Summary
Summary Asset Liability Management Part II EOR
- Course
- Institution
Summary of the second part of ALM at Tilburg University.
[Show more]
Seller
Follow
rickprive611
Content preview
Tilburg UniversityQFAS
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller rickprive611. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $9.21. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75759 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now