Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

286 Optimizing Optimization

on the distribution of scenarios. We can, for example, include nonlinear posi-
tions as they would typically arise from call or put options. 4 This is a major
advantage relative to mean – variance optimization that cannot deal with these

instruments. Suppose VaR   20% and i wriis
n
∑ 1 ^25 %.^ We then get
e s  max[0,  20% –  25%]  5%, i.e., a 5% excess. In practice, we will
need many scenarios to approximate a continuous portfolio distribution from
a grid of discrete scenarios. We will come back to this point when we discuss
approximation error in CVaR optimization. CVaR will always be larger than
VaR (because it represents the average of losses larger than value at risk) and,
hence, we can write it as:

CVaR VaR S s es

S

α^11 ()∑ 1

(12.2)

where
1
S s 1 s

S
∑e^ denotes the average loss across all scenarios and^ α^ the prob-
ability of a loss larger than VaR (i.e., 5% for a 95% confidence level), which
leverages this average up to adjust for the fact that we are looking for a loss
conditional on being in the tail. A complete optimization problem looks like
the following linear program, where μi denotes the average return for asset i
and μ * stands for the targeted portfolio return.

max

VaR w e,, S s s

S

is

VaR^11 α()∑ 1 e

(12.3)

i wii

n

∑ 1 μμ *

(12.4)

eVaRsii wris

n

∑ 1

(12.5)

es^0 (12.6)

wi^0 (12.7)

4 Option positions need to be path independent as CVaR optimization still remains a one-period
model. We can, for example, not include the payoff from a look-back call on a stock index as
end of period option payoffs do not relate to end of period index payoffs.