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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
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
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
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
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
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