It is complex to price American options since they can be
exercised at any time point up to expiry date. The time the holder
chooses to exercise the options depends on the spot price of the
underlying asset . In this sense the exercising time is a
random variable itself. It is obvious that the Black-Scholes
differential equations still hold as long as the options are not
exercised. However the boundary conditions are so complicated that
an analytical solution is not possible. In this section we study
American options in more detail. The numerical procedures of
pricing will also be discussed in the next section.
As shown in Section 2.1, the right to early exercise
implies that the value of an American option can never drop below
its intrinsic value. For example the value of an American put
should not go below
with the exercise price
.
In contrast this condition does not hold for European options.
Thus American puts would be exercised before expiry date if the
value of the option would drop below the intrinsic value.
Let's consider an American put on a stock with expiry date . If
the stock price
at time
is zero, then
holds for
since the price process follows a
geometric Brownian motion. It is then not worth waiting for a
later exercise any more. If the put holder waits, he will loose
the interests on the value
that can be received from a bond
investment for example. If
, the value of the put at
is
which is the same as the intrinsic value. Since the
respective European put cannot be exercised early, e.g. at time
, we can only get
on the expiry date. If we discount it
to time
with
, we only get
that is the value of the European put at time
. Obviously
this value is smaller than the value of an American put and its
intrinsic value. Figure 8.1 shows the put value with
a continuous cost of carry
.
As we can see an early exercise of the put is maybe necessary even
before . For a certain critical stock price
, the
lost of interests on the intrinsic value, which the holder can
receive by exercising it immediately, is higher than the possible
increase of the option value due to the eventual underlying
fluctuations. That is one of the reasons why the critical
underlying price is dependent on time:
.
From the derivation of the Black-Scholes differential equations it
follows that they are valid as long as the option is not
exercised. Given that there are no transaction costs in perfect
markets, a revenue can be realized from an early exercise, which
equals to the intrinsic value of the option. One says in this
case that the option falls back to its intrinsic value by early
exercising. Thus the pricing of American options is an open
boundary problem. The Black-Scholes differential equations are
valid where the underlying is either higher than the critical
put-price
or lower than the critical
call-price
. The boundaries defined through
and
are unknown.
Figure 8.2 shows the regions where the option price
for an American call satisfies the Black-Scholes
differential equations.
The numerical solution for such boundary problems is described in
the next section. Based on the assumptions of perfect markets and
arbitrage free argument in Section 2.1 we derive some
properties of American options without considering any specific
mathematical models for the price process .
Proof:
Let denote the exercise price,
the expiry date,
the time to maturity of a call and
the price of the
underlying asset.
and
denote
the value of the respective American and European calls at time
with time to maturity
and the spot price
.
Using the put-call parity for European options we find
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(9.1) |
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|
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(9.2) |
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(9.3) |
Figure 8.3 shows a graphical representation of the first part of the theorem.
It is also possible to derive a formula similar to the put call
parity for American options. Given that the model is unknown for
the critical price ,
and consequently the time
point for early exercise, the formula is just an inequality.
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(9.4) |
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|
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||
if ![]() |
(9.5) | ||
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|
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Proof:
Supposing without any restriction that the underlying asset is a
stock
paying dividends at time
.
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1. We show first the left inequality. We consider a portfolio consisting of the following four positions:
Therefore it holds for every time as mentioned:
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(9.6) |
|
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(9.7) |
2. For continuous cost of carry we first consider the case where
. We prove the left inequality at first
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(9.8) |
If, on the other hand, the call is exercised early at time
, the whole portfolio is then liquidated and we get:
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(9.9) |
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(9.10) |
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(9.11) |
We show now the right inequality for the case
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(9.12) |
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(9.13) |
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(9.14) |
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(9.15) |
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(9.16) |