Mathematical Modeling in Finance with Stochastic Processes

3.4. A STOCHASTIC PROCESS MODEL OF CASH MANAGEMENT 119

We can write this asWskp =−2 max(s−k,0)
Then solving for the boundary conditions, the full solution is

Wsk= 2 [s(1−k/S)−max(s−k,0)].

Expected Duration and Expected Total Cash in a Cycle

Consider thefirst passage timeN when the reserves first reach 0 orS, so
that cycle ends and the bank intervenes to change the cash reserves. The
value ofN is a random variable, it depends on the sample path. We are
first interested inDs =E[N], the expected duration of a cycle. From the
previous section we already knowDs=s(S−s).
Next, we are interested in the mean cost of holding cash on hand during
a cyclei, starting from amounts. Call this meanWs. Letrbe the cost per
unit of cash, per unit of time. We then obtain the cost by weightingWsk,
the mean number of times the cash is at number of unitskstarting from
s, multiplying byk, multiplying by the factorr and summing over all the
available amounts of cash:

Ws=

S∑− 1

k=1

rkWsk

= 2

[

s

S

S∑− 1

k=1

rk(S−k)−

∑s−^1

k=1

rk(s−k)

]

= 2

[

s

S

[

rS(S−1)(S+ 1)

6

]

−

rs(s−1)(s+ 1)

6

]

=r

s

3

[

S^2 −s^2

]

.

These results are interesting and useful in their own right as estimates of
the length of a cycle and the expected cost of cash on hand during a cycle.
Now we use these results to evaluate the long run behavior of the cycles. Upon
resetting the cash at hand toswhen the amount of cash reaches 0 orS, the
cycles are independent of each of the other cycles because of the assumption
of independence of each step. LetKbe the fixed cost of the buying or selling
of the treasury bonds to start the cycle, letNibe the random length of the
cyclei, and letRibe the total opportunity cost of holding cash on hand
during cyclei. Then the cost overncycles isnK+R 1 +···+Rn. Divide by