American Options - Computational Methods for Quantitative Finance - page 66

Fig. 5.1 American put option

Denoting by T t,T the set of all stopping times for S t with values in the interval

(t, T ) , the value of an American option is given by

τ ∈ T t,T E e − r(τ − t) g(S τ )

s .

V(t,s) := sup | S t = (5.1)

As for the European vanilla style contracts, there is a close connection between
the probabilistic representation ( 5.1 ) of the price and a deterministic, PDE based
representation of the price. We have

Theorem 5.1.1 Let v(t,x) be a sufficiently smooth solution of the following system
of inequalities

BS v

∂ t v − A + rv ≥ 0 in J × R ,

g(e x ) v(t,x) ≥ in J × R ,

(5.2)

BS v

(∂ t v − A + rv)(g − v) = 0 in J × R ,

g(e x )

v( 0 ,x) = in R .

Then , V(T − t,e x ) = v(t,x) .

A proof can be found in [15], we also refer to [98] for further details. For each

J there exists the so-called optimal exercise price s ∗ (t) t ∈ ∈ ( 0 ,K) such that for all

s ∗ (t) the value of the American put option is the value of immediate exercise,
i.e. V(t,s) = g(s) , while for s>s ∗ (t) the value exceeds the immediate exercise
value, see Fig. 5.1 .

s ≤

s>s ∗ (t) The region C := { (t, s) | } is called the continuation

c of

region and the complement is the exercise region . Since the optimal exercise

price is not known a priori, it is called a free boundary for the associated pricing PDE
and the problem of determining the option price is then a free boundary problem .
Note that the inequalities ( 5.2 ) do not involve the free boundary s ∗ (t) .

C C

Remark 5.1.2 In the Black-Scholes model, the derivative of V at x = s ∗ (t) is con-
tinuous which is known as the smooth pasting condition. This does not hold for pure
jump models.