Such a compound option gives the owner the right at time for
the price
to buy a Call that has a maturity date
and a
strike price
.
The value of this option at time
with an actual price
can be calculated by applying the Black-Scholes formula
twice:
Thus it holds that
Calculate for
using the Black-Scholes equation
with these restrictions at
analog to the normal Call.
A Chooser Option is a form of the
compound option, where the buyer can decide at a later date which
type of option he would like to have. To illustrate consider a
regular Chooser Option with the parameters:
This option gives one the right at time for the price
to buy a Call or a Put (as one likes), which has a maturity
and a strike price
: in the language of compound options this
is referred to as a Call-on-a-Call or Put.
The value can be found by applying the Black-Scholes
formula three times:
gives the holder the right to buy a stock at time for the
price
provided that
For example, a European Knock-in-Call consists of the right to buy stock provided that
The value of a barrier Option is no longer dependent on a stock
price at a specific point in time, but on the overall development
of the stock price during the option's life span. Thus in
principle it does not fulfill the Black-Scholes differential
equation. The dependence however, is essentially simple enough to
work with the conventional Black-Scholes application. As an
example consider a Down-and-out-Call with . As long as
,
fulfills the
Black-Scholes equation with the restriction:
With an American Average Strike Option this is also the payoff
when the option is exercised ahead of time at some arbitrary time
To calculate the value of an Asian Option consider a general class
of European Options with a payoff at time that is dependent on
and
with
Analogous to the Black-Scholes equation we derive an equation for
the value at time of such a path dependent Option
. At time
with a
stock price
this results in
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Using Itôs Lemma it follows for
that:
Analogous to the derivation of the Black-Scholes formula
continuous Delta hedging produces a risk free portfolio from an
option and
sold stocks. Together with the restriction of no
arbitrage it follows for the case of no dividends
that:
For European options an explicit analytical solution of the differential equation exists in which complicated, specialized functions appear, the so called confluent hypergeometric functions. The numerical solution, however, is easier and faster to obtain.
The integral
in practice is calculated as the
sum over all quoted prices, for example, at 30 second time
intervals. Discrete time Asian Options use in place of this a
substantially larger time scale.
changes only once a day or
once a week:
Such a discrete time Asian Option is largely consistent with a
common option with discrete dividend payments at time periods
From the assumption of no arbitrage follows
a continuity restriction at
:
The value of a lookback Option
depends on the maximum or minimum of the stock price over the
entire life span of the option, for example, a lookback put
over the time period
has at maturity the payoff
With
it holds that:
This is the normal Black-Scholes equation. only appears as an
argument of
and in the boundary conditions:
The solution is for a remaining time period of
:
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