Dynamic Protection for Bayesian Optimal Portfolio

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

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

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

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

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

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

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

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

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

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

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

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

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

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) 1. 13 / 24

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

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

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

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

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

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

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

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

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

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

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

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