CASE STUDY: Hedging Portfolio of Options with Expectile (XVaR)

PROBLEM 1: problem_cs_hedging_portfolio_of_options_with_XVaR
Minimize XVaR(x) (minimizing hedging error)
subject to
P(x) ≤ Const (constraint on initial portfolio value)
Box constraints (non-negativity constraints on positions)
——————————————————————–
XVaR = Expectile of residual
P(x) = initial price of the hedging portfolio (linear function)
Box constraints = constraints on individual decision variables
——————————————————————–

 
parameter values: q in range (0.5, 1.0)
 
PROBLEM 2: problem_equivalence_of_CVaR_and_XVaR_portfolio
Minimize XVaR(x) (minimizing expectile risk)
subject to
avg_g(x) ≥ 0 (nonnegative portfolio return constraint)
linear(x) = 1 (portfolio budget constraint)
Box constraints (non-negativity constraints on positions)

Calculate pr_pen(x) (calculate confidence level alpha corresponding to q)
Calculate CVaR(x) (calculate CVaR at expectile optimal point)

Minimize CVaR(x) (minimizing CVaR risk)
subject to
avg_g(x) ≥ 0 (nonnegative portfolio return constraint)
linear(x) = 1 (portfolio budget constraint)
Box constraints (non-negativity constraints on positions)
Value:
XVaR(x) (Calculate expectile at optimal CVaR point)
——————————————————————–
XVaR(x) = Expectile of linear loss
CVaR(x) = CVaR of linear loss
pr_pen(x) = probability of exceedance
avg_g(x) = Expected value of linear loss (linear function)
linear(x) = Sum of portfolio weights (linear function)
Box constraints = constraints on individual decision variables
——————————————————————–

 
parameter values: q in range (0.55, 1.0)