- Option Pricing 183
The price of the American put will be denoted byPA(n)forn=0, 1 ,2. At
expiry the payoff will be positive only in the scenario with two downward stock
price movements,
n 012
0. 00
? <
PA(n)? < 0. 00
? <
7. 80At time 1 the option writer can choose between exercising the option immedi-
ately or waiting until time 2. In the up state at time 1 the immediate payoff
and the value of waiting are both zero. In the down state the immediate payoff
is 4 dollars, while the value of waiting is 1. 05 −^1 ×^13 × 7. 8 ∼= 2 .48 dollars. The
option holder will choose the higher value (exercising the option in the down
state at time 1). This gives the time 1 values of the American put,
n 012
0. 00
0. 00 <
PA(n)? < 0. 00
4. 00 <
7. 80At time 0 the choice is, once again, between the payoff, which is zero, or the
value of waiting, which is 1. 05 −^1 ×^13 × 4 ∼= 1 .27 dollars. Taking the higher of
the two completes the tree of option prices,
n 012
0. 00
0. 00 <
PA(n) 1. 27 < 0. 00
4. 00 <
7. 80For comparison, the price of a European put isPE(0) = 1. 05 −^1 ×^13 × 2. 48 ∼= 0. 79
dollars, clearly less than the American put pricePA(0)∼= 1 .27 dollars.
This can be generalised, leading to the following definition.Definition 8.1
AnAmerican derivative securityorcontingent claimwith payoff functionf
expiring at timeN is a sequence of random variables defined by backward