To keep a balance with the relevant literature cited, the notations unavoidably change somewhat from one chapter to another. We will try our best to minimize notational disruption. The models are somewhat similar so we highlight the important notations used by considering a dynamic investor.
The risky assets follow a geometric Brownian motion (GBM) described by
dSi
Si
= midt+σidZti (2.10)
where the vector (dZt) represents Brownian motion vector,Sirepresents the asset price,mi
represents the drift andσi the volatility.
The processes Zi
t;t > 0 are standard Brownian motions on a filtered probability space
2.4. Notation used in the thesis 13
EP[dZtidZ j
t] = ρi jdt. We also assume that the filtration Ft is right-continuous and each Ft
contains all P-null sets ofF∞. The correlation co-efficients imply a correlation structure for
multi-dimensional GBM.
We could also represent the above via aCholeskydecomposition as
dSi Si = midt+ M X l=1 qildBlt (2.11)
where (dBt) represent an independent Brownian motion vector and qilare extracted as en-
tries of the variance-covariance matrix of the vector (dZt) .
We will follow the notation given below. By ‘time node’ we mean the node value as shown in Figure 2.1. ‘Time node’ is different from the actual value of time.
EP: represents the expectation under the natural probability measure3
Wk−: total wealth of the investor at time nodekimmediately before re-balancing4 A−
i,k : fraction of wealth in the risky assetiat time nodekimmediately before re-balancing
If there is only one risky asset in our model we would suppress one subscript and useA−
k
as the fraction of wealth in the risky asset.
Xi−,k : amount of wealth in the risky assetiat time nodekimmediately before re-balancing If there is only one risky asset in our model we would suppress one subscript and use X−
k
for the amount of wealth in the risky asset.
Yk−: amount of wealth in the risk-free asset at time nodekimmediately before re-balancing Analogous variables for the above exist for immediately after re-balancing.
T : length of the time horizon
N: number of observation points available for re-balancing
∆k : transfer of wealth under a control strategy at time nodek
ri,k: risky growth for assetiover a period∆Tk defined as the ratio Si,k+1
Si,k sk : risk-free interest rate over period∆Tk
γ`: the deformation parameter whereγ` −→0 for convergence (discussed in chapter 5) V : is the co-efficient of risk aversion in CRRA utility.
z: is the co-efficient in CARA utility.
The structure of transaction cost in chapter six5 is different but in most of the chapters we
will have :
λk : transaction cost factor in buy-side direction
µk : transaction cost factor in sell-side direction
If the state space vector at time k is denoted by ~φk and control vector by ~ζk then value
function under utility functionU(WN) is :
J(~φk)= maxE~ζP(U(WN)|Fk) (2.12)
$: probability density of risky growth over an interval.
ηi,j,+
k : fraction of wealth transferred from assetito asset j.
3We will not invoke thenumeraireconcept anywhere in the thesis so as to use martingale methods with respect
to a risk-neutral probability measure, as portfolio optimization is intrinsically a real world measure topic.
4Wealth is measured in monetary terms invested in total in individual assets. It is not measured as a result of
liquidation in risk-free asset or by using one asset as a numeraire.
14 Chapter2. Literature review and notations
Mk: maximum wealth attained at a time nodek.
Zk : denotes spread value at time nodek.
πk: cumulative profits at time nodek.
δk: profit in periodk.
We must emphasize some notational confusion that might occur. ByW(t) we would mean the wealth at timetand this is usually the case for continuous time exposition of problems. In discretized form byWk we would mean the wealth at nodekwhere there are sayNnodes over
the time horizon. Such notational differences are applicable to all relevant dynamic portfolio variables.
Note: The labeling of axes might have fonts not perfectly aligned because this was done manually in the Mathematica generated graphics.
In the next chapter we will present the basic modeling framework of the thesis.