Practical Financial Optimization: Decision Making for Financial Engineers

Notation

Sets and Indices

U = {1, 2, …, n} index set of available financial instruments or asset classes.

T = {0, 1, …, τ,… T} set of time periods, from today (0) until maturity (T). Unless stated otherwise in the text all time periods are of equal duration which is typically taken to be one month.

K = {1, 2, …, κ, …, K} index set of risk factors.

∑_t = {1, 2, …, S_t} index set of states at period t.

Ω = {1, 2, …, N} index set of scenarios.

i index of instrument or asset class from the set U.

t index of time periods from the set T.

j index of risk factor from the set K.

l index of scenario from the set Ω.

Variables and Parameters

x n-dimensional vector of investments in assets, with elements x_i. The units are in percentages of the total asset value or amounts in face value; the choice of units depends on the model and is made clear in the text.

b₀ n-dimensional vector of initial holdings in assets, with elements b_0i.

cash invested in short-term deposits at period t.

cash borrowed at short-term rates at period t.

ν₀ initial holdings in risk-free asset (cash).

p^l statistical probability assigned to scenario l.

n-dimensional random vector of asset returns, with elements .

n-dimensional ...