Dynamic Protection for Bayesian Optimal Portfolio

Transcription

1 Dynamic Protection for Bayesian Optimal Portfolio Hideaki Miyata Department of Mathematics, Kyoto University Jun Sekine Institute of Economic Research, Kyoto University Jan. 6, 2009, Kunitachi, Tokyo 1 / 24

2 Problem Optimal stopping problem, with the running maximum, sup 0 τ T EN τ X τ, N t := 1 max s [0,t] K s X s, X t := f(t, w t )e δt, and 1-dim BM w, where f C 1,2 ([0, T] R) that satisfies ( t + 1 ) 2 xx f = 0, f, x f > 0, δ R 0, and K C 1 ([0, T], R >0 ) are given. 2 / 24

3 Problem Optimal stopping problem, with the running maximum, sup 0 τ T EN τ X τ, N t := 1 max s [0,t] K s X s, X t := f(t, w t )e δt, and 1-dim BM w, where f C 1,2 ([0, T] R) that satisfies ( t + 1 ) 2 xx f = 0, f, x f > 0, δ R 0, and K C 1 ([0, T], R >0 ) are given. (f(t, x) := x 0 exp {ax (a 2 t)/2}: BS-case.) 2 / 24

4 Outline (Financial) Dynamic protection for Bayesian optimal portfolio Russian option pricing for local volatility model (Mathematical) /Contributions Extension of Peskir (2005) s analysis for BS-model. 3 / 24

5 Outline (Financial) Dynamic protection for Bayesian optimal portfolio Russian option pricing for local volatility model (Mathematical) /Contributions Extension of Peskir (2005) s analysis for BS-model. Characterize with a free-boundary problem for 3-dim. Markov (t, X, N). 3 / 24

6 Outline (Financial) Dynamic protection for Bayesian optimal portfolio Russian option pricing for local volatility model (Mathematical) /Contributions Extension of Peskir (2005) s analysis for BS-model. Characterize with a free-boundary problem for 3-dim. Markov (t, X, N).(Reduction to 2-dim. Markov (t, NX) is known in BS-case.) 3 / 24

7 Outline (Financial) Dynamic protection for Bayesian optimal portfolio Russian option pricing for local volatility model (Mathematical) /Contributions Extension of Peskir (2005) s analysis for BS-model. Characterize with a free-boundary problem for 3-dim. Markov (t, X, N).(Reduction to 2-dim. Markov (t, NX) is known in BS-case.) Free-boundary is a unique sol. of a nonlinear integral equation. 3 / 24

8 Outline (Financial) Dynamic protection for Bayesian optimal portfolio Russian option pricing for local volatility model (Mathematical) /Contributions Extension of Peskir (2005) s analysis for BS-model. Characterize with a free-boundary problem for 3-dim. Markov (t, X, N).(Reduction to 2-dim. Markov (t, NX) is known in BS-case.) Free-boundary is a unique sol. of a nonlinear integral equation. A numerical computation scheme. 3 / 24

9 DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks Financial 4 / 24

10 Application (1): Dynamic Fund Protection In a complete market with the risk-neutral P, DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks 5 / 24

11 Application (1): Dynamic Fund Protection DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks In a complete market with the risk-neutral P, X: one unit of (the discounted) investment fund, δ: dividend rate, paid to customers (not re-invested in the fund). 5 / 24

12 Application (1): Dynamic Fund Protection DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks In a complete market with the risk-neutral P, X: one unit of (the discounted) investment fund, δ: dividend rate, paid to customers (not re-invested in the fund). N t : minimal aggregate number n t of the fund units in a customer s account s.t. (i) n 0 = 1, (ii) n t n s for t s 0, and (iii) n t X t K t (: floor) for t 0. 5 / 24

13 Application (1): Dynamic Fund Protection DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks In a complete market with the risk-neutral P, X: one unit of (the discounted) investment fund, δ: dividend rate, paid to customers (not re-invested in the fund). N t : minimal aggregate number n t of the fund units in a customer s account s.t. (i) n 0 = 1, (ii) n t n s for t s 0, and (iii) n t X t K t (: floor) for t 0. The protection scheme is called the dynamic fund protection (Gerber-Pafumi, 2000 and Gerber-Shiu, 2003). 5 / 24

14 Dynamic Protetion vs. European-OBPI DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks A simple European-OBPI (i.e., buy European put written on X with strike k) controls a down-side-risk of X at the terminal T: X T + (k X T ) + = X T k k. DFP controls a down-side-risk of the process X (or NX) N t X t K t for t [0, T]. 6 / 24

15 Attractive Feature of Dynamic Fund Protetion DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks DFP may be more attractive than European-OBPI for customers... Suppose X T < K T (= k). Then, E-OBPI: X T + (K T X T ) + = K T, : floor value, ( ) K t DFP: N T X T = 1 + max X T 0 t T X ( t 1 + K ) T X T = X T + K T. X T 7 / 24

16 Attractive Feature of Dynamic Fund Protetion DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks DFP may be more attractive than European-OBPI for customers... Suppose X T < K T (= k). Then, E-OBPI: X T + (K T X T ) + = K T, : floor value, ( ) K t DFP: N T X T = 1 + max X T 0 t T X ( t 1 + K ) T X T = X T + K T. X T This comparison is not fair, of course.. 7 / 24

17 Pricing Dynamic Fund Protection DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks DFP-scheme is not self-financing. To compute the fair value of DFP-option, compute the Snell envelope, V t := ess sup t τ T E [N τ X τ F t ], (t [0, T]), i.e., the minimal superreplicating self-financing portfolio of NX. Apply a standard no-arbitrage pricing argument in complete market. 8 / 24

18 Extension: Beyond BS DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks BS: X t := x 0 exp {aw t (a 2 /2 + δ)t} (δ = 0 by Gerber-Pafumi; δ > 0 and T = by Gerber Shiu) 9 / 24

19 Extension: Beyond BS DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks BS: X t := x 0 exp {aw t (a 2 /2 + δ)t} (δ = 0 by Gerber-Pafumi; δ > 0 and T = by Gerber Shiu) CEV (with δ = 0) by Imai-Boyle. 9 / 24

20 Extension: Beyond BS DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks BS: X t := x 0 exp {aw t (a 2 /2 + δ)t} (δ = 0 by Gerber-Pafumi; δ > 0 and T = by Gerber Shiu) CEV (with δ = 0) by Imai-Boyle. Dynamically invested fund, X :=X (x 0,π,δ), dx t X t where =π t ds t S t + (1 π t )rdt δdt, X 0 = x 0. 9 / 24

21 Extension: Beyond BS DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks BS: X t := x 0 exp {aw t (a 2 /2 + δ)t} (δ = 0 by Gerber-Pafumi; δ > 0 and T = by Gerber Shiu) CEV (with δ = 0) by Imai-Boyle. Dynamically invested fund, X :=X (x 0,π,δ), dx t X t where =π t ds t S t + (1 π t )rdt δdt, X 0 = x 0. π t := σ 2 (µ r): growth-optimal fund, 9 / 24

22 Extension: Beyond BS DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks BS: X t := x 0 exp {aw t (a 2 /2 + δ)t} (δ = 0 by Gerber-Pafumi; δ > 0 and T = by Gerber Shiu) CEV (with δ = 0) by Imai-Boyle. Dynamically invested fund, X :=X (x 0,π,δ), dx t X t where =π t ds t S t + (1 π t )rdt δdt, X 0 = x 0. π t := σ 2 (µ r): growth-optimal fund, π t := σ 2 (E[µ F t ] r): Bayesian growth optimal fund. 9 / 24

23 Application (2): DP for Bayesian Optimal Portfolio DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks Financial market with a bank-account B 1 and a stock(-index) ds t = S t σ(t, S)(dz t + λdt), S 0 > 0 on (Ω, F, P 0, (G t ) t [0,T] ), where G t := σ(z u ; u [0, t]) σ(λ), 1-dim. BM z and r.v. λ ν, independent of z. Another filtration, S t := σ(s u ; u [0, t]), is regarded as the available information for investors. 10 / 24

24 DP for Dynamically Invested Fund DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks Treat DP of where dx t X t X := X (x 0,π,δ), where π := (π t ) t [0,T] belongs to A T := = π t ds t S t δdt, X 0 = x 0, { (f t ) t [0,T] : S t -prog. m ble, T 0 f t σ t 2 dt < a.s. In particular, we are interested in the optimally invested fund, ˆX, so that ( ) ( ) sup EU X x 0,π,δ T = EU ˆXT. π A T }. 11 / 24

25 Reference Measure (EMM under Partial Information) DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks Introduce to see dp dp 0 is a (P, G t )-BM and since and = exp Gt ( λz t 1 ) 2 λ2 t w t := z t + λt S t := σ(s u ; u t) = σ(w u ; u t) =: F t for t [0, T] ds t = S t σ(t, S)dw t dw t = : unique strong sol. ds t S t σ(t, S). 12 / 24

26 Bayesian Optimal Portfolio (Karatzas-Zhao, 2000) DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks Here, define where F(t, y) := exp R 0 X (t, x, y) :=e δ(t t) ˆX t = ˆX (t, w t ) (t [0, T]). ˆX (t, y) := X(t, Y(x 0 ), y), R ( xy 1 ) 2 x 2 t ν(dx), I ( e δt x F(T, y + T tz) ) 1 2π e z 2 2 dz, I := (U ) 1, and Y( ) := X(0,, 0) / 24

27 Russian Option for Local-volatility Model DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks Optimal stopping problem is rewritten as f(0, 0) sup 0 τ T Ẽ Here, P on (Ω, F T ) is defined by d P dp { }] [e δτ 1 max Y s s [0,τ] := f(t, w t) Ft f(0, 0), Y t := K t /X t satisfies the Markovian SDE,. dy t = Y t [a(t, Y t )d w t + (κ t + δ)dt] with a(t, y) := x log f (t, f 1 (t, K t /y)), κ t := t log K t, and the P-BM, { w t := w t } t 0 x log f(u, w u )du. 14 / 24

28 Related Works on Russian Options DFP vs. E-OBPI Feature Pricing Extension DP-BOP(1) DP-BOP(2) EMM BOP Russian Option Related Woks BS-case, i.e., X t := x 0 exp {aw t (δ + a 2 /2)t}, K t := K 0 e αt, and dy t = Y t {ad w t + (α + δ)dt}. T = : explicit results by Shepp-Shiryaev, Duffie-Harrison, Salminen, etc. T < : studied by Duistermaat-Kyprianou-van Schaik, Ekström, Peskir, etc. Some extenstions: Guo-Shepp, Pedersen, Gapeev, etc. General treatment with viscocity of variational inequality: Barles-Daher-Romano. 15 / 24

29 Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics Mathematical 16 / 24

30 Dynamic Version of Optimal Stopping Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics By dynamic-programming and the Markov property of (t, X, N), or (t, Y, N) := (t, K/X, N), deduce ess sup t τ T where E [N τ X τ F t ] = V (t, X t, N t ) = e δt X t V (t, K t /X t, N t ), V (t, y, z) := N (t,y,z) s sup Ẽe δτ N τ (t,y,z), 0 τ T t :=z max u [0,s] Y (t,y) u, and Y (t,y) solves dy s = Y s {a(t + s, Y s )d w s + (κ t+s + δ)ds}, Y 0 = y. 17 / 24

31 Free Boundary Problem Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics ( t + L δ)v (t, y, z) =0 for (t, y, z) C, V (t, y, z) y=b(t,z)+ =z, (instantaneous stopping), y V (t, y, z) y=b(t,z)+ =0, (smooth-pasting), z V (t, y, z) z=y+ =0, (normal-reflection), V (t, y, z) >z for (t, y, z) C, V (t, y, z) =z for (t, y, z) D, where L := 1 2 y2 a(t, y) 2 yy + y(κ t + δ) y, C := {(t, y, z) [0, T) E; b(t, z) < y z}, D := {(t, y, z) [0, T) E; 0 < y b(t, z)}, E := { (y, z) R 2 ; 0 < y z, z 1 }. 18 / 24

32 Main Theorem (1) Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics (1) For V, b : [0, T) [1, ) R >0, s.t. (V, b) solves the Free-boundary Prob. (2) b is the unique solution to [ z = e δ(t t) Ẽ so that N (t,b(t,z),z) T t ] + δ Ẽ[ N s (t,b(t,z),z) I ( Y s (t,b(t,z)) T t (i) b: continuous, nondecreasing w.r.t. t, (ii) b(t, z) < z and b(t, z) = z. 0 dse δs < b ( t + s, N (t,b(t,z),z) s ))] 19 / 24

33 Main Theorem (2) Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics (3) Optimal stopping time: (4) Closed-form expression: ˆτ t := inf {0 t T; Y t b(t, N t )}. [ V (t, y, z) = e δ(t t) Ẽ N (t,y,z) T t ] Ẽ[ N s (t,y,z) I ( Y s (t,y) + δ T t 0 dse δs < b ( ))] t + s, N s (t,y,z). 20 / 24

34 Remark The result may not be surprising to some extent... Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics 21 / 24

35 Remark Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics The result may not be surprising to some extent... Point: the integral equation for b is the characteristic of the problem in the sense that free boundary b is determined as a unique sol. of the equation, and, value function V is represented in closed-form with b. 21 / 24

36 Remark Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics The result may not be surprising to some extent... Point: the integral equation for b is the characteristic of the problem in the sense that free boundary b is determined as a unique sol. of the equation, and, value function V is represented in closed-form with b. Peskir s change of variable formula for continuous semimartingale with local-time on curve, (an extension of Itô-Meyer-Tanaka formula), plays an important rôle. 21 / 24

37 How to Numerically Compute b and V? Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics Using we see [ Ẽ q(s, dy, dz ; t, y, z) := P ( Y (t,y) s dy, N (t,y,z) dz ), N (t,y,z) T t Ẽ [ N (t,y,z) s = K t+sy K t ] = K ( ) Ty z q(t t, dy, dz ; t, y, z), K t E y I ( Y s (t,y) < b ( ))] t + s, N s (t,y,z) ( ) z I(y < b(t + s, z ))q(s, dy, dz ; t, y, z). E y So, the integral equation for b is represented with q. 22 / 24

38 Boundary Crossing Probability Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics We see where N (t,y,z) s d z d and w u l(u) for all u [0, s]. l(u) = l(u; d, t, y, k) := f 1 ( t + u, K t+u d ) K t+u y. Approximation schema to compute the joint probability P (w s c 1, w u l(u) for all u [0, s]) have been studied by several works. 23 / 24

39 Related Topics for Future Works Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics More general (multi-dimensional, possibly) model with numerics. Large-time asymptotics (as T ). 24 / 24

40 Related Topics for Future Works Dynamic Prob. Free-bdry Prob. Main Theorem (1) Main Theorem (2) Remark How to compute? Bdry Cross Prob. Related Topics More general (multi-dimensional, possibly) model with numerics. Large-time asymptotics (as T ). Comparison with American-OBPI strategy (by El Karoui-Jeanblanc-Lacoste, El Karoui-Meziou, etc.) 24 / 24