The aforegoing mixture of formal and informal analysis relies heavily on
the assumptions that:
(i) the variance rate of return on the stock is known;
(ii) the daily stock returns are normally distributed (strictly speaking we mean
log-normally distributed; however, practically speaking there is no
perceptible difference between the two on a daily basis).
Relaxing these assumptions will change our concludions to some extent.
For example, suppose that investors and the overall market are learning
about the variance rate of return on the stock by observing realised daily
returns. Then, assuming some type of Bayesian updating of variance estimates,
a large rise or fall in stock price will lead to an upward revision in
the variance estimate and hence to an upward revision of the option pricing
function, as this function increases with variance. (Note, of course, that in
this alternative world the option pricing function need not be the Black-
&holes function.) Then the hedge value, in addition to rising because of the
large value of !; will also rise because of the revision of variance estimates.
278 PP. Boyle and D. Emanuel. Discrrlely adjusted option hedges