[PPT] - Regulation, Div ersit y and Arbitrage Winslo w Strong Ph.D. PowerPoint Presentation

SLIDE 1 Regulation, Div ersit y and Arbitrage Winslo w Strong Ph.D. Student at UC Santa Ba rba ra A dviso r: Jean-Pierre F

uque

Third W estern Conferen e in Mathemati al Finan e San ta Barbara, California No v em b er 14, 2009 1

SLIDE 2 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 (⇒

ntinuous),

volatilit y is b

unded,
ntinuous

trading, no transa tion

sts,

no dividends, numb er

f
mpanies

is

nstant.

➡

Under these assumptions [FKK05℄ the

nly

w a y diversit y an b e maintained is fo r the drifts to b e ome unb

undedly

negative as sto ks b e ome la rge.

➡

Motivating question: an diversit y and no a rbitrage

exist

if diversit y is maintained b y a w ealth- onserving redistribution

f

apital amongst

mpanies?

4

SLIDE 3 Mo del Overview

➡

Sta rt with a strongly Ma rk

vian

sto k mo del. Here w e

nly
nsider

solutions to SDEs.

➡

Regulation is imp

sed

as a deterministi p ro edure

urring

at the random time when relative apitalizations exit a p ermissible region.

➡

A regulato ry event redistributes apital amongst

mpanies.

T

tal

ma rk et value is

nserved.

➡

The sto k p ro ess fo rgets the past at regulation and its dynami s a re

mpletely

determined b y the p

st-regulation

sta rting p

int.

➡

P

rtfolio

ash

ws

a re p rop

rtional

to sto k ash

ws

at a regulation event. Thus p

rtfolio

value is

nstant

up

n

regulation even though sto ks jump.

➡

This assumption is designed to mimi equit y

ws

when a

mpany

is b rok en up into smaller pa rts (e.g. Bell A tlanti 1984). 5

SLIDE 4 The Unregulated (Pre)Mo del Consider a ma rk et mo del (Ω, F, F = {Ft}t≥0, P, W, X) whi h is the unique strong solution to the SDE

dXi,t = Xi,t (b(Xt)dt + σ(Xt)dWt) , X0 = x, 1 ≤ i ≤ n

living in the p

sitive
rthant

a.s. ∀t ≥ 0. W is a d -dimensional Bro wnian motion, d ≥ n ≥ 2, and F is the

mpleted

Bro wnian ltration.

➡

There is a money ma rk et a ount, B , and furthermo re w e assume fo r simpli it y that Bt = 1,

∀t ≥ 0

rresp
nding

to a risk-free rate

f

interest

r ≡ 0. ➡

W e require that the volatilit y matrix, σ(x) ∈ Rn×d , have full rank (n ),

∀x ∈ Υ .

6

SLIDE 5

➡

Assume:

➥

T rading ma y

ur

in

ntinuous

time.

➥

Sto ks pa y no dividends

➥

There a re no transa tion

sts.

➡

Ma rk et apitalization p ro ess: M . Ma rk et w eight p ro ess: µ .

Mt :=

n

i=1

Xi,t, µi,t := Xi,t Mt Xt ∈ Υ := {(x1, . . . , xn) ∈ Rn | x1 > 0, . . . , xn > 0} ∀t ≥ 0 µt ∈ ∆n

+ :=

(π1, . . . , πn) ∈ Rn | π1 > 0, . . . , πn > 0,

n

i

πi = 1

∀t ≥ 0

7

SLIDE 6 Regulation Pro edure Conne ma rk et w eights to U µ b y redistribution

f

apital amongst the sto ks via a deterministi mapping, Rµ , up

n

exit from U µ . T

tal

apital is

nserved.

Denition 1. A regulation rule, Rµ , with resp e t to the

p

en, nonempt y set, U µ ⊂ ∆n

+

, is a Bo rel fun tion

Rµ : ∆n

+ \ U µ → U µ

This indu es

U x := µ−1(U µ) = {x ∈ Υ | µ(x) ∈ U µ} Rx : Υ \ U x → U x Rx(x) := Rµ(µ(x))

n

i=1

xi

8

SLIDE 7

➡

Either

f (U µ, Rµ)
r (U x, Rx)

determine the same regulation rule, so w e

ften

refer to it as (U, R) .

➡ U x

is a

ni

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

ς := inf {t > 0 | µ(Xt) / ∈ U µ} = inf {t > 0 | Xt / ∈ U x} ➡

After ς the regulated ma rk et mo del resets as if sta rting fresh from initial p

int Rx(Xς)

until exit from U x again.

➡

Applying this p ro edure indu tively denes the la w

f

the regulated sto k p ri e p ro ess

n

sto hasti intervals via referen e to the p remo del la w. 9

SLIDE 8 Regulated Ma rk et Mo del

τ0 = 0, W 1 := W, X1 = X, τ1 := ς1 := inf

t > 0 | X1

t /

∈ U x

By indu tion dene the follo wing k ≥ 2,

n {τk−1 < ∞},

W k

t := Wτk−1+t − Wτk−1,

∀t ≥ 0 dXk

i,t = Xk i,t

b(Xk

t )dt + σ(Xk t )dW k t

,

1 ≤ i ≤ n, Xk

0 = Rx(Xk−1 ςk−1)

ςk := inf

t > 0 | Xk−1

t

/ ∈ U x , τk :=

k

j=1

ςj Xk

is dened

n {τk−1 < ∞}

as the unique strong solution to the SDE ab

ve.

10

SLIDE 9 Explosions? There is a p

ssibilit

y

f

explosion, i.e.

f limk→∞ τk < ∞

. T

ha

ra terize this p

ssibilit

y dene the follo wing p ro esses and va riables

Nt :=

∞

k=1

1{t>τk} ∈ Ft, N∞ := lim

t→∞ Nt

The event {N∞ = k}

rresp
nds

to exa tly k exits

urring

eventually in whi h ase no further regulation is needed after the k th, and τk+1 = ∞ .

τ∞ :=

∞
n

{N∞ < ∞}

limk→∞ τk

n

{N∞ = ∞} 11

SLIDE 10 Regulated Sto k Pri e Pro ess Denition 2. F

r

regulation rule (U, R) and initial p

int y0 ∈ U x

, the regulated sto k p ri e p ro ess is dened as

Yt(ω) := X1

01{0}(t) + ∞

k=1

1(τk−1,τk](ω, t)Xk

t−τk−1(ω),

(ω, t) ∈ [0, τ∞).

(1)

Y0 = X1

0 = y0

a.s. If P(τ∞ = ∞) = 1 then w e all the triple (y0, U, R) viable. 12

SLIDE 11 P

rtfolios

in the Regulated Ma rk et

➡

P

rtfolio

values a re unae ted b y a regulation event, mimi king a sto k split.

➡

W e w ant to re over the useful to

l
f

rep resenting the apital gains p ro ess as a sto hasti integral.

➡

Dene an ee tive sto k p ro ess, ˆ

Y

, ree ting

nly

the non-regulato ry movements

f Y

. Re alling that Yτ+

k = Rx(Y k

τk) = Xk+1

n {τk < ∞},

ˆ Yt := Yt −

Nt

k=1

(Yτ+

k − Yτk)

(2)

= X1

0 + ∞

k=1

(Xk

(0∨(t−τk−1))∧ςk − Xk 0)

(3) 13

SLIDE 12 Denition 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

Y

integrable,

that is, the sto hasti integral

H · ˆ Y = (H · ˆ Y )t≥0 := ( t

0 Hsd ˆ

Ys)t≥0

is w ell-dened in the sense

f

sto hasti integration theo ry fo r semima rtingales. 2. There is a

nstant, K

, not dep ending

n t

su h that

(H · ˆ Y )t ≥ −K,

a.s., ∀t ≥ 0 Denition 4. A self-nan ing w ealth p ro esses in the regulated mo del is any

V H

whi h satises:

V H

t

= V H

0 + (H · ˆ

Y )t ∀t ≥ 0.

14

SLIDE 13 EMMs in the Regulated Mo del Assume that (y0, 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

f

risk, θ = σ′

t(σtσ′ t)−1bt

. When

T |θ(Yt)|2dt < ∞

a.s. ∀T > 0 then w e ma y dene the lo al ma rtingale and sup erma rtingale,

Zt := E(−(θ(Y ) · W))t = exp

−

t θ(Ys)dWs + 1 2 t |θ(Ys)|2ds

15

SLIDE 14 Prop

sition

1. If Z is a ma rtingale then the measure Q generated from

dQ dP := ZT

is a lo al ma rtingale measure fo r ˆ

Y

n

ho rizon [0, T] .

➡

The usual to

ls,

e.g. the Kazamaki and Novik

v

riteria p rovide su ient (although not ne essa ry)

nditions

fo r Z to b e a ma rtingale. 16

SLIDE 15 Diversit y Denition 5. A ma rk et mo del is diverse

n

ho rizon T if there exists

δ ∈ (0, 1)

su h that max1≤i≤n{µi,t} < 1 − δ, ∀t : 0 ≤ t ≤ T a.s. A ma rk et mo del is w eakly diverse

n

ho rizon T if there exists δ ∈ (0, 1) su h that

1 T T max

1≤i≤n{µi,t}dt < 1 − δ

a.s.

➡

The regulato ry p ro edure

nnes

the ma rk et w eights to ¯

U µ

, so it is easy to engineer diverse regulated ma rk ets.

➡

F

r

example, x any δ < n−1

n

and let

U µ = {ν ∈ ∆n

+ : νi < 1 − δ, 1 ≤ i ≤ n}

(4) 17

SLIDE 16 Regulated V

latilit

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

rtfolio

[FB08 ℄.

dXi,t = Xi,t

1 + α

2µi,t dt +

1

µi,t dWi,t

,

1 ≤ i ≤ n

fo r any

nstant α ≥ 0

with µi,t = Xi,t/Mt , Mt = n

i=1 Xi,t

. This implies

bi,t = 1 + α 2µi,t , σi,ν,t = δiν

1

µi,t , rt = 0

and

θν,t = 1 + α 2√µν,t 1 ≤ ν, i ≤ n

(5) This system has a w eak solution that is unique in la w [BP02 ℄. 18

SLIDE 17 Sin e w e do not have strong existen e

r

uniqueness, the

nstru tion

p resented herein an't b e used. Nevertheless b y a mo re general

nstru tion,

there exists

(Ω, F, F, P, W, {Xk}∞

1 )

satisfying the ne essa ry p rop erties to dene regulated sto k p ro ess Y as b efo re. Fixing some ε ∈ (0, 1

n) ,

and some 0 < δ < (1 − ε) ∧ n−1

n

w e ho

se

U µ := {µ : ε < µi < 1 − δ,

n

i=1

µi = 1} ⊂ ∆n

+

Rµ(µ) = µ0 := µ(y0), ∀µ ∈ ∆n

+ \ U µ

This simple regulation rule is viable. The rule ab

ve

implies that

µt ∈ ¯ U µ, ∀t ≥ 0,

a.s. and so the regulated sto k p ri e p ro ess, Y , is diverse.

t |θ(Ys)|2ds < Ct ε , ∀t ≥ 0,

a.s. 19

SLIDE 18 This implies that the Novik

v
ndition

E

exp
1

2 T |θ(Ys)|2ds

< ∞

is satised here, and so the exp

nential

lo al ma rtingale Z is a ma rtingale and generates an ELMM

n FT

b y dQ

dP := ZT

. In fa t, Q is a ma rtingale measures here rather than merely a lo al ma rtingale measure b e ause

µi,t > ε ⇒ σi,ν,t = δiν

1

µi,t < 1 √ε.

20

SLIDE 19 Con lusions

➡

EMMs (with resp e t to ˆ

Y

) and diversit y (with resp e t to Y ) a re

mpatible

in this regulato ry mo del.

➡

The k ey

ndition

here is that the unregulated mo del b e a rbitrage free up until exit from U . In general additional regula rit y is needed.

➡

Do the

n lusions

fo r this t yp e

f

regulato ry mo del a rry

ver

to a mo del where

mpanies

a re fo r ed to split, with total ma rk et apital

nserved?

21

SLIDE 20 Referen es [BP02℄ Ri ha rd F. Bass and Edwin A. P erkins, Degenerate sto hasti dierential equations with hlder

ntinuous
e ients

and sup er-ma rk

v

hains, T ransa tions

f

the Ameri an Mathemati al So iet y 355 (2002), 373405. [FB08℄ Daniel F ernholz and A drian Banner, Sho rt-term relative a rbitrage in volatilit y-stabilized ma rk ets, Annals

f

Finan e 4 (2008), 445454. [F er99℄ Rob ert F ernholz, On the diversit y

f

equit y ma rk ets, Journal

f

Mathemati al E onomi s 31 (1999), 393417. [F er02℄ , Sto hasti p

rtfolio

theo ry, rst ed., Sp ringer, 2002. 22

SLIDE 21 [FKK05℄ Rob ert F ernholz, Ioannis Ka ratzas, and Constantinos Ka rda ras, Diversit y and relative a rbitrage in equit y ma rk ets, Finan e and Sto hasti s 9 (2005), 127. 23