An elementary introduction to mathematical finance

Описание

Introduction and Preface
An option gives one the right, but not the obligation, to buy or sell a
security under specified terms. A call option is one that gives the right
to buy, and a put option is one that gives the right to sell the security.
Both types of options will have an exercise price and an exercise time.
In addition, there are two standard conditions under which options operate: European options can be utilized only at the exercise time, whereas
American options can be utilized at any time up to exercise time. Thus,
for instance, a European call option with exercise price K and exercise
time t gives its holder the right to purchase at time t one share of the
underlying security for the price K, whereas an American call option
gives its holder the right to make the purchase at any time before or at
time t.
A prerequisite for a strong market in options is a computationally efficient way of evaluating, at least approximately, their worth; this was
accomplished for call options (of either American or European type) by
the famous Black–Scholes formula. The formula assumes that prices
of the underlying security follow a geometric Brownian motion. This
means that if S( y) is the price of the security at time y then, for any
price history up to time y, the ratio of the price at a specified future time
t + y to the price at time y has a lognormal distribution with mean and
variance parameters tμ and tσ2, respectively. That is,
logS(t + y)
S( y)
will be a normal random variable with mean tμ and variance tσ2. Black
and Scholes showed, under the assumption that the prices follow a geometric Brownian motion, that there is a single price for a call option that
does not allow an idealized trader – one who can instantaneously make
trades without any transaction costs – to follow a strategy that will result in a sure profit in all cases. That is, there will be no certain profit
(i.e., no arbitrage) if and only if the price of the option is as given by
the Black–Scholes formula. In addition, this price depends only on the

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