Dynamic Protection for Bayesian Optimal Portfolio · Dynamic protection for Bayesian optimal...

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Dynamic Protection for Bayesian Optimal Portfolio

Hideaki MiyataDepartment of Mathematics, Kyoto University

Jun SekineInstitute of Economic Research, Kyoto University

Jan. 6, 2009, Kunitachi, Tokyo

Problem

⊲ Problem

⊲ Outline

Applications

Results

2 / 24

Optimal stopping problem,

sup0≤τ≤T

ENτXτ ,

with the running maximum,

Nt := 1 ∨ maxs∈[0,t]

Ks

Xs

, Xt := f(t, wt)e−δt,

and 1-dim BM w, where f ∈ C1,2([0, T ] × R) that satisfies

(

∂t +1

2∂xx

)

f = 0, f, ∂xf > 0,

δ ∈ R≥0, and K ∈ C1([0, T ], R>0) are given.

Problem

⊲ Problem

⊲ Outline

Applications

Results

2 / 24

Optimal stopping problem,

sup0≤τ≤T

ENτXτ ,

with the running maximum,

Nt := 1 ∨ maxs∈[0,t]

Ks

Xs

, Xt := f(t, wt)e−δt,

and 1-dim BM w, where f ∈ C1,2([0, T ] × R) that satisfies

(

∂t +1

2∂xx

)

f = 0, f, ∂xf > 0,

δ ∈ R≥0, and K ∈ C1([0, T ], R>0) are given.(f(t, x) := x0 exp ax − (a2t)/2: BS-case.)

Outline

⊲ Problem

⊲ Outline

Applications

Results

3 / 24

(Financial) Applications

Dynamic protection for Bayesian optimal portfolio Russian option pricing for local volatility model

(Mathematical) Results/Contributions

Extension of Peskir (2005)’s analysis for BS-model.

Outline

⊲ Problem

⊲ Outline

Applications

Results

3 / 24





Characterize with a free-boundary problem for 3-dim.Markov (t, X, N).

Outline

⊲ Problem

⊲ Outline

Applications

Results

3 / 24





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.)

Outline

⊲ Problem

⊲ Outline

Applications

Results

3 / 24






Free-boundary is a unique sol. of a nonlinearintegral equation.

Outline

⊲ Problem

⊲ Outline

Applications

Results

3 / 24






Free-boundary is a unique sol. of a nonlinearintegral equation.

A numerical computation scheme.

Financial Applications

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

4 / 24

Application (1): Dynamic Fund Protection

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

5 / 24

In a complete market with the risk-neutral P,


⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

5 / 24


X: one unit of (the discounted) investment fund,δ: dividend rate, paid to customers (not re-invested in thefund).


⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

5 / 24



Nt: minimal aggregate number nt of the fund units in acustomer’s account s.t.

(i) n0 = 1,

(ii) nt ≥ ns for t ≥ s ≥ 0, and

(iii) ntXt ≥ Kt (: floor) for ∀t ≥ 0.


⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

5 / 24



Nt: minimal aggregate number nt of the fund units in acustomer’s account s.t.

(i) n0 = 1,

(ii) nt ≥ ns for t ≥ s ≥ 0, and

(iii) ntXt ≥ Kt (: floor) for ∀t ≥ 0.

The protection scheme is called the dynamic fund protection

(Gerber-Pafumi, 2000 and Gerber-Shiu, 2003).

Dynamic Protetion vs. European-OBPI

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

6 / 24

A simple European-OBPI (i.e., buy European put written onX with strike k) controls a down-side-risk of X at theterminal T :

XT + (k − XT )+ = XT ∨ k ≥ k.

DFP controls a down-side-risk of the process X (or NX)

NtXt ≥ Kt for ∀t ∈ [0, T ].

“Attractive” Feature of Dynamic Fund Protetion

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

7 / 24

DFP may be more “attractive” than European-OBPI forcustomers...

Suppose XT < KT (= k). Then,

E-OBPI: XT + (KT − XT )+ = KT , : floor value,

DFP: NT XT =

(

1 + max0≤t≤T

Kt

Xt

)

XT

≥(

1 +KT

XT

)

XT = XT + KT .

“Attractive” Feature of Dynamic Fund Protetion

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

7 / 24

DFP may be more “attractive” than European-OBPI forcustomers...

Suppose XT < KT (= k). Then,

E-OBPI: XT + (KT − XT )+ = KT , : floor value,

DFP: NT XT =

(

1 + max0≤t≤T

Kt

Xt

)

XT

≥(

1 +KT

XT

)

XT = XT + KT .

This comparison is not “fair”, of course..

Pricing Dynamic Fund Protection

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

8 / 24

DFP-scheme is not self-financing. To compute the fair value of “DFP-option”, compute the

Snell envelope,

Vt := ess supt≤τ≤T

E [NτXτ |Ft] , (t ∈ [0, T ]),

i.e., the minimal superreplicating self-financing portfolio ofNX.

Apply a standard no-arbitrage pricing argument in completemarket.

Extension: Beyond BS

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

9 / 24

BS: Xt := x0 exp awt − (a2/2 + δ)t (δ = 0 byGerber-Pafumi; δ > 0 and T = ∞ by Gerber–Shiu)


⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

9 / 24


CEV (with δ = 0) by Imai-Boyle.


⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

9 / 24


CEV (with δ = 0) by Imai-Boyle. Dynamically invested fund,

X :=X(x0,π,δ), where

dXt

Xt

=πt

dSt

St

+ (1 − πt)rdt − δdt, X0 = x0.


⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

9 / 24




dXt

Xt

=πt

dSt

St

+ (1 − πt)rdt − δdt, X0 = x0.

πt := σ−2(µ − r): growth-optimal fund,


⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

9 / 24




dXt

Xt

=πt

dSt

St

+ (1 − πt)rdt − δdt, X0 = x0.

πt := σ−2(µ − r): growth-optimal fund, πt := σ−2 (E[µ|Ft] − r): Bayesian growth optimal fund.

Application (2): DP for Bayesian Optimal Portfolio

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

10 / 24

Financial market with a bank-account B ≡ 1 and astock(-index)

dSt = Stσ(t, S)(dzt + λdt), S0 > 0

on (Ω,F , P0, (Gt)t∈[0,T ]), where

Gt := σ(zu; u ∈ [0, t]) ∨ σ(λ),

1-dim. BM z and r.v. λ ∼ ν, independent of z. Another filtration,

St := σ(Su; u ∈ [0, t]),

is regarded as the available information for investors.

DP for Dynamically Invested Fund

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

11 / 24

Treat DP ofX := X(x0,π,δ),

wheredXt

Xt

= πt

dSt

St

− δdt, X0 = x0,

where π := (πt)t∈[0,T ] belongs to

AT :=

(ft)t∈[0,T ] : St-prog. m’ble,

∫ T

0

|ftσt|2dt < ∞ a.s.

.

In particular, we are interested in the optimally invested fund,X, so that

supπ∈AT

EU(

Xx0,π,δT

)

= EU(

XT

)

.

Reference Measure (EMM under Partial Information)

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

12 / 24

IntroducedP

dP0

∣

∣

∣

∣

Gt

= exp

(

−λzt −1

2λ2t

)

to seewt := zt + λt

is a (P,Gt)-BM and

St := σ(Su; u ≤ t) = σ(wu; u ≤ t) =: Ft for ∀t ∈ [0, T ]

sincedSt = Stσ(t, S)dwt : unique strong sol.

and

dwt =dSt

Stσ(t, S).

Bayesian Optimal Portfolio (Karatzas-Zhao, 2000)

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

13 / 24

Xt = X (t, wt) (t ∈ [0, T ]).

Here, defineX (t, y) := X (t,Y(x0), y),

where

F (t, y) :=

∫

R≥0

exp

(

xy − 1

2|x|2t

)

ν(dx),

X (t, x, y) :=eδ(T−t)

∫

R

I(

eδT x

F (T, y +√

T − tz)

)

1√2π

e−|z|2

2 dz,

I := (U ′)−1, and Y(·) := X (0, ·, 0)−1.

Russian Option for Local-volatility Model

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

14 / 24

Optimal stopping problem is rewritten as

f(0, 0) × sup0≤τ≤T

E

[

e−δτ

1 ∨ maxs∈[0,τ ]

Ys

]

.

Here, P on (Ω,FT ) is defined by

dP

dP

∣

∣

∣

∣

Ft

:=f(t, wt)

f(0, 0),

Yt := Kt/Xt satisfies the Markovian SDE,

dYt = Yt [a(t, Yt)dwt + (κt + δ) dt]

with a(t, y) := ∂x log f (t, f−1 (t, Kt/y)), κt := ∂t log Kt, and

the P-BM, wt := −

wt −∫ t

0∂x log f(u, wu)du

.

Related Works on Russian Options

⊲ Problem

⊲ Outline

Applications

⊲ DFP

⊲ vs. E-OBPI

⊲ Feature

⊲ Pricing

⊲ Extension

⊲ DP-BOP(1)

⊲ DP-BOP(2)

⊲ EMM

⊲ BOP

⊲ Russian Option

⊲ Related Woks

Results

15 / 24

BS-case, i.e., Xt := x0 exp awt − (δ + a2/2)t, Kt := K0eαt,

anddYt = Yt adwt + (α + δ)dt .

T = ∞: explicit results by Shepp-Shiryaev, Duffie-Harrison,Salminen, etc.

T < ∞: studied by Duistermaat-Kyprianou-van Schaik,Ekstrom, Peskir, etc.

Some extenstions: Guo-Shepp, Pedersen, Gapeev, etc.

General treatment with viscocity of variational inequality:Barles-Daher-Romano.

Mathematical Results

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.

⊲ Main Theorem (1)


⊲ Remark

⊲ How to compute ?

⊲ Bdry Cross Prob.

⊲ Related Topics

16 / 24

Dynamic Version of Optimal Stopping

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

17 / 24

By dynamic-programming and the Markov property of (t, X, N),or (t, Y, N) := (t, K/X, N), deduce

ess supt≤τ≤T

E [NτXτ |Ft] = V (t, Xt, Nt) = e−δtXtV (t, Kt/Xt, Nt) ,

where

V (t, y, z) := sup0≤τ≤T−t

Ee−δτN (t,y,z)τ ,

N (t,y,z)s :=z ∨ max

u∈[0,s]Y (t,y)

u ,

and Y (t,y) solves

dYs = Ys a(t + s, Ys)dws + (κt+s + δ)ds , Y0 = y.

Free Boundary Problem

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

18 / 24

(∂t + L − δ)V (t, y, z) =0 for (t, y, z) ∈ C,

V (t, y, z)|y=b(t,z)+ =z, (instantaneous stopping),

∂yV (t, y, z)|y=b(t,z)+ =0, (smooth-pasting),

∂zV (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 := 12y2a(t, y)2∂yy + y(κt + 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) ∈ R2; 0 < y ≤ z, z ≥ 1

.

Main Theorem (1)

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

19 / 24

(1) For V , ∃b : [0, T ) × [1,∞) → R>0, s.t. (V, b) solves theFree-boundary Prob.

(2) b is the unique solution to

z = e−δ(T−t)E

[

N(t,b(t,z),z)T−t

]

+ δ

∫ T−t

0

dse−δs

× E[

N (t,b(t,z),z)s I

(

Y (t,b(t,z))s < b

(

t + s, N (t,b(t,z),z)s

))]

so that

(i) b: continuous, nondecreasing w.r.t. t,(ii) b(t, z) < z and b(T−, z) = z.

Main Theorem (2)

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

20 / 24

(3) Optimal stopping time:

τt := inf 0 ≤ t ≤ T ; Yt ≤ b(t, Nt) .

(4) “Closed-form” expression:

V (t, y, z) = e−δ(T−t)E

[

N(t,y,z)T−t

]

+ δ

∫ T−t

0

dse−δs

× E[

N (t,y,z)s I

(

Y (t,y)s < b

(

t + s, N (t,y,z)s

))]

.

Remark

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

21 / 24

The result may not be surprising to some extent...

Remark

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

21 / 24


Point: the integral equation for b is the “characteristic” ofthe problem in the sense that

free boundary b is determined as a unique sol. of theequation, and,

value function V is represented “in closed-form” with b.

Remark

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

21 / 24


Point: the integral equation for b is the “characteristic” ofthe problem in the sense that

free boundary b is determined as a unique sol. of theequation, and,

value function V is represented “in closed-form” with b.

Peskir’s change of variable formula for continuoussemimartingale with local-time on curve, (an extension ofIto-Meyer-Tanaka formula), plays an important role.

How to Numerically Compute b and V ?

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

22 / 24

Using

q(s, dy′, dz′; t, y, z) := P(

Y (t,y)s ∈ dy′, N (t,y,z) ∈ dz′

)

,

we see

E

[

N(t,y,z)T−t

]

=KT y

Kt

∫

E

(

z′

y′

)

q(T − t, dy′, dz′; t, y, z),

E[

N (t,y,z)s I

(

Y (t,y)s < b

(

t + s, N (t,y,z)s

))]

=Kt+sy

Kt

∫

E

(

z′

y′

)

I(y′ < b(t + s, z′))q(s, dy′, dz′; t, y, z).

So, the integral equation for b is represented with q.

Boundary Crossing Probability

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

23 / 24

We see

N (t,y,z)s ≤ d

⇔ z ≤ d and wu ≥ ℓ(u) for all u ∈ [0, s].

where

ℓ(u) = ℓ(u; d, t, y, k) := f−1

(

t + u,Kt+u

d

)

− Kt+u

y.

Approximation schema to compute the joint probability

P (ws ≤ c1, wu ≥ ℓ(u) for all u ∈ [0, s])

have been studied by several works.

Related Topics for Future Works

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

24 / 24

More general (multi-dimensional, possibly) model withnumerics.

Large-time asymptotics (as T → ∞).

Related Topics for Future Works

⊲ Problem

⊲ Outline

Applications

Results

⊲ Dynamic Prob.

⊲ Free-bdry Prob.



⊲ Remark



⊲ Related Topics

24 / 24

More general (multi-dimensional, possibly) model withnumerics.

Large-time asymptotics (as T → ∞).

Comparison with American-OBPI strategy (by ElKaroui-Jeanblanc-Lacoste, El Karoui-Meziou, etc.)

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