Tilburg University
QFAS
Summary ALM Part II
Author: Supervisor:
Rick Smeets Schweizer, N
April 2, 2024
,Table of Contents
1 Numerical Solution of Optimal Investment Problems 2
1.1 Discrete Time Framework . . . . . . . . . . . . . . . . . . . . 2
1.2 Numerical Ingredients . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Computing and storing unknown functions . . . . . . . 3
1.2.3 Computation of conditional expectations . . . . . . . . 5
2 The Final Algorithm 6
2.1 Preperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Backward Recursion . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Remarks on the Algorithm . . . . . . . . . . . . . . . . . . . . 7
3 Hedging in Incomplete Markets 8
3.1 Basis Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Pure and Static Hedging . . . . . . . . . . . . . . . . . . . . . 9
3.3 Time-Inconsistency of Variance-Minimization . . . . . . . . . . 10
3.4 Dealing with Time-Inconsistency . . . . . . . . . . . . . . . . 11
3.5 The Setting of Basak and Chabakauri . . . . . . . . . . . . . . 12
3.6 The Hedge-Neutral World . . . . . . . . . . . . . . . . . . . . 13
3.7 Comparison With The Precommitment Strategy . . . . . . . . 13
3.8 Proof Of The Main Result . . . . . . . . . . . . . . . . . . . . 15
1
, 1 Numerical Solution of Optimal Investment
Problems
Solving optimal investment problems is difficult manually and remains chal-
lenging with computers. Quick, precise solutions are harder to achieve than
often assumed, and universally reliable automated solvers for complex invest-
ments do not yet exist.
1.1 Discrete Time Framework
Numerical methods require discretization, so we turn to a discrete time frame-
work. Two main differences in notation between Delage et al. and Munk are
that portfolio weights are called x rather than π and the indirect utility
function is called V rather than J.
We denote by St a (possibly multi-dimensional) state process, by rt+∆t the
vector of excess returns of the risky assets over [t, t + ∆t], and by Rf the
(constant) risk-free return over [t, t + ∆t], i.e., Rf = erf ∆t or Rf = 1 + rf ∆t.
For portfolio weights xt invested in the risky assets, this results in the wealth
dynamics
Wt+∆t = Wt (Rf + x′t rt+∆t ).
We abstract from labor income and consumption (y = c = 0) and thus do not
need to care about time discounting either. This gives us the maximization
problem
V0 (W0 , S0 ) = max E0 [u(WT )]
(xt )∈A
under the constraint
Wt+∆t = Wt (Rf + x′t rt+∆t ).
Here A denotes the set of admissible portfolio strategies. Portfolio strategies
must be adapted, and possibly fulfill further constraints like short-selling or
borrowing.
Exercise 1 What do the short-selling and borrowing constraints look like as
conditions on x?
2
QFAS
Summary ALM Part II
Author: Supervisor:
Rick Smeets Schweizer, N
April 2, 2024
,Table of Contents
1 Numerical Solution of Optimal Investment Problems 2
1.1 Discrete Time Framework . . . . . . . . . . . . . . . . . . . . 2
1.2 Numerical Ingredients . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Computing and storing unknown functions . . . . . . . 3
1.2.3 Computation of conditional expectations . . . . . . . . 5
2 The Final Algorithm 6
2.1 Preperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Backward Recursion . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Remarks on the Algorithm . . . . . . . . . . . . . . . . . . . . 7
3 Hedging in Incomplete Markets 8
3.1 Basis Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Pure and Static Hedging . . . . . . . . . . . . . . . . . . . . . 9
3.3 Time-Inconsistency of Variance-Minimization . . . . . . . . . . 10
3.4 Dealing with Time-Inconsistency . . . . . . . . . . . . . . . . 11
3.5 The Setting of Basak and Chabakauri . . . . . . . . . . . . . . 12
3.6 The Hedge-Neutral World . . . . . . . . . . . . . . . . . . . . 13
3.7 Comparison With The Precommitment Strategy . . . . . . . . 13
3.8 Proof Of The Main Result . . . . . . . . . . . . . . . . . . . . 15
1
, 1 Numerical Solution of Optimal Investment
Problems
Solving optimal investment problems is difficult manually and remains chal-
lenging with computers. Quick, precise solutions are harder to achieve than
often assumed, and universally reliable automated solvers for complex invest-
ments do not yet exist.
1.1 Discrete Time Framework
Numerical methods require discretization, so we turn to a discrete time frame-
work. Two main differences in notation between Delage et al. and Munk are
that portfolio weights are called x rather than π and the indirect utility
function is called V rather than J.
We denote by St a (possibly multi-dimensional) state process, by rt+∆t the
vector of excess returns of the risky assets over [t, t + ∆t], and by Rf the
(constant) risk-free return over [t, t + ∆t], i.e., Rf = erf ∆t or Rf = 1 + rf ∆t.
For portfolio weights xt invested in the risky assets, this results in the wealth
dynamics
Wt+∆t = Wt (Rf + x′t rt+∆t ).
We abstract from labor income and consumption (y = c = 0) and thus do not
need to care about time discounting either. This gives us the maximization
problem
V0 (W0 , S0 ) = max E0 [u(WT )]
(xt )∈A
under the constraint
Wt+∆t = Wt (Rf + x′t rt+∆t ).
Here A denotes the set of admissible portfolio strategies. Portfolio strategies
must be adapted, and possibly fulfill further constraints like short-selling or
borrowing.
Exercise 1 What do the short-selling and borrowing constraints look like as
conditions on x?
2
