Numerical methods for the pricing of Swing options: a ...

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Submitted on 1 Dec 2006

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Numerical methods for the pricing of Swing options: astochastic control approach

Christophe Barrera-Esteve, Florent Bergeret, Charles Dossal, EmmanuelGobet, Asma Meziou, Rémi Munos, Damien Reboul-Salze

To cite this version:Christophe Barrera-Esteve, Florent Bergeret, Charles Dossal, Emmanuel Gobet, Asma Meziou, et al..Numerical methods for the pricing of Swing options: a stochastic control approach. Methodology andComputing in Applied Probability, Springer Verlag, 2006, Methodology and Computing in AppliedProbability, 8 (4), pp.517-540. �10.1007/s11009-006-0427-8�. �inria-00117175�

https://hal.inria.fr/inria-00117175

https://hal.archives-ouvertes.fr

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TK 467H? ;I? <: 9; :B6H? 9o ;I? 8A< J9 ?< 79; H9BB?<: 97J

;9 A ;BAJAFC? 67<;BED ?7; = ;I? {ACEA;697 A7J ;I? I?J8678 9o H97;BAH;< 97 8A< AB? : ?Bo9BD ?J;IB9E8I ;I? E<? 9o o9B5ABJ H97;BAH;< 5 6;I J6n?B?7; D A;EB6;6?<

t ∈ [0, T ]w<?? ;I? A::B9AHI

86{?7 67 tA6CC?; = u977 A7J v9D:A6J6< w)^^x yy s ;I?6B :B6H?< AB? J?79;?J Fl(F (s, t))0≤s≤t

KvI? 67;?B?<; BA;?< AB? A<<ED ?J ;9 F ? H97<;A7; = ?pEAC ;9

rA7J o9B ;I? <Am? 9o <6D:C6H6;l= 5?

H97<6J?B A 97?GoAH;9B l6?CJ = A< 67 tA6CC?; = u977 A7J v9D:A6J6< w)^^x y s

dF (s, t)

F (s, t)= σe−α(t−s)dWs, 0 ≤ s ≤ t.

wÊ KÊy

q7 ;I? CA<; ?b:B?<<697 = ;I? 67 6;6AC H97J6;697 6< J?r7?J Fl ;I? o9B5ABJ HEB{?(F (0, t))0≤t≤T

� 68EB? Ê s �A< �9B5ABJ @EB{?

wH9BB?<: 97J678 ;9 ;I? <?A<97AC6;l= A< <I957 97 � 68EB? Êy K q; 59ECJ F ? ?A<l ;9 67H9B: 9BA;??b;BA oAH;9B< 9B A H97{?76?7H? l6?CJ ;?BD = 5 6;I9E; D A~ 9B D 9J6rHA;697< ;9 5IA; o9CC95 < K� 6;I67 9EB oBAD ?59Bm = ;I? <: 9; :B6H? A; ;6D ?

t6< 86{?7 Fl

Ft = F (t, t)K q7 apEA;697

wÊ KÊy = ;I? Jl7AD 6H< 6< 86{?7 J6B?H;Cl E7J?B ;I? E<EAC B6<mG7?E;BAC :B9FAF6C6;l D ?A<EB?Q

K

WJ?r7?< A C67?AB `B9576A7 D 9;697 A7J <9 = ;I? :B9FAF6C6;l

Q;BA7<o9BD < ;I? o9B5ABJ

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QFE; E7J?B ;I?

I6<;9B6HAC :B9FAF6C6;l � I95?{?B = ABF6;BA8? AB8ED ?7;< <I95 ;IA; ;I? oA6B {ACE? 9o A r7A7H6ACH97;BAH; IA< ;9 F ? ?{ACEA;?J A< ;I?

QG?b: ?H;A;697 9o 6;< J6<H9E7;?J HA<IG|95 w<?? �ABA;�A<

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QK

} 95 = C?; E< J?<HB6F ? A 8?7?BAC 45 678 H97;BAH; K �? A<<ED ? ;IA; ?7?B8l 6< :EBHIA<?J A;<9D ? rb?J JA;?<

0 = t0 < · · · < ti < · · · < tN = TK vI? {9CED ? A; ;6D ?

ti6< J?79;?J Fl

qtiA7J <EF~ ?H; ;9 ;I? H97<;BA67;

qmin ≤ qti ≤ qmax.wÊ K)y

vI? HEDECA;6{? {9CED ? :EBHIA<?J E: ;9 ;6D ?ti

6< 86{?7 FlQti =

∑i−1j=0 qtj

K q; DE<; <A;6<ol;I? ;?BD 67AC H97<;BA67;

Qmin ≤ QT ≤ Qmax.wÊ K_y

v9 <6D:C6ol= ;I? :EBHIA<? :B6H? 6< A<<ED ?J ;9 F ? H97<;A7; = ?pEAC ;9K

K �95?{?B 6o 7?H?<<ABl=6; HA7 F ? ?b;?7J?J ;9 A J?;?BD 676<;6H ;6D ? oE7H;697 K �?7H? = ;I? FEl?B 9o ;I? H97;BAH; 8?;<

qti(Fti −K)A; ;6D ?

tiK �EB;I?BD 9B? = 5I?7 ;I? 8C9FAC :EBHIA<? H97<;BA67;< AB? {69CA;?J = I?

D Al A::Cl <9D ? : ?7AC;6?< A; D A;EB6;lK vI?<? HA7 F ? :B9: 9B;697AC ;9 ;I? <: 9; :B6H?FT

A;D A;EB6;l A7J ;9 ;I? 9{?B 9B E7J?BGH97<ED:;697 s −A FT [QT−Qmin]−−B FT [QT−Qmax]+

=5I?B?

AA7J

BAB? : 9<6;6{? H97<;A7;< K v9 F ? H97H6<? = 5? J?79;? ;I6< : ?7AC;l Fl PT (FT , QT )

67 ;I? <?pE?C K �9B A 86{?7 H97<ED:;697 <;BA;?8lq = (qti)0≤i≤N

= ;I? oA6B :B6H? 6< ;IE< 86{?7Fl

J (q) = E

(N−1∑

i=0

e−rtiqti(Fti −K) + e−rTPT (FT , QT )

)wÊ Kx y

w;I? ?b: ?H;A;697 6< : ?Bo9BD ?J E7J?B ;I? B6<mG7?E;BAC :B9FAF6C6;l D ?A<EB?Q

y K

VX2, � �� σ = 70%

�α = 4

�r = 0%

�qmin = 0

�qmax = 6

�Qmin = 1300

�Qmax = 1900

�K = 20

�T = 1 �� N = 365 ��

�� F (0, ti) = 20,∀0 ≤ i ≤ N

vI? :B?H?J678 J?<HB6:;697 IA< ;9 F ? <C68I;Cl D 9J6r?J o9B �� K q7 ;I6< HA<? =A7 ?D:;l ;A7m 6< B?7; ;9 ;I? 957?B 9o ;I? H97;BAH; A7J ACC95 < I6D ;9 ?6;I?B FEl <9D ? 8A<97 ;I? D ABm?; A7J 67~ ?H; 6; 67 ;I? ;A7m = 9B 5 6;IJBA5 6; A7J <?CC 6; 97 ;I? D ABm?; K z; ?AHI

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JA;? = ;I? :B9r; 6< 86{?7 Fl

ψ(Fti , qti) =

−qti(Fti + cI)6oqti ≥ 0

wq7~ ?H;697 y =

−qti(Fti − cW )6oqti ≤ 0

w� 6;IJBA5ACy =

06oqti = 0

w4AD ? C?{?C o9B ;I? <;9BA8?y =

5I?B?cI > 0

wB?<: KcW > 0

y J?79;?< <9D ? <;B6H;Cl : 9<6;6{? 67~ ?H;697 wB?<: K 5 6;IJBA5ACyH9<; K > 9B?9{?B = ;I? ;A7m HA:AH6;l IA< <9D ? {9CED ? C6D 6;A;697< = 5I6HI 6D: 9<?< H97<;BA67;<C6m?

0 ≤ f1(t) ≤ Qt ≤ f2(t)o9B <9D ? J?;?BD 676<;6H oE7H;697<

f1A7J

f2K �9B ;I? 7?b;

9:;6D 6�A;697 :B9FC?D = 5? HA7 o9BDECA;? ;I?<? :Il<6HAC H97<;BA67;< 67 ;?BD < 9o A <D 99;I: ?7AC6�A;697 s −C∑N

i=0 e−rti [(Qti −f1(ti))−]2−C∑N

i=0 e−rti [(Qti −f2(ti))+]2

= 5 6;I A CAB8?H97<;A7;

CK qo 5? H97<6J?B ;I6< : ?7AC6�A;697 A< A r7A7H6AC : ?7AC;l= ;I? oA6B :B6H? 9o ;I6<

�� 6< ;I?7 86{?7 Fl

J (q) = E

(N∑

i=0

Ψti(qti , Fti , Qti)

),

wÊ K�y

5I?B?Ψt(q, F,Q) = e−rt

(ψ(F, q) −C[(Q− f1(t))−]2 − C[(Q− f2(t))+]2

) 6o

t < tNA7J

ΨtN (q, F,Q) = e−rtN(−C[(Q− f1(tN ))−]2 − C[(Q− f2(tN ))+]2

) K

VX2, O � � �� Qmin = Qmax = 0 ��

f2 � � ��

σ = 70%�α = 4

�r = 0%

�qmin = −0.2

�qmax = 0.4

�f1 ≡ 0

�f2(ti) =

min(qmaxi, 20, qmin(i − N))�cI = 0.6

�cW = 0.2

�K = 20

�T = 1 ��

�N = 365 � � � �� F (0, ti) = 20 vI?<? J?<HB6:;697< 9o 45 678 9:;697< A7J <;9BA8? H97;BAH;< HA7 F ? E76r?J 5 6;I67 ;I? o9CGC95 678 o9BDECA;697 9o ;I? :B6H?

J (q) = E

(N−1∑

i=0

Ψti(qti , Fti , Qti) + PT (FT , QT )

).

wÊ K�y

q7 ;I? AF 9{? =q

6< A 86{?7 H97<ED:;697 <;BA;?8l=Ψ·(·)

J?79;?< <9D ? A::B9:B6A;? 67<;A7;AG7?9E< :B9r; A7J PT (·) <9D ? ;?BD 67AC : ?7AC;lKv9 :B6H? ;I?<? 67<;BED ?7;< = 5? D Am? ;I? HBEH6AC A<<ED:;697 ;IA; ;I? H97<ED:;697 J?H6<697<

�� ! � "�#$�% � & � "%'�� &� �%��(��#� �

AB? ;Am?7 AHH9BJ678 ;9 ;I? <9C? ?{9CE;697 9o ;I? E7J?BCl678 A<<?; = A7J 79; ;I? B?AC 7??J< 678A< K 49 = ;I? :B6H? 6< 86{?7 Fl

supq

J (q).

q; 6< A <;9HIA<;6H 9:;6D AC H97;B9C :B9FC?D K q7 ;I? 7?b; <?H;697 = 5? <;A;? ;I? A<<9 H6A;?JJl7AD 6H :B98BADD 678 ?pEA;697 w<?? `?B;<?mA< A7J 4IB?{? wÊ]��y = `?B;<?mA< wÊ]]�yy oB9D5I6HI 5? J?B6{? <9D ? 7ED ?B6HAC D ?;I9J< K

� �i��fkj� ��e�g h ��h gd h�d�� kj �fg�f�� kd� ��i��

ekgd

� 6B<; = o9B F 9;I ;I?9B?;6HAC A7J 7ED ?B6HAC :EB: 9<?< = 6; 6< F ?;;?B ;9 B?5B6;?(Ft)0≤t≤T

E<678A7 ?b;BA <;A;? {AB6AFC?

X = (Xt)0≤t≤T

J?r7?J FlXt = σe−αt

∫ t

0 eαudWu.

}9;? ;IA;X

6<A7 � B7<;?67G�IC?7F ?Hm :B9 H?<< = 5I6HI <9C{?<

dXt = −αXtdt + σdWtK �?; = ;I 6< :B9: ?B;l

5 6CC 79; F ? B?ACCl E<?J 67 ;I? <?pE?C K > 9B?9{?B = A7 6DD ?J6A;? A::C6HA;697 9o q;� �< o9BDECAl6?CJ<

F (s, t) = F (0, t) exp[e−α(t−s)Xs − σ2

4α(e−2α(t−s) − e−2αt)]

K �? :AB;6HECABCl IA{?

Fti = F(0, ti

)exp

(Xti −

σ2

4α

(1 − e−2αti

)).

w) K�y

� � ��

z< ACB?AJl D ?7;697?J = ;I? {ACEA;697 9o 45 678 H97;BAH;< HA7 F ? ?DF?JJ?J 67;9 A <;9 HIA<;6HH97;B9C oBAD ?59Bm K q7J??J = ;I? Jl7AD 6H< 9o ;I? <l<;?D 6< J?<HB6F ?J Fl A J6<HB?;?G;6D ?> ABm9{ HIA67

(Fti , Qti)0≤i≤N

;IA; 6< H97;B9CC?J Fl ;I? C9 HAC H97<ED:;697(qti)0≤i≤N−1

K� 6{?7 A ;6D ?

tiA7J A <;A;?

(F,Q)= ;I? :B6H? 6< 86{?7 Fl

P (ti, F,Q) = sup(qtj )i≤j≤N−1

E(N−1∑

j=i

Ψtj (qtj , Ftj , Qtj ) + PT (FT , QT )|Fti = F,Qti = Q).

q7 ;I? CA<; ?b:B?<<697 = ;I? <E: 6< ;Am?7 9{?B ACC AJD 6<<6F C? H97<ED:;697< K z <;A7JABJ B?<EC;<;A;?< ;IA; = E7J?B D 6CJ A<<ED:;697< = ;I?B? ?b6<;< A7 9:;6D AC > ABm9{6A7 H97<ED:;697 = 86{?7Fl ;I? Jl7AD 6H :B98BADD 678 ?pEA;697 wAC<9 m7957 A< `?CCD A7 �< ?pEA;697 y K

�� ! � "�#$�% � & � "%'�� &� �%��(��#� �

� 0/�/2YUY/[ O �� p ��

C� � � � � � ��

�� ti ∈ [0, T ]�F > 0

�q ∈ [qmin, qmax]

��(Q,Q′) ∈ [−N(qmin)−, N(qmax)+]

�

|Ψti(q, F,Q)| + |PT (F,Q)| ≤ C(1 + F p),

|Ψti(q, F,Q′) − Ψti(q, F,Q)| ≤ C(1 + F p)|Q′ −Q|.

� �� q∗(t, F,Q)

� � ��

P (ti, F,Q) = maxq∈[qmin,qmax]

[Ψti(q, F,Q) + E

(P (ti+1, Fti+1

, Q+ q)|Fti = F)], (ti < T ),

P (T, F,Q) = PT (F,Q).w) K�y

}9;? ;IA; ;I6< B?<EC; I9CJ< o9B F 9;I 45 678 A7J <;9BA8? H97;BAH;< K�� vI6< 6< A7 6DD ?J6A;? A::C6HA;697 9o � B9: 9<6;697 � K� K 67 `?B;<?mA< A7J 4IB?{?

wÊ]��y K � 6B<; = ;I? H97J6;697< 97 ;I? 8B95;I 9o 67<;A7;A7?9E< A7J ;?BD 67AC :B9r;< ;98?;I?B5 6;I ;I? HCA<<6HAC ?<;6D A;?<

E(sup0≤t≤T Fpt ) < ∞ ?7AFC? E< ;9 HI?Hm A<<ED:;697< w�+y

A7J w�−y 67 � B9: 9<6;697 � K� K vI?7 = 6; B?D A67< ;9 :B9{? ;IA; ;I? <E:B?DED 67 w) K�y 6<A;;A67?J K vI6< 6DD ?J6A;?Cl o9CC95 < oB9D ;I? H97;67E6;l 9o

P (ti+1, F,Q)5 KB K; K

Qw5I6HI 6<

A ?A<l H97<?pE?7H? 9o 9EB �6:<HI6;� H97;67E6;l Il: 9;I?<6<y K � 67ACCl= <67H? ;I? Jl7AD 6H< 9o;I? o9B5ABJ H97;BAH;< J9 ?< 79; J?: ?7J 97

(qti)i= ;I? E<EAC H97J6;697AC ?b: ?H;A;697 5 KB K; K

(Fti , Qti)HA7 F ? <6D:C6r?J E<678 ;I? B?CA;697

Qti+1= Qti + qti

K q; :B9{6J?< ;I? Jl7AD 6H:B98BADD 678 ?pEA;697 A< ?b:B?<<?J 67 w) K�y K

�

�ABA8BA:I< ) K) A7J ) K_ B?<: ?H;6{?Cl 6D:C?D ?7; ;I?maxq∈[qmin,qmax]

9: ?BA;697 A7J ;I? H9D G:E;A;697 9o

E(·|Fti = F )o9B <9C{ 678 A::B9b6D A;?Cl ;I? Jl7AD 6H :B98BADD 678 ?pEA;697

w) K�y K

��

z< E<EAC = <9C{ 678 ;I? Jl7AD 6H :B98BADD 678 ?pEA;697 w) K�y 6< oAB oB9D F?678 ;B6{ 6AC F ?HAE<?9o ;I? H97J6;697AC ?b: ?H;A;697 A7J ;I? 9:;6D 6�A;697 5 KB K; K

qA; ?AHI ;6D ? K �9B A 7ED ?B6HAC

B?<9CE;697 = ;I6< o9BDECA;697 DE<; F ? <6D:C6r?J K v9 J9 <9 = 5? rB<; J6<HB?;6�? ;I? <?; 9oAJD 6<<6F C? {ACE?< 9o

qtisqti ∈ {qmin, qmin + ∆q, . . . , qmax − ∆q, qmax}

5I?B?∆q

J?79;?< A

�� ! � "�#$�% � & � "%'�� &� �%��(��#� ]

� 68EB? ) s u ?<;B6H;697 9o ;I? <?; 9o AJD 6<<6F C? :EBHIA<?J {9CED ?< 5I?7 ;I? 8C9FAC H97<;BA67;<HA7 79; F ? {69CA;?J K

: 9<6;6{? H97<ED:;697 <;?: K@97<?pE?7;Cl= ;I? :EBHIA<?J HEDECA;6{? {9CED ?Qti =

∑i−1j=0 qtj

F ?C978< ;9 A <?; 9o : 9<<6F C? {ACE?< {Q1ti, Q2

ti, . . . , Q

Mtiti

} = 5I?B?Mti

J?79;?< ;I? 7EDF ?B 9o: 9<<6F C? HEDECA;6{? {9CED ?< ;IA; AB? :EBHIA<?J E: ;9 ;6D ?

tiK qo 6; 6< 79; ACC95?J ;9 FB?Am

;I? 8C9FAC :EBHIA<? H97<;BA67;<Qmin

A7JQmax

w6 K? K ;I? A::C6?J : ?7AC;6?< AB? 67r76;?y = ;I?<?; A = {(ti, Qti) : 0 ≤ i ≤ N} 9o AJD 6<<6F C? HEDECA;6{? H97<ED:;697< HA7 F ? ?� H6?7;ClC6D 6;?J K vI6< 5 6CC F ? ?b;B?D ?Cl E<?oEC ;9 <: ??J E: ;I? B?<9CE;697 9o w) K�y K �9B ?bAD:C? = C?;E< H97<6J?B A H97;BAH; 5 6;I ;I? HIABAH;?B6<;6H<

Qmax = 1=Qmin = 0

=∆q = qmax = 1

A7J

qmin = −2= 9{?B ;I? : ?B69J {t0 = 0, t1, . . . , t4 = T} K vI? <?; A 6< B?:B?<?7;?J 97 � 68EB? ) K

�9B A 86{?7 H9E:C?(ti, Qti)

= J?79;? FlAdm(ti, Qti)

;I? <?; 9o J6<HB?;? AJD 6<<6F C? {ACE?<9oqti

K qo ;I? H97<;BA67;< HA7 F ? FB9m?7 = ;I?7Adm(ti, Qti) = {qmin, qmin + ∆q, . . . , qmax −

∆q, qmax}K � ;I?B5 6<? 6; HA7 F ? DEHI <D ACC?B w<?? � 68EB? )y K vIE< = o9B A7l

(ti, Qti) ∈ A =w) K�y F ?H9D ?<

P (ti, F,Qti) = maxq∈Adm(ti,Qti

)

[Ψti(q, F,Qti) + E

(P (ti+1, Fti+1

, Qti + q)|Fti = F)].

w) K]y

q7 C68I; 9o w) K]y = 5? <I9ECJ H9D:E;? A< D A7l oE7H;697<P (ti, ·, Qti)

A< ?C?D ?7;< 67 ;I? <?; A K� 95?{?B = ;I6< CA;;?B HA7 F ? pE6;? CAB8? w<?? abAD:C? Êy = A7J ;IE< DEHI ;6D ? H97<ED 678 KvI6< 6< A7 67I?B?7; JBA5FAHm 9o ;I6< A::B9AHI A7J A< 5? 5 6CC CA;?B <?? = 5? J9 79; D ??;;I?<? 6<<E?< 5 6;I ;I? A::B9AHI?< FA<?J 97 :ABAD ?;?B6�?J H97<ED:;697< K

��X\� W, � ��

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Ê^

�� A �� qti �� ∆q = qmax − qmin

� 68EB? _ s 4;A;?< 5I?B? ;I? :B6H? oE7H;697P (ti, Qti , ·)

<I9ECJ F ? H9D:E;?J

v9 H9D:C?;? ;I? B?<9CE;697 9o ;I? Jl7AD 6H :B98BADD 678 ?pEA;697 w) K]y = 5? J?{?C9: 67 ;I?<?pE?C ;59 J6n ?B?7; <;BA;?86?< K

Ê K � <678 A ;B679D 6AC ;B?? ;9 J?<HB6F ? ;I? ?{9CE;697 9o(Fti)0≤i≤N

K q; C?AJ< ;9 ;I? <9GHACC?J o9B?<; 9o ;B??< AC89B6;ID w<?? �AB6G�A{A<<A76 = 46D HI6 A7J �AB? w)^^ Êy � tA6CC?; =u 977 A7J v9D:A6J6< w)^^x yy K

) K � <678 > 97;? @ABC9 <6DECA;697< 9o(Fti)0≤i≤N

= 5I6HI AB? DEHI D9B? AJA:;?J ;9DEC;6GoAH;9B D 9J?C< K vI?l ?7AFC? ;9 H9D:E;? H97J6;697AC ?b: ?H;A;697< 67{9C{?J 67w) K]y w5 6;I A79;I?B B?:B?<?7;A;697 y E<678 B?8B?<<697 ;?HI76pE?< wA7AC989E< ;9 ;I?�978<;AnG4HI5AB;� A::B9AHI w)^^ Êy o9B `?BDEJA 9:;697<y K vI6< A::B9AHI IA< 79;F ??7 E<?J 67 ;I? ?b6<;678 C6;?BA;EB? 97 45 678 9:;697< K

�� ! � "�#$�% � & � "%'�� &� �%��(��#� ÊÊ

� ��

O �� ,2T0Y�UY/[ / � U�, XW�/0YU�\

q7 C68I; 9o w) K�y = 5? 7??J ;9 H97<;BEH; A ;B679D 6AC ;B?? = 67 9BJ?B ;9 J?<HB6F ? I95(Xti)0≤i≤N

wA7J ;I?B?o9B?(Fti)0≤i≤N

y ?{9C{?< K �? FB6?|l B?D 67J ;I? H97<;BEH;697 9o <EHI A ;B?? wo9BJ?;A6C< = <?? �AB6G�A{A<<A76 = 46D HI6 A7J �AB? w)^^ Êyy K aAHI 79J? HA7 ?6;I?B 89 E: = 9B 89J957 9B ?{?7 <;Al 67 ;I? <AD ? :Il<6HAC <;A;? K vIB?? m67J< 9o FBA7HI?< AB? ;I?7 A{A6CAFC?;9 r; ;I? D ?A7GB?{?B;678 F ?IA{69B 9o

XK 4?? � 68EB?

4= 5I?B? HA<? ) A7J _ H9BB?<: 97J

B?<: ?H;6{?Cl ;9 ;I? ;9: A7J ;I? F 9;;9D 9o ;I? ;B679D 6AC ;B?? K vI? <:AH678 F ?;5??7 79J?<

� 68EB? x s �7 ;I? C?o; = ;IB?? : 9<<6F C? FBA7HI?< K �7 ;I? B68I; = ;B679D 6AC ;B?? K

6< ?pEAC ;9σ√

3∆t

= 5I?B?∆t

J?79;?< ;I? J6<HB?;6�A;697 ;6D ? <;?: K vAFC? Ê 86{?< ;I?:B9FAF6C6;6?< 97 ?AHI AC;?B7A;6{? FBA7HI678 K vI?<? CA;;?B< AB? H9D:E;?J 67 9BJ?B ;9 D A;HI;I? rB<; ;59 D 9D ?7;< = A7J ;I?6B <ED 6< 7A;EBACCl ?pEAC ;9 97? K z < :B9FAF6C6;6?< DE<; 79;F ? 7?8A;6{? = 5? J?JEH? ;IA; 7?H?<<AB6Cl

jmax =⌈

0.184α∆t

⌉ = A7J ;I6< F 9E7J< ;I? ;B?? �< 5 6J;I<67H?

j ∈ [−jmax, jmax]K � ?79;? Fl

Xjti

;I? {ACE? 9oXti

97 79J?j

67 ;I? ;B679D 6AC ;B?? = A7J

@A<? Ê @A<? ) wj = jmax

y @A<? _ wj = −jmax

y

pup16 − 1

2αj∆t + 12 [αj∆t]

2 76 − 3

2αj∆t + 12 [αj∆t]

2 16 + 1

2αj∆t + 12 [αj∆t]

2

pmiddle23 − [αj∆t]

2 −13 + 2αj∆t − [αj∆t]

2 −13 − 2αj∆t − [αj∆t]

2

pdown16 + 1

2αj∆t + 12 [αj∆t]

2 16 − 1

2αj∆t + 12 [αj∆t]

2 76 + 3

2αj∆t + 12 [αj∆t]

2

vAFC? Ê s � B9FAF6C6;6?< J?: ?7J678 97 ;I? {?B;6HAC : 9<6;697j

67 ;I? ;B?? K

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Ê)

FlLti

;I? 7EDF ?B 9o 79J?< A; ;6D ?ti

K v9 ?AHI : 967; (ti, Q

jtiti

)

jti=1,...,Mti

97 ;I? 8BA:I6H =

H9BB?<: 97J< A BA78? (X

jti

)

j=1,...,Lti

9o 79J?< 67 ;I? ;B?? wA; ;6D ?ti

y K �9BDECA w) K�y 86{?< ;I?

H9BB?<: 97J678 <: 9; :B6H?< 9o 8A< (F

jti

)

j=1,...,Lti

K @97<6J?B 795 ;I? AC89B6;ID o9B ;I? o9B?<;9o ;B??< K `l apEA;697 w) K]y = ;I? :B6H?

P (t0, F1t0, Q1

t0)

9o ;I? 9:;697 6< H9D:E;?J FAHm5ABJ<67 ;I? o9B?<; = 67 ;I? o9CC95 678 5AlK

• �[YUYX WY�XUY/[ K @9D:E;? ;I? ;?BD 67AC {ACE?<P (tN , F

jtN, Qk

tN)

o9Bj = 1, . . . , LtN

A7J

k = 1, . . . ,MtN

K

• �U,0XUY/[ K z <<ED ? 795 ;IA; ;I? {ACE?<P (ti+1, F

jti+1

, Qkti+1

)o9B

j = 1, . . . , Lti+1

A7J

k = 1, . . . ,Mti+1

AB? m7957 K �?7H? = o9B ?AHI : 9<<6F C? HEDECA;6{? {9CED ? (ti, Q

kti

) 56;IA <: 9; :B6H? ?pEAC ;9

Fjti

= 5? H9D:E;? ;I? {ACE?Ψti(q, F

jti, Qk

ti)+E(P (ti+1, Fti+1

, Qkti

+

q)|F jti)= o9B A7l

q67

Adm(ti, Qkti)= E<678 ;I? :B9FAF6C6;6?< 67 ;I? ;B?? w<?? vAFC? Êy K

vI? D Ab6DED 9{?Bq

86{?<P (ti, F

jti, Qk

ti)K

vI?7 = 6;?BA;678 ;6CC ;6D ?t0

:B9{6J?< ;I? :B6H? 9o ;I? H97;BAH; K

O �� O �Q\,0YTXW 0,2QWU2

�? 86{? 7ED ?B6HAC B?<EC;< o9B ;I? ;59 HA<?< J?<HB6F ?J 67 4?H;697 Ê K �? AC<9 :C9; ;I?9:;6D AC 8A< H97<ED:;697 5 6;I B?<: ?H; ;9 A 7A;EBAC 8A< <AD:C? :A;I K �? 9F<?B{? A FA78GFA78 F ?IA{69B o9B ;I? 9:;6D AC H97<ED:;697 = A< 6; 6< ;I?9B?;6HACCl ~ E<;6r?J 67 4?H;697 _ KÊ K��X\� W, � 3 N. Y[� T/[U0XTU � vI? {ACE?< 9o :ABAD ?;?B< AB? <?; ;9 ;I9<? 9o @A<? Ê K q7;I? ;AFC? F ?C95 = F 9;I o9B5ABJ HEB{?< AB? H97<6J?B?J K v9 IA{? H9D:ABAFC? B?<EC;< 5 6;IJ6n ?B?7; H97<ED:;697 oB?pE?7H6?< = 5? AJ~E<;

qminA7J

qmaxK �9B 67<;A7H? = 5 6;I A 5??mCl

H97<ED:;697 = 5? <?;qmin = 0

A7Jqmax = 6 × 7

K

@97<ED:;697 : ?B69J � CA; o9B5ABJ HEB{? u?AC o9B5ABJ HEB{?

� JAl< )�^^ xx_]_ JAl< )�] Ê x��) JAl< )�^x x�^�Ê JAl )�Ê� x� ÊÊ

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Ê_

� 68EB? � s 45 678 H97;BAH; K �:;6D AC 8A< J?H6<697 o9B A 7A;EBAC 8A< <AD:C? :A;I K �7 ;I? C?o; =5 6;I A |A; o9B5ABJ HEB{? K �7 ;I? B68I; = 5 6;I ;I? B?AC o9B5ABJ HEB{? 9o � 68EB? Ê K

@97<ED:;697 : ?B69J � CA; o9B5ABJ HEB{? u?AC o9B5ABJ HEB{?

� JAl< �� K�] ))x K]x JAl< �� K_� )_]

) JAl< �� K�� )x) K)Ê JAl �� K]) )x) K_

��X\� W, O 3 2U/0X�, T/[U0XTU � vI? {ACE?< 9o :ABAD ?;?B< AB? <?; ;9 ;I9<? 9o @A<? ) K�9B F 9;I HA<?< = ;I? :B6H? H97{?B8? A< ;I? H97<ED:;697 oB?pE?7Hl 67HB?A<?< K vI? H97{?BG

8?7H? 6< <C95?B 5 6;I ;I? B?AC o9B5ABJ HEB{? F ?HAE<? 9o ;I? I68I <?A<97AC6;lK q7 ;I? <?pE?C =;I? AF 9{? {ACE?< AB? ;Am?7 A< B?o?B?7H?< o9B ;I? 9;I?B 7ED ?B6HAC D ?;I9J< K �? AC<9 79;?;IA; ;I? B?AC6�?J 9:;6D AC H97<ED:;697 6< 9o;?7 9o FA78GFA78 ;l: ? w;I6< 6< ;I?9B?;6HACClH97rBD ?J Fl vI?9B?D _ KÊy K

� � ��

O �� ,2T0Y�UY/[ / � U�, XW�/0YU�\

�? 7??J ;9 <6DECA;?NMC

67J?: ?7J?7; B?AC6�A;697< 9oF = (Fti)0≤i≤N

K v9 J9 <9 A7J67 {6?5 9o w) K�y = 6; <E� H?< ;9 <6DECA;?

(Xti)0≤i≤N

= 5I6HI HA7 F ? J97? Fl 5B6;678Xti =

σe−ati∑i−1

j=0 gj

= 5I?B?gj =

∫ tj+1

tjeαudWu

K vI?7 = <67H?(gj)j

AB? 67J?: ?7J?7; �AE<<6A7BA7J9D {AB6AFC?< 5 6;I A �?B9 D ?A7 A7J A {AB6A7H? ?pEAC ;9 ∫ tj+1

tje2αudu = e

2αtj+1−e2αtj

2α

=

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Êx

� 68EB? � s 4;9BA8? H97;BAH; K �:;6D AC 8A< J?H6<697 o9B A 7A;EBAC 8A< <AD:C? :A;I K �7 ;I?C?o; = 5 6;I A |A; o9B5ABJ HEB{? K �7 ;I? B68I; = 5 6;I ;I? B?AC o9B5ABJ HEB{? 9o � 68EB? Ê K

;I? <6DECA;697 9o(Xti)0≤i≤N

6< <;A7JABJ K }9;? ;IA; A7 ?b;?7<697 ;9 DEC;6GoAH;9B D 9J?C<6< <;BA68I;o9B5ABJ 5 6;I ;I6< A::B9AHI w?{?7 6o 5? J9 79; :B?<?7; B?CA;?J B?<EC;< I?B?y K

�? 795 ?b:CA67 I95 ;I? AC89B6;ID 6< J?{6<?J K �7H? ;I? 9:;6D AC H97<ED:;697q∗,mti

o9B;I?

mG;I :A;I J?;?BD 67?J = ;I? :B6H? 6< H9D:E;?J {6A A >97;? @ABC9 A::B9b 6D A;697 9o ;I?

o9BD1

NMC

NMC∑

m=1

[N−1∑

i=0

Ψti(q∗,mti

, Fmti, Q

∗,mti

) + PT (FmT , Q

∗,mT )

].

�9B ;I6< = 5? rB<; IA{? ;9 r7J A; A7l 86{?7 JA;?ti

= ;I? 9:;6D AC <;BA;?8l o9B ACC <6DECA;?J:A;I<

(Fm)1≤m≤NMC

A7J ACC AJD 6<<6F C? HEDECA;6{? H97<ED:;697<Q

w6 K? K(ti, Q) ∈ Ay K �?

;I?B?o9B? 7??J ;9 D ?A<EB? ;I? pEA7;6;l

R(ti, F,Q+ q) = E( N−1∑

j=i+1

Ψtj (q∗tj, Ftj , Q

∗tj

) + PT (FT , Q∗T )|Fti = F,Qti+1

= Q+ q),

5I?B?(q∗tj )j>i

J?79;?< ;I? 9:;6D AC H97;B9C A; ;6D ?tj

A7J(Q∗

tj)j>i

;I? H9BB?<: 97J678 HEGDECA;?J H97<ED:;697 K vI6< 6< ;I? F ?<; ?b: ?H;?J B?;EB7 oB9D JA;?

ti+1;6CC ;I? ?7J 9o

;I? H97;BAH; K z <<ED ? = o9B A 5I6C? = ;IA;R(ti, F,Q + q)

6< 7ED ?B6HACCl A::B9b6D A;?J Fl

R(ti, F,Q + q)= E<678

NMC

>97;? @ABC9 <6DECA;697< K vI?7 ;I? 9:;6D AC <;BA;?8l= A; ;6D ?

ti= o9B ;I?

mG;I :A;I A7J ;I? AJD 6<<6F C? HEDECA;6{? H97<ED:;697

Q6< J?;?BD 67?J Fl s

q∗ti(Fmti, Q) = arg max

q∈Adm(ti,Q)

[Ψti(q, F

mti, Q) + R(ti, F

mti, Q+ q)

].

�?7H? = Fl ?<;6D A;678 ;I? H97J6;697AC ?b: ?H;A;697R(ti, F,Q + q)

o9B ?AHI JA;? A7J ?AHI: 9<<6F C? HEDECA;6{? H97<ED:;697

Q= ;I? 9:;6D AC ?b?BH6<? <;BA;?8l 6< H9D:C?;?Cl <: ?H6r?J

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Ê�

AC978 ?AHI :A;I K �9CC95 678 ;I? �978<;AnG4HI5AB;� D?;I9J9C98l= 5? rb(ti, Q)

A7J A::B9bG6D A;? ;I? BA7J9D {AB6AFC?

R(ti, Fti , Q)A< ;I?

L2G:B9~ ?H;697 9o ∑N−1

j=i+1 Ψtj (q∗tj, Ftj , Q

∗tj

) +

PT (FT , Q∗T )

= 97 ;I? <:AH? <:A77?J Fl A <?; 9o <?C?H;?J FA<6< oE7H;697<[Φl(Fti)]0≤l≤L

K q;C?AJ< ;9

R(ti, F,Q) =∑L

l=0 αl(ti, Q)Φl(F )= 5I?B? ;I? H9 ?� H6?7;<

[αl(ti, Q)]0≤l≤L

<9C{? ;I?C?A<;G<pEAB?< :B9FC?D

arg min(αl)l

NMC∑

m=1

N−1∑

j=i+1

Ψtj (q∗,mtj

, Fmtj, Q

∗,mtj

) + PT (FmT , Q

∗,mT ) −

L∑

l=0

αlΦl(Fmti

)

2

56;I A7 676;6AC H97<ED:;697Qti+1

= Q+ qK �AB69E< D ?;I9J< AB? A{A6CAFC? ;9 <9C{? 7ED ?BG

6HACCl ;I6< :B9FC?D = J?: ?7J678 97 ;I? {9CA;6C6;l A7J ;I? ?b?BH6<? oB?pE?7HlK �9B A 8?7?BACB?o?B?7H? 97 C?A<;G<pEAB?< :B9FC?D < = <?? �9CEF A7J �A7 �9A7 wÊ]]�y K q;?BA;678 ;I? :B9 H?GJEB? = 5? 9F;A67 FAHm5ABJ< A7 A::B9b6D A;697

q∗ti(Fmti, Q)

AC978 ACC <6DECA;?J :A;I< A7J ACC: 9<<6F C? HEDECA;6{? H97<ED:;697< K vI? :B6H? 9o ;I? H97;BAH; ?A<6Cl o9CC95 < K

O �� O �Q\,0YTXW 0,2QWU2

vI? :B6H678 IA< F ??7 : ?Bo9BD ?J 5 6;I �^^^ <6DECA;697< w:CE< A7;6;I?;6Hy = 5 6;I ;I? rB<; o9EBHA7976HAC : 9Cl79D 6AC< w

Φ(x) = xi = 0 ≤ i ≤ 3y A7J 5 6;I {AB69E< H97<ED:;697 oB?pE?7H6?< K

�? IA{? 79; B?: 9B;?J A7l B?<EC; o9B : ?B69J< <D ACC?B ;IA7 x JAl< = <67H? ;I? B?<9CE;697 9o;I? C?A<;G<pEAB?< :B9FC?D < E<EACCl 8?7?BA;?< 67<;AF6C6;6?< o9B <D ACC 67 6;6AC ;6D ?<

tiK

��X\� W, � 3 N. Y[� T/[U0XTU � �9B ;I? o9CC95 678 B?<EC;< = ;I? <;A;6<;6HAC ?BB9B< JE? ;9

@97<ED:;697 : ?B69J � CA; o9B5ABJ HEB{? u?AC o9B5ABJ HEB{?Ê� JAl< )x�^ xx Ê^Ê^ JAl< )� Ê^ xx�^� JAl< )��^ xx�^x JAl< )�x^ x� Ê^

�9B?<; 9o ;B??< wJA6Cl :EBHIA<?y )�Ê� x� ÊÊ

>97;? @ABC9 <6DECA;697< AB? AF 9E; Ê^^ E76;< A; ;I? ]�%

C?{?C K � ?7H? = ;I6< A::B9AHI 86{?<899J B?<EC;< K��X\� W, O 3 2U/0X�, T/[U0XTU � vI? {ACE?< 9o :ABAD ?;?B< AB? <?; ;9 ;I9<? 9o @A<? ) K �?J6B?H;Cl :C9; ;I? B?<EC;< 5 KB K; K ;I? oB?pE?7HlK

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Ê�

� 68EB? � s � CA; o9B5ABJ HEB{? 97 ;I? C?o; K u?AC o9B5ABJ HEB{? 97 ;I? B68I; K

@97H?B7678 ;I? oE7H;697< FA<6< = 5? IA{? 9F<?B{?J ;IA; ;I? 7ED ?B6HAC {ACE?< AB? pE6;?67<?7<6;6{? ;9 ;I? 7EDF?B 9o : 9Cl79D 6AC< oB9D ;I? D 9D ?7; ;IA; ;I? J?8B?? _ 6< AHI6?{?J KvI? <;A;6<;6HAC ?BB9B JE? ;9 <6DECA;697< C6D 6;< A7l : 9<<6F C? 6D:B9{?D ?7; w5I?7 5? ;Am?D 9B? FA<6< oE7H;697<y K> 9B?9{?B = 67 9EB 7ED ?B6HAC ?b: ?B6D ?7;< w<6D 6CABCl ;9 ;I? o9B?<; D ?;I9Jy = 6; A:: ?AB< ;IA; ;I?H97<ED:;697 6< 9o FA78GFA78 ;l: ? K v9 :B?:AB? ;I? H977?H;697 5 6;I ;I? oEB;I?B H97<ED:;697:ABAD ?;?B6�A;697 A::B9AHI = 5? :C9; 97 � 68EB? � ;I? 9:;6D AC 67~ ?H;697 ;IB?<I9CJ o9B ;I?<;9BA8? H97;BAH; w<?? �ABA8BA:I _ K_ KÊy K � ?B? = 5? ;Am? ;I? |A; HEB{? B?<HAC?J A; C?{?C Êo9B H97{?76?7H? K vI? ;59 rB<; Ab?< <;A7J B?<: ?H;6{?Cl o9B ;I? ;6D ? A7J ;I? {9CED ? 9o<;9B?J 8A< K vI? <EBoAH? <I95 < ;I? J?: ?7J?7H? 9o ;I? 8A< :B6H? 5 KB K; ;I? {9CED ? A7J ;I?9:;6D AC ;6D ? o9B 67~ ?H;697 K }9;? ;IA; ;I? :B6H? o9B 5I6HI 5? IA{? ;9 67~ ?H; J?HB?A<?< A< ;I?

� 68EB? � s

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Ê�

{9CED ? 9o ;I? <;9B?J 8A< 67HB?A<?< K � I?7 ;I? ;A7m 6< ?D:;l= 5? 67~ ?H; A; �� H9<; K �7 ;I?H97;BABl= 5I?7 ;I? ;A7m 6< ACD 9<; oECC = ;I? :B6H? DE<; F ? DEHI C95?B ;9 67~ ?H; <9D ? D9B?8A< K vI?<? 9F<?B{A;697< AB? 67;E6;6{?Cl <A;6<ol 678 K qBB?8ECAB6;6?< 9o ;I? 9:;6D AC oB97;6?B AB?JE? ;9 <;A;6<;6HAC ?BB9B< 67 ;I? >97;? @ABC9 AC89B6;ID K �95?{?B = ;I? 9{?BACC <IA: ? <??D <pE6;? <D 99;I K vIE< = D 9J?C678 ;I6< 9:;6D AC oB97;6?B 5 6;I A <D 99;I :ABAD ?;?B6�A;697 D AlC?AJ ;9 ?� H6?7; :B9 H?JEB?< K vI6< 6< J?{?C9: ?J 67 ;I? <?pE?C K

� ��f��efkj ��fg� k��ekgd

q7 ;I6< <?H;697 = 5? J?{?C9: 7ED ?B6HAC :B9H?JEB?< FA<?J 97 :ABAD ?;?B6�?J H97<ED:;697< KvI? :EB: 9<? 6< ;9 A{96J E<678 ;I? Jl7AD 6H :B98BADD 678 ?pEA;697 w) K�y = 5I9<? B?<9CEG;697 HA7 F ? DEHI ;6D ? H97<ED 678 K vI? A::B9AHI F ?C95 C?AJ< ;9 AHHEBA;? ?<;6D A;697< 9o

supq J (q)= :B9{6J?J ;IA; ;I? :ABAD ?;?B6�A;697 6< B?C?{A7; ?79E8I K v9 ;I6< ?7J = 5? rB<;

J?;?BD 67? <9D ? 67;B67<6H :B9: ?B;6?< 9o ;I? 9:;6D AC H97<ED:;697 = 5I6HI 5 6CC I?C: 67 <?;;678;I? :ABAD ?;?B6�A;697 K

� � ��

�?B?Ao;?B = 5? <;A;? A B?D ABmAFC? ;I?9B?;6HAC B?<EC; H97H?B7678 ;I? 7?H?<<ABl 9:;6D AC6;l H97GJ6;697< <A;6<r?J Fl ;I? 9:;6D AC H97<ED:;697 = 67 ;I? 45 678 H97;BAH; w5 6;I A : ?7AC;l A C6;;C?<D 99;I?B ;IA7 PT (F,Q) = −AF [Q−Qmin]− −BF [Q−Qmax]+

y K vI? :B99o 6< B?C?8A;?J;9 z:: ?7J6b z K q; <;A;?< ;IA; ;I? 9:;6D AC H97<ED:;697 6< 9o FA78GFA78 ;l: ? E7J?B <9D ?A<<ED:;697< K vI6< IA< F ??7 7ED ?B6HACCl 9F<?B{?J 5 6;I ;I? :B9 H?JEB?< :B?<?7;?J F ?o9B? K

��,/0,\ � �� wÊ K�y � �� Ψti(q, F ) = e−rtiq(F − K) �� PT (F,Q) =

−e−rTFP (Q)�P

��

P[e−rti (Fti −K) + E

(e−rTFTP

′(Q∗T )|Fti , Q

∗ti

)= 0]

= 0,w_ KÊ^y

�� ~ ��{Ä~ ��{�� ~ �{�~ �� Æ ~~� {Æ�~ �� }~��~ {�{��Ä�� Ä�� ~�� Æ ~�{��~ � ��~ �{�� ¢ ��~�� Ψti

(q, F ) ¡� ¡� ¡ q ¡ ��~�~� � ��¢ {� ��¢½��{�� {� ��~ �½��¢ {� ��¢½��~��~��{��º �{�~� ��~~ �{��~� {qmin, 0, qmax}.

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Ê�

� � �� ti��

q∗(ti, Fti , Q∗ti) =qmax1

e−rti(Fti−K)+E

“

e−rT FT P ′(Q∗T

)|Fti,Q∗

ti

”

>0

+ qmin1e−rti(Fti

−K)+E

“

e−rT FT P ′(Q∗T

)|Fti,Q∗

ti

”

<0.

@97J6;697 w_ KÊ^y A:: ?AB< ;9 F ? IABJ ;9 HI?Hm <67H? 6; 67{9C{?< ;I? E7m7957 9:;6D AC H97G<ED:;697 K �95?{?B = 6; HA7 F ? ?b: ?H;?J w?{?7 6o 5? IA{? 79; F ??7 AFC? ;9 :B9{? 6;y ;IA; o9B

ti < T= ;I? CA5 9o ;I? BA7J9D {AB6AFC?

e−rti (Fti −K) + E(e−rTFTP

′(Q∗T )|Fti , Q

∗ti

) IA< AJ?7<6;l 5 KB K; K ;I? �?F ?<8E? D ?A<EB? K vI?7 = ^ 5 6CC F ? A 7ECC : 967; A7J @97J6;697 w_ KÊ^y5 6CC F ? <A;6<r?J K}9;? ;IA; vI?9B?D _ KÊ 6< <;B978Cl B?CA;?J ;9 ;I? <;9 HIA<;6H D Ab6DED :B67H6: C? w�97;BlA867:B67H6: C?y o9B 9:;6D AC H97;B9C :B9FC?D < K �95?{?B = ;I6< CA;;?B <;B6H;Cl I9CJ< 67 ;I? H97;67GE9E< HA<? wF 9;I 67 ;6D ? A7J <:AH?y K 467H? ;I? HEDECA;6{? H97<ED:;697 IA< A H97;67E9E<<;A;? <:AH? = 6; IA< F ??7 : 9<<6F C? ;9 J?B6{? 9:;6D AC6;l H97J6;697< 67 9EB H97;?b; K> 9B?9{?B = o9B C?<< <D 99;I : ?7AC;6?< 9o ;I? o9BD

P (Q) = −A [Q−Qmin]−−B [Q−Qmax]+=

97? HA7 ?A<6Cl <?? ;IA; ;I?B? AB? H?B;A67 B?8697< 5I?B? ;I? J?B6{A;6{? 9o ;I? ;?BD 67AC : ?7AC;lJ9 ?< 79; J?: ?7J A7l D 9B? 97 ;I? 5I9C? H97<ED:;697 K vI? :B9: 9<6;697 F ?C95 ?b:C6H6;ClHIABAH;?B6�?< ;I? H97<ED:;697 ;IB?<I9CJ 67 ;I?<? <: ?H6AC HA<?< K

� 0/�/2YUY/[ � �O ��

� ��Qti ∈ [Qmin − qmin (N − i), Qmax − qmax (N − i)]

� � � ��

(QT ∈ [Qmin, Qmax]) � � �� K �� 1

��

qti = qmax 1Fti>K + qmin 1Fti

≤K .

� ��Qti + qmin (N − i) ≥ Qmax

� ��

P ′(QT ) = −A � �� F (ti, T ) = E (FT |Fti)

� � �� 2

��

qt = qmax 1e−rti(Fti−K)−A e−rT F (ti,T )>0 + qmin 1e−rti(Fti

−K)−A e−rT F (ti,T )≤0.

�� ! � "�#$�% � & � "%'�� &� �%��(��#� Ê]

� ��Qti + qmax (N − i) ≤ Qmin

� ��

P ′(QT ) = B � � �� 3��

��qt = qmax 1e−rti(Fti

−K)+B e−rT F (ti,T )>0 + qmin 1e−rti(Fti−K)+B e−rT F (ti,T )≤0.

q7 9BJ?B ;9 6D:B9{? 9:;6D 6�A;697 :B9 H?JEB?< = ;I?<? <: ?H6rH HA<?< IA{? F ??7 ;Am?7 67;9AHH9E7; 67 ;I? :ABAD ?;?B6�A;697 9o ;I? 8A< H97<ED:;697 K

� � ��

� �O �� ,2T0Y�UY/[ X[1 \/UY�XUY/[2

�? rB<; :B9: 9<? A 797C67?AB :ABAD ?;?B6�A;697 9o ;I? H97<ED:;697 oE7H;697qt = q(t, Ft, Qt)

=E<678 A 7?EBAC 7?;59Bm w<?? � Alm67 wÊ]]xy o9B 8?7?BAC B?o?B?7H?< 97 7?EBAC 7?;59Bm<y K�? IA{? <?C?H;?J o9B 9EB <;EJl A 97?GI6JJ?7GCAl?B 7?EBAC 7?;59Bm = 5I9<? ABHI6;?H;EB? 6<J?<HB6F ?J 67 � 68EB? ] K vI? 67:E; CAl?B

(xi)1≤i≤36< H977?H;?J ;9 ;I? <;A;? A7J ;6D ? {AB6AFC?<

t

1

Qt

Ft

��

y

θ��kl

θ��l

� 68EB? ] s z BHI6;?H;EB? 9o ;I? 7?EBAC 7?;59Bm K

sx1 = t

=x2 = Ft

A7Jx3 = Qt

K vI?B? 6< 97? I6JJ?7 CAl?B 5 6;IL

7?EB97< = A7J ;I? 9E;:E;CAl?B B?;EB7< ;I? C?{?C 9o 9:;6D AC H97<ED:;697 A; ;6D ?

tK zH;EACCl= ;I? 7EDF ?B 9o I6JJ?7

E76;< ;9 E<? 6< oAB oB9D HC?AB K q; 6< B?CA;?J ;9 ;I? H9D:C?b6;l 9o ;I? E7J?BCl 678 oE7H;697 ;IA;;I? 7?;59Bm 6< ;Bl 678 ;9 D 9J?C K z 899J <;AB;678 : 967; 6< ;9 HI99<? A 7EDF ?B 9o E76;< ?pEAC;9 IACo ;I? <ED 9o ;I? 7EDF ?B 9o 67:E; A7J 9E;:E; E76;< K �? IA{? H97JEH;?J A H?B;A677EDF?B 9o ?b: ?B6D ?7;< 5 6;I J6n?B?7; H97r8EBA;697< A7J r7ACCl <?C?H;?J A 7?;59Bm 5 6;I

4

I6JJ?7 7?EB97< = 67 :B?o?B?7H? ;9 A CAB8?B 97? 5 6;I A 7?8C686F C? 6D:B9{?D ?7; K vI? 7?;59Bm

�� ! � "�#$�% � & � "%'�� &� �%��(��#� )^

6< J?r7?J Fl A D A;B6b A7J A {?H;9B 9o 5?68I;< w;I? :ABAD ?;?B<y s ;I? 67:E; 5?68I;< {θ ��kl}A7J ;I? 9E;:E; 5?68I;< {θ��l } K vI? 9E;:E; 9o ;I? 7?;59Bm 6< 86{?7 Fl

y =

4∑

l=1

θ��l s

(3∑

k=0

θ ��

kl xk

),

5I?B? (θ �

�0l

)l

AB? ;I? F6A< 5?68I;< w5? <?;x0 = 1

y A7Js

6< ;I? 468D 96J oE7H;697 ss(x) =

11+e−x

K vI? C9 HAC H97<ED:;697 6< ;I?7 J?r7?J Fl

q = qmin + (qmax − qmin) s(y),w_ KÊÊy

5I6HI <;6CC F ?C978< ;9 ;I? BA78?[qmin, qmax]

KvI? D 9;6{A;697 9o <EHI A :ABAD ?;?B6�A;697 6< ;I? o9CC95 678 K � 6B<; = 7?EBAC 7?;59Bm< AB?{?Bl <9:I6<;6HA;?J ;?HI76pE?< HA:AFC? 9o D 9J?C678 ?b;B?D ?Cl H9D:C?b oE7H;697< w67 :ABG;6HECAB = 797C67?AB 97?< 5 6;I CAB8? 7EDF ?B< 9o {AB6AFC?<y = A7J A; ;I? <AD ? ;6D ? = ;I? C?{?C9o ;I? E<?B m795 C?J8? 7??J?J ;9 <EHH?<<oE CCl A::Cl 7?EBAC 7?;59Bm< 6< DEHI C95?B ;IA759ECJ F ? ;I? HA<? E<678 = o9B ?bAD:C? = <9D ? D9B? ;BAJ6;697AC 797C67?AB <;A;6<;6HAC D ?;IG9J< K q7 AJJ6;697 = ;I? ?b: ?H;?J :Al9n J (q)

J?r7?J 67 wÊ K�y F ?H9D ?< J (θ)= 5 6;I

θ =(θ �

�kl, θ

��l ; 0 ≤ k ≤ 3 ; 1 ≤ l ≤ 4

) = A7J 67 ;I? <?pE?C ;I? 9:;6D 6�A;697 5 KB K; Kθ

56CC F ? : ?BGo9BD ?J E<678 A7 ?<;6D A;697 9o ;I? 8BAJ6?7; ∇J (θ)

K vI6< ~ E<;6r?< 5Il 5? I?B? H97<6J?B A<D 99;I :ABAD ?;?B6�A;697 K �95?{?B = ;IA7m< ;9 vI?9B?D _ KÊ = 5? D Al ?b: ?H; ;I? 9:;6D ACH97<ED:;697 ;9 F ? 9o FA78GFA78 ;l: ? K vI6< F ?IA{69B HA7 F ? <??7 A< A C6D 6; HA<? 5I?B? ;I?9E;:E; 5?68I;< ;?7J ;9 67r76;lK q; D ?A7< ;IA;

yB?:B?<?7;< 67 A 797C67?AB 5Al ;I? FA78G

FA78 ;IB?<I9CJ s o9B CAB8? : 9<6;6{? {ACE?< 9oy=q ≈ qmax

A7J o9B CAB8? 7?8A;6{?< {ACE?< 9o

y=q ≈ qmin

K

� �O �O ��, �0X1Y,[U XW�/0YU�\

�B9D A7 676;6AC BA7J9D H97r8EBA;697 9o 5?68I;< = ;I? <;9HIA<;6H 8BAJ6?7; AC89B6;ID D Am?< =67 ?b: ?H;A;697 = A D 9{? ;95ABJ< : 967;< 9o I68I?B {ACE?< 67 ;I? <?ABHI <:AH? = 67 9BJ?B ;9C9 HA;? A C9 HAC D Ab6DED w<?? `?7{?76<;? = > ?;6{ 6?B A7J � B69EB?; wÊ]]^y = �E<I7?B A7J � 67wÊ]]�y o9B 8?7?BAC B?o?B?7H?< 97 <;9 HIA<;6H A::B9b 6D A;697 AC89B6;ID <y K �? ;IE< J?r7? AB?HEB<6{? <?pE?7H? 9o :ABAD ?;?B<

(θn)Fl <?;;678

θn+1 = θn + γn∇θJ (θn)

= 5I?B?∇θJ (θn)

6< A 8BAJ6?7; ?<;6D A;9B 9o ;I? D A::678θ 7→ J (θ)

A7J(γn)

6< A <E6;AFC? <?pE?7H? 9o

�� ! � "�#$�% � & � "%'�� &� �%��(��#� ) Ê

7977?8A;6{? <;?: <6�?< K �? IA{? HI9<?7 ;9 rbγn

;9 A H97<;A7; ;IB9E8I9E; 9EB 7ED ?B6HAC?b: ?B6D ?7;< = FE; ;I? J6� HEC; :AB; 6< ;9 J?H6J? I95 CAB8? ;I? <;?:< <I9ECJ F ? K �AB8? <;?:<D Al H97{?B8? D 9B? pE6HmCl= FE; D Al AC<9 9{?B<;?: ;I? <9CE;697 9B 89 9n 67 ;I? 5B978J6B?H;697 K q7 H97;BA<; = {?Bl <D ACC <;?:< D Al 89 67 ;I? H9BB?H; J6B?H;697 = FE; ;I?l AC<9B?pE6B? A CAB8? 7EDF?B 9o 6;?BA;697< K vI? H9BB?H; <?;;678 o9B ;I? <;?: <6�? 6< A::C6HA;697GJ?: ?7J?7; = A7J ;l:6HACCl HI9<?7 Fl ?b: ?B6D ?7;< K vI? AC89B6;ID <;9:< 67 A I68I : 967; =5I6HI 6< A C9 HAC D Ab6DED K vIB9E8I9E; ;I? 7ED ?B6HAC ?b: ?B6D ?7;< 5? IA{? : ?Bo9BD ?J =;I? {ACE?< ;9 5I6HI ;I? AC89B6;ID IA< H97{?B8?J 5?B? ACC {?Bl HC9<? ;9 ;I? 8C9FAC 9:;6DEDw:B?<ED AFCl 86{?7 Fl ;I? D ?;I9J 9o 4?H;697 ) K_y K z < o9B ∇θJ (θ)

= 6; IA< F ??7 H9D:E;?J FlA HCA<<6H > 97;?G@ABC9 A::B9AHI = E<678 A :A;IG5 6<? D ?;I9J w<?? �E<I7?B A7J �A78 wÊ]] ÊyA7J B?H?7;Cl �9F ?; A7J >E79< w)^^�yy K q; 6< ;IE< ?b:B?<<?J A< A7 ?b: ?H;A;697 s

∇θJ (θ) =E

(N−1∑

i=0

∇θq(θ, ti, Fti , Qti)∂qΨti(qti , Fti , Qti)

+ ∂θQti∂QΨti(qti , Fti , Qti)

)+ E

(e−rT∂θQT∂QPT (FT , QT )

).

w_ KÊ)y

vI? pEA7;6;l ∇θq(θ, ti, F,Q)B?<EC;< 67 J6n ?B?7;6A;678 ;I? :ABAD ?;?B6�A;697 9o ;I? H97<ED:G

;697 J?r7?J 67 w_ KÊÊy = 5I6HI 6< <;BA68I;o9B5ABJ K �9B ;I? >97;? @ABC9 ?{ACEA;697< 9o ∇θJ (θ)=

5? 7??J ;9 <6DECA;?(∂θQti)0≤i≤N

K q; 6< 9F;A67?J E<678 ;I? ?pEA;697Qti+1

= Qti + qti=

oB9D 5I6HI 5? J?JEH?

∂θQti+1= ∂θQti + ∂θq(θ, ti, Fti , Qti) + ∂Qq(θ, ti, Fti , Qti) ∂θQti ,

56;I A7 676;6AC H97J6;697∂θQ0 = 0

K

� �O �� Q\,0YTXW 0,2QWU2

��X\� W, � 3 N. Y[� T/[U0XTU � z< HA7 F ? <??7 67 <?H;697 _ KÊ = 5? ?b: ?H; ;I? 9:;6D ACH97<ED:;697 ;9 F ? 9o FA78GFA78 ;l: ? K vIE< 67 9BJ?B ;9 ;Am? 67;9 AHH9E7; ;I6< wB?D ABmAFC?yB?<EC; = A ;IB?<I9CJ oE7H;697 HA7 F ? AJJ6;697ACCl A::C6?J ;9 ;I? 9E;:E; E76; s

q�� = qmax 1q'��>

qmax+qmin2

+ qmin 1q'��≤ qmax+qmin

2

.

q7 ;I? AF 9{? ?b:B?<<697 =q�� A7J

q�� <;A7J o9B ;I? 9E;:E; 7?EB97 A7J ;I? ;IB?<I9CJ{ACE? = B?<: ?H;6{?ClK �?B? = ;I? :ABAD ?;?B< AB? ;I? 97?< 86{?7 67 @A<? Ê K �? AC<9 H97<6J?B

�� ! � "�#$�% � & � "%'�� &� �%��(��#� ))

0 1000 2000 3000500 1500 2500

0

1000

2000

3000

500

1500

2500

Before thresholdingAfter thresholding

0 1000 2000 3000500 1500 2500

2000

3000

4000

5000

2500

3500

4500

Before thresholdingAfter thresholding

� 68EB? Ê^ s � B6H? 9o ;I? 45 678 H97;BAH; 5 KB K; K ;I? 7EDF ?B 9o 6;?BA;697< K �7 ;I? C?o; = 5 6;IA |A; o9B5ABJ HEB{? K �7 ;I? B68I; = 5 6;I ;I? o9B5ABJ HEB{? 9o � 68EB? Ê K

} ?EBAC 7?;59Bm }?EBAC 7?;59Bm �9B?<; 9o ;B??<F ?o9B? ;IB?<I9CJ678 Ao;?B ;IB?<I9CJ678

� CA; o9B5ABJ2659 ± 28 2681 ± 28 )�Ê�

HEB{?u?AC

4555 ± 31 4584 ± 31x� ÊÊ

o9B5ABJ HEB{?

vAFC? ) s � 67AC :B6H? 9o ;I? 45 678 H97;BAH; = F ?o9B? A7J Ao;?B ;IB?<I9CJ678 ;I? H97<ED:;697w;I? {ACE?< Ao;?B ± ?pEAC ;I? IACo 9o ;I? ]�� H97rJ?7H? 67;?B{AC 5 6J;I y K

A = B = 1K vI? <;?: <6�? o9B ;I? <;9 HIA<;6HG8BAJ6?7; AC89B6;ID 6< H97<;A7; A7J ?pEAC ;9

3K

� ?79;? Fl

• NMC

;I? B?pE6B?J 7EDF ?B 9o > 97;? @ABC9 <6DECA;697< ;9 HACHECA;? A7 ?<;6D A;9B 9o;I? 8BAJ6?7; = o9B A 86{?7 <?; 9o :ABAD ?;?B< =

• NIT

;I? 7EDF ?B 9o 6;?BA;697< 97 ;I? :ABAD ?;?B< 67 ;I? 9:;6D 6�A;697 <;A8? K

vI? AC89B6;ID HA7 ;I?B?o9B? F ? <:C6; 67;9 ;59 <;A8?< s

Êy A rB<; 9:;6D 6�A;697 <;A8? = 5I?B? 5? <?; o9B ?bAD:C?NMC = 100

A7J A CAB8? 7EDF?B9o 6;?BA;697<

NIT = 3000= <9 ;IA; ;I? AC89B6;ID :B?<ED AFCl H97{?B8?< K

)y A <?H97J :B6H678 <;A8? 5 6;I ;I? CA<; <?; 9o :ABAD ?;?B< A7J A DEHI I68I?BNMC

w6;HA7 F ? <?; ?pEAC ;9

100000o9B ?bAD:C?y = 67 9BJ?B ;9 B?JEH? ;I? H97rJ?7H? 67;?B{AC

�� ! � "�#$�% � & � "%'�� &� �%��(��#� )_

97 ;I? :B6H? K

u?<EC;< AB? B?: 9B;?J 67 vAFC? ) K }9;? ;IA; o9B F 9;I o9B5ABJ HEB{?< = ;I? B?<EC;< AB? oAB Fl

1%oB9D ;I? 9:;6D AC :B6H? H9D:E;?J Fl ;I? ;B?? D ?;I9J = A7J ;IB?<I9CJ678 A; ;I? ?b 6; 9o

;I? 7?EBAC 7?;59Bm <??D < ;9 F ? B?C?{A7; K �EB;I?BD 9B? = 5? HA7 <?? ;IA; ;I?<? ;59 ;?<;<:B9{6J? : 95?BoEC 67<68I;< H97H?B7678 ;I? ;l: ? 9o H97{?B8?7H? ;IA; D Al F ? 9F<?B{?J 67 ;I?9:;6D 6�A;697 AC89B6;ID < w67J?: ?7J?7;Cl 9o ;I? <?C?H;?J :ABAD ?;?B6�A;697 y K �9B ;I? rB<;o9B5ABJ HEB{? = 5? HA7 <?? ;IA; ;I? AC89B6;ID B?pE6B?< 97Cl

10006;?BA;697< ;9 H97{?B8? =

A7J ;IE< 6; 6< E<?C?<< ;9 89 E7;6C3000

6;?BA;697< K �7 ;I? H97;BABl= o9B ;I? <?H97J o9B5ABJHEB{? = 5? IA{? A <C95?B 9:;6D 6�A;697 = 5I6HI 6< ?{?7 : ?Bo9BD ?J 67 ;59 <;?:< wA< 6o A C9 HAC9:;6DED 5?B? B?AHI?J A; A7 67;?BD ?J6A;? <;A8?y K vI? pEAC6;l 9o B?<EC;< A7J ;I? ;l: ? 9oH97{?B8?7H? AB? ;IE< <;B978Cl B?CA;?J ;9 ;I? <?C?H;?J o9B5ABJ HEB{? K��X\� W, O 3 2U/0X�, T/[U0XTU � vI? {ACE?< 9o :ABAD ?;?B< AB? <?; ;9 ;I9<? 9o @A<? ) K vI?<;?: <6�? o9B ;I? <;9HIA<;6HG8BAJ6?7; AC89B6;ID 6< H97<;A7; A7J ?pEAC ;9

0.1K vI? ;?BD 67AC

: ?7AC;6?< AB? AC5Al< <A;6<r?J 67 ;I6< HA<? K q7 oAH; = A< ACB?AJl D ?7;697?J = ;I? H97<;BA67;< 67;I? <;9BA8? HA<? HA7 F ? o9BDECA;?J A< o9CC95 <

f1(t) ≤ Qt ≤ f2(t),∀t ∈ [0, T ].

� ?79;? Flqparams(t)

;I? 9E;:E; 9o ;I? 7?EBAC 7?;59Bm A; ;6D ?tK

� Qt ≥ f1(t)o9B ?{?Bl

t ∈ [0, T ]K vIE< A< <997 A< |Qt − f1(t)| ≤ |qmin| dt

= 5? B?:CAH?;I? 9E;:E; 9o ;I? 7?;59Bm Fl s

q1t =

qparams(t),

6oqparams(t) +Qt dt ≥ f1(t)

f1(t)−Qt

dt,

9;I?B5 6<?.

� Qt ≤ f2(t)o9B ?{?Bl

t ∈ [0, T ]K vIE< 6o

q1t dt+Qt > f2(t)= 5? B?:CAH?

q1tFl f2(t)−Qt

dt

K

� 67ACCl= ;I? B?;A67?J H97;B9C A; ;6D ?t

;Am?< ;I? o9BD 9o

qt =D67 [

q1t ,f2(t) −Qt

dt

]=

D67 [DAb (qparams(t),

f1(t) −Qt

dt

),f2(t) −Qt

dt

].

}9;? ;IA; ;I? AF 9{? H97<ED:;697 6< 79; 9o HCA<< C1A7l D 9B? K �95?{?B = ;I? <;9 HIA<;6H

8BAJ6?7; AC89B6;ID 6< <;6CC A::C6HAFC? = <67H? ;I?B? 6< 97Cl A r76;? <?; 9o : 967;< 5I?B? ;I?oE7H;697 6< 79; J6n ?B?7;6AFC? K

�� ! � "�#$�% � & � "%'�� &� �%��(��#� )x

}?EBAC 7?;59Bm �9B?<; 9o ;B??<� CA; o9B5ABJ

64.7 ± 0.4 �� =]HEB{? w

NIT = 3000y

u?AC218.7 ± 0.6 )x) K_

o9B5ABJ HEB{? wNIT = 6000

y

vAFC? _ s � 67AC :B6H? 9o ;I? <;9BA8? H97;BAH; w;I? {ACE?< Ao;?B ± ?pEAC ;I? IACo 9o ;I? ]��H97rJ?7H? 67;?B{AC 5 6J;I y K

vI? B?<EC;< B?: 9B;?J 67 vAFC? _ AB? H9D:E;?J 5 6;INIT = 3000

wB?<: KNIT = 6000

y6;?BA;697< o9B ;I? |A; o9B5ABJ HEB{? wB?<: K o9B ;I? o9B5ABJ HEB{? oB9D � 68EB? Êy = <;6CCE<678

NMC = 100000:A;I< o9B ;I? r7AC {ACE? K }9;? ;IA; ;I? 7?EBAC 7?;59Bm 86{?< A C?<<

0 1000 2000 3000500 1500 2500

-20

-10

0

10

20

30

40

50

60

70

0 1000 2000 3000 4000 5000 6000

180

190

200

210

220

230

� 68EB? ÊÊ s � B6H? 9o ;I? <;9BA8? H97;BAH; 5 KB K; K ;I? 7EDF ?B 9o 6;?BA;697< K �7 ;I? C?o; = 5 6;IA |A; o9B5ABJ HEB{? K �7 ;I? B68I; = 5 6;I ;I? o9B5ABJ HEB{? 9o � 68EB? Ê K

<A;6<ol 678 :B6H? 67 ;I? HA<? 9o ;I? B?AC o9B5ABJ HEB{? K vI? <;B978 <?A<97AC6;l H9D: 97?7;9o ;I? HEB{? 6< E7J9EF;?JCl A7 6D: 9B;A7; ?b:CA67 678 oAH;9B K zJJ678 D 9B? I6JJ?7 7?EB97<H?B;A67 Cl 6D:B9{?< ;I? pEAC6;l 9o ;I? :B6H? = FE; AC<9 6D:C6?< C978?B H9D:E;678 ;6D ?< K�?; E< r7ACCl <;B?<< ;IA; 79 ;IB?<I9CJ678 A; ;I? 9E;:E; 9o ;I? 7?;59Bm IA< F ??7 A::C6?JI?B? K �7C6m? ;I? 45 678 H97;BAH; = 5? IA{? 79; F ??7 AFC? 67 ;I6< HA<? ;9 r7J A B?C?{A7;;IB?<I9CJ678 BEC? A; ;I? 9E;:E; 9o ;I? 7?;59Bm K

�� ! � "�#$�% � & � "%'�� &� �%��(��#� )�

� � ��

� �� /0 U�, N. Y[� T/[U0XTU

�7C6m? ;I? :B?{69E< A::B9AHI 5 6;I 7?EBAC 7?;59Bm< = I?B? ;I? :ABAD ?;?B6�A;697 9oq(ti, F,Q)

J9?< 79; 7??J ;9 F ? <D 99;I 5 6;I B?<: ?H; ;9 :ABAD ?;?B< K �? BA;I?B o9 HE< 97 ;I? FA78GFA78F ?IA{69B 5I6HI HA7 F ? ?b: ?H;?J oB9D vI?9B?D _ KÊ K vIE< = 5? I?B? A<<ED ? ;IA;

• q∗(ti, Fti , Qti);Am?< 97Cl ;59 {ACE?<

qminA7J

qmax=

• q∗(ti, Fti , Qti)6< A7 67HB?A<678 oE7H;697 5 KB K;

Fti

K

vI? <?H97J :B9: ?B;l 6< I?EB6<;6H s 5? ?b: ?H; ;IA; A I68I?B :B6H? 9o 8A< 5 6CC C?AJ ;9 A I68I?BH97<ED:;697 A; A rb?J :B6H?

KK � B?{69E< AC89B6;ID < FA<?J 97 ;I? Jl7AD 6H :B98BADD 678

?pEA;697 w<?? 4?H;697 )y 86{? B?<EC;< ;IA; H97rBD ;I6< I?EB6<;6H K �B9D ;I?<? ;59 :B9: ?B;6?<5? J?JEH? ;IA; 6o

q6< 9:;6D AC = ;I?B? ?b6<;< A ;IB?<I9CJ

S∗ J?: ?7J678 97 F 9;Iti

A7JQti

<EHI ;IA;

qti = qmax1{Fti>S∗(ti,Qti

)} + qmin1{Fti≤S∗(ti,Qti

)}.

�EB A::B9AHI 6< ;9 D 9J?C ;I6< 9:;6D AC ;IB?<I9CJ :B6H? A< A oE7H;697 9ot

A7JQt

K �? HA7J6{6J? ;I? <?; 9o H9E:C?<

(ti, Qti)67;9 o9EB �97?< K

• �97? Ê H9BB?<: 97J< ;9 A H97<ED:;697 <EHI ;IA; ;I? H97<ED ?B 6< <EB? ;9 B?<: ?H; ;I?8C9FAC H97<;BA67;< K vI? 9:;6D AC ;IB?<I9CJ 6< ?pEAC ;9 ;I? <;B6m?

KK

• �97? ) wB?<: K_y H9BB?<: 97J< ;9 A7 9{?B wB?<: K E7J?By H97<ED:;697 K q7 ;I?<? ;59 HA<?<;I? 9:;6D AC ;IB?<I9CJ 6< A7 ?b:C6H6; oE7H;697 9o

tiK qo ;I? 8C9FAC : ?7AC;6?< AB? 67r76;? =

;I?<? ;59 �97?< AB? E<?C?<< K

• �97? x 6< ;I? 97Cl 97? 5I?B? ;I? ;IB?<I9CJ IA< ;9 F ? o9E7J K

a{?7 6oFti

|EH;EA;?< A C9; = 5? ?b: ?H; ;IA; ;I? ;IB?<I9CJS∗ HIA78?< A C6;;C? 5 6;I B?<: ?H;

;9ti

A7JQti

K �9B 67<;A7H? = 67 ;I? <: ?H6AC HA<? 5I?B?Fti

6< J?;?BD 676<;6H 5 6;Ir = 0

=S∗ 6<

H97<;A7; K @97<?pE?7;Cl=S∗(ti, Qti)

DAl F ? A::B9b 6D A;?J Fl A <D 99;I oE7H;697 9o(ti, Qti)

K� 68EB? � AF 9E; ;I? 67~ ?H;697 9:;6D AC J?H6<697 H97rBD < ;IA; ;I? ;IB?<I9CJ <EBoAH? D Al F ?

�� ! � "�#$�% � & � "%'�� &� �%��(��#� )�

t

1

2

3

4

Qt

Qmin

Qmax

t = T

Qt = qmax × t

Qt = qmin × t

.

.

.

.

.

.

..

.

.

. .

.

....

. .

..

. .

..

1

2

3

4

t = T

Qt − qmin × t

t −Qt−qmin×t

qmax−qmin

Qmax − qmin × t

Qmin − qmin × t

� 68EB? Ê) s �ABAD ?;?B6�A;697 9o ;I? ;IB?<I9CJ A< A oE7H;697 9o F 9;I(t,Qt)

A7J(t −

Qt−qmin

qmax−qmin, Qt − qmin × t)

K

A::B9b 6D A;?J Fl A <D 99;I 97? o9B ;I?<? H97;BAH;< K zH;EACCl= 6; 6< D 9B? ?� H6?7; ;9 ?b:B?<<

S∗ A< A oE7H;697 9o(i− Qti − qmin × i

qmax − qmin, Qti − qmin × i

),

;I? <E:: 9B; 9o ;I? oE7H;697S∗ F ?678 795 A B?H;A78C? K vI? :ABAD ?;?B<

θ9o ;I? oE7H;697

S∗ AB? {ACE?< 9o ;I? ;IB?<I9CJ A; A rb?J : 9<6;697 <:B?AJ = 97 A E76o9BD 8B6J 5 6;I A HI9<?7<6�? w

5 × 5o9B ?bAD:C?y K vI? {ACE? 9o

S∗ 9E;<6J? ;I? 8B6J 6< ?pEAC ;9 ;I? <;B6m? 6o ;I?H9E:C?

(ti, Qti)6< 67 �97? Ê = A7J ;9 A F6C67?AB 67;?B: 9CA;697 9o ;I? o9EB 7?AB?<; 7?68IF 9B< =

9;I?B5 6<? K

� �� O ��UY\ Y�XUY/[ / � �X0X\,U,02

� 6;I ;I6< :ABAD ?;?B6�A;697 = ;I? oE7H;697 J (θ);9 D Ab6D 6�? 6< 79; <D 99;I = A7J A <: ?H6rH

9:;6D 6�A;697 :B9 H?JEB? IA< ;9 F ? HABB6?J 9E; K467H? :ABAD ?;?B<

θAB? ;IB?<I9CJ {ACE?< = ;I?l HA7 F ? 676;6AC6�?J 7?AB ;I? {ACE?

KK vI?

6J?A 9o ;I? 9:;6D 6�A;697 6< ;I? o9CC95 678 K � 6;I A 86{?7 <?; 9o <6DECA;697< = 5? HA7 ?{ACEA;?

J (θ)o9B A7l {ACE? 9o

θK �? AB? C99m678 o9B ;I? 9:;6D AC :ABAD ?;?B Fl A::Cl678 H99BJ67A;?G

5 6<? 6;?BA;6{? : ?B;EBFA;697< 9o ;I? :ABAD ?;?B< K vI? <6�? 9o ;I? : ?B;EBFA;697 ?pEAC<ε

A7J:ABAD ?;?B< AB? 6;?BA;6{?Cl HI9<?7 AHH9BJ678 ;9 A <;A;6<;6HAC ;?<; FA<?J 97 ;I? ?D:6B6HACD ?A7 A7J {AB6A7H? 9o ;I? 6D:B9{?D ?7; K v9 8?; F ?;;?B A7J F ?;;?B :ABAD ?;?B< 5? 67HB?A<?;I? <6�? 9o ;I? <?; 9o <6DECA;697< 67 <?{?BAC <;A8?< K �? <;AB; 5 6;I

l = 0A7J

NMC = 10000K

�? J?r7?θ

+j0

= (θ1, . . . , θj0 + ε, . . . , θd)A7J

θ−j0

= (θ1, . . . , θj0 − ε, . . . , θd)K vI? AC89B6;ID

HA7 F ? J?<HB6F ?J A< o9CC95 < K

�� ! � "�#$�% � & � "%'�� &� �%��(��#� )�

• 4;A8?l

�46DECA;? A <?; 9o

2lNMC

67J?: ?7J?7; :A;I<(Fti)i

K

�� ?79;? Fl

σ(X);I? ?D:6B6HAC <;A7JABJ J?{6A;697 9o

XH9D:E;?J 5 6;I ;I6< <?;

9o :A;I< K

�� I6C? J (θ)

6< 6D:B9{?J

∗ @9D:E;? ;I? ?D:6B6HAC D ?A7 9o J (θ)E<678 ;I6< <?; 9o :A;I< K

∗ �9B ?AHI :ABAD ?;?B H99BJ67A;?θj0

· 6o J (θ+j0

) − J (θ) > 1.65 × σ(J (θ+j0

) − J (θ))9B

· 6o J (θ−j0

) − J (θ) > 1.65 × σ(J (θ−j0

) − J (θ))

HIA78? ;I? {ACE? 9oθj0

A7J H97;67E? ;I? 9:;6D 6�A;697 5 6;I ;I? F ?<; 9o ;I?<?;IB?? <?;< 9o :ABAD ?;?B< K

• l → l + 1K

q;?BA;697< AB? <;9:: ?J 5I?72lNMC

6< ;9 9 CAB8? o9B H9D:E;A;697< 9B 5I?7 ;I?B? 6< 79 D 9B?6D:B9{?D ?7; K

� �� L�� WYTXUY/[2 U/ 2U/0X�, T/[U0XTU

z <6D 6CAB <;BA;?8l HA7 F ? E<?J o9B ;I? <;9BA8? H97;BAH; = <67H? 5? ?b: ?H; ;IA; ;I? 9:;6D ACH97<ED:;697

q(t,Qt, Ft);Am?< 97Cl ;IB?? {ACE?< K q7 ;I6< HA<? 5? FE6CJ ;59 <EBoAH?< A<<9 H6G

A;?J ;9 ;I? ;59 ;IB?<I9CJ< = E<678 ;I? <AD ? D ?;I9J K �? HI99<? A <6�? 9o 8B6J A7J HIA78?;I? {ACE?< 9o 79J?< = 97? Fl 97? = A; ?AHI 9:;6D 6�A;697 <;?: s 79J?< 9o ;I? rB<; <EBoAH? = ;I?779J?< 9o ;I? <?H97J 97? K 467H? ;I? :B?{69E< HIA78? 9o {AB6AFC?< 6< E<?C?<< = 5? :ABAD ?;?B6�?;I? ;59 <EBoAH?< 5 6;I

tiA7J

Qti

K

� �� ,2QWU2

��X\� W, � 3 N. Y[� T/[U0XTU � �9B J6n?B?7; o9B5ABJ HEB{?< = ;I? <9CE;697 A7J ;I? 9:;6D AC:ABAD ?;?B< H9D:E;?J 5 6;I ;I6< AC89B6;ID J9 79; J?: ?7J 97 ;I? 676;6AC <?; 9o :ABAD ?;?B< K�EB;I?BD 9B? = 79 C9 HAC D Ab6DED :I?79D ?797 IA< F ??7 79;6H?J K vI? pEAC6;l 9o B?<EC;<D A67Cl J?: ?7J< 97 ;I? o9B5ABJ HEB{?< K � 6;I A

4 × 48B6J 5? HA7 B?AHI = 67 ;I? F ?<; HA<?< =

�� ! � "�#$�% � & � "%'�� &� �%��(��#� )�

]�� 9o ;I? : ?Bo9BD A7H? 9F;A67?J Fl A Jl7AD 6H :B98BADD 678 A::B9AHI K vI? o9CC95 678B?<EC;< AB? H9D:E;?J 5 6;I

ε = 0.1=l = 5

<;?:< wH9BB?<: 97J678 ;9 A D Ab6DED <?; 9o _)^^^^ J6n ?B?7; <6DECA;697<y A7J ]�� H97rJ?7H? 67;?B{AC< K

�9B5ABJ HEB{? 46�? 9o 8B6J J �9B?<; 9o ;B??<B?AC

4 × 4 4565 ± 25x� ÊÊ

B?AC10 × 10 4578 ± 25

x� ÊÊ|A;

4 × 4 2696 ± 23 )�Ê�|A;

10 × 10 2692 ± 23 )�Ê��X\� W, O 3 2U/0X�, T/[U0XTU � vI?<? B?<EC;< D A67Cl J?: ?7J 97 ;I? HI96H? 9o

εA7J ;I?

<6�? 9o 8B6J< Kε

7??J< ;9 F ? <D ACC ?79E8I ;9 IA{? B?C6AFC? B?<EC;< FE; A ;99 <D ACC {ACE?9oε

DAl AC<9 8?7?BA;? I?A{l H9D:E;A;697< K q7 9EB HA<? 8B6J< 5?B? ;9 9 H9AB<? ;9 E<? A{?Bl <D ACC {ACE? 9o

εK zJA:;6{? 8B6J< = 5I6HI F ?H9D ? D9B? :B?H6<? A<

t8B95 < = D Al F ?

D 9B? ?� H6?7; ;IA7 E76o9BD 97?< K q7 ;I? <;9BA8? H97;BAH; = A 8B6J ;IA; ?7<EB?< H97;67E6;l9o ;I? ;IB?<I9CJ <EBoAH? A; ?J8?< 9o �97? Ê = 86{?< 899J B?<EC;< A< 5?CC K q; A:: ?AB< ;IA;67 6;6AC6�A;697 A7J 9BJ?B 9o :ABAD ?;?B<

θj

67 ;I? 9:;6D 6�A;697 :B9H?JEB? AB? 79; HBEH6AC K

�9B5ABJ HEB{? 46�? 9o 8B6J J �9B?<; 9o ;B??<B?AC

4 × 4 225.3 ± 1.2 )x) K�B?AC

10 × 10 230 ± 1.2 )x) K�

� �gdj ikgd

q7 ;I? :A: ?B = 5? IA{? :B?<?7;?J <?{?BAC 7ED ?B6HAC D ?;I9J< ;9 :B6H? 45 678 9:;697< K vI?5?CCGm7957 o9B?<; 9o ;B??< D ?;I9J ACC95 < ;9 J?AC ?� H6?7;Cl 5 6;I 97? oAH;9B D 9J?C< K �?IA{? ?b;?7J?J ;I? �978<;AnG4HI5AB;� D?;I9J w67 6;6ACCl E<?J o9B `?BDEJA 9:;697<y ;9 9EBoBAD ?59Bm K vI6< AE;I9B6�?< DEC;6GoAH;9B D 9J?C< K vI?<? ;59 :B9H?JEB?< ;Am? AJ{A7;A8?9o ;I? Jl7AD 6H :B98BADD 678 ?pEA;697 A7J ;IE< = C?AJ ;9 AHHEBA;? {ACEA;697< 9o ;I? 45 678H97;BAH; K }ED ?B6HAC ?b: ?B6D ?7;< H97rBD ;I6< o?A;EB? K �? J?{?C9: AC;?B7A;6{? AC89B6;ID <FA<?J 97 B?C?{A7; :ABAD ?;?B6�A;697< 9o ;I? H97<ED:;697 K vI?<? 7?5 D ?;I9J< HA7 AC<9IA7JC? DEC;6GoAH;9B D 9J?C< K > 9B?9{?B = E7C6m? ;I? :B?{69E< 97?< = ;I?l IA{? ;I? AJ{A7;A8?;9 86{? 67;?BD ?J6A;? :B6H?< ;IB9E8I9E; ;I? 9:;6D 6�A;697 <;A8? Kv9 8E6J? 9EB HI96H? 9o :ABAD ?;?B6�A;697 = 5? IA{? :B9{?7 ;IA; ;I? 9:;6D AC :EBHIA<? 6<

�� ! � "�#$�% � & � "%'�� &� �%��(��#� )]

9o FA78GFA78 ;l: ? K vI? rB<; :ABAD ?;B6H D ?;I9J 6< FA<?J 97 A 7?EBAC 7?;59Bm = 5I6HIHA7 A<lD:;9;6HACCl B?:B9JEH? ;I? FA78GFA78 F ?IA{69B K vI? 9F~ ?H;6{? oE7H;697 6< <D 99;I5 KB K; K :ABAD ?;?B< A7J 8BAJ6?7; <;9 HIA<;6H D ?;I9J< HA7 F ? E<?J o9B ;I? 9:;6D 6�A;697 K vI?B?<EC;< AB? pE6;? <A;6<ol678 K vI? <?H97J :ABAD ?;B6H D ?;I9J J6B?H;Cl D 9J?C< ;I? FA78GFA78;IB?<I9CJ A< A oE7H;697 9o ;I? ;6D ? A7J ;I? HEDECA;6{? H97<ED:;697 K vI6< <??D < ;9 F ?:AB;6HECABCl AJA:;?J ;9 ;I? :B9FC?D A7J ;IE< C?AJ< ;9 F ?;;?B B?<EC;< K

� � fg g � g � ��gf��

�?;q(.)

F ? A7 9:;6D AC H97;B9C A7Jq(.)

A7 AJD 6<<6F C? H97;B9C 67[qmin, qmax]

K �9Bε ∈ [0, 1]

=5? J?r7? A ;I6BJ AJD 6<<6F C? H97;B9C

qεtj

=

qti + ε (qti − qti) j = i,

qtj j 6= i.

vI? 9F~ ?H;6{? oE7H;697 H9BB?<: 97J678 ;9qε 6< 86{?7 Fl

J (qε) = E

N−1∑

j=0

e−rtj qεtj

(Ftj −K) + e−rTFTP (QεT )

.

467H?q

6< 9:;6D AC = 5? 8?; J (qε) ≤ J (q)= o9B ?{?Bl

ε ∈ [0, 1]= A7J ;I?B?o9B? ∂J (qε)

∂ε|ε=0 ≤ 0.

�? HA7 ?A<6Cl H9D:E;? ;I6< J?B6{A;6{? A7J 9F;A67

∂J (qε)

∂ε|ε=0 = lim

ε↓0

J (qε) − J (q)

ε

= E{(qti − qti)

[e−rti(Fti −K) + e−rTFTP

′(QT )]}

= E{(qti − qti)

[e−rti(Fti −K) + E

(e−rTFTP

′(QT )|Fti , Qti

)]}≤ 0.

vI? AF 9{? 67?pEAC6;l I9CJ< ;BE? o9B ACC AJD 6<<6F C? H97;B9C(qtj )j

= ;IE< 67 :AB;6HECAB o9B

qtj = qtj + (q − qti)1λti>01i=j

= 5I?B?

Hti = e−rti(Fti −K) + E(e−rTFTP

′(QT )|Fti , Qti

)

A7Jλti = (q− qti)Hti

wq ∈ [qmin, qmax]

F ?678 rb?Jy K vI6< 86{?<E(1λti

>0λti

)≤ 0.

q7 9;I?B59BJ< =

λti

6< 7?8A;6{? 9E;<6J? <9D ? 7?8C686F C? <?;< K vIE< 5? IA{? :B9{?7 ;I? o9CC95 6789:;6D AC6;l H97J6;697

(q − qti)[e−rti(Fti −K) + E

(e−rTFTP

′(QT )|Fti , Qti

)]≤ 0,∀ q ∈ [qmin, qmax],

A7J ;I? J?<6B?J B?<EC; o9CC95 < A; 97H? K

�� ! � "�#$�% � & � "%'�� &� �%��(��#� _^

� ��f�dj�

�Ê� z K `?7{?76<;? = > K > ?;6{ 6?B = A7J � K � B69EB?; K �� K 4:B678?BG�?BCA8 = } ?5 �9Bm = Ê]]^ K

�)� � K� K ` ?B;<?mA< K � �� `?CGD 97; = >z s z;I?7A 4H6?7;6rH = Ê]]� K

�_� � K� K ` ?B;<?mA< A7J 4 Ka K 4IB?{? K � �� = {9CED ? Ê_] 9o

� �� K zHAJ?D 6H � B?<< q7H K ��ABH9EB; `BAH?t9{A79{6HI �EFC6<I?B<� = } ?5 �9Bm = Ê]�� K vI? J6<HB?;? ;6D ? HA<? K

�x � � K @C?5 C95 = @ K 4;B6HmCA7J = A7J � K � AD 67<m6 K �ACEA;697 9o 45 678 H97;BAH;< Kv?HI76HAC B?: 9B; = a7?B8l �95?B u 6<m > A7A8?D ?7; = u 6<m �A;?B< � B9E: = tECl

)^^ Ê K

�� K �9CEF A7J @ K� K �A7 �9A7 K � �� _BJ ?J K `AC;6D 9B? = >� svI? t9I7< �9:m67< �76{ K � B?<< K bb{66 = �]x : K = Ê]]� K

�� a K �9F ?; A7J u K >E79< K 4?7<6;6{6;l A7ACl<6< E<678 q;�G> ACC6A{67 HACHECE< A7JD AB;678AC?< K z::C6HA;697 ;9 <;9HIA<;6H H97;B9C :B9FC?D K �� = x_ w�y sÊ�� Ê�Ê_ = )^^� K

�� 4 K � Alm67 K �� K > H> 6CCA7 = } ?5�9Bm = Ê]]x K

�� K tA6CC?; = a Kq K u977 = A7J 4 K v9D:A6J6< K �ACEA;697 9o H9DD9J6;lGFA<?J 45 6789:;697< K � �� = �^ s]^]�]) Ê = )^^x K

�]� q K � ABA;�A< K �7 ;I? :B6H678 9o zD ?B6HA7 9:;697< K �� = Ê� wÊy s_��^ = Ê]�� K

�Ê^� � Kt K �E<I7?B A7J t K �A78 K z >97;? @ABC9 D ?;I9J o9B <?7<6;6{ 6;l A7ACl<6<A7J :ABAD ?;B6H 9:;6D 6�A;697 9o 797C67?AB <;9 HIA<;6H <l<;?D < K �� = )] w�y sÊ) Ê�� Ê)x] = Ê]] Ê K

�� ! � "�#$�% � & � "%'�� &� �%��(��#� _ Ê

�ÊÊ� � Kt K �E<I7?B A7J � K � 67 K � �� K 4:B678?BG�?BCA8 = `?BC67 A7J }?5 �9Bm = Ê]]� K

�Ê)� z K �AB6G�A{A<<A76 = > K 46D HI6 = A7J z K �AB? K z J6<HB?;? {ACEA;697 9o 45 6789:;697< K �� = ] wÊy s_�� x = )^^ Ê K

�Ê_� � K �978<;An A7J a K4 K 4HI5AB;� K �ACE678 AD ?B6HA7 9:;697< Fl <6DECA;697 s z<6D:C? C?A<; <pEAB?< A::B9AHI K � � � �� = Êx sÊÊ_� Êx� =

)^^ Ê K

�Êx � 4 K u A6mAB A7J > K qC6� K q7;?BBE:;6F C? :Il<6HAC ;BA7<D 6<<697 H97;BAH;< o9B H97G8?<;697 D A7A8?D ?7; K � �� =_ sÊ�_]� Ê�x� = )^^ Ê K

�Ê�� z K@ K vI9D:<97 K �ACEA;697 9o :A;IGJ?: ?7J?7; H97;678?7; HCA6D < 5 6;I DEC;6: C??b?BH6<? J?H6<697< 9{?B ;6D ? s ;I? HA<? 9o vAm?G9BG�AlK �� = _^ s)�Ê�)]_ = Ê]]� K

Numerical methods for the pricing of Swing options: a ...

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