Valuing contingent exotic options: a discounted density approach


 Άλκηστις Ζαφειρόπουλος
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1 Title Valuing contingent exotic options: a discounted density approach Author(s) Yang, H; Gerber, HU; Shiu, ESW Citation The 2012 International Conference on Actuarial Science and Risk Management (ASRM 2012), Xiamen, China, June Issued Date 2012 URL Rights This work is licensed under a Creative Commons Attribution NonCommercialNoDerivatives 4.0 International License.
2 Valuing contingent exotic options: a discounted density approach Department of Statistics and Actuarial Science The University of Hong Kong Hong Kong Based on a paper with Hans Gerber and Elias Shiu
3 Brownian motion (Wiener process) X (t) = µt + σw (t) {W (t)}: notation: standard Wiener process D = σ2 2 running minimum: running maximum: m(t) = min 0 s t X (s) M(t) = max 0 s t X (s)
4 2 Exponential stopping of Brownian motion τ: exponential random variable independent of {X (t)} f τ (t) = λe λt, t > 0 We are interested in X (τ), M(τ),... δ: force of interest used for discounting
5 Three discounted density functions f δ X (τ) (x) = 0 e δt f X (t) (x)f τ (t)dt f δ M(τ) (m) = 0 e δt f M(t) (m)f τ (t)dt f δ X (τ),m(τ) (x, m) = 0 e δt f X (t),m(t) (x, m)f τ (t)dt
6 α < 0 and β > 0 solutions of the quadratic equation Dξ 2 + µξ (λ + δ) = 0 1). f δ X (τ),m(τ) (x, m) = λ D e αx (β α)m = λ D eα(m x) βm, < x m, m 0 2). f δ M(τ) (m) = λ λ+δ βe βm, m 0 3). f δ X (τ) (x) = { λ D(β α) e αx, if x < 0, λ D(β α) e βx, if x > 0.
7 f δ X (τ),m(τ) X (τ) (x, z) = λ D e βx (β α)z, z max( x, 0). f δ M(τ),M(τ) X (τ) (y, z) = λ D e βy+αz, y 0, z 0. f δ M(τ) X (τ) (z) = λ βd eαz = λ λ + δ ( α)eαz, z 0.
8 m(t) = max{ X (s); 0 s t} If E[e δτ g(x (τ), M(τ))] = h(α, β), then we can translate it to E[e δτ g( X (τ), m(τ))] = h( β, α).
9 f δ X (τ),m(τ) (x, y) = λ D e βx+(β α)y, y min(x, 0), f δ X (τ),x (τ) m(τ) (x, z) = λ D e αx (β α)z, z max(x, 0), f δ m(τ),x (τ) m(τ) (y, z) = λ D e αy βz, y 0, z 0, fm(τ) δ (y) = λ βd e αy = f δ X (τ) m(τ) (z) = λ αd e βz = λ λ + δ ( α)e αy, y 0, λ λ + δ βe βz, z 0.
10 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.
11 3. Financial applications S(t): stock price S(t) = S(0)e X (t) = S(0)e µt+σw (t), t 0 a contingent option provides a payoff at time τ Example: τ: time of death GMDB (Guaranteed Minimum Death Benefits)
12 3. Lookback options Many equityindexed annuities credit interest using a high water mark method or a low water mark method
13 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 ]
14 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 }.
15 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)].
16 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)].
17 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.
18 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) ] +
19 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
20 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)],
21 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)
22 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)
23 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 ( )
24 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] +
25 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) ].
26 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)].
27 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
28 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.
29 Parity relation Knockout option + Knockin option = Ordinary option. Notation: L denotes the barrier and l = ln[l/s(0)]
30 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) )
31 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;
32 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) )
33 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
34 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)
35 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)
36 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
37 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.
38 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
39 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.
40 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.
41 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.
42 Double barrier option Payoff: π(s(τ))i {a < m(τ), M(τ) < b}
43 Double barrier option Using a formula from B & S (2002), we have = E[b(S(0)e X (τ) )I {a < m(τ), M(τ) < b}] [ κ 0 0 Ξ Ξ 1 Ξ b(s(0)e z )e αz dz + Ξ 1 b(s(0)e z )e βz dz b a b +Ξ b(s(0)e z )e βz dz + Ξ 1 b(s(0)e z )e αz dz 0 0 b b ] Υ 1 b(s(0)e z )e αz dz Υ b(s(0)e z )e βz dz, a a a where Ξ = e 1 2 (b a)(β α) and Υ = e 1 2 (a+b)(β α).
44 Double barrier option When δ > 0, we have E[e δτ b(s(0)e X (τ) )I {a < m(τ), M(τ) < b}] = λ e (λ+δ)t b(s(0)e X (t) )I {a < m(t), M(t) < b }dt 0 [ κ 0 0 = Ξ Ξ 1 Ξ b(s(0)e z )e αz dz + Ξ 1 b(s(0)e z )e βz dz b a b +Ξ b(s(0)e z )e βz dz + Ξ 1 b(s(0)e z )e αz dz 0 0 b b ] Υ 1 b(s(0)e z )e αz dz Υ b(s(0)e z )e βz dz. a a a
45 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
46 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.
47 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.
48 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)
49 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
50 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)] +,
51 [ ] 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).
52 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).
53 Hence α = 1 β and β = 1 α. Thus, κ = κ