VaR as the CVaR sensitivity: Applications in risk optimization

Full text

(1)VaR as the CVaR sensitivity: Applications in risk optimization Alejandro Balbása , Beatriz Balbásb , Raquel Balbásc a. University Carlos III of Madrid. C/ Madrid, 126. 28903 Getafe (Madrid, Spain). alejan-. [email protected]. b. University of Castilla La Mancha. Avda. Real Fábrica de Seda, s/n. 45600 Talavera. (Toledo, Spain). [email protected]. c. University Complutense of Madrid. Somosaguas. 28223 Pozuelo de Alarcón (Madrid,. Spain). [email protected]. Abstract VaR minimization is a complex problem playing a critical role in many actuarial and …nancial applications of mathematical programming. The usual methods of convex programming do not apply due to the lack of sub-additivity. The usual methods of di¤erentiable programming do not apply either, due to the lack of continuity. Taking into account that the CVaR may be given as an integral of VaR, one has that VaR becomes a …rst order mathematical derivative of CVaR. This property will enable us to give accurate approximations in VaR optimization, since the optimization VaR and CVaR will become quite closely related topics. Applications in both …nance and insurance will be given.. Key words VaR Optimization, CVaR Sensitivity, Approximation Methods, Optimality Conditions, Actuarial and Financial Applications.. A.M.S. Classi…cation 90C59, 90C30, 91B30, 91G10, 90B50. J.E.L. Classi…cation C02, C61, G11, G22.. 1. Introduction. V aR has many applications in …nance and insurance. Risk management, capital requirements, …nancial reporting, asset allocation, bonus-malus systems, optimal reinsurance, etc. just compose a brief list of topics closely related to V aR. Beyond V aR, risk measurement 1.

(2) is an open problem provoking a growing interest and discussion in recent years. Since Artzner et al. ( 1999) introduced their coherent measures of risk much more approaches have been proposed. Very important examples are the expectation bounded measures of risk (Rockafellar et al., 2006), consistent risk measures (Goovaerts et al., 2004 ), actuarial risk measures (Goovaerts and Laeven, 2008), indices of riskiness (Aumann and Serrano, 2008, Foster and Hart, 2009, Bali et al., 2011), etc. The existence of alternative risk measures implies that many risk-linked problems may be studied without dealing with V aR. Moreover, V aR is not sub-additive (Artzner et al., 1999), it is di¢ cult to optimize (Gaivoronski and P‡ug, 2005) and it presents some more drawbacks which may recommend to deal with other risk measures such as CV aR (Rockafellar and Uryasev, 2000). Nevertheless, for several reasons V aR still plays a critical role for many practitioners, institutions and researchers. Firstly, regulation (Basel for banks, Solvency for insurers, etc.) still assigns a vital role to V aR. Secondly, V aR never becomes in…nity, while the rest of usual risk measures may attain this value. For instance, CV aR becomes in…nity for random risks whose expected losses equal in…nity too (for instance, positive random variables with unbounded expectation). In…nite values may provoke analytical and mathematical problems quite di¢ cult to overcome, specially if several heavy tails are simultaneously involved (Chavez-Demoulin et al., 2006). Heavy tails are usual in some actuarial topics (Zajdenwebe, 1996), some operational risk topics (Mitra et al., 2015) and other issues. Thirdly, sub-additivity may be undesirable for some actuarial and …nancial problems, as pointed out by Dhaene et al. (2008), who suggested the use of V aR for some merger-linked problems, for instance. Fourthly, for very important …nancial problems V aR often provides valuable solutions from both theoretical (Basak and Shapiro, 2001, Assa, 2015) and empirical (Annaert et al., 2009) viewpoints, and V aR also facilitates the use probabilities in both the objective function and/or the constraints of several …nancial optimization problems (Dupacová and Kopa, 2014, Zhao and Xiao, 2016, etc.). The optimization of V aR is much more complicated than the optimization of other risk measures (Rockafellar and Uryasev, 2000, Larsen et al., 2002, Gaivoronski and P‡ug, 2005, Shaw, 2011, Wozabal, 2012, etc.). Since V aR is neither convex nor di¤erentiable, one may face the existence of many local minima, and they may become undetectable by means of the standard optimization methods. There are many and quite di¤erent approaches addressing the optimization of V aR (Larsen et al., 2002, Gaivoronski and P‡ug, 2005, 2.

(3) Shaw, 2011, Wozabal, 2012, etc.). All of them yield interesting algorithms or optimality conditions allowing us to …nd adequate solutions under di¤erent assumptions, but non of them solves the problem in an exhaustive manner. There are many cases which cannot be treated with the existent methodologies. A very interesting approach may be found in Wozabal et al. (2010) and Wozabal (2012). The authors deal with discrete probability spaces composed of …nitely many atoms, and they prove that V aR equals the di¤erence of two convex functions. This property allows them to provide e¢ cient optimizing algorithms. Nevertheless, it is easy to show that the property above does not hold for general probability spaces. Since there are many problems involving V aR and continuous random variables (Shaw, 2011, Zhao and Xiao, 2016, etc.), further extensions containing general probability spaces should be welcome. This paper deals with a very simple idea. If the CV aR (also called AV aR, or average value at risk) may be given as an integral of V aR, then V aR must become a …rst order mathematical derivative of CV aR. Consequently, an approximation of V aR must be given by the change in CV aR over the change in level of con…dence (or, in other words, by a quotient of increments). Hence, an approximation of V aR must be given by the di¤erence of two convex functionals, and the result of Wozabal (2012) will become true in general probability spaces if one takes a limit. Ideas above will be formalized in Section 2, where it will be proved that V aR is the limit of the di¤erence of convex functionals. We will also explain why one does not need to take any limit in the discrete case. In Section 3 we will consider a sequence of optimization problems whose objective function has a limit, and we will analyze the relationship between the sequence of solutions and the solution optimizing the limit. As a consequence, we will establish conditions under which the optimization of V aR may be solved by optimizing the di¤erence of two convex functionals. In Section 4 we will focus on a methodology proposed in Balbás et al. ( 2010a) and we will address the minimization of the di¤erence of two convex functionals in arbitrary probability spaces. Several optimality conditions will be found. Applications in …nance (optimal investment) and insurance (optimal reinsurance) will be given in Section 5. Though the purpose of Section 5 is merely illustrative, these examples will be general enough, since they will apply in both static and dynamic frameworks and for discrete or continuous price/claim processes. Section 6 will summarize the paper.. 3.

(4) . )$"%"&)"* & &'++"'&*. +6 H:== 562= H:E9 E96 AC@323:=:EJ DA246 , F, 4@>A@D65 @7 E96 D6E E96 σ−2=863C2 F. 2?5 E96 AC@323:=:EJ >62DFC6 +6 42? 4@?D:56C ≤ p < ∞ 2?5 E96 DA246 Lp 2=D@ 56?@E65 3J Lp @C Lp , F, @7 C62=G2=F65 C2?5@> G2C:23=6D y DF49 E92E |y|p < ∞ . C6AC6D6?E:?8 E96 >2E96>2E:42= 6IA64E2E:@? &642== E92E Lq :D E96 5F2= DA246 @7 Lp H96C6. < q ≤ ∞ /p /q 2?5 L :D 4@>A@D65 @7 E96 6DD6?E:2==J 3@F?565 C2?5@> G2C:23=6D. &:6DK &6AC6D6?E2E:@? (96@C6> &F5:? &642== 2=D@ E92E E96 FDF2= ?@C> @7 Lp :D y :7 ≤ p < ∞ 2?5 y. . /p. p. |y|p . . Ess1Sup |y| Ess1Sup 56?@E:?8 O6DD6?E:2= DFAC6>F>P . @C ≤ p ≤ p ≤ ∞ H6 92G6 E92E Lp ⊃ Lp ? A2CE:4F=2C L ⊃ Lp ⊃ L 7@C 6G6CJ ≤ p ≤. ∞ &642== 2=D@ E92E 7@C ≤ p ≤ ∞ H6 92G6 E92E Lp >2J 36 6?5@H65 H:E9 E96 E@A@=@8J. σ Lp , Lq H9:49 :D H62<6C E92? E96 ?@C> E@A@=@8J FCE96C>@C6 :7 < p < ∞ E96? 6G6CJ. 4@?G6I 4=@D65 2?5 3@F?565 DF3D6E @7 Lp :D σ Lp , Lq −4@>A24E 2?9?2?249ND (96@C6>. 2?5 =2@8=FND (96@C6> 7 :D 2 L?:E6 D6E E96? Lp 364@>6D 2 L?:E65:>6?D:@?2= DA246 7@C . 6G6CJ ≤ p ≤ ∞ Lp Lp 7@C 6G6CJ ≤ p ≤ p ≤ ∞ 2?5 2== @7 E96 :?EC@5F465 E@A@=@8:6D. @7 Lp 4@:?4:56 FCE96C 56E2:=D 23@FE 2?249 DA246D @7 C2?5@> G2C:23=6D >2J 36 7@F?5 :?. &F5:? 2?5 @AA

(5) (96 DA246 L 4@?E2:?:?8 6G6CJ C62=G2=F65 C2?5@> G2C:23=6 >2J 36 6?5@H65 H:E9 E96 FDF2= 4@?G6C86?46 :? AC@323:=:EJ :? H9:49 42D6 L 364@>6D 2 >6EC:4 3FE ?@E 2?249 DA246 (96 FDF2= 5:DE2?46 :? L :D 8:G6? 3J dy, z M in , |y − z| 2?5 :E :D <?@H? E92E L ⊂ L &F5:? . :?2==J H6 H:== 562= H:E9 >2?J E@A@=@8:42= AC@A6CE:6D == @7 E96> >2J 36 7@F?5 :? 6==J 6E FD LI 2 4@?L56?46 =6G6= − μ ∈ , D FDF2= 7@C 2 C2?5@> G2C:23=6 y ∈ L E96. :D E96 FDF2= 56L?:E:@? @7 V aR1−μ (y) :7 y C6AC6D6?ED 2 7FEFC6 C2?5@> H62=E9 @C :?4@>6 ? >2?J. 24EF2C:2= 2?5 L?2?4:2= 2AA=:42E:@?D y C6AC6D6?ED C2?5@> 42A:E2= =@DD6D :? H9:49 42D6 :D C6A=2465 3J V aR1−μ (y) := Sup {x ∈ IR; IP (y ≤ x) < 1 − μ} . (9C@F89@FE E9:D A2A6C H6 H:== 562= H:E9 3FE 2 A2C2==6= 2?2=JD:D 4@F=5 36 :>A=6>6?E65 7@C .

(6). .

(7) *2=F6 2E &:D< V aRμ y @7 y :D 8:G6? 3J V aRμ y −Inf {x ∈ y ≤ x > μ} ,. . 2?5 7@C y ∈ L ⊂ L E96 @?5:E:@?2= *2=F6 2E &:D< CV aRμ y :D CV aRμ y . μ. μ. V aRt y dt..

(8) . . 44@C5:?8 E@ &@4<276==2C '6 #. CV aRμ y >2J 36 2=D@ 8:G6? 3J CV aRμ y M ax {− yz ≤ z ≤ /μ, z } ,. . μ {z ∈ L ≤ z ≤ /μ, z } ,. . 2?5 E96 D6E. H9:49 5@6D ?@E 56A6?5 @? y :D 42==65 E96 CV aRμ −DF38C25:6?E :E :D :?4=F565 :? Lq 7@C. 6G6CJ ≤ q ≤ ∞ 2?5 :E :D 4@?G6I 2?5 σ Lq , Lp −4@>A24E 7@C 6G6CJ < q ≤ ∞ ?. @3G:@FD :>A=:42E:@? @7 :D E96 6BF2=:EJ −CV aRμ y M in { yz ≤ z ≤ /μ, z }. . 7@C 6G6CJ y ∈ L D64@?5 :>A=:42E:@? @7 :D E96 L −?@C> 4@?E:?F:EJ @7 E96 7F?4E:@? L. y → CV aRμ y ∈ ,. . 2=@?8 H:E9 :ED σ L , L −=@H6C D6>:4@?E:?F:EJ. :I y ∈ L E :D <?@H? E92E E96 7F?4E:@? , . E :D 62DJ E@ D66 E92E Lp. t → V aRt y ∈ . . y → V aR1−μ (y) ∈ IR :D ?@E 4@?E:?F@FD :7 p = 0 @C 1 ≤ p < ∞ ?5665 E2<6. μ = 0.1 Ω = (0, 1) F E96 @C6= σ−2=863C2 2?5 IP E96 636D8F6 >62DFC6 (2<6 E96 D6BF6?46 @7 C2?5@> G2C:23=6D (0, 1) n = 1, 2, ... 2?5 E2<6. ⎧ ⎨ −1, ω → yn (ω) = ⎩ 0,. (0, 1). if 0 < ω < 0.1 + 1/(2n) otherwise. ⎧ ⎨ −1, ω → y0 (ω) = ⎩ 0,. if 0 < ω < 0.1 otherwise. (96? Limn→∞ (yn ) = y0 :? E96 ?@C> E@A@=@8J @7 Lp 7@C 1 ≤ p < ∞ 2?5 :? E96 >6EC:4 E@A@=@8J @7 L0 6D:56D V aR1−μ (y0 ) = 0 2?5 V aR1−μ (yn ) = 1 7@C n > 0.

(9) :D ?@?:?4C62D:?8 C:89E4@?E:?F@FD 2?5 636D8F6 :?E68C23=6 :? , (9FD :7 @?6 4@?D:5 6CD E96 7F?4E:@? , . μ → ϕy μ μCV aRμ y ∈ ,. . H9:49 >2J 36 2=D@ 8:G6? 3J D66

(10) μ. ϕy μ . V aRt y dt, . E96? E96 :CDE F?52>6?E2= (96@C6> @7 2=4F=FD 8F2C2?E66D E92E ϕ y μ V aRμ y. . 7@C 6G6CJ μ ∈ , ϕ y 56?@E:?8 E96 C:89E92?5 D:56 56C:G2E:G6 @7 ϕy @C n ∈ O=2C86 6?@F89P IAC6DD:@?D 2?5 DF886DE E96 2AAC@I:>2E:@? V aRμ y ≈. μ /n CV aRμ/n y − μCV aRμ y ,. /n. i.e. V aRμ y ≈ nμ CV aRμ/n y − nμCV aRμ y .. . !@C6 244FC2E6=J V aRμ y Limn nμ CV aRμ/n y − nμCV aRμ y. . 9@=5D 7@C 6G6CJ y ∈ L @?D6BF6?E=J :7 Y ⊂ L E96 @AE:>:K2E:@? AC@3=6>D M in {V aRμ y y ∈ Y }.

(11) . 2?5 nμ . CV aRμ/n y − CV aRμ y y ∈ Y nμ 4@F=5 92G6 OD:>:=2C D@=FE:@?DP '64E:@?

(12) H:== 36 56G@E65 E@ 2?2=JK:?8 D6G6C2= C6=2E:@?D9:AD M in. 36EH66? $C@3=6>D

(13) 2?5 2?5 D@>6 >6E9@5D D@=G:?8 $C@3=6> H:== 36 AC6D6?E65 :? '64E:@? . . '&&+"& +! '(+"%"1+"'& ' & . !2?J C6=2E:@?D9:AD 36EH66? E96 D@=FE:@? @7

(14) 2?5 E96 D@=FE:@? @7 >2J 36 AC@G65 :? 2 >@C6 86?6C2= D6EE:?8 (9FD =6E FD 8:G6 EH@ 86?6C2= =6>>2D E92E H:== 2AA=J :? @FC A2CE:4F=2C 7C2>6H@C< .

(15) %% 105+&'4 # 5'6 A #0& # 5'37'0%' fn n 1( 4'#. 8#.7'& (70%6+105 10 A i.e. fn A → n , , , ... 57%* 6*#6 fn n → f 21+069+5' %108'4)'0%' 10 A 105+&'4 xn ∈ A 51.8+0) M in {fn x x ∈ A} (14 '8'4; n ≥ a f x ≥ Lim"Supn fn xn (14 '8'4; x ∈ A Lim"Supn fn xn &'016+0) 6*' .+/+6 572'4+14 1( 6*' 5'37'0%' fn xn n . b 105+&'4 # 8'%614 52#%' E #0& # %108': %10' C 57%* 6*#6 A ⊂ C ⊂ E 72215' 6*#6 fn A → %#0 $' ':6'0&'& 61 C n , , ... #0& $'%1/'5 215+6+8'.; *1/1)'0'175 i.e. fn λx λfn x (14 x ∈ C λ ≥ #0& n , , , ... 72215' =0#..; 6*#6 Inf f x x ∈ λ. λ. λA. > −∞ *'0 f x ≥ Lim"Supn fn xn (14 '8'4; x ∈. λA. )'' a :I x ∈ A 7 ε > :E :D DFR 4:6?E E@ AC@G6 E96 6IAC6DD:@? f x ≥ −ε Lim1Supn fn xn @?D:56C n ∈ DF49 E92E |fn x − f x| < ε 9@=5D 7@C 6G6CJ n ≥ n (96? f x ≥ fn x − ε ≥ fn xn − ε 9@=5D 7@C 6G6CJ n ≥ n b @?D:56C x ∈ A 2?5 =6E FD AC@G6 E92E f x ≥ ?5665 @E96CH:D6 H6 H@F=5 92G6 Inf. f z z ∈. λA λ. ≥ Inf {λf x λ > } −∞,. 4@?EC25:4E:?8 E96 2DDF>AE:@?D ':?46 f :D A@D:E:G6=J 9@>@86?6@FD H6 92G6 E92E f x ≥ 9@=5D 7@C 6G6CJ x ∈. λ. λA 7 Lim1Supn fn xn < E96 2DD6CE:@? 364@>6D @3G:@FD. D@ =6E FD 2DDF>6 E92E Lim1Supn fn xn ≥ @C λ ≥ H6 92G6 λLim1Supn fn xn ≥ Lim1Supn fn xn . @?D:56C z λx H:E9 x ∈ A DD6CE:@? a :>A=:6D E92E f x ≥ Lim1Supn fn xn 6?46 f z λf x ≥ λLim1Supn fn xn ≥ Lim1Supn fn xn . . %% 105+&'4 # 5'6 A #0& # 5'37'0%' fn n 1( 4'#. 8#.7'& (70%6+105 10 A 57%* 6*#6 fn n → f 70+(14/.; 10 A 105+&'4 xn ∈ A 51.8+0) M in {fn x x ∈ A} (14 '8'4; n ≥ a 72215' 6*#6 x ∈ A #0& 6*'4' ':+565 # 6121.1); 10 A 57%* 6*#6 x +5 #0 #)).1/'4#6+10 21+06 1( xn n +0 2#46+%7.#4 +( x Limn xn #0& f +5 .19'4 5'/+%106+07175 #6 x *'0 f x Lim"Supn fn xn #0& x 51.8'5 M in {f x x ∈ A} .

(16) b 25'7&1%108'45' 1( a #.51 *1.&5 72215' 6*#6 M in {f x x ∈ A} +5 51.8#$.'. *'0 6*'4' ':+565 # 6121.1); 10 A 57%* 6*#6 f +5 .19'4 5'/+%106+07175 10 A #0& xn n. *#5 #0 #)).1/'4#6+10 21+06 x ∈ A 51.8+0) M in {f x x ∈ A} #0& 57%* 6*#6 f x Lim"Supn fn xn 0 2#46+%7.#4 +( 6*' 126+/+<#6+10 241$.'/ M in {f x x ∈ A} +5 51.8#$.' 6*'0 Lim"Supn fn xn +5 +65 126+/#. 8#.7' c 105+&'4 # 8'%614 52#%' E #0& # %108': %10' C 57%* 6*#6 A ⊂ C ⊂ E 72215' 6*#6 fn A → %#0 $' ':6'0&'& 61 C n , , ... #0& $'%1/'5 215+6+8'.; *1/1)'0'175 f x x ∈. 72215' 6*#6 Inf. λ. > −∞ 72215' =0#..; 6*#6 x ∈ A #0&. λA. 6*'4' ':+565 # 6121.1); 10 A 57%* 6*#6 x +5 #0 #)).1/'4#6+10 21+06 1( xn n #0& f +5 .19'4 5'/+%106+07175 #6 x *'0 f x Lim"Supn fn xn #0& x 51.8'5 M in f x x ∈. λ>. λA . )'' a @?D:56C ε > (96C6 6I:DE 2 ?6:893@C9@@5 V @7 x 2?5 n ∈ DF49 E92E f x > f x − ε 9@=5D 7@C 6G6CJ x ∈ V 2?5 |fn x − f x| < ε 9@=5D 7@C 6G6CJ x ∈ A 2?5 6G6CJ n ≥ n @?D6BF6?E=J :7 x ∈ V 2?5 n ≥ n f x < f x ε < fn x ε.. . 6D:56D E96C6 6I:DED 2 ?2EFC2= ?F>36C DE:== 56?@E65 3J n DF49 E92E 7@C 6G6CJ m ≥ n E96C6 6I:DED k ≥ m DF49 E92E xk ∈ V 2?5 E96C67@C6 f x < fk xk ε (96 @3G:@FD :>A=:42E:@? :D E92E f x ≤ Lim1Supn fn xn ε 2?5 E96C67@C6 f x ≤ Lim1Supn fn xn 6?46 E96 4@?4=FD:@? EC:G:2==J 7@==@HD 7C@> 6>>2 a b E :D 62DJ E@ D66 E92E E96 72>:=J {∅} ∪ f ∪ f x, ∞ x ∈ . @7 DF3D6ED. @7 A :D 2 E@A@=@8J @? A >2<:?8 f =@H6C D6>:4@?E:?F@FD 'FAA@D6 E92E A :D 4@>A24E (96? xn n H:== 92G6 2? 288=@>6C2E:@? A@:?E x ∈ A 6==J H9:49 H:== D2E:D7J f x Lim1Supn fn xn 2?5 H:== D@=G6 M in {f x x ∈ A} 5F6 E@ 6>>2 a ? @C56C E@ D66 E92E A :D 4@>A24E 4@?D:56C 2 72>:=J @7 @A6? D6ED D2E:D7J:?8 A⊂. x. f x, ∞ f. x. x, ∞. f Infx x, ∞ ,. H96C6 Infx x :D E96 @3G:@FD :?L>F> 7 a ∈ A 2?5 f a M in {f x x ∈ A} E96?. x, ∞ :? E96 8:G6? 72>:=J a ∈ f Infx x, ∞ :>A=:6D E92E f a > Infx x D@ E96C6 6I:DED x, ∞ 2?5 4@?D6BF6?E=J A ⊂ @7 @A6? D6ED DF49 E92E f a > x (96C67@C6 a ∈ f x, ∞ 3642FD6 f x ≥ f a > x H:== 9@=5 7@C 6G6CJ x ∈ A f .

(17) c 62C:?8 :? >:?5 6>>2 b :E :D DFR 4:6?E E@ AC@G6 E92E f x ≤ Lim1Supn fn xn ε 7@C 6G6CJ ε > D :? E96 AC@@7 @7 a E96C6 6I:DE 2 ?6:893@C9@@5 V ⊂ A @7 x 2?5 n ∈ DF49 E92E f x > f x − ε 9@=5D 7@C 6G6CJ x ∈ V 2?5 |fn x − f x| < ε 9@=5D 7@C 6G6CJ x ∈ A 2?5 6G6CJ n ≥ n @?D6BF6?E=J :7 x ∈ V 2?5 n ≥ n E96? 9@=5D D :? a 7@C 6G6CJ m ≥ n E96C6 6I:DED k ≥ m DF49 E92E xk ∈ V 2?5 E96C67@C6 . f x < fk xk ε "6IE =6E FD D66 E92E D@=FE:@?D @7 2=H2JD J:6=5 2 =@H6C 3@F?5 7@C

(18) . )'('*"+"'& 105+&'4 Y ⊂ L #0& yn ∈ Y 51.8+0) (14 '8'4; n ∈ V aRμ y ≥ Lim"Supn nμ CV aRμ/n yn − nμCV aRμ yn . . *1.&5 (14 '8'4; y ∈ Y . )'' (9:D :D 2? @3G:@FD 4@?D6BF6?46 @7 2?5 6>>2 a 6D:56D F?56C 255:E:@?2= 4@?5:E:@?D D@=FE:@?D @7 =625 E@ D@=FE:@?D @7

(19) . !')% 105+&'4 Y ⊂ L #0& 572215' 6*#6 *1.&5 70+(14/.; 10 Y 105+&'4 n ∈ #0& yn ∈ Y 51.8+0) (14 '8'4; n ≥ n a 105+&'4 y ∈ Y #0& 572215' 6*#6 6*'4' ':+565 # 6121.1); 10 Y 57%* 6*#6 y +5 #0 #)).1/'4#6+10 21+06 1( yn nn0 #0& Y. y → V aRμ y ∈ +5 .19'4 5'/+%106+07175 #6. y *'0 y 51.8'5

(20) #0& V aRμ y '37#.5 6*' 4+)*6 *#0& 5+&' 1( b ( < p < ∞ Y ⊂ Lp +5 %108': %.15'& #0& $170&'& #0& Y .19'4 σ Lp , Lq −5'/+%106+07175 10 Y 6*'0. yn nn0. y → V aRμ y ∈ +5. *#5 #0 #)).1/'4#6+10 21+06 y ∈ Y. 51.8+0)

(21) #0& V aRμ y '37#.5 6*' 4+)*6 *#0& 5+&' 1( c ( yn nn0 *#5 #0 #)).1/'4#6+10 21+06 y ∈ Y +0 6*' L −014/ 6121.1); 1( Y 6*'0 y. 51.8'5

(22) #0& V aRμ y '37#.5 6*' 4+)*6 *#0& 5+&' 1( . )''. a :D 2 4@?D6BF6?46 @7 6>>2 a b 7@==@HD 7C@> a :7 @?6 362CD :? >:?5 E92E. Y :D σ Lp , Lq −4=@D65 2?9?2?249ND (96@C6> &F5:? 2?5 σ Lp , Lq −4@>A24E. .

(23) p q =2@8=FND (96@C6> 2?5 E96C67@C6 yn nn0 ⊂ Y 92D 2 σ L , L −288=@>6C2E:@? A@:?E. y ∈ Y 6==J :?2==J c 7@==@HD 7C@> a :7 @?6 362CD :? >:?5 E92E Y. y → nμ CV aRμ/n y − nμCV aRμ y ∈ . :D L −?@C> 4@?E:?F@FD D66 2?5 E96C67@C6 D@ :D Y. y → V aRμ y ∈ 5F6 E@ E96 . F?:7@C> 4@?G6C86?46 @7 @? Y 6>>2D b 2?5 c 2=D@ 92G6 :?E6C6DE:?8 :>A=:42E:@?D :? V aR @AE:>:K2E:@?. !')% 105+&'4 Y ⊂ L #0& 572215' 6*#6 Inf. V aRμ y y ∈. λ. λY . >. −∞ 105+&'4 n ∈ #0& yn ∈ Y 51.8+0) (14 '8'4; n ≥ n a 41$.'/ M in V aRμ y y ∈. . λY λ. +5 $170&'& #0& 6*' 4+)*6 *#0& 5+&' 1( +5 # .19'4 $170& 1( +65 126+/#. 8#.7' b 72215' 6*#6 *1.&5 70+(14/.; 10 Y 72215' 6*#6 y ∈ Y #0& 6*'4' ':+565 # 6121.1); 10 Y 57%* 6*#6 y +5 #0 #)).1/'4#6+10 21+06 1( yn n #0& Y. y → V aRμ y ∈ +5. .19'4 5'/+%106+07175 #6 y *'0 y 51.8'5 #0& V aRμ y '37#.5 6*' 4+)*6 *#0& 5+&' 1( c 72215' 6*#6 *1.&5 70+(14/.; 10 Y ( < p < ∞ Y ⊂ Lp +5 %108': %.15'& #0& $170&'& #0& Y. y → V aRμ y ∈ +5 .19'4 σ Lp , Lq −5'/+%106+07175 10 Y 6*'0. yn nn0 *#5 #0 #)).1/'4#6+10 21+06 y ∈ Y 9*+%* 51.8'5 #0& V aRμ y '37#.5 6*' 4+)*6 *#0& 5+&' 1( d 72215' 6*#6 *1.&5 70+(14/.; 10 Y ( yn nn0 *#5 #0 #)).1/'4#6+10 21+06 y ∈ Y. +0 6*' L −014/ 6121.1); 1( Y 6*'0 y 51.8'5 #0& V aRμ y '37#.5 6*' 4+)*6 *#0& 5+&' 1( . )''. @E9 V aRμ 2?5 nμ CV aRμ/n − nμCV aRμ 2C6 56L?65 @? L 2?5. 2C6 A@D:E:G6=J 9@>@86?6@FD &@4<276==2C 2?5 )CJ2D6G 6?46 a :D 2 A2CE:4F=2C 42D6 @7 6>>2 b 2?5 b :D 2 A2CE:4F=2C 42D6 @7 6>>2 a 6D:56D c 2?5 d 7@==@H 7C@> b :7 @?6 362CD :? >:?5 E96 D2>6 2C8F>6?ED 2D :? E96 AC@@7D @7 (96@C6> b 2?5 c . .

(24) 44@C5:?8 E@ (96@C6>D 2?5 :E :D :>A@CE2?E E@ 8:G6 4@?5:E:@?D 8F2C2?E66:?8 E96 F?:7@C> 4@?G6C86?46 @7 . )'('*"+"'& a 105+&'4 # 010+0%4'#5+0) #0& 6*'4'(14' '$'5)7' +06')4#$.' (70%6+10 f a, b → 14 < h ≤ b − a. ≤ f a − h. ah a. f t dt ≤ f a − f a h .. b 105+&'4 y ∈ L *'0 (14 n , ,

(25) , ... ≤ V aRμ y − nμ CV aRμ/n y − nμCV aRμ y. . ≤ V aRμ y − V aRμ/n y . c 105+&'4 Y ⊂ L ( . Limn V aRμ/n y V aRμ y 70+(14/.; 10 y ∈ Y 6*'0 *1.&5 70+(14/.; 10 y ∈ Y )'' a hf a ≥ h. ah a. ah a. f t dt ≥ hf a h 3642FD6 f :D ?@?:?4C62D:?8 6?46 f a ≥. f t dt ≥ f a h (96 LCDE :?6BF2=:EJ :>A=:6D E92E ≤ f a −. H9:=6 E96 D64@?5 @?6 :>A=:6D E92E h. ah a. − h. ah a. h. ah a. f t dt. f t dt ≤ −f a h 2?5 E96C67@C6 f a −. f t dt ≤ f a − f a h. b (96 C6DF=E EC:G:2==J 7@==@HD 7C@> a 3642FD6 7@C h . h. n. H6 92G6 E92E

(26) =625D E@. μh. V aRt y dt nμ CV aRμ/n y − nμCV aRμ y .. μ. . c (96 C6DF=E EC:G:2==J 7@==@HD 7C@> . %)# 72215' 6*#6 +5 # =0+6' 5'6 72215' 6*#6 6*'4' #4' 01 '.'/'065 1( 9+6* 07.. 241$#$+.+6; !' %#0 %105+&'4 6*' 5'6 C 1( %172.'5 , ω 57%* 6*#6 ω ∈ /. . j. ω j ≤ μ #0& ω . #0& =0+6' 016+%' 6*#6. j. ⊂ ω ∈. ω j > μ $8+175.; C +5 010 81+&. ∅ +5 #%%'26'& 105+&'4 # 4#0&1/ 8#4+#$.' y %%14&+0). 61 V aRμ y +5 %*#4#%6'4+<'& $; #0 '.'/'06 , ω ∈ C 0&''& %105+&'4 #0 14&'4 .

(27) {ω y , ω y , ..., ω k y} 10 57%* 6*#6 y ω y ≤ y ω y ≤ .... ≤ y ω k y 6#-'. {ω y , ω y , ..., ω j y} 9+6*. j i. ω i y ≤ μ #0&. j i. ω i y > μ. #0& 6#-' ω ω j y *'0 −V aRμ y y ω 14'18'4 (14 '8'4; μ /n < j i. ω i y 10' 56+.. *#5 −V aRμ/n y y ω i.e. V aRμ y V aRμ/n y. +0%' C +5 =0+6' 6*'4' ':+565 n ∈ 57%* 6*#6 (14 '8'4; n ≥ n #0& '8'4; , ω ∈ C 10' *#5 ω . j. ω j > μ /n *'4'(14' V aRμ y V aRμ/n y (14 '8'4;. n ≥ n #0& '8'4; 4#0&1/ 8#4+#$.' y ∈ L 016+%' 6*#6 L L +0 6*+5 2#46+%7.#4 %#5' 105'37'06.; +/2.+'5 6*#6 *1.&5 #5 #0 '37#.+6; (14 n ≥ n #0& '8'4; 4#0&1/. 8#4+#$.' y ∈ L 0 14&'4 914&5 6*' 5'37'0%' 1( 4'/#+05 %1056#06 (14 '8'4; y #0& '8'4; n ≥ n 6 +5 1$8+175 6*#6 6*' 70+(14/ %108'4)'0%' 1( *1.&5 +0 6*' 9*1.' 52#%' L #0& 6*'4'(14' *'14'/5 #0& #22.; 14'18'4 9' &1 016 *#8' 61 6#-' #0; .+/+6. $'%#75' 41$.'/

(28) #0& 41$.'/ #4' ':#%6.; 6*' 5#/' 241$.'/ (14 '8'4; Y ⊂ L #0& '8'4; n ≥ n 5+/+.#4 4'57.6 +5 2418'& +0 !1<#$#. '6 #. #0& !1<#$#. (14 2146(1.+1 %*1+%' 241$.'/5 #0& 16*'4 126+/+<#6+10 241$.'/5 +081.8+0) V aRμ #0& # =0+6' 5'6 746*'4 ':6'05+105 #22.;+0) (14 +0=0+6'.; /#0; 56#6'5 1( 0#674' 4'37+4' 6*' 75' 1( . .+/+65 #0& 6*' #0#.;5+5 #$18'. . (+"%"1"& +! $"&# (()'/"%+"'&*. (9:D D64E:@? H:== 36 56G@E65 E@ D@=G:?8 $C@3=6> !@C6 244FC2E6=J 7@C k > ν > . − μ − ν ∈ , 2?5 Y ⊂ L H6 H:== DEF5J $C@3=6> M in {kCV aRμν y − CV aRμ y y ∈ Y } .. . ':?46 &@4<276==2C 2?5 )CJ2D6G AC6D6?E65 E96:C 72>@FD >6E9@5 E@ @AE:>:K6 E96 CV aR >2?J 2FE9@CD 92G6 6IE6?565 E96 5:D4FDD:@? 7@C @E96C C:D< >62DFC6D 2?5 7C2>6H@C<D &FDK4KJ?D<: 2?5 '92A:C@ 2=3TD '6 #. a 6E4 +:E9 C6DA64E E@ $C@3=6> +@K232= '6 #. 2?5 +@K232= AC@A@D65 ?6H AC@465FC6D 2AA=J:?8 F?56C 5:D 4C6E6 AC@323:=:EJ DA246D H:E9 L?:E6=J >2?J 2E@>D ? @C56C E@ L?5 @AE:>2=:EJ 4@?5:E:@?D 7@C $C@3=6> 2?5 86?6C2= AC@323:=:EJ DA246D H6 H:== 7@==@H E96 2AAC@249 @7 2=3TD '6 #. a D:?46 :E 92D AC@G65 E@ 36 G6CJ 6R 4:6?E :? 3@E9 24EF2C:2= 2=3TD '6 #. 2?5 L?2?4:2= 2=3TD '6 #. b 2=3TD '6 #. a @C 2=3TD '6 #. b 2AA=:42E:@?D %% 105+&'4 y ∈ Y y 51.8'5 +( #0& 10.; +( 6*'4' ':+56 θ ∈ #0& z ∈ L.

(29) 57%* 6*#6 y , θ , z 51.8'5 5'' ⎧ ⎪ ⎪ θ k yw ≥ , ∀w ∈ μν ⎪ ⎪ ⎪ ⎪ ⎪ z . ⎪ ⎪ ⎪ ⎪ ⎨ z≥ M in θ yz ⎪ z ≤ /μ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y∈Y ⎪ ⎪ ⎪ ⎪ ⎩ y, θ, z ∈ L × × L. . . y, θ, z $'+0) 6*' &'%+5+10 8#4+#$.' ( 51 6*'0 9' *#8' 6*#6 θ kCV aRμν y y z −CV aRμ y #0& 6*' 126+/#. 8#.7'5 1( $16* #0& . %1+0%+&' )''. 'FAA@D6 E92E y D@=G6D 2?5 4@?D:56C y, θ, z . 762D:3=6 (2<6 θ . kCV aRμν y 2?5 z ∈ μ DF49 E92E D66 2?5 y z −CV aRμ y . 2?5 D9@H E92E y , θ , z :D . 762D:3=6 ':?46 y, θ, z :D . 762D:3=6 2?5 D9@H E92E θ ≥ kCV aRμν y 2?5 −CV aRμ y ≤ yz 6?46 θ yz ≥ kCV aRμν y − CV aRμ y ≥. kCV aRμν y − CV aRμ y θ y z . @?G6CD6=J DFAA@D6 E92E y , θ , z D@=G6D . 7 y ∈ Y E96? 2?5 D9@H 6I:DE6?46 @7 z ∈ μ H:E9 yz −CV aRμ y 2?5 E96 . 762D:3:=:EJ @7 y, θ kCV aRμν y , z . 6?46 kCV aRμν y − CV aRμ y θ yz ≥ θ y z .. . 6E FD AC@G6 E92E θ kCV aRμν y ..

(30) . ?5665 @E96CH:D6 θ 4@F=5 36 C6A=2465 3J kCV aRμν y < θ 2?5 H6 H@F=5 DE:== 92G6 2 . 762D:3=6 D@=FE:@? 5F6 E@ (9FD θ y z > kCV aRμν y y z H@F=5 :>A=J 2 4@?EC25:4E:@? "6IE =6E FD AC@G6 E92E y z −CV aRμ y .. . .

(31) ?5665 @E96CH:D6 H@F=5 :>A=J y z > −CV aRμ y 2?5 H6 4@F=5 L?5 z ∈ μ. H:E9 y , θ , z . 762D:3=6 2?5 y z −CV aRμ y < y z (9FD θ y z > θ y z H@F=5 36 2 4@?EC24E:@? 282:? :?2==J

(32) 2?5 :>A=J E92E y D@=G6D . . "@E:46 E92E $C@3=6> . :D =:?62C :? E96 θ−G2C:23=6 2?5 3:=:?62C :? E96 y, z −G2C:23=6 ? A2CE:4F=2C :7 @?6 LI6D y @C z E96? . 364@>6D =:?62C 2?5 E96C67@C6 :E :D 62DJ E@ L?5 :ED @AE:>2=:EJ 4@?5:E:@?D. !')% 26+/#.+6; %10&+6+105 72215' 6*#6 +5 Y %108': #0& y ∈ Y 51.8'5 . *'4' ':+565 w , z , α, α , αμ ∈ μν × μ × × L × L 57%* 6*#6 ⎧ ⎪ ⎪ α z ⎪ ⎪ ⎪ ⎪ ⎪ αμ /μ − z ⎪ ⎪ ⎪ ⎪ ⎨ y α α − α . μ. ⎪ α ≥ , αμ ≥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y z − kw ≤ y z − kw , ∀y ∈ Y ⎪ ⎪ ⎪ ⎪ ⎩ y w ≤ y w , ∀w ∈ μν. . 746*'4/14' +( ≤ p ≤ ∞ #0& y ∈ Lp 6*'0 α , αμ ∈ Lp × Lp . )'' (96C6 6I:DED y , θ , z D@=G:?8 . (9FD z ∈ μ 2?5 y , θ D@=G6D E96 =:?62C AC@3=6>. ⎧ ⎪ ⎪ ⎨ θ k yw ≥ , ∀w ∈ μν M in θ yz y∈Y ⎪ ⎪ ⎩ y, θ ∈ L × . D :? 2=3TD '6 #. a E96 5F2= AC@3=6> @7 :D ⎧ ⎨ M ax w Inf { y z − kw y ∈ Y } , ⎩ w ∈ μν. . . E96C6 :D ?@ 5F2=:EJ 82A 36EH66? 2?5 2?5 E96 4@>A=6>6?E2CJ D=24<?6DD 4@?5:E:@?D 36EH66? 2?5 =625 E@ E96 L7E9 2?5 D:IE9 4@?5:E:@? @7 .

(33).

(34) 6D:56D θ , z D@=G6D ⎧ ⎪ θ k y w ≥ , ∀w ∈ μν ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ z . M in θ y z z≥ ⎪ ⎪ ⎪ ⎪ z ≤ /μ ⎪ ⎪ ⎪ ⎪ ⎩ θ, z ∈ × L. . D :? 2=3TD '6 #. a E96 28C2?8:2? 7F?4E:@? @7 :D L z, w, α , αμ y z − k y w −. zαμ dω ,. zα dω . . H96C6 α ≥ 2?5 αμ ≥ 36=@?8 E@ E96 5F2= DA246 @7 L z, w, α , αμ H:== 36 5F2= 762D:3=6 :7 2?5 @?=J :7 L z, w, α , αμ 92D 2 L?:E6 =@H6C 3@F?5 :? E96 2R ?6 DA246. {z ∈ L z } H9:49 :D 6BF:G2=6?E E@ E96 6I:DE6?46 @7 α ∈ DF49 E92E y z −. zα dω . zαμ dω α z . . 7@C 6G6CJ z ∈ L (9FD y − α αμ α AC@G6D E96 E9:C5 4@?5:E:@? @7 2?5 >@C6@G6C E96 4@>A=6>6?E2CJ D=24<?6DD 4@?5:E:@?D 36EH66? 2?5 :ED 5F2= =625 E@ E96 LCDE 2 D64@?5 @?6 :?2==J E96 LCDE D64@?5 2 E9:C5 4@?5:E:@? :? :>A=J E92E ⎧ ⎧ ⎨ y α, z < /μ ⎨ α − y, z > and αμ , α ⎩ , ⎩ , otherwise otherwise. 2?5 E96C67@C6 α , αμ ∈ Lp × Lp :7 y ∈ Lp . . . /%($*. &:D< @AE:>:K2E:@? A=2JD 2 4C:E:42= C@=6 :? L?2?46 2?5 :?DFC2?46 (96C6 2C6 >2?J 4=2DD:42= AC@3=6>D G6CJ 7C6BF6?E=J G:D:E65 2?5 C6G:D:E65 :? E96 =:E6C2EFC6 +6 92G6 D6=64E65 EH@ 6I2>A=6D (9:D :D ?@E 2E 2== 2? 6I92FDE:G6 =:DE 2?5 H6 2C6 2H2C6 @7 E92E 3FE H6 ;FDE 92G6 2? :==FDEC2E:G6 AFCA@D6 +6 H@F=5 =:<6 E@ D9@H 9@H E96 E96@CJ @7 D64E:@?D 23@G6 >2J 36 FD67F= :? 3@E9 24EF2C:2= 2?5 L?2?4:2= 2AA=:42E:@?D +6 H:== ?@E 4@>A=6E6=J D@=G6 E96 D6=64E65 6I2>A=6D 3642FD6 :E :D 36J@?5 E96 D4@A6 @7 E9:D A2A6C H9@D6 7@4FD :D @? E96 V aR >:?:>:K2E:@? "6G6CE96=6DD H6 H:== D66 9@H E96 56G6=@A65 E96@CJ >2J 6?23=6 FD E@ L?5 256BF2E6 D@=FE:@?D .

(35) (96 LCDE 6I2>A=6 :D 24EF2C:2= +6 92G6 49@D6? E96 @AE:>2= C6:?DFC2?46 AC@3=6> ORP 3642FD6 :E H2D 2>@?8 E96 >@DE DEF5:65 24EF2C:2= @AE:>:K2E:@? AC@3=6>D 5FC:?8 >2?J J62CD 2=FDK<2 2: 2?5 (2? 9: 2?5 (2? 2=3TD '6 #. .9F2?8 '6 #.. 6E4 ':>:=2C=J @FC D64@?5 49@:46 A@CE7@=:@ D6=64E:@? 2?5 2DD6E 2==@42E:@? P CAA H2D E96 7@4FD @7 >2?J A2A6CD 5FC:?8 >2?J J62CD E@@ '92H FA24@GT 2?5 @A2.

(36) .92@ 2?5 ,:2@ 2=3TD '6 #. a 6E4. . (+"%$ )"&*,)&. @?D:56C 2? :?DFC2?46 4@>A2?J 92G:?8 E@ A2J E96 C2?5@> :?56>?:L42E:@? u ≥ 2E 2 7FEFC6 52E6 T (96 4@>A2?J 42? 3FJ 2 C6:?DFC2?46 H9@D6 C6E2:?65 C:D< ur 2?5 46565 C:D< uc H:== D2E:D7J u ur uc (96 49@:46 @7 ur ≥ 2?5 uc ≥ :D E96 7@4FD @7 E96 ORP ':?46 E96 D@=FE:@? :D @7E6? 249:6G65 H:E9 2 stop1loss 4@?EC24E uc u − U M ax {u − U, }. 7@C D@>6 U ≥ H9:49 4@F=5 AC@G@<6 C6:?DFC6C >@C2= 92K2C5 H6 H:== AC6G6?E E96 762D:3:=:EJ @7 E9:D 4@?EC24E 3J 7@==@H:?8 E96 2AAC@249 @7 2=3TD '6 #. D66 2=D@ .9F2?8 '6 #. 6?46 4@?D:56C E96 2?249 DA246 X 4@>A@D65 @7 E96 @FE @7 2 4@F?E23=6 D6E 4@?E:?F@FD 7F?4E:@?D x , ∞ → H:E9 L?:E6 =@H6C 2?5 FAA6C 3@F?5 6?5@H65 H:E9 :ED FDF2= ?@C> x. . Sup {|x t| t ≥ } 7 E96 6IA64E2E:@? u 2?5 G2C:2?46 σ u @7 u. D2E:D7J u < ∞ 2?5 σ u < ∞ :D E96 AC@323:=:EJ >62DFC6 86?6C2E65 3J u @? , ∞ i.e. B :D E96 AC@323:=:EJ @7 E96 6G6?E u ∈ B 7@C 6G6CJ @C6= D6E B ⊂ , ∞ 2?5 J X → L :D 8:G6? 3J. t. x s ds,. J x t . E96? :E :D 62DJ E@ D66 E92E J :D H6== 56L?65 =:?62C 2?5 4@?E:?F@FD ?5665 H6 H:== 92G6 E92E. t. |J x t| ≤. x . 7@C 6G6CJ x ∈ X 2?5 6G6CJ t ≥ . . J x dx ≤ x. . . . ds x. . t dx x. . t. . σ u u. . 7@C 6G6CJ x ∈ X 2?5 E96 C6DF=E H:== EC:G:2==J 7@==@H 7C@> AC@A6CE:6D G6CJ DE2?52C5 :? 7F?4E:@?2= 2?2=JD:D &F5:? ? AC24E:46 2?5 @FE @7. 636D8F6 ?F== D6ED x >2J 36. 7 E96 2AAC@249 @7 2=3TD '6 #. :D ?@E :>A=6>6?E65 2?5 @?6 562=D H:E9 >@C6 4=2DD:42= 7C2>6. H@C<D 2: 2?5 (2? 9: 2?5 (2? 6E4 E96? F?56C D@>6 DEC2:89E7@CH2C5 >@5:L42E:@?D E96 C6DE @7 E96 6I2>A=6 6DD6?E:2==J C6>2:?D E96 D2>6. .

(37) F?56CDE@@5 2D E96 D6?D:E:G:EJ @C LCDE @C56C 56C:G2E:G6 @7 E96 C6E2:?65 C:D< H:E9 C6DA64E E@ 4=2:>D 2=3TD '6 #. ? 724E @?6 42? :56?E:7J u H:E9 u t J t t 7@C 6G6CJ t ≥ (9FD :7 x ∈ X :D E96 49@D6? C6:?DFC2?46 4@?EC24E E96? ur J x 2?5 uc J − x H:== 36 E96 49@D6? C6E2:?65 2?5 46565 C:D< C6DA64E:G6=J (96 C6:?DFC6C >2J AC6G6?E 96C9:D >@C2= 92K2C5 3J :>A@D:?8 E96 4@?DEC2:?E x ≥ h 7@C 2 D6=64E65 OE9C6D9@=5 @7 E96 C6E2:?65 D6?D:E:G:EJP h ∈ X h ≥ #3G:@FD=J :7 E96 C6:?DFC6C 2446AED stop − loss 4@?EC24ED E9:D 4@?DEC2:?E 364@>6D :CC6=6G2?E 3J D6=64E:?8 h #? E96 4@?EC2CJ :7 stop − loss 4@?EC24ED 2C6 ?@E 2446AE65 E96J H:== 364@>6 :?762D:3=6 3J 49@@D:?8 h ≥ ε 7@C D@>6 ε > 6E C > 36 2 =@25:?8 C2E6 2?5 4@?D:56C E96 C6:?DFC2?46 AC:46 C uc C J − x 4@>AFE65 H:E9 E96 6IA64E65 G2=F6 AC6>:F> AC:?4:A=6 =E6C?2E:G6 AC6>:F> AC:?4:A=6D >2J 36 4@?D:56C65 E@@ 2=FDK<2 2=3TD '6 #. 6E4 3FE 2D D2:5 23@G6 H6 @?=J 2EE6>AE E@ :==FDEC2E6 E96 :?E6C6DE @7 '64E:@?D

(38) 2?5 (96 L?2= H62=E9 @7 E96 :?DFC6C H:== 36 W x − J x − C J − x. . 36:?8 E96 2>@F?E @7 >@?6J A2:5 3J E96 :?DFC6C 4=:6?ED 7 V aRμ W x C6M64ED E96 :?DFC6C C:D< E96? E96 ORP >2J 364@>6 E96 G64E@C @AE:>:K2E:@? AC@3=6> ⎧ ⎪ ⎪ ⎨ M ax W x M in V aRμ W x ⎪ ⎪ ⎩ x ∈ X, h ≤ x ≤. ':?46 C > ∈ C J ∈ 2?5 V aRμ :D EC2?D=2E:@? :?G2C:2?E CEK?6C '6 #. :>A=:6D E96 6BF:G2=6?46 36EH66? E9:D AC@3=6> 2?5 ⎧ ⎪ ⎪ ⎨ M in − J x M in V aRμ −J x − C J x ⎪ ⎪ ⎩ x ∈ X, h ≤ x ≤. D FDF2= :? G64E@C @AE:>:K2E:@? E9:D AC@3=6> >2J 36 D@=G65 3J >62?D @7 A@D:E:G6 H6:89ED 7 W > :D E96 H6:89E @7 J x 2?5 :D E96 H6:89E @7 V aRμ −J x− C J x E96? E96 @3;64E:G6 7F?4E:@? @7 ORP H:== 36 V aRμ −J x − C W J x 2?5 E96 ORP L?2= G6CD:@? H:== 364@>6 E2<6 W C W > 2?5 C642== 282:? E92E V aRμ. .

(39) :D EC2?D=2E:@? :?G2C:2?E ⎧ ⎨ M in V aR μ W J x − J x ⎩ x ∈ X, h ≤ x ≤. . "6IE =6E FD D66 E92E E96 AC@A@D65 >6E9@5@=@8J 2AA=:6D E@ D@=G6 ?5665 LCDE @7 2== ?@E:46 E92E :D 2 A2CE:4F=2C 42D6 @7

(40) :7 Y {W J x − J x x ∈ X, h ≤ x ≤ } . '64@?5=J Y ⊂ L 2?5 :E :D 4@?G6I 2?5 3@F?565 5F6 E@ 'FAA@D6 E92E 9@=5D F?:7@C>=J @? Y (96? E96 4=@DFC6 @7 Y H:== D2E:D7J E96 4@?5:E:@?D @7 (96@C6> b 2?5 6G6CJ 288=@>6C2E:@? A@:?E @7 E96 D6BF6?46 @7 D@=FE:@?D @7 H:== D2E:D7J E96 4@?5:E:@?D @7 (96@C6> c ? @C56C E@ D66 E92E 9@=5D F?:7@C>=J @? Y =6E FD 5C2H @? $C@A@D:E:@? c 2?5 +6 @?=J 92G6 E@ AC@G6 E92E Limn V aRμ/n W J x − J x V aRμ W J x − J x F?:7@C>=J @? x ∈ X, h ≤ x ≤ ':?46 V aRν :D EC2?D=2E:@? :?G2C:2?E 7@C 6G6CJ − ν ∈ , :E :D DFR 4:6?E E@ D9@H E92E Limn V aRμ/n −J x V aRμ −J x. . F?:7@C>=J @? x ∈ X, h ≤ x ≤ "@E:46 E92E h ≤ x ≤ :>A=:6D E92E J x 2?5 J − x 2C6 4@>@?@E@?6 DD2 2?5 2C2: ':?46 V aRν :D 4@>@?@E@?6 255:E:G6 7@C 6G6CJ. − ν ∈ , DD2 2?5 2C2: H6 92G6 E92E V aRμ/n −J V aRμ/n −J x V aRμ/n −J − x V aRμ −J V aRμ −J x V aRμ −J − x 2?5 E96C67@C6 V aRμ −J x − V aRμ/n −J x V aRμ −J − V aRμ −J − x − V aRμ/n −J − V aRμ/n −J − x V aRμ −J − V aRμ/n −J − V aRμ −J − x − V aRμ/n −J − x ≤ V aRμ −J − V aRμ/n −J .

(41) 3642FD6 :D ?@?:?4C62D:?8 7F?4E:@? (9FD E96 F?:7@C> 4@?G6C86?46 @7 EC:G:2==J 7@==@HD 7C@> E96 C:89E4@?E:?F:EJ @7 7@C y J #?46 H6 <?@H E92E $C@A@D:E:@?

(42) 2?5 (96@C6>D b 2?5 c 2AA=J :E @?=J C6>2:?D E@ G6C:7J (96@C6> 2?5 (96J =625 E@ w , z , α, α , αμ ∈ μν × μ × × L × L z z − kw 2?5 ⎧ ⎪ ⎪ α z ⎪ ⎪ ⎪ ⎪ ⎪ αμ /μ − z ⎪ ⎪ ⎪ ⎪ ⎨ −J x W J x α α − α μ ⎪ α ≥ , αμ ≥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∀h ≤ x ≤. W J x − J x z ≤ W J x − J x z , ⎪ ⎪ ⎪ ⎪ ⎩ W J x − J x w ≤ W J x − J x w , ∀w ∈ μν.

(43) . +6 H:== ?@E D@=G6 'JDE6>

(44) 3642FD6 :E H@F=5 D:8?:L42?E=J 6?=2C86 E96 A2A6C "6G6CE96=6DD D:>:=2C DJDE6>D 92G6 366? D@=G65 :? 2=3TD '6 #. 2?5 2=3TD '6 #. b H96C6 E96 2FE9@CD @AE:>:K6 E96 CV aR 3J >62?D @7 4=@D6=J C6=2E65 6BF2E:@?D. . (+"%$ "&-*+%&+. 6E FD :?EC@5F46 E96 P CAA 3J >62?D @7 E96 2=3TD '6 #. b 2AAC@249 E :D G6CJ 86?6C2= 3642FD6 :E 2AA=:6D 7@C 3@E9 DE2E:4 2?5 5J?2>:4 7C2>6H@C<D 2?5 :E D:>A=:L6D D@>6 2DA64ED 3J >62?D @7 E96 DE@492DE:4 5:D4@F?E 724E@C SDF @?D:56C 2 E:>6 :?E6CG2= , T 2?5 DFAA@D6 E92E >2C<6E65 4=2:>D 2E T 2C6 8:G6? 3J C2?5@> G2C:23=6D @? E96 AC@323:=:EJ DA246 , F, 'FAA@D6 E92E 6G6CJ >2C<6E65 4=2:> y :D :? L. 2?5 E92E E96 >2C<6E :D 4@>A=6E6 i.e. 6G6CJ y ∈ L :D 2 >2C<6E65 4=2:> @C C624923=6 A2J @S @>A=6E6?6DD :D ?@E ?646DD2CJ 2=3TD '6 #. b 3FE :E D:>A=:L6D E96 6IA@D:E:@? 2?5 2D D2:5 23@G6 H6 @?=J ECJ E@ :==FDEC2E6 D6G6C2= A@DD:3:=:E:6D @7 AC6G:@FD D64E:@?D FCC6?E AC:46D 2C6 8:G6? 3J E96 =:?62C 2?5 4@?E:?F@FD 7F?4E:@? AC:4:?8 CF=6 L. y → y . z y ∈ z ∈ L :D E96 SDF 2?5 >FDE D2E:D7J z > :? @C56C E@ AC6G6?E E96 2C3:EC286 +6 H:== 2=D@ :>A@D6 z @C 6BF:G2=6?E=J E96 C:D<=6DD C2E6 G2?:D96D #?46 282:? E9:D 2DDF>AE:@? >2J 36 C6>@G65 3FE :E D:>A=:L6D D@>6 ?@E2E:@?D (96 P CAA H:== 7@4FD @? 3@E9 C:D< 2?5 6IA64E65 A2J@S A6C :?G6DE65 5@==2C (9FD :7 R >. :D E96 56D:C65 6IA64E65 C6EFC? ?@E:46 E92E R 42? 36 C624965 H:E9 E96 C:D<=6DD D64FC:EJ .

(45) E96 P CAA H:== 364@>6. ⎧ ⎨ y ≥ R, z y ≤. M in V aRμ y ⎩ y ∈ L. . ':?46 y y − R R V aRμ y −R V aRμ y − R y ≥ R ⇔ y − R ≥ 2?5 z y ≤ ⇔ z y − R ≤ − R C6A=24:?8 y H:E9 y − R 2?5 56?@E:?8 282:? 3J y E96 564:D:@? G2C:23=6 364@>6D. ⎧ ⎨ y ≥ , z y ≤ −α M in V aRμ y ⎩ y ∈ L. . H:E9 α R − > $C@3=6> :D 762D:3=6 F?56C G6CJ H62< 4@?5:E:@?D 3FE :D :D @7E6? F?3@F?565 2=3TD '6 #. b @C :?DE2?46 :E :D F?3@F?565 7@C E96 =24< 2?5 '49@=6D AC:4:?8 >@56= 2?5 7@C >2?J DE@492DE:4 G@=2E:=:EJ AC:4:?8 >@56=D 7 :E :D 3@F?565 2?5 D@=G23=6 E96? 7@C M =2C86 6?@F89 E96 D@=FE:@? y H:== D2E:D7J y . . ≤ M D66 . (96C67@C6 :E H:== 2=D@ D@=G6. ⎧ ⎨ y ≥ , z y ≤ −α M in V aRμ y ⎩ y∈L , y ≤M. . . #3G:@FD=J E96 762D:3=6 D6E @7 :D 4=@D65 3@F?565 2?5 σ L , L −4@>A24E (96C67@C6 D:?46 λy :D EC:G:2==J 762D:3=6 :7 λ ≥ 2?5 y :D 762D:3=6 (96@C6> a H:== 2AA=J FCE96C>@C6 244@C5:?8 E@ $C@A@D:E:@? c :7 9@=5D F?:7@C>=J @? E96 762D:3=6 D6E E96? (96@C6>D c 2?5 d H:== 2AA=J E@@ (96 F?:7@C> 7F=L==>6?E @7 :? E96 762D:3=6 D6E H:== ?@E 9@=5 :? 86?6C2= "6G6CE96=6DD E96 D@=G23:=:EJ @7 H:== @7E6? 72:= 2D H6== 7 2AAC@AC:2E6 4@?DEC2:?ED 2C6 25565 :? D@ 2D E@ C64@G6C D@=G23:=:EJ 7@C :?DE2?46 :7 D@>6 3@F?5D 7@C E96 FDF2= 6=E2 @C @E96C C66<D 2C6 :>A@D65 E96? E96 F?:7@C> 4@?G6C86?46 @7 H:== 36 AC@G65 H:E9 D:>:=2C 2C8F>6?ED E@ E9@D6 FD65 :? I2>A=6 +:E9 C6DA64E E@ (96@C6> 2D 2=C625J 5@?6 :?

(46) 7@C $C@3=6> 'JDE6> :D 62D:=J 252AE65 E@ $C@3=6> D 2=C625J D2:5 E9:D D64E:@? @?=J 92D :==FDEC2E:G6 AFCA@D6D 2?5 H6 H:== ?@E AC6D6?E 2 AC@7@F?5 2?2=JD:D @7 3642FD6 :E H@F=5 D:8?:L42?E=J 6?=2C86 E96 A2A6C 4@?E6?E. '&$,*"'&. (96 @AE:>:K2E:@? @7 V aR :D DE:== G6CJ :>A@CE2?E :? L?2?46 2?5 :?DFC2?46 2>@?8 >2?J @E96C L6=5D (9@F89 E96C6 2C6 2=E6C?2E:G6 C:D< >62DFC6D H:E9 G2=F23=6 AC@A6CE:6D D6G6C2=. .

(47) authors have justi…ed the usefulness of V aR in many applications. The optimization of V aR is much more complicated than the optimization of other risk measures. Since V aR is neither convex nor di¤erentiable, the standard methods of mathematical programming are frequently di¢ cult to apply. There are many and quite di¤erent approaches addressing the optimization of V aR. All of them yield interesting algorithms or optimality conditions, but non of them solves the problem in an exhaustive manner. There are many cases which cannot be treated with the existent methodologies. This paper has proved that a V aR approximation may be given with a linear combination of two CV aRs with di¤erent con…dence level. More accurately, V aR is a CV aR derivative, and therefore it is the limit of a sequence of linear combination of CV aRs with di¤erent con…dence level. This property has been used in order to provide new methods to optimize both V aR and linear combinations of CV aRs in general probability spaces. Applications in …nance (optimal investment) and insurance (optimal reinsurance) have been given. They show the practical e¤ectiveness of the provided new methodologies. Acknowledgments. This research was partially supported by “Ministerio de Economía” (Spain, Grant ECO2012. 39031. C02. 01). The usual caveat applies.. References [1] Annaert, J., S. Van Osselaer and B. Verstraete, 2009. Performance evaluation of portfolio insurance strategies using stochastic dominance criteria. Journal of Banking & Finance, 33, 272-280. [2] Artzner, P., F. Delbaen, J.M. Eber and D. Heath, 1999. Coherent measures of risk. Mathematical Finance, 9, 203-228. [3] Assa, H., 2015. Trade-o¤ between robust risk measurement and market principles. Journal of Optimization Theory and Applications, 166, 306-320. [4] Assa, H. and K.M. Karai, 2013. Hedging, Pareto optimality and good deals. Journal of Optimization Theory and Applications, 157, 900-917.. 21.

(48) [5] Aumann, R.J. and R. Serrano, 2008. An economic index of riskiness. Journal of Political Economy, 116, 810-836. [6] Balbás, A., B. Balbás and R. Balbás, 2010a. Minimizing measures of risk by saddle point conditions. Journal of Computational and Applied Mathematics, 234, 2924-2931. [7] Balbás, A., B. Balbás and R. Balbás, 2010b. CAP M and AP T like models with risk measures. Journal of Banking & Finance, 34, 1166–1174. [8] Balbás, A., B. Balbás and R. Balbás, 2016a. Good deals and benchmarks in robust portfolio selection. European Journal of Operational Research, 250, 666 - 678. [9] Balbás, A., Balbás, B. and R. Balbás, 2016b. Outperforming benchmarks with their derivatives: Theory and empirical evidence. The Journal of Risk, forthcoming. [10] Balbás, A., B. Balbás, R. Balbás and A. Heras, 2015. Optimal reinsurance under risk and uncertainty. Insurance: Mathematics and Economics, 60, 61 - 74. [11] Basak, S. and A. Shapiro, 2001. Value at risk based risk management. Review of Financial Studies, 14, 371-405. [12] Bali, T.G., N. Cakici and F. Chabi-Yo, 2011. A generalized measure of riskiness. Management Science, 57, 8, 1406-1423. [13] Cai, J. and K.S. Tan, 2007. Optimal retention for a stop loss reinsurance under the V aR and CT E risk measures. ASTIN Bulletin, 37, 1, 93-112. [14] Chavez-Demoulin, V., P. Embrechts and J. Neslehová, 2006. Quantitative models for operational risk: Extremes, dependence and aggregation. Journal of Banking & Finance, 30, 2635–2658. [15] Chi, Y. and K.S. Tan, 2013. Optimal reinsurance with general premium principles. Insurance: Mathematics and Economics, 52, 180-189. [16] Dhaene, J., R.J. Laeven, S. Vandu¤el, G. Darkiewicz and M.J. Goovaerts, 2008. Can a coherent risk measure be too subadditive? Journal of Risk and Insurance, 75, 365-386. [17] Dupacová, J. and M. Kopa, 2014. Robustness of optimal portfolios under risk and stochastic dominance constraints. European Journal of Operational Research, 234, 434 - 441. 22.

(49) [18] Foster, D.P. and S. Hart, 2009. An operational measure of riskiness. Journal of Political Economy, 117, 785-814. [19] Gaivoronski, A. and G. P‡ug, 2005. Value at risk in portfolio optimization: Properties and computational approach. The Journal of Risk, 7,2, 1 - 31. [20] Goovaerts, M.J. and R. Laeven, 2008. Actuarial risk measures for …nancial derivative pricing. Insurance: Mathematics and Economics, 42, 540-547. [21] Goovaerts, M.J., R. Kaas, J. Dhaene and Q. Tang, 2004. A new classes of consistent risk measures. Insurance: Mathematics and Economics, 34, 505-516. [22] Kaluszka, M., 2005. Optimal reinsurance under convex principles of premium calculation. Insurance: Mathematics and Economics, 36, 375 - 398. [23] Kelly, J.L., 1955. General topology. Springer. [24] Kopp, P.E., 1984, Martingales and stochastic integrals. Cambridge University Press. [25] Larsen, N., H. Mausser and S. Uryasev, 2002. Algorithms for optimization of valueat-risk. In: Pardalos P, Tsitsiringos V (eds) Financial Engineering, e-Commerce and Supply Chain, Kluwer Academic Publishers, Dordrecht, Netherlands, 129–157. [26] Mitra, S., A. Karathanasopoulos, G. Sermpinis, C. Dunis and J. Hood, 2015. Operational risk: Emerging markets, sectors and measurement. European Journal of Operational Research, 241, 122-132. [27] Rockafellar, R.T. and S. Uryasev, 2000. Optimization of conditional-value-at-risk. The Journal of Risk, 2, 21 –42. [28] Rockafellar, R.T., S. Uryasev and M. Zabarankin, 2006. Generalized deviations in risk analysis. Finance & Stochastics, 10, 51-74. [29] Rudin, W., 1973. Functional analysis. McGraw-Hill. [30] Rudin, W., 1987. Real and complex analysis. Third Edition. McGraw-Hill, Inc. [31] Ruszczynski, A. and A. Shapiro, 2006. Optimization of convex risk functions. Mathematics of Operations Research, 31, 3, 433-452.. 23.

(50) [32] Shaw, W.T., 2011. Risk, VaR, CVaR and their associated portfolio optimizations when Asset returns have a multivariate Student T distribution. Available at SSRN: http://ssrn.com/abstract=1772731 or http://dx.doi.org/10.2139/ssrn.1772731. [33] Wozabal, D., 2012. Value-at-Risk optimization using the di¤erence of convex algorithm. OR Spectrum, 34, 861-883. [34] Wozabal, D., R. Hochreiter and G. P‡ug, 2010. A D.C. formulation of value-at-risk constrained optimization. Optimization, 59, 377–400. [35] Zajdenwebe, D., 1996. Extreme values in business interruption insurance. Journal of Risk and Insurance, 63, 95-110. [36] Zhao, P. and Q. Xiao, 2016. Portfolio selection problem with Value-at-Risk constraints under non-extensive statistical mechanics. Journal of Computational and Applied Mathematics, 298, 74-91. [37] Zhuang, S.C., C. Weng, K.S. Tan and H. Assa, 2016. Marginal indemni…cation function formulation for optimal reinsurance. Insurance: Mathematics and Economics, 67, 6576.. 24.

(51)

No results found