Regulation, Div ersit y and Arbitrage Winslo w Strong Ph.D. Student at UC Santa Ba rba ra A dviso r: Jean-Pierre F ouque Third W estern Conferen e in Mathemati al Finan e San ta Barbara, California No v em b er 14, 2009 1
Ba kground R. F ernholz [F er99 , F er02 ℄: diversit y and equivalent ma rtingale measures (EMMs) a re in ompatible. His mo del: sto ks a re Ito p ro esses ( ⇒ ontinuous), volatilit y is b ounded, ontinuous trading, no transa tion osts, no dividends, numb er of ompanies ➡ is onstant. Under these assumptions [FKK05℄ the only w a y diversit y an b e maintained is ➡ fo r the drifts to b e ome unb oundedly negative as sto ks b e ome la rge. Motivating question: an diversit y and no a rbitrage o exist if diversit y is maintained b y a w ealth- onserving redistribution of apital amongst ➡ ompanies? ➡ 4
Mo del Overview Sta rt with a strongly Ma rk ovian sto k mo del. Here w e only onsider solutions to SDEs. Regulation is imp osed as a deterministi p ro edure o urring at the random time when relative apitalizations exit a p ermissible region. ➡ A regulato ry event redistributes apital amongst ompanies. T otal ma rk et value is onserved. ➡ The sto k p ro ess fo rgets the past at regulation and its dynami s a re ompletely determined b y the p ost-regulation sta rting p oint. ➡ P o rtfolio ash �o ws a re p rop o rtional to sto k ash �o ws at a regulation event. Thus p o rtfolio value is onstant up on regulation even though sto ks jump. ➡ This assumption is designed to mimi equit y �o ws when a ompany is b rok en up into smaller pa rts (e.g. Bell A tlanti 1984). ➡ 5 ➡
The Unregulated (Pre)Mo del Consider a ma rk et mo del (Ω , F , F = {F t } t ≥ 0 , P, W, X ) whi h is the unique strong solution to the SDE living in the p ositive o rthant a.s. ∀ t ≥ 0 . W is a d -dimensional Bro wnian motion, d ≥ n ≥ 2 , and F is the ompleted Bro wnian �ltration. dX i,t = X i,t ( b ( X t ) dt + σ ( X t ) dW t ) , X 0 = x, 1 ≤ i ≤ n There is a money ma rk et a ount, B , and furthermo re w e assume fo r simpli it y that B t = 1 , o rresp onding to a risk-free rate of interest r ≡ 0 . W e require that the volatilit y matrix, σ ( x ) ∈ R n × d , have full rank ( n ), ➡ ∀ x ∈ Υ . ∀ t ≥ 0 6 ➡
Assume: T rading ma y o ur in ontinuous time. Sto ks pa y no dividends There a re no transa tion osts. ➡ Ma rk et apitalization p ro ess: M . Ma rk et w eight p ro ess: µ . ➥ ➥ ➥ ➡ n µ i,t := X i,t � M t := X i,t , M t i =1 X t ∈ Υ := { ( x 1 , . . . , x n ) ∈ R n | x 1 > 0 , . . . , x n > 0 } ∀ t ≥ 0 7 � n � ( π 1 , . . . , π n ) ∈ R n | π 1 > 0 , . . . , π n > 0 , � µ t ∈ ∆ n + := π i = 1 ∀ t ≥ 0 i
Regulation Pro edure Con�ne ma rk et w eights to U µ b y redistribution of apital amongst the sto ks via a deterministi mapping, R µ , up on exit from U µ . T otal apital is onserved. De�nition 1. A regulation rule , R µ , with resp e t to the op en, nonempt y set, U µ ⊂ ∆ n , is a Bo rel fun tion This indu es + R µ : ∆ n + \ U µ → U µ U x := µ − 1 ( U µ ) = { x ∈ Υ | µ ( x ) ∈ U µ } R x : Υ \ U x → U x 8 n � R x ( x ) := R µ ( µ ( x )) x i i =1
Either of ( U µ , R µ ) o r ( U x , R x ) determine the same regulation rule, so w e often refer to it as ( U, R ) . is a oni region, i.e. x ∈ U x ⇒ λx ∈ U x , ∀ λ > 0 , allo wing any total ma rk et value, M , fo r a given µ ∈ U µ . ➡ The regulation rule is �rst applied at the exit and stopping time ➡ U x ➡ After ς the regulated ma rk et mo del �resets� as if sta rting fresh from initial p oint R x ( X ς ) until exit from U x again. ∈ U µ } = inf { t > 0 | X t / ∈ U x } ς := inf { t > 0 | µ ( X t ) / Applying this p ro edure indu tively de�nes the la w of the regulated sto k p ri e p ro ess on sto hasti intervals via referen e to the p remo del la w. ➡ 9 ➡
Regulated Ma rk et Mo del By indu tion de�ne the follo wing k ≥ 2 , on { τ k − 1 < ∞} , W 1 := W, X 1 = X, t > 0 | X 1 ∈ U x � � τ 0 = 0 , τ 1 := ς 1 := inf t / W k t := W τ k − 1 + t − W τ k − 1 , ∀ t ≥ 0 dX k i,t = X k b ( X k t ) dt + σ ( X k t ) dW k X k 0 = R x ( X k − 1 � � , 1 ≤ i ≤ n, ς k − 1 ) i,t t k is de�ned on { τ k − 1 < ∞} as the unique strong solution to the SDE ab ove. � t > 0 | X k − 1 ∈ U x � � ς k := inf / , τ k := ς j t j =1 10 X k
Explosions? There is a p ossibilit y of explosion, i.e. of lim k →∞ τ k < ∞ . T o ha ra terize this p ossibilit y de�ne the follo wing p ro esses and va riables The event { N ∞ = k } o rresp onds to exa tly k exits o urring eventually in ∞ whi h ase no further regulation is needed after the k th, and τ k +1 = ∞ . � N t := 1 { t>τ k } ∈ F t , N ∞ := lim t →∞ N t k =1 on { N ∞ < ∞} on { N ∞ = ∞} � ∞ 11 τ ∞ := lim k →∞ τ k
Regulated Sto k Pri e Pro ess De�nition 2. F o r regulation rule ( U, R ) and initial p oint y 0 ∈ U x , the regulated sto k p ri e p ro ess is de�ned as (1) a.s. ∞ � Y t ( ω ) := X 1 1 ( τ k − 1 ,τ k ] ( ω, t ) X k 0 1 { 0 } ( t ) + t − τ k − 1 ( ω ) , ( ω, t ) ∈ [0 , τ ∞ ) . k =1 If P ( τ ∞ = ∞ ) = 1 then w e all the triple ( y 0 , U, R ) viable . Y 0 = X 1 0 = y 0 12
P o rtfolios in the Regulated Ma rk et P o rtfolio values a re una�e ted b y a regulation event, mimi king a sto k split. W e w ant to re over the useful to ol of rep resenting the apital gains p ro ess as a sto hasti integral. ➡ De�ne an e�e tive sto k p ro ess, ˆ , re�e ting only the non-regulato ry movements of Y . Re alling that Y τ + on { τ k < ∞} , ➡ (2) ➡ Y τ k ) = X k +1 k = R x ( Y k 0 (3) N t ˆ � Y t := Y t − ( Y τ + k − Y τ k ) k =1 13 ∞ � = X 1 ( X k (0 ∨ ( t − τ k − 1 )) ∧ ς k − X k 0 + 0 ) k =1
De�nition 3. A dmissible trading strategies in the regulated mo del a re p redi table p ro esses H su h that 1. H is ˆ -integrable, that is, the sto hasti integral is w ell-de�ned in the sense of sto hasti integration theo ry fo r semima rtingales. 2. There is a onstant, K , not dep ending on t su h that Y � t H · ˆ Y = ( H · ˆ 0 H s d ˆ Y ) t ≥ 0 := ( Y s ) t ≥ 0 a.s., ∀ t ≥ 0 De�nition 4. A self-�nan ing w ealth p ro esses in the regulated mo del is any whi h satis�es: ( H · ˆ Y ) t ≥ − K, V H 14 0 + ( H · ˆ V H = V H Y ) t ∀ t ≥ 0 . t
EMMs in the Regulated Mo del Assume that ( y 0 , U, R ) is viable, that is τ ∞ = ∞ a.s. W e assumed that σ has full rank ( n ) so there exists a ma rk et p ri e of risk, θ = σ ′ . When a.s. ∀ T > 0 t ( σ t σ ′ t ) − 1 b t then w e ma y de�ne the lo al ma rtingale and sup erma rtingale, � T | θ ( Y t ) | 2 dt < ∞ 0 �� t � t � �� θ ( Y s ) dW s + 1 | θ ( Y s ) | 2 ds 15 Z t := E ( − ( θ ( Y ) · W )) t = exp − 2 0 0
Prop osition 1. If Z is a ma rtingale then the measure Q generated from is a lo al ma rtingale measure fo r ˆ on ho rizon [0 , T ] . The usual to ols, e.g. the Kazamaki and Novik ov riteria p rovide su� ient (although not ne essa ry) onditions fo r Z to b e a ma rtingale. dQ dP := Z T Y ➡ 16
Diversit y De�nition 5. A ma rk et mo del is diverse on ho rizon T if there exists su h that max 1 ≤ i ≤ n { µ i,t } < 1 − δ, ∀ t : 0 ≤ t ≤ T a.s. A ma rk et mo del is w eakly diverse on ho rizon T if there exists δ ∈ (0 , 1) su h that a.s. δ ∈ (0 , 1) The regulato ry p ro edure on�nes the ma rk et w eights to ¯ , so it is easy to � T engineer diverse regulated 1 ma rk ets. 1 ≤ i ≤ n { µ i,t } dt < 1 − δ max T 0 F o r example, �x any δ < n − 1 and let U µ ➡ (4) ➡ n 17 U µ = { ν ∈ ∆ n + : ν i < 1 − δ, 1 ≤ i ≤ n }
Regulated V olatilit y-Stabi li ze d Ma rk ets A non diverse ma rk et admitting relative a rbitrage with resp e t to the ma rk et p o rtfolio [FB08 ℄. fo r any onstant α ≥ 0 with µ i,t = X i,t /M t , M t = � n . This implies � � � 1 + α 1 dX i,t = X i,t dt + dW i,t , 1 ≤ i ≤ n 2 µ i,t µ i,t and i =1 X i,t (5) � b i,t = 1 + α 1 θ ν,t = 1 + α This system has a w eak solution that is unique in la w [BP02 ℄. , σ i,ν,t = δ iν , r t = 0 1 ≤ ν, i ≤ n 2 √ µ ν,t 2 µ i,t µ i,t 18
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