Summary Asset Liability Management Part I EOR

Tilburg University

QFAS


Summary ALM

Author: Supervisor:
Rick Smeets Schweizer, N

April 2, 2024

,Table of Contents
1 Introduction to Dynamic Asset Allocation 3

2 Theory Warm-up: The discrete time case 3
2.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Optimization Problem . . . . . . . . . . . . . . . . . . . . 6
2.4 The Dynamic Programming Principle . . . . . . . . . . . . . . 6
2.5 Markovian Framework . . . . . . . . . . . . . . . . . . . . . . 7
2.6 Backward solution approach . . . . . . . . . . . . . . . . . . . 7

3 Cont. Time Portfolio Choice: the HJB approach 8
3.1 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Price and Wealth Dynamics . . . . . . . . . . . . . . . . . . . 9
3.4 Towards the HJB . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5 Solution Approach . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Cont. Time Portfolio Choice: the HJB approach with con-
stant investment opportunities 13
4.1 The Simplified Problem . . . . . . . . . . . . . . . . . . . . . 14
4.2 Consumption and Portfolio Choice . . . . . . . . . . . . . . . 14
4.3 Two Fund Separation . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 CRRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.5 Conjecture and final solution . . . . . . . . . . . . . . . . . . . 17

5 Cont. Time Portfolio Choice: a second look outside the
Black-Scholes world 19
5.1 Recalculating the HJB . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Portfolio Choice . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Three Fund Separation . . . . . . . . . . . . . . . . . . . . . . 20

6 Cont. Time Portfolio Choice: the Martingale Method 21
6.1 The Martingale Method in a Nutshell . . . . . . . . . . . . . . 22
6.2 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3 Option Pricing Theory - a Reminder . . . . . . . . . . . . . . 23
6.4 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 24
6.5 A Candidate Solution . . . . . . . . . . . . . . . . . . . . . . . 25

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,6.6 Investment Strategies . . . . . . . . . . . . . . . . . . . . . . . 26
6.7 The CRRA Case . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.8 The Black-Scholes Setting . . . . . . . . . . . . . . . . . . . . 29
6.9 Beyond Black-Scholes . . . . . . . . . . . . . . . . . . . . . . . 30
6.10 Logarithmic Utility . . . . . . . . . . . . . . . . . . . . . . . . 31




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, 1 Introduction to Dynamic Asset Allocation
Definition 1.1. An incomplete market is a market where not all risks
can be hedged, i.e., a market where we cannot let risk disappear by holding
a well-chosen portfolio of liquidity traded assets.


2 Theory Warm-up: The discrete time case
Dynamic Asset Allocation can be referred to as optimal portfolio choice
as well as optimal investment. The problem is to find a long-term consump-
tion and investment strategy that maximizes expected utility, where we have
to take several trade-offs into consideration such as risk versus return, or
consuming now or later.

2.1 The Setting
We consider the following statements with respect to the discrete setting in
Dynamic Asset Allocation:

1. We have a time horizon [0, T ] partitioned into N time intervals [tn , tn+1 ]
of length ∆t, that means tn = n∆t and tN = N ∆t = T .

2. An agent can invest at each time point t ∈ T = {t0 , ..., tN −1 } into d + 1
assets with price processes Pti , i = 0, ..., d.

3. Asset 0 is locally risk-free, that means that at time t it is known that
0
Pt+∆t = (1 + rt ∆t)Pt0 . ’Locally’ means that the risk free interest rate
is only known for the upcoming period.

4. The remaining assets i > 0 are risky and their returns are denoted by
 i
Pt+∆t − Pti

i
Rt+∆t = ,
Pti

moreover the column vector of the returns of the risky assets equals
1 d

′
Rt+∆t = Rt+∆t , . . . , Rt+∆t




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