STOCHASTIC

CALL OPTIONS AND TIMING STRATEGY IN THE STOCK MARKET 113
60 to 90 days hence. The range of price changes is relatively small and there
should be little difference between the models [20]. Recently there have been
some conjectures on martingale models but little published as yet. For further
discussion on random walk models for stock prices the reader should consult
references [3, 6, 7, 8, 10, 14, 16, 17, 18, 19, 20],
Let p0 be the price guaranteed in the call.
For the investor who assumes an absolute random walk model the main results
are:
(al) For the case where E[Zk\ g 0, the optimal strategy for the call holder
is to do nothing until the last day the call is effective. On that day the
purchase is made at the lower of the call price or the market price. (For
the geometric case the same result holds if E\Zk\ g 1.)
(a2) If E[Zt] > 0 then there exists a decreasing sequence of numbers Oo =
Po £ Oi S OJI S • • • such that if Y„ is the market price with n days to
go to the expiration date of the call, then an open market purchase
should be made if Y„ < a„ . Otherwise the investor should wait one
more day before deciding. The critical numbers a„ may be found as
a„ = sup {y: J P„_i( y) where the functions P„(-) are
defined recursively as Po(y) = min (p0, y], and
P,(y) =mm{y,Jpn-1(t)dF(t\y)\
for all y.
(a3) If the price changes are normally distributed with mean n g 0 and
variance