Dynamic Protection for Bayesian Optimal Portfolio · Dynamic protection for Bayesian optimal...
Transcript of Dynamic Protection for Bayesian Optimal Portfolio · Dynamic protection for Bayesian optimal...
1 / 24
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
(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.
Characterize with a free-boundary problem for 3-dim.Markov (t, X, N).
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.
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
(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.
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 nonlinearintegral equation.
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.
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 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,
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,
X: one unit of (the discounted) investment fund,δ: dividend rate, paid to customers (not re-invested in thefund).
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,
X: one unit of (the discounted) investment fund,δ: dividend rate, paid to customers (not re-invested in thefund).
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.
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,
X: one unit of (the discounted) investment fund,δ: dividend rate, paid to customers (not re-invested in thefund).
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)
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)
CEV (with δ = 0) by Imai-Boyle.
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)
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.
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)
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.
πt := σ−2(µ − r): growth-optimal fund,
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)
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.
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)
⊲ Main Theorem (2)
⊲ 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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ 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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ 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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ 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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ 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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ Related Topics
21 / 24
The result may not be surprising to some extent...
Remark
⊲ Problem
⊲ Outline
Applications
Results
⊲ Dynamic Prob.
⊲ Free-bdry Prob.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ Related Topics
21 / 24
The result may not be surprising to some extent...
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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ Related Topics
21 / 24
The result may not be surprising to some extent...
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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ 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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ 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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ 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.
⊲ Main Theorem (1)
⊲ Main Theorem (2)
⊲ Remark
⊲ How to compute ?
⊲ Bdry Cross Prob.
⊲ 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.)