This document discusses pricing models for American option contracts. It begins by outlining the standard model, which values American options based on the moment that guarantees maximum option value. However, the author proposes an alternative view, where the optimal exercise time is when the underlying asset reaches its maximum value on [0,T]. Exercising at this maximum value ensures a payoff equal to the selling price, avoiding arbitrage. The document formalizes this idea using concepts like risk-neutral probabilities and derivations of put-call parity relationships to define fair option prices.

1. 1
Pricing of American Options.
Ilya I. Gikhman
6077 Ivy Woods Court
Mason, OH 45040, USA
Ph. 513-573-9348
Email: ilya_gikhman@yahoo.com
Classification code. G13.
Key words. European options, American options.
Abstract. In this paper, we present somewhat alternative point of view on early exercised American
options. The standard valuation of the American options the exercise moment is defined as one, which
guarantees the maximum value of the option. We discuss the standard approach in the first two sections of
the paper. The standard approach was initially presented in the papers [3] - [7]. Our idea is that the
exercise moment of the American call / put options is defined by maximum / minimum value of
underlying. It was shown that at this moment exercise and sell prices are equal.
I. An option gives a contract holder the right to buy or to sell an underlying security such as a
share of stock on a specified future date called maturity for a known exercise price called also strike price.
Option to buy is a call option whereas an option to sell is known as a put option. Such options are called
European options. American option gives its holders the right to exercise the option on or before the
expiration date.
Consider a stock which dynamics follows ordinary SDE
d S ( t ) = µ ( t ) S ( t ) dt + σ ( t ) S ( t ) dw ( t ) (1)
with a standard Wiener process w ( t ), t  0 defined on a complete probability space {  , F , P }. Here
µ ( t ) and σ ( t ) > 0 are known nonrandom continuous function on t.
European Call and Put options are defined by its payoff
C E ( T , S ) = max { S ( T ) – K , 0 }
P E ( T , S ) = max { K – S ( T ) , 0 }

2. 2
at maturity date T. The valuation (pricing) problem is to define fair price of the options at any time prior
to maturity. First step is to define notion ‘fair’ price of an option. Introduce a risk free traded instrument
known as bond. As far as bonds are risk free assets its rate of return is riskless either for deposits or for
borrowing transactions. The rule used for determining fair pricing is no arbitrage principle. It states that
there is no way to get profit in the market starting with zero value of initial investment. Briefly recall
European option pricing following Black-Scholes concept [1].
Denote V ( t , S ( t )) the option price at t. Then in case of European call and put options
V ( t , S ( t )) = C E ( t , S ( t ))
V ( t , S ( t )) = P E ( t , S ( t ))
correspondingly. Date-t instantaneous hedge portfolio is defined as
Π ( u , S ( u ) ; t ) = V ( u , S ( u ) ) +  ( t , S ( t ) ) S ( u ) (2)
where
 ( t , S ( t ) ) =
S
))t(S,t(V


Here u, u  t is a variable and t , t  [ 0 , T ] is a fixed parameter. Formula (2) defines change in the
value of the portfolio at moment t  0. Differential of the value Π ( u , t ) with respect to variable u at t is
equal to
d Π ( u , S ( u ) ) | u = t = d V ( t , S ( t ) ) +  ( t , S ( t ) ) d S ( t ) (3)
Bearing in mind Ito formula the change in the value of the portfolio at the moment u = t is equal to
d Π ( u , S ( u ) ) | u = t = [ 2
222
S
))t(S,t(V
2
)t(σ)t(S
t
))t(S,t(V





] d t
Latter formula does not contain the risk term proportional to d w ( t ). In order to exclude arbitrage
opportunity: borrow at risk free financing portfolio at t and return borrowing plus risk free interest at
t + dt one needs to assume that
d Π ( t , S ( t ) ) = r Π ( t , S ( t ) ) d t (4)
Here r denotes the risk free interest rate. In other words, rates of return of the riskless portfolio and
riskless borrowing rate should be equal. Equality (4) leads to Black Scholes equation
2
222
x
)x,t(V
2
σx
t
)x,t(V





+ r
x
)x,t(V


– r V ( t , x ) = 0 (BSE)
t < T. For European call and put options boundary conditions are equal to
max { S ( T ) – K , 0 } , max { K – S ( T ) , 0 }

3. 3
correspondingly. Denote C E ( t , x ; T , K ), P E ( t , x ; T , K ) the European call and put prices at t. These
functions are solutions of the (BSE) equation. Note that underlying of the BSE solution is the risk neutral
heuristic random process S r ( t ), which follows stochastic equation
d S r ( t ) = r S r ( t ) dt + σ S r ( t ) dw ( t ) (5)
Indeed, the process S r ( t ) does not represent a traded asset. Pricing formula (BSE) shows that underlying
of the option is the random process Sr ( u ) on original probability space{ Ω , F , P } while according to
the general definition of the derivative underlying should be the random process (1). Risk-neutral world
was presented as a solution of the confusion. Of course, it does not eliminate the fact that underlying of
the Black-Scholes pricing formula is the random process (5). Note that random process S ( t ) is always
defined on original probability space { Ω , F , P } regardless whether options on this stock exist or not.
Risk neutral valuation suggests to consider equation (1) on the risk-neutral probability space { Ω , F , Q }
where ‘risk-neutral’ probability measure Q is defined by the formula
Q ( A ) = A
{ exp 
T
0
[
σ
rμ 
d w Q ( t ) –
2
1

T
0
(
σ
rμ 
) 2
d t ] } P ( d ω )
for a set A  F. Here w Q ( t ) is a Wiener process on { Ω , F , Q }. Then the random process
S r ( t ) = S r ( t , ω ) is a solution of the risk-neutral equation (5) on probability space { Ω , F , P } with a
Wiener process
w ( t ) = w Q ( t ) + 
t
0
σ
rμ 
d u
on { Ω , F , P }. Thus the essence of the risk neutral valuations is to consider diffusion equation (1) on
risk-neutral probability space { Ω , F , Q }. Note that usually one incorrectly states that stock is
considered on risk neutral probability space while stock is defined on real space { Ω , F , P }. Actually we
considered equation that corresponding stock on { Ω , F , Q } and there is no traded asset which follows
(1) on { Ω , F , Q } that is equal to (5) on { Ω , F , P }.
II. Value of the American options give its holders the right to exercise it at any time prior to
maturity date. Denote C A ( 0 , x ; T , K ) value of American call option at t = 0 given that S ( 0 ) = x
Here T and K are maturity and exercise price of the option correspondingly. Primary approach to
American option pricing uses no arbitrage arguments was outlined in [6]. The well known idea is that
with no dividends on underlying stock American call option should not be exercised prior to maturity. We
present other idea related to American options valuation.
The standard valuation suggests do not exercises American call options prior to maturity based on the fact
that selling options gives higher than exercise price at the same moments of time. Recall put-call parity
relationships. We present this relationship for Black Scholes European option prices. Indeed if we use
notations for call and put options without its formal definitions derivation of the put-call parity looks
rather like a heuristic statement. Then

4. 4
C E ( t , x ; T , K ) = B ( t , T ) E [ S r ( T ; t , x ) – K ]  { S r ( T ; t , x ) > K } ,
P E ( t , x ; T , K ) = B ( t , T ) E [ K – S r ( T ; t , x ) ]  { S r ( T ; t , x ) < K } =
= B ( t , T ) E [ K – S r ( T ; t , x ) ] [ 1 –  { S r ( T ; t , x ) > K } ]
where S ( t ) = x and B ( 0 , T ) the value of the risk free bond at t = 0 with expiration at T. The function
u ( t , x ) = E S r ( T ; t , x ) satisfies the same risk free bond equation and boundary condition
u ( T , x ) = x. Therefore
P E ( t , x ; T , K ) – C E ( t , x ; T , K ) = B ( t , T ) [ K – E S r ( T ; t , x ) ] =
= B ( t , T ) [ K – x B – 1
( t , T ) ]
Hence
P E ( t , x ; T , K ) – C E ( t , x ; T , K ) = B ( t , T ) K – S ( t ) (PCP)
Taking into account that values of the put and call options are positive from (PCP) it follows that
max { S ( t ) – K , 0 } ≤ max { S ( t ) – B ( t , T ) K , 0 } ≤ C E ( t , S ( t ) ; T , K ) ≤ S ( t ) (6)
Inequality (6) shows that exercise price is always less than the price of the European call option.
Suppose that American call option is exercised at a random moment τ ( ω )  [ 0 , T]. If the call option is
exercised at the moment t , t ≤ T then payoff is equal to the value S ( t ) – K which is the European call
payoff with maturity at t , given that S ( t ) > K.
On the other hand it might be confusion to observe that exercising option prior to maturity implies a
strictly positive payoff while exercising at maturity should be also equal to zero. We study American call
option pricing more accurately than it is suggested by equality (PCP) and (6).
III. Introduce value of the American call option C A ( 0 , x ; τ ( ω ) ; T , K ) , which is exercised at
a random moment τ ( ω ) = τ 0 T ( ω )  [ 0 , T ]. It is obvious that
C A ( 0 , x ; τ ( ω ) ; T , K ) = C E ( 0 , x ; τ ( ω ) , K ) (7)
Indeed,
C A ( 0 , x ; τ ( ω ) ; T , K ) { τ ( ω ) = t } = C E ( 0 , x ; t , K ) { τ ( ω ) = t }
for any t  [ 0 , T ]. It follows from definition of the call option that
C E ( τ , S ( τ ) ;  , K ) = max { S ( τ ) – K , 0 }
Value C E ( 0 , x ; τ ( ω ) , K ) defines no arbitrage price of the European call option at t = 0 with
maturity at a random moment τ ( ω ). Define American option price at t = 0 by the formula

5. 5
C A ( 0 , x ; T , K ) = E C A ( 0 , x ; τ ( ω ) ; T , K ) = E C E ( 0 , x ; τ ( ω ) , K ) (8)
Note that distribution of the exercise moment is an explicit parameter which specifies American option
price. From (7) and (PCP) it follows that for each 
C A ( 0 , x ; τ ( ω ) ; T , K ) = C E ( 0 , x ; τ ( ω ) , K ) ≥ max { S ( 0 ) – B ( 0 , τ ( ω )) K , 0 }
and therefore
C A ( 0 , x ; T , K ) ≥ E max { S ( 0 ) – E B ( 0 , τ ) K , 0 } > max { S ( 0 ) – K , 0 }
Hence, we could state that selling option for C A ( 0 , x ; T , K ) looks better than exercise it for
max { S ( 0 ) – K , 0 } at t = 0. If the price of the option is larger than exercise price prior to a moment t
then American option does not exercised on [ 0 , t ].
Hence, a sufficient condition for does not exercise American option on [ 0 , t ] is
C A ( u , S ( u ) ; T , K ) = E C A ( 0 , x ; τ ( ω ) ; T , K ) > S ( u ) – K = C E ( u , S ( u ) ; t , K ) (9)
u  [ 0 , t ]. Indeed, inequality (9) does not admits arbitrage opportunity during [ 0 , t ]. If for some t ,
S ( t ) > 0 and we observe the inverse relationship
C A ( t , S ( t ) ; T , K ) < S ( t ) – K
then at the moment t there exists an arbitrage opportunity. Indeed, buying option for C A ( t , S ; T , K ) at
t and immediately exercising it leads to a riskless positive profit
S – K – C A ( t , S ; T , K ) > 0
Therefore the necessary and sufficient condition to exercise American call at a moment t is the equality
C A ( t , S ; T , K ) = S ( t ) – K (10)
Equality (10) is equivalent to construction of the moment(s) for which
C E ( τ T ( ω ) , S ( τ T ( ω ) ,  ) ; τ T ( ω ) ; T , K ) = S ( τ T ( ω ) ,  ) – K
Bearing in mind (9), (10) one can state that
P { τ T ( ω ) = T } = 1
and therefore with probability 1
C A ( t , S ; τ T ( ω ) ; T , K ) = C E ( t , S ; T , K )
This is the essence of the American call benchmark pricing.
On the other hand it is obvious that if American call option is exercised at the random moment
 =  [ 0 , T ] ( ω ) which represents maximum value of the underlying stock on [ 0 , T ] , i.e.
[ S (  ( ω ) ,  ) – K ] =
]T,0[t
max

[ S ( t ,  ) – K ]  { S ( t ,  ) > K }

6. 6
then it guarantees a higher payoff value than exercising option at the maturity date T. We also should see
that moment  ( ω ) does not a Markov stopping time. Therefore investors could never state whether this
moment is already realized or not. This practical difficulty suggests a reduction of the maximum exercise
moment to the set of Markov stopping times. Such a reduction simplifies the problem and could not
guarantee optimal return of American options.
Bearing in mind equality (9) and excluding arbitrage opportunity taking place if
C A ( t , S ( t ) ; T , K ) < S ( t ) – K
We note that exercise price of the American call option is equal to the selling price at the moment  ( ω ).
Indeed, let  be arbitrary random variable taking values on [ 0 , T ]. Then bearing in mind that  is the
moment of the maximum S ( t ) on [ 0 , T ] we conclude that for any random moment of time τ = τ ( ω )
S (  ) – K ≤ S (  ) – K
Therefore
C A (  , S (  ) ; τ ( ω ) , K ) ≤ S (  ) – K
This equality shows that either prior or later than at  ( ω ) exercise of the option , i.e. τ ≥  leads to the
lower value payoff of the option. This remark also takes place for τ = T too. Thus
C A (  ( ω ) , S (  ( ω ) ) ;  ( ω ) ; T , K ) = S (  ( ω ) ,  ) – K
Bearing in mind that  does not a stopping time we can also conclude that for some market scenarios
C A (  ( ω ) , S (  ( ω ) ) may not reach the level S (  ( ω ) ,  ) – K. From (7) it follows that
C A ( 0 , x ; T , K ) = E C E ( 0 , x ;  ( ω ) , K ) = E B ( 0 ,  ( ω ) ) [ S r (  ( ω ) ) – K ] (11)
where B ( 0 , t ) is the value of the risk free bond at time 0 with expiration date t. Formula (11) represents
Black Scholes pricing and uses risk neutral underlying process S r ( t ). The maturity of this underlying is
specified by the moment  that is specified by the real underlying process S ( t ). Now let us look at the
problem how to apply this idea in practice. Denote
F ( y ) = P {
]T,0[t
max

S ( t ,  ) < y }
Choose a number Q  ( 0 , 1 ) such that the value [ y Q – K ] / C A ( 0 , x ; T , K ) from investor point of
view is sufficient return over ( 0 , T ) on investment C A ( 0 , x ; T , K ) at t = 0 and probability 1 – Q
does not a small chance to expect that the price S ( t ) will reach the level y Q . Here Q and y Q are
defined as
P {
]T,0[t
max

S ( t ,  ) < y Q } = Q
Note that the moment when S ( t ) reaches the level y Q is a Markov stopping time defined on filtration
generated by the observations S ( t ). Option buyer can also get a table with barrier levels y 1 < … < y n

7. 7
and correspondent probabilities Q 1 < … < Q n and profits [ y j – K ] / C A ( 0 , x ; T , K ) , j = 1, … , n
which corresponds to a particular trading strategies.
IV. Consider now American put pricing. From equality (PCP) it follows that
max { K – S ( t ) , 0 } ≤ max { B ( t , T ) K – S ( t ) , 0 } ≤ P E ( t , x ; T , K ) ≤ B ( t , T ) K
Let  (  )  [ 0 , T ] be arbitrary random moment at which investor exercises put option. We do not
suppose here that  (  ) is a Markov moment. Denote  t a market scenario for which
 (  t ) = t  [ 0 , T ]
Given scenario  t the value of American put option P A ( 0 , x ;  (  t ) ; T , K ) should be equal to the
correspondent value P E ( 0 , x ; t , K ) of the European put option with the same strike price K and
maturity t. Thus the fair price of the put option for each market scenario  is defined by equality
P A ( t , x ; τ (  ) ; T , K ) = P E ( t , x ;  (  ) , K ) (12)
For a fixed  random function P E ( t , x ;  (  ) , K ) is equal to European Put option and therefore
P E ( t , x ;  (  ) , K ) = { e – r ( T – t )
E max [ K – S r ( T ; t , x ) , 0 ] } T =  (  )
Taking expectation we arrived at a statistical estimate which we interpret as American put option price
P A ( t , x ; T , K ) = E P A ( t , x ;  (  ) ; T , K ) = E P E ( t , x ;  (  ) , K )
The value of American put could be decomposed into European put and early exercise premium
P A ( t , x ; T , K ) = P E ( t , x ; T , K ) – [ E P E ( t , x ;  (  ) , K ) – P E ( t , x ; T , K ) ]
The difference
P A ( t , x ; T , K ) – P A ( t , x ;  (  ) ; T , K ) = P A ( t , x ; T , K ) – P E ( t , x ;  (  ) , K )
defines market risk of the American put. The value
P { P A ( t , x ; T , K ) – P A ( t , x ;  (  ), T , K ) > 0 }
specifies the chance of overpricing, when buyer of the American put pays higher price than it is implied
by the market scenario. Similarly, the value
P { P A ( t , x ; T , K ) – P A ( t , x ;  (  ), T , K ) < 0 }
specifies seller’s overpaid market risk. Following the idea introduced for American call option pricing
define random moment  P (  ) which guarantees maximum value of the put payoff
]T,0[t
max

[ K – S ( t ) ]  { S ( t ) < K } = [ K – S (  P (  )) ]  { S (  P (  )) < K }

8. 8
Bearing in mind that exercise moment  P (  ) of the American put coincides with
]T,0[t
min

S ( t ) we can
transform the optimal exercise time problem for the American put to the optimal exercise problem for the
American call option. Indeed, taking into account that
P {
]T,0[t
min

S ( t ) > 0 } = 1
we enable to apply Ito formula to f ( S ( t ) ) = S – 1
( t ). Then applying Ito formula we can easy verify
that
d S – 1
( t ) = [ – µ ( t ) + σ 2
( t ) ] S – 1
( t ) dt + σ ( t ) S – 1
( t ) dw ( t )
Latter formula demonstrates the fact that the random process S – 1
( t ) is a Geometric Brown Motion
with drift and diffusion coefficients σ 2
( t ) – µ ( t ) , σ ( t ) correspondingly. Therefore the optimal time
to exercise American put for each market scenario 
]T,0[t
max

{ K – S ( t ,  ) , 0 }
is reduced to find minimum of the process S ( t ,  ) which coincides with the problem of finding
maximum of the process S – 1
( t ) on [ 0 , T ]. Thus the problem of American put exercising is reduced to
similar problem for the American call option.

9. 9
References.
1. Black, F., Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. The Journal of
Political Economy 81 637-659.
2. Carr, P. , Jarrow, R. , Myneni, R., (1992), Alternative Characterizations of American Put Options,
Mathematical Finance 2, 87-106.
3. Jacka, S.D., (1991). Optimal Stopping and the American Put, Journal of Mathematical Finance,
Volume 1 1–14.
4. Kim, I. J., (1990). The Analytic Valuation of American Options, Review of Financial Studies,
Volume 3 547–72.
5. McKean, H. P., Jr. (1965), Appendix: A free boundary problem for the heat equation arising from
a problem mathematical economics. Indust. Manage. Rev. 6 32-39.
6. Merton, R., (1973), The theory of rational option pricing, Bell Journal of Economics 4, 141–183.
7. van Moerbeke, P.L.J. (1976), On optimal stopping and free boundaries problems. Arch. Rational
Mech. Anal. 60 101-148.