Optimization in Financial Engineering

Yuriy ZinchenkoDepartment of Mathematics and Statistics

University of Calgary

December 02, 2009

Why?

Objective has never been so clear:– maximize

Nobel prize winners:– L. Kantorovich

linear optimization

– H. Markowitz “Efficient Portfolio”, foundations of modern

Capital Asset Pricing theory

Talk layout

(Convex) optimization Portfolio optimization

– mean-variance model– risk measures– possible extensions

Securities pricing– non-parametric estimates– moment problem and duality– possible extensions

Optimization

Optimization

convex set

convex optimization

}],{[},{: SyxSyxS n

SxxcT

x:sup

S

x

y

c

Optimization

prototypical optimization problem –

Linear Programming (LP)

– any convex set admits “hyperplane representation”

),,( :max nmbAbAxxc mnmT

x

SAx ≤ b

x1

x2

c

Optimization

LP duality– re-write LP

as

and introduce

– optimal values satisfy weak duality:

– since

strong duality:

bAxxcT

x:max

(D) ,:max mT

xsbsAxxc

(P) ,:min mTT

zzczAzb

(D)alal(P) *vv xcszAxzsAxzbz TTTTT )(

(D)alal(P) *vv

Optimization

conic generalizations

where K is a closed convex cone, K* – its dual

– strong duality frequently holds and always

w.l.o.g. any convex optimization problem is conic

(D) ,:max *KsbsAxxcT

x

(P) ,:min KzczAzb TT

z

)(alal(P) * Dvv

(D) ,:max mT

xsbsAxxc

(P) ,:min mTT

zzczAzb

Optimization

conic optimization instances– LP:– Second Order Conic Programming (SOCP):

– Positive Semi-Definite Programming (SDP):

powerful solution methods and software exists– can solve problems with hundreds of thousands constraints

and variables; treat as black-box

mKK *(P) ,:min KzczAzb TT

z

||}||:),{( 1* xtxtKK m

}0:{closure* XSXKK k

Portfolio optimization

Mean-variance model

Markowitz model– minimize variance

– meet minimum return

– invest all funds

– no short-selling

where Q is asset covariance matrix,

r – vector of expected returns from each asset

0

,11

,

:min

min

x

x

rxr

Qxx

T

T

T

x

0

,11

,

,

:min

min

),(

x

x

rxr

tQxx

t

T

T

T

tx

SzzcT

x:sup

Markowitz model– explicit analytic solution given rmin

– interested in “efficient frontier” set of non-dominated portfolios

can be shown to be a “convex set”

Mean-variance model

Expectedreturn

A

Standard deviation

?B

Risk measures

mean-variance model minimizes variance– variance is indifferent to both up/down risks

coherent risk measures:– “portfolio” = “random loss” – given two portfolios X and Y, is coherent if

(X+Y) (X) + (Y) “diversification is good” (t X) = t (X) “no scaling effect” (X) (Y) if X Y a.s. “measure reflects risk” (X + ) = (X) - “risk-free assets reduce risk”

Risk measures

VaR (not coherent):– “maximum loss for a given confidence 1-”

CVaR (coherent):– “maximum expected loss for a given confidence 1-”– CVaR may be approximated using LP,

so may consider

}1)(:inf{ xXPx

)](VaR|[ XXE

0,11,:);(CVaRmin min xxrxrx TT

x

Probabilitydensity

Loss X

Possible extensions

risk vs. return models:

– portfolio granularity likely to have contributions from nearly all assets

– robustness to errors or variation in initial data Q and r are estimated

0

,11

,

:)""other (or min

min

x

x

rxr

riskQxx

T

T

T

x

Securities pricing

Non-parametric estimates

European call option:– “at a fixed future time may purchase a stock X at price k”– present option value (with 0 risk-free rate)

know moments of X; to bound option price consider

)],0[max( kXE

0)( ,1)(

... ,][ ,][

:)],0[max( min/max

0

2

xdxx

XVarXE

kXE

Moment problem and duality

option pricing relates to moment problem– given moments, find measure

intuitively, the more moments more definite answer semi-formally, substantiate by moment-generating function extreme example: X supported on {0,1}, let

– E[X]=1/2,– E[X2]=1/2,…

note objective and

constraints linear w.r.t. – duality?

0)( ,1)(

... ,][ ,][

:)],0[max( min/max

0

2

xdxx

XVarXE

kXE

Moment problem and duality

duality indeed (in fact, strong!)– constraints A() is linear transform

look for adjoint A*(), etc.

0)( ,,...,1,0 ,)(][

:)]),0[max((min

),0(

xnimdxxxXE

kXE

iii

Na

aa

Naaa

xkxxy

ym

,...,0

,...,0

),,0max(

:min

0)( ,,...,1,0 ,)(][

:)],0[max(max

),0(

xnimdxxxXE

kXE

iii

(D) ,:max *KsbsAxxcT

x

(P) ,:min KzczAzb TT

z

Na

aa

Naaa

xkxxy

ym

,...,0

,...,0

),,0max(

:max

Moment problem and duality

duality indeed (in fact, strong!)– constraints of the dual problem: p (x) ≥ 0, p – polynomial

– nonnegative polynomial SOS SDP representable

xxp ,0)(

Na

aa

Naaa

xkxxy

ym

,...,0

,...,0

),,0max(

:min

j i

ij

jj

ji

i txxacxtxxp 22222 )()()()( 0),,...,,,1(),...,,,1()( 2/22/2 QxxxQxxxxp NTN 0),,...,,,1(),...,,,1()( 2/22/2 TNTTN LLQxxxLLxxxxp

Moment problem and duality

due to well-understood dual, may solve efficiently

– and so, find bounds on the option price

0)( ,1)(

... ,][ ,][

:)],0[max( min/max

0

2

xdxx

XVarXE

kXE

Possible extensions

– exotic options– pricing correlated/dependent securities– moments of risk neutral measure given securities– sensitivity analysis on moment information

Few selected references

References

Portfolio optimization– (!) SAS Global Forum: Risk-based portfolio optimization using

SAS, 2009– J. Palmquist, S. Uryasev, P. Krokhmal: Portfolio optimization with

Conditional Value-at-Risk objective and constraints, 2001– S. Alexander, T. Coleman, Y. Li: Minimizing CVaR and VaR for a

portfolio of derivatives, 2005 Option pricing

– D. Bertsimas, I. Popescu: On the relation between option and stock prices : a convex optimization approach, 1999

– J. Lasserre, T. Prieto-Rumeau, M. Zervos: Pricing a Class of Exotic Options Via Moments and SDP Relaxations, 2006

Thank you