Optimization in Financial Engineering Yuriy Zinchenko Department 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

convex set convex optimization S x y c

Optimization prototypical optimization problem – Linear Programming (LP) – any convex set admits “hyperplane representation” S Ax ≤ b x1x1 x2x2 c

Optimization LP duality – re-write LP as and introduce – optimal values satisfy weak duality: – since strong duality:

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

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

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

Markowitz model – explicit analytic solution given r min – interested in “efficient frontier” set of non-dominated portfolios can be shown to be a “convex set” Mean-variance model Expected return A Standard deviation ?B

Mean-variance model consider two uncorrelated assets…

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 Probability density 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

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

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[X 2 ] =1/2,… note objective and constraints linear w.r.t.  – duality?

Moment problem and duality duality indeed (in fact, strong!) – constraints A(  ) is linear transform look for adjoint A * (  ), etc.

Moment problem and duality duality indeed (in fact, strong!) – constraints of the dual problem: p (x) ≥ 0, p – polynomial – nonnegative polynomial  SOS  SDP representable

Moment problem and duality due to well-understood dual, may solve efficiently – and so, find bounds on the option price

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