Optimization in Financial Engineering

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Optimization in Financial Engineering

• Yuriy Zinchenko
• Department of Mathematics and Statistics
• University of Calgary
• December 02, 2009

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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

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Talk layout

• (Convex) optimization
• Portfolio optimization
• mean-variance model
• risk measures
• possible extensions
• Securities pricing
• non-parametric estimates
• moment problem and duality
• possible extensions

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Optimization
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Optimization

• convex set
• convex optimization

x
S
y
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Optimization

• prototypical optimization problem
• Linear Programming (LP)
• any convex set admits hyperplane representation

x2
S
Ax b
x1
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Optimization

• LP duality
• re-write LP
• as
• and introduce
• optimal values satisfy
• weak duality
• since
• strong duality

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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

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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

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Portfolio optimization
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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

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Mean-variance model

• Markowitz model
• explicit analytic solution given rmin
• interested in efficient frontier
• set of non-dominated portfolios
• can be shown to be a convex set

Expected return
Standard deviation
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Mean-variance model

• consider two uncorrelated assets

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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
• ? (XY) ? ? (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

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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
a
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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

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Securities pricing
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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

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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
• EX1/2,
• EX21/2,
• note objective and
• constraints linear w.r.t. ?
• duality?

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Moment problem and duality

• duality indeed (in fact, strong!)
• constraints A(?) is linear transform
• look for adjoint A(?), etc.

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Moment problem and duality

• duality indeed (in fact, strong!)
• constraints of the dual problem p (x) 0, p
  polynomial
• nonnegative polynomial ? SOS ? SDP representable

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Moment problem and duality

• due to well-understood dual, may solve
  efficiently
• and so, find bounds on the option price

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Possible extensions

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

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Few selected references
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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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Thank you