Valuing contingent exotic options: a discounted density approach

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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- NonCommercial-NoDerivatives 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 equity-indexed annuities credit interest using a high water mark method or a low water mark method

13 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 ]

14 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 }.

15 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)].

16 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)].

17 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.

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) )]. Time-0 value E[e δτ [γ max 0 t τ S(t) S(τ)] +] = γ1 α α E[e δτ S(τ)],

21 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)

22 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)

23 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 ( )

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 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

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. 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.

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

30 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) )

31 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;

32 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) )

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 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.

38 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

39 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.

40 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.

41 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.

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)) 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

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) 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)

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