Investment into HighYield Bonds
On Some Aspects of Investment into HighYield Bonds Helen Kovilyanskaya Vom Fachbereich Mathematik der Universität Kaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften Doctor
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationCol.lecció d Economia E14/310. Some optimization and decision problems in proportional reinsurance. Anna Castañer M. Mercè Claramunt Maite Mármol
Col.lecció d Economia E14/310 Some optimization and decision problems in proportional reinsurance Anna Castañer M. Mercè Claramunt Maite Mármol UB Economics Working Papers 2014/310 Some optimization and
More informationOptimal Reinsurance Strategy under Fixed Cost and Delay
Optimal Reinsurance Strategy under Fixed Cost and Delay Masahiko Egami Virginia R. Young Abstract We consider an optimal reinsurance strategy in which the insurance company 1 monitors the dynamics of its
More informationMath 526: Brownian Motion Notes
Math 526: Brownian Motion Notes Definition. Mike Ludkovski, 27, all rights reserved. A stochastic process (X t ) is called Brownian motion if:. The map t X t (ω) is continuous for every ω. 2. (X t X t
More informationHedging with options in models with jumps
Hedging with options in models with jumps ama Cont, Peter Tankov, Ekaterina Voltchkova Abel Symposium 25 on Stochastic analysis and applications in honor of Kiyosi Ito s 9th birthday. Abstract We consider
More informationNonLife Insurance Mathematics. Christel Geiss and Stefan Geiss Department of Mathematics and Statistics University of Jyväskylä
NonLife Insurance Mathematics Christel Geiss and Stefan Geiss Department of Mathematics and Statistics University of Jyväskylä July 29, 215 2 Contents 1 Introduction 5 1.1 Some facts about probability..................
More informationPricing and Hedging of Credit Risk: Replication and MeanVariance Approaches
Contemporary Mathematics Pricing and Hedging of Credit Risk: Replication and MeanVariance Approaches Tomasz R. Bielecki, Monique Jeanblanc, and Marek Rutkowski Abstract. The paper presents some methods
More informationONEDIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK
ONEDIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK Definition 1. A random walk on the integers with step distribution F and initial state x is a sequence S n of random variables whose increments are independent,
More informationAsset Allocation and AnnuityPurchase Strategies. to Minimize the Probability of Financial Ruin. Moshe A. Milevsky 1. Kristen S.
Asset Allocation and AnnuityPurchase Strategies to Minimize the Probability of Financial Ruin Moshe A. Milevsky 1 Kristen S. Moore 2 Virginia R. Young 3 Version: 18 October 25 1 Schulich School Of Business,
More informationunder Stochastic Interest Rates Dipartimento di Matematica Applicata alle Scienze University of Trieste Piazzale Europa 1, I34127 Trieste, Italy
Design and Pricing of EquityLinked Life Insurance under Stochastic Interest Rates Anna Rita Bacinello Dipartimento di Matematica Applicata alle Scienze Economiche, Statistiche ed Attuariali \Bruno de
More informationSome Research Problems in Uncertainty Theory
Journal of Uncertain Systems Vol.3, No.1, pp.310, 2009 Online at: www.jus.org.uk Some Research Problems in Uncertainty Theory aoding Liu Uncertainty Theory Laboratory, Department of Mathematical Sciences
More informationFinancial Risk Management of Guaranteed Minimum Income Benefits Embedded in Variable Annuities
Financial Risk Management of Guaranteed Minimum Income Benefits Embedded in Variable Annuities by Claymore James Marshall A thesis presented to the University of Waterloo in fulfillment of the thesis requirement
More informationarxiv:math/0503556v1 [math.pr] 24 Mar 2005
The Annals of Applied Probability 2004, Vol. 4, No. 4, 2090 29 DOI: 0.24/0505604000000846 c Institute of Mathematical Statistics, 2004 arxiv:math/0503556v [math.pr] 24 Mar 2005 NUMBER OF PATHS VERSUS NUMBER
More informationAN INSTITUTIONAL EVALUATION OF PENSION FUNDS AND LIFE INSURANCE COMPANIES
AN INSTITUTIONAL EVALUATION OF PENSION FUNDS AND LIFE INSURANCE COMPANIES DIRK BROEDERS, AN CHEN, AND BIRGIT KOOS Abstract. This paper analyzes two different types of annuity providers. In the first case,
More informationTWO L 1 BASED NONCONVEX METHODS FOR CONSTRUCTING SPARSE MEAN REVERTING PORTFOLIOS
TWO L 1 BASED NONCONVEX METHODS FOR CONSTRUCTING SPARSE MEAN REVERTING PORTFOLIOS XIAOLONG LONG, KNUT SOLNA, AND JACK XIN Abstract. We study the problem of constructing sparse and fast mean reverting portfolios.
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationNoarbitrage conditions for cashsettled swaptions
Noarbitrage conditions for cashsettled swaptions Fabio Mercurio Financial Engineering Banca IMI, Milan Abstract In this note, we derive noarbitrage conditions that must be satisfied by the pricing function
More informationOption Pricing by Transform Methods: Extensions, Unification, and Error Control
Option Pricing by Transform Methods: Extensions, Unification, and Error Control Roger W. Lee Stanford University, Department of Mathematics NYU, Courant Institute of Mathematical Sciences Journal of Computational
More information2 Right Censoring and KaplanMeier Estimator
2 Right Censoring and KaplanMeier Estimator In biomedical applications, especially in clinical trials, two important issues arise when studying time to event data (we will assume the event to be death.
More informationThe Monte Carlo Framework, Examples from Finance and Generating Correlated Random Variables
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh The Monte Carlo Framework, Examples from Finance and Generating Correlated Random Variables 1 The Monte Carlo Framework Suppose we wish
More informationQuantitative Strategies Research Notes
Quantitative Strategies Research Notes March 999 More Than You Ever Wanted To Know * About Volatility Swaps Kresimir Demeterfi Emanuel Derman Michael Kamal Joseph Zou * But Less Than Can Be Said Copyright
More informationCOMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS
COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic meanvariance problems in
More informationMath 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 3 Solutions
Math 37/48, Spring 28 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 3 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,
More information