Exchange Rate Economics: Theories and Evidence

Market microstructure approach 355
the level of allowable inventories. When inventory follows a random walk it will
reach the upper or lower bounds in a finite number of trades,with a probability of
one (see Ross 1983). Dynamic optimisation models (see,for example,Amihud and
Mendelson 1980 and Ho and Stoll 1981) resolve this problem. In such models the
market maker faces a stochastic order flow and will optimise his bid–ask spread
over time by shifting both the bid and ask downward (upward) and increase the
width of the spread when a positive (negative) inventory has accumulated.
For example,in the price-inventory model of Amihud and Mendelson (1980) a
market maker is faced with buy and sell order flows and these flows are assumed
to arrive as independent Poisson processes. The buy and sell arrival rates,defined
as d and x,respectively,are assumed to be a function of the bid,B,and ask,A,
prices quoted by the broker:
d = D(A) and x = X (B). (14.22)
The inventory level is denoted by k:
k ∈{−, ..., },
where and denote the largest allowable short and long positions,respectively,
and d k an x k denote the order arrival rates when prices are set as functions of the
inventory level:
d k = D(A(k)) and x k = X (B(k)). (14.23)
The expected time at position k is given by the Poisson process known as
1/(d k + x k ). Given this,the probability that the next order will be a buy (sell)
order is d k /(d k + x k )(x k /(d k + x k )). Hence the expected cash flow per unit of time
at position k is given by:
[
Q (k) =
x ]
k
· B(k) · (d k + x k )
d k + x k
d k
· A(k) −
d k + x k
= d k · A(k) − x k · A(k). (14.24)
The objective of the market maker is to maximise the expected profit per unit
of time,as given by:
π =
∑
k=−
k Q (k),
where is the probability of being at inventory level k. The solution to the optimising
problem gives values for A(k) and B(k). The market maker controls inventory
by adjusting prices up (down) to make an investor sale (purchase) more likely when
the inventory level is low (high). As the inventory nears its bounds the spread must
widen to avoid the issue of inventory following a random walk.