Structure of optimal stopping domains for American options with knock out domains
American options give us the possibility to exercise them at any moment of time up to maturity. An optimal stopping domain for American type options is a domain that, if the underlying price process enters we should exercise the option. A knock out option is a American barrier option of knock out ty...
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| Цитувати: | Structure of optimal stopping domains for American options with knock out domains / R. Lundgren // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 98–129. — Бібліогр.: 22 назв.— англ. |
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irk-123456789-45162009-11-25T12:00:37Z Structure of optimal stopping domains for American options with knock out domains Lundgren, R. American options give us the possibility to exercise them at any moment of time up to maturity. An optimal stopping domain for American type options is a domain that, if the underlying price process enters we should exercise the option. A knock out option is a American barrier option of knock out type, but with more general shape structure of the knock out domain. An algorithm for generating the optimal stopping domain for American type knock out options is constructed. Monte Carlo simulation is used to determine the structure of the optimal stopping domain. Results of the structural, and stability of studies are presented for different models of payoff functions and knock out domains. 2007 Article Structure of optimal stopping domains for American options with knock out domains / R. Lundgren // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 98–129. — Бібліогр.: 22 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4516 en Інститут математики НАН України |
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American options give us the possibility to exercise them at any moment of time up to maturity. An optimal stopping domain for American type options is a domain that, if the underlying price process enters we should exercise the option. A knock out option is a American barrier option of knock out type, but with more general shape structure of the knock out domain. An algorithm for generating the optimal stopping domain for American type knock out options is constructed. Monte Carlo simulation is used to determine the structure of the optimal stopping domain. Results of the structural, and stability of studies are presented for different models of payoff functions and knock out domains. |
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Lundgren, R. |
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Lundgren, R. Structure of optimal stopping domains for American options with knock out domains |
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Lundgren, R. |
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Lundgren, R. |
| title |
Structure of optimal stopping domains for American options with knock out domains |
| title_short |
Structure of optimal stopping domains for American options with knock out domains |
| title_full |
Structure of optimal stopping domains for American options with knock out domains |
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Structure of optimal stopping domains for American options with knock out domains |
| title_full_unstemmed |
Structure of optimal stopping domains for American options with knock out domains |
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structure of optimal stopping domains for american options with knock out domains |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4516 |
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Structure of optimal stopping domains for American options with knock out domains / R. Lundgren // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 4. — С. 98–129. — Бібліогр.: 22 назв.— англ. |
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AT lundgrenr structureofoptimalstoppingdomainsforamericanoptionswithknockoutdomains |
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2025-07-02T07:44:34Z |
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2025-07-02T07:44:34Z |
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1836520335257632768 |
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Theory of Stochastic Processes
Vol.13 (29), no.4, 2007, pp.98–129
ROBIN LUNDGREN
STRUCTURE OF OPTIMAL STOPPING
DOMAINS FOR AMERICAN OPTIONS WITH
KNOCK OUT DOMAINS
American options give us the possibility to exercise them at any mo-
ment of time up to maturity. An optimal stopping domain for Amer-
ican type options is a domain that, if the underlying price process
enters we should exercise the option. A knock out option is a Amer-
ican barrier option of knock out type, but with more general shape
structure of the knock out domain. An algorithm for generating the
optimal stopping domain for American type knock out options is con-
structed. Monte Carlo simulation is used to determine the structure
of the optimal stopping domain. Results of the structural, and sta-
bility of studies are presented for different models of payoff functions
and knock out domains.
1. Introduction
American options give us the possibility to exercise them at any moment
of time up to maturity. An optimal stopping domain for American type
options is a domain that, if the underlying price process enter it, then the
option should be exercised in order to get the maximal payoff. Studies on op-
timal stopping for Markov type processes were done in Snell (1952), Chow,
Robbins and Siegmund (1971), Van Moerbeke (1976), Shiryaev (1978),
and Peskir and Shiryaev (2006), optimal stopping for American options in
Kukush and Silvestrov (1999, 2004), Jönsson, Kukush and Silvestrov (2004,
2005) and Jönsson (2005). Other papers about optimal stopping and Amer-
ican put options are Jacka (1991) and Salminen (1999).
Similar studies on the structure of optimal stopping domains, as con-
sidered in this paper but for American options without knockout domains
have been done in Jönsson (2001). In this paper we extend these studies
to discrete time knock out American options. An option with knock out
2000 Mathematics Subject Classifications. 60G40, 91B28.
Key words and phrases. Markov process, optimal stopping, discrete time, American
option, barrier option, knock out option, Monte Carlo simulation.
98
OPTIMAL STOPPING FOR AMERICAN OPTIONS 99
domain can be seen as a barrier option of knock out type, but with region
that knocks out the option of a more general shape than for ordinary barrier
options.
Some preceding works about barrier options are Boyle and Lau (1994),
Broadie, Glasserman and Kou (1997), Baldi, Caramellino and Ivono (1999)
and Lo, Lee and Hui (2003). The use of Monte Carlo simulation in finance
was introduced by Boyle (1977). It has been extended in several papers, we
like to mention Boyle, Broadie and Glasserman (1997) and Longstaff and
Schwartz (2001). A nice survey is presented in the book by Glasserman
(2004).
We first describe an algorithm that generates optimal stopping domain
for American type knock out options. Monte Carlo simulation is used to
determine the structure of the optimal stopping domain. This paper is an
extension of Lundgren (2007) and includes a more detailed description of
the considered model, proof of the corresponding results about structure of
optimal stopping domains, several additional numerical examples of optimal
stopping domains for different payoff functions and knock out domains, as
well as of the algorithm for estimation of the probabilities of classification
errors.
2. American Options with knock out domains
We consider a price process to be of Markov type with multiplicative incre-
ments
Sx,n(k + 1) = Sx,n(k)Yk, k = n, n + 1 . . . , (1)
where Yk, k = n, n + 1, . . . are independent non-negative random variables,
x and n are starting points for the price process in prices and time, repres-
sively, with the initial value Sx,n(n) = x ∈ R+ = [0,∞).
A payoff is representated by a measurable function gn(x) : N×R+ �→ R+.
We assume the following integrability condition
Egk(Sx,n(k)) < ∞, 0 ≤ n ≤ k < ∞, x ∈ R+. (2)
Introduce a knockout domain H , and define the rules such that if the
underlying process enter the domain, the contract will be worthless. The
knock out domain H has the following discrete structure
H = {H0, H1, H2 . . .}, (3)
where Hn, n = 0, 1 . . . are some measurable subsets of R+.
Let us introduce the random time τH
x,n, the first time when the price
process enters the knock out domain,
τH
x,n = min{k ≥ n : Sx,n(k) ∈ Hk}.
100 ROBIN LUNDGREN
Let MN
x,n be the class of all Markov moments τ for the process Sx,n(k),
k = n, n + 1 . . . such that n ≤ τ ≤ N , where N is a non negative integer
number interpreted as an expiration date. Our goal is to maximize the
expected payoff
Φx,n(τ) = Ee−Rτ gτ (Sx,n(τ))χ(τ < τH
x,n) (4)
over all Markov moments τ ∈ MN
x,n, where χ(τ < τH
x,n) is the indicator
making sure that the time τ is less than the first knock out entry time τH
x,n,
and where Rn = r0 + r1 + . . . + rn, R0 = 0 and rn ≥ 0 is a non random risk
free interest rate between moments n and n + 1.
The time we are looking for is the optimal stopping time τopt i.e.,
Φx,n(τopt) = sup
τ∈MN
x,n
Φx,n(τ) (5)
The optimal stopping time for the American knock out options have
a structure similar with those for ordinary American options. Define the
operator Tn acting on a non-negative measurable function f(x) as
Tnf(x) = Ee−rnf(x · Yn+1)χ(x · Yn+1 /∈ Hn+1).
To determine the optimal stopping moment τopt the method of backward
induction is used, this gives for n = N − 1, N − 2, . . . , 0 the following
recursion {
ω0(x) = gN(x)χ(x /∈ HN)
ωN−n(x) = max{gn(x), TnωN−n−1(x)}χ(x /∈ Hn)
(6)
The set of all asset prices at moment n such that it is optimal strategy
to stop form a optimal stopping domain at moments n = 0, . . . , N ,
Γn = {x ∈ R+ : gn(x) = ωN−n(x)},
consequently the complement to Γn form the set
ΓC
n = R+\Γn = {x ∈ R+ : gn(x) < ωN−n(x)},
which is the continuation domain at moment n = 0, 1, . . . , N . Thus the
optimal stopping domain has the following discrete structure
Γ = {Γ0, Γ1, . . . , ΓN}.
The following theorem is an analogue of the general results for optimal
stopping theory of Markov processes given in the books Chow, Robbins and
Siegmund (1971), Shiryaev (1978), and Peskir and Shiryaev (2006). The
slight difference with the classical results mentioned above, is that we need
OPTIMAL STOPPING FOR AMERICAN OPTIONS 101
to take into account the possibility for the price process to enter the knock
out domain. This makes the stopping rules and the payoff dependent of the
whole past trajectory of the price process.
Theorem 1.The optimal stopping time maximizing (4) is given for every
x ∈ R+, n = 0, . . . , N by
τopt = min{k ≥ n : Sx,n(k) ∈ Γk} ∧ N (7)
and
Φx,n(τopt) = ωN(x). (8)
Proof. It is divided into two parts. The first part is to prove that the reward
function e−RnωN−n(x) is an upper bound for the functional Φx,n(τ) for any
Markov moment τ ∈ MN
x,n. Secondly we prove that this upper bound is
archived if the stopping time τ = τopt is taken according to formula (7).
Let τx,n be some Markov moment from the class MN
x,n and define a
random reward corresponding to stopping at moment τx,n as
ξx,n(τx,n) = e−Rτx,n gτx,n(Sx,n(τx,n))χ(τx,n < τH
x,n).
We start by showing that the upper bound for Eξx,n(τx,n) is given by
e−RnωN−n(x). This will be shown by the following recursive procedure.
For moment n = N we have ω0(x) = gN(x)χ(x /∈ HN). On the other
hand, in this case, τx,N = N for any Markov moment τx,N ∈ MN
x,N and
sx,N(N) = x. Therefore the expected reward
Eξx,N(τx,N) = e−RN gN(x)χ(x /∈ HN) = e−RN ω0(x).
This equality holds also for the Markov moment τopt.
For moment n = N − 1 the recursion (6) yields, ω1(x) = max{gN−1(x),
TN−1ω0(x)}χ(x /∈ HN−1), and the random reward is given by
ξx,N−1(τx,N−1) = e−RN−1gN−1(x)χ(x /∈ HN−1)χ(τx,N−1 = N − 1) +
+ e−RN gN(sx,N−1(N))χ(x /∈ HN)χ(τx,N−1 = N).
By taking the expectation and using the Markov property, which makes
event {sx,N−1(N) ∈ A} and {τx,N−1 > N−1} independent, and the structure
of the recursion (6) we have
Eξx,N−1(τx,N−1) = e−RN−1gN−1(x)χ(x /∈ HN−1)P{τx,N−1 = N − 1} +
+ e−RN−1
∫ ∞
0
e−rN gN(y)χ(x /∈ HN−1)
· χ(y /∈ HN)P{sx,N−1(N) ∈ dy, τx,N−1 > N − 1} =
= e−RN−1gN−1(x)χ(x /∈ HN−1)P{τx,N−1 = N − 1} +
102 ROBIN LUNDGREN
+ e−RN−1
∫ ∞
0
e−rN gN(y)χ(x /∈ HN−1)
· χ(y /∈ HN)P{τx,N−1 > N − 1}P{sx,N−1(N) ∈ dy} =
= e−RN−1gN−1(x)χ(x /∈ HN−1)P{τx,N−1 = N − 1} +
+ e−RN−1TN−1ω0(x)P{τx,N−1 > N − 1} ≤ e−RN−1ω1(x).
By following the same recurrent procedure we get, for n = 0, 1 . . . , N
and x ∈ R+.
Φx,n(τx,n) = Eξx,n(τx,n) ≤ e−RnωN−n(x)
At moment n = N−1 we see that the optimal stopping time τopt defined
in equation (7) has two possible values, τopt = N−1 if x ∈ ΓN−1 and τopt = N
otherwise. That is P{τopt = N − 1} = χ(x ∈ ΓN−1) and P{τopt = N} =
χ(x /∈ ΓN−1). By using these facts and the form of the recurrent formula
(6) it is seen that τopt will give the expected reward
Eξx,N−1(τopt) = e−RN−1gN−1(x)χ(x /∈ HN−1)χ(x ∈ ΓN−1) +
+ e−RN−1TN−1ω0(x)χ(x /∈ ΓN−1) = e−RN−1ω1(x).
By follow this procedure in a recurrent way backwards we see that (7)
will give the maximum reward, we get, for every n = 0, 1 . . .N, x ∈ X,
Φx,n(τopt) = e−RnωN−n(x).
The proof above is given to make this paper self readable. There exists
an alternative approach to prove Theorem 1. One can try to embed the
model in the Markov process setting and then to follow the classical proof
given in Shiryaev (1978).
This actually can be done by introducing a two dimensional Markov
process Zx,n(k) = (Sx,n(k), Ix,n(k)) where the additional component indicate
by non-knock out is be given by Ix,n(k) =
∏k
i=n χ(Sx,i /∈ Hi). �
3. Payoff functions
Theorem 1 is used to construct and analyze the algorithm for finding optimal
stopping domains.
In our experimental studies we restrict our studies to put options, the
result for call options are similar.
We also assume that the starting time is 0 and assume that the model
is homogenous in time. That is rn = r, gn(x) = g(x) and Yk, k = 0, 1, . . .
are i.i.d random variables.
A put option is a contract that gives the holder the right but not the
obligation to sell an asset to a predetermined price K know as the strike
price. The right to sell expires after a predetermined expiration time N ,
OPTIMAL STOPPING FOR AMERICAN OPTIONS 103
called maturity. Since there is a right but not an obligation to exercise the
option. The holder needs to pay a premium p for the option. But since the
holder always pays the premium p for the option, the optimal stopping time
and optimal stopping domain will not be affected of the option price p. So
we may let p = 0 for this studies of optimal stopping domains. In this paper
the option is of American type, then the holder has the right to exercise the
option at any moment of time 0 ≤ τ ≤ N . The function that determines
the value of the contract is called the payoff function. A payoff function is
a function g : R+ �→ R+. In this paper we consider several different types
of payoff functions. The first type we consider is a payoff function.
g(x) = a[K − x]+ =
{
0 if x ≥ K,
a(K − x) if x < K,
(9)
where a will determine the slope of the payoff function. If a is equal to one,
the payoff function corresponds to the payoff for the standard put option.
We then consider the piecewise linear payoff function. That payoff has
two different strike price 0 < K2 < K1 and two different slopes a1, a2 ≥ 0,
that is the scale parameters for the price intervals [K1, K2) and [K2, 0],
respectively,
g(x) =
⎧⎨
⎩
0 if x > K1,
a1(K1 − x) if K1 ≥ x > K2,
a1(K1 − K2) + a2(K2 − x) if x ≤ K2.
(10)
The piecewise payoff function can be replicated by a portfolio of two stan-
dard put options, first with strike price K1 and portfolio weight a1 and the
second with strike price K2 and portfolio weight ã2.
Next payoff function considered is the stepwise payoff function. It will
yield a constant payoff for a given interval of price in underlying. In this
paper a two step B1 < B2, stepwise payoff function is considered. The
function has the form
g(x) =
⎧⎨
⎩
0 if x ∈ [K1,∞),
B1 if x ∈ [K2, K1),
B2 if x ∈ [0, K2).
(11)
The stepwise payoff function can be replicated by a portfolio of two digital
options with strike prices K1 and K2 and with weights B1 and B̃2.
The next payoff function considered is the quadratic put. The payoff
functions has one strike price K, a scale parameter a and is given by
g(x) =
{
0 if x > K,
a(K − x)2 if x ≤ K.
(12)
104 ROBIN LUNDGREN
We also consider the logarithmic payoff function. The payoff function
also have a scale parameter a, a strike price K, and is given by
g(x) =
{
0 if x ≥ K,
a log(K − x − 1) if x < K.
(13)
4. The Monte Carlo algorithm
The algorithm presented below is similar with those described in Jönsson
(2001) for usual American options. But with the difference that the algo-
rithm below takes into account effects coming from knock out domain.
First define an upper and a lower boundary for the stock prices. Denote
su as the upper level of stock prices and sl as the lower level. Then define all
prices by sn,j = sl + jΔ, j = 0, 1, . . . , J, where sn,j is the price at moment n
with level j. Note that sn,0 = sl, define Δ in such way that sn,J = su where
n = 0, 1, . . . , N denote each moment of time. So define a mesh in time and
stock prices with discrete points (n, sn,j). In this study of optimal stopping
domains. Note that the algorithm will generate is a quasi-optimal stopping
domain, not the exact true optimal stopping domain. Further we will refer
to the quasi-optimal stopping domain as just stopping domain.
the algorithm starts from expiration date N and for each point (n, sn,j)
on the grid compare the profit we make by exercising the option at time
n with keeping the option until time n + 1. Investigate for each moment
n = 0, 1, . . . , N if the stock price are in the knock out domain sn,j ∈ Hn, and
also if the stock price between two adjacent points (n, sn,j) and (n, sn,j+1)
belongs to the stopping domain or not. That is done by constructing an
interval
In,j =
[
sn,j − Δ/2, sn,j + Δ/2
)
and if In,j ∩ Γn
= ∅ say that sn,j ∈ Γn. Also note that since the option
matures at n = N all prices at that moment belong to the stopping domain,
that is ΓN = [sl, su]. We use backward induction and next consider the
moment n = N−1, at this moment we have the choice to exercise the option
at the moment n = N −1 or to keep holding the contract until the moment
n = N . So for each j we make the comparison if g(sN−1,j) ≥ Tω0(sN−1,j)
then In−1,j ∩Γn−1,j
= ∅. So for the moment n = N −1 the stopping domain
is given by,
ΓN−1 =
⋃
j:g(sn−1,j)≥Tω0(sN−1,j )
IN−1,j.
Monte Carlo simulation is used to determine Tω0(sN−1,j). For simplicity
we introduce the short notation S
(i)
sj,n,n(k) = S(i)(k), where {S(i)
x,n(k), k ≥ n}
are independent realisations of the price process Sx,n(k), k ≥ n. For M
OPTIMAL STOPPING FOR AMERICAN OPTIONS 105
independent simulations the expected continuation profit will be given by
T̂
(M)
N−1g(sN−1,j) =
1
M
M∑
i=1
e−rg(S(i)(N))χ(S(i)(N) /∈ HN)
The approximated stopping domain will be given by
Γ̂N−1 =
⋃
j:g(sn−1,j)≥T
(M)
N−1g(sN−1,j)
IN−1,j. (14)
For moment n = N − 2 and each j the comparison g(sN−2,j) ≥ Tω0(sN−2,j)
is made, and then In−1,j ⊂ Γn−1,j. For the moment n = N − 2 the optimal
stopping domain is given as in (14). At the moment when we determine the
continuation profit, we also need to consider the possibility of entering the
stopping domain and having an early exercise at the moment n = N−1. So,
the already estimated structure of the stopping domain is considered in the
backward procedure. We also need to consider the possibility of entering
the knock out domain and getting a zero payoff. The continuation profit is
determined by
T̂
(M)
N−2g(sN−2,j) =
1
M
∑M
i=1 (e−rg(S(i)(N − 1))χ(S(i)(N − 1) ∈ Γ̂N−1)
χ(S(i)(N − 1) /∈ HN−1) + e−2rg(S(i)(N))
χ(S(i)(N − 1) /∈ Γ̂N−1)χ(S(i)(N − 1) /∈ HN−1)χ(S(i)(N) /∈ HN)).
For every moment n = 0, 1, . . . , N − 3 we have to determine the contin-
uation profit TωN−n−1(sn,j) of every stock price sn,j, j = 0, 1, . . . , J on the
grid, and use the fact that we know the structure of the stopping domain
for each moment of time n + 1, n + 2, . . . , N − 1, N and that we know the
structure of the knock out domain when making the estimates.
5. Classification errors
When we study the algorithm we study the probabilities of classification
errors. We will consider two types of classification errors. But there are two
more errors occurring, one due to the fact that we create an interval around
each point in the mesh In,j, and say that the interval belongs to Γn, or not.
In worst case this can lead to that we classify almost entire In,j wrong. The
second error occurring is due to the fact that we using already estimated
structure of the stopping domain. Since we only have an estimate of the
stopping domain used in the estimator T̂
(M)
n g(sn,j), the estimator will be
biased. But in this paper analysis of these both errors are neglected.
First we have the classification error when the algorithm indicates that
the stock price sn,j belongs to the stopping domain g(sn,j) ≥ T̂
(M)
n g(sn,j),
106 ROBIN LUNDGREN
but instead belongs to the continuation domain g(sn,j) < TωN−n−1(sn,j).
From the central limit theorem we get that the probability pn,j is
pn,j = P
{
T̂
(M)
n g(sn,j) < g(sn,j)
}
= P
{
T
(M)
n g(sn,j)−TωN−n−1(sn,j)
σn,j
√
M <
g(sn,j)−TωN−n−1(sn,j)
σn,j
√
M
}
� 1 − Φ
(
TωN−n−1(sn,j)−g(sn,j)
σn,j
√
M
)
where the standard deviation of the estimate T̂
(M)
n is given by σn,j/
√
M and
σn,j is the standard deviation of one component.
The second type of classification error we can make is when the algorithm
indicates that the price does not belong to the stopping domain g(sn,j) <
T̂
(M)
n (sn,j), but the stock price does g(sn,j) ≥ TωN−n−1(sn,j), we get by
the central limit theorem the probability of make such kind of classification
error to be
qn,j = P
{
T̂
(M)
n g(sn,j) > g(sn,j)
}
= P
{
T
(M)
n g(sn,j)−TωN−n−1(sn,j)
σn,j
√
M >
g(sn,j)−TωN−n−1(sn,j)
σn,j
√
M
}
� 1 − Φ
(
TωN−n−1(sn,j)−g(sn,j )
σn,j
√
M
)
So the probability of making such error qn,j are the same as for the
first type of error pn,j. One can use a second moment estimate similar with
T̂
(M)
n g(sn,j) to get a good estimate of σn,j . Define the estimate of the second
moment of the moment n = N − 1 as
T̃
(M)
N−1 =
1
M
M∑
i=1
e−2rg2(S(i)(N))χ(S(i)(N) /∈ HN).
For moments n = 0, 1 . . . , N − 2 the formulas are similar. Then the es-
timate of the variance is σ̃2
n,j = T̃
(M)
n g(sn,j) − (T̂
(M)
n g(sn,j))
2. Let Econt
n,j =
max(Econt
n,j , sn,j) be the continuation profit. Then define the operator L and
the dimensionless measure of the variance d2
n,j as
Ln,j =
Econt
n,j − g(sn,j)
g(sn,j)
, d2
n,j =
σ2
n,j
g(sn,j)2
.
Note that if Ln,j ≤ 0 the strategy is to exercise. Then the probability of
making classification error is given by
pn,j � 1 − Φ
(√
M
|Ln,j|
dn,j
)
.
OPTIMAL STOPPING FOR AMERICAN OPTIONS 107
sN−1,j g(sN−1,j) HN Ẽcont
N−1,j σ̃2
N−1,j L̃N−1,j d̃2
N−1,j
99.00 1.00 85.00 1.21216 1.37511 0.21216 1.37511
98.00 2.00 85.00 2.03980 1.85832 0.01990 0.46458
97.00 3.00 85.00 2.98171 2.06349 -0.00610 0.22928
96.00 4.00 85.00 3.96526 2.09163 -0.00869 0.13073
95.00 5.00 85.00 4.95841 2.06295 -0.00832 0.08252
94.00 6.00 85.00 5.95165 2.01901 -0.00806 0.05608
93.00 7.00 85.00 6.94492 1.97731 -0.00787 0.04035
92.00 8.00 85.00 7.93844 1.93460 -0.00770 0.03023
91.00 9.00 85.00 8.93252 1.89355 -0.00750 0.02338
90.00 10.00 85.00 9.92549 1.85841 -0.00745 0.01858
89.00 11.00 85.00 10.90246 1.93013 -0.00887 0.01595
88.00 12.00 85.00 11.74688 3.17346 -0.02109 0.02204
87.00 13.00 85.00 11.94892 10.83933 -0.08085 0.06414
86.00 14.00 85.00 10.46743 32.00194 -0.25233 0.16328
Table 1: Estimated values of the expected continuation profit Ẽcont
N−1,j , the
variance σ̃2
N−1,j and for the measures L̃N−1,j , d̃2
N−1,j. Note that the measure
L̃N−1,j changes sign when entering the optimal stopping domain.
Let us estimate the classification errors described above and lets con-
sider moments n = N − 1 and n = N − 2. The reason why moments so
early in the recursion are studied are, because of when using the stopping
domain for estimation of the price of the option contract, the price is more
sensitive to perturbations of the stopping domain close to maturity. We
now use Monte Carlo simulation to determine an estimate of Ẽcont
n,j .
In all simulations as was mentioned in section 3, we have assumed that
the underlying process follows a geometrical random walk, i.e as in equation
(1) where Yn are described by Yn = eμ+σWn , and Wn is a sequence of in-
dependent random variables having standard normal distribution. We will
consider a constant yearly interest rate of r = 4% a strike price of K = 100,
Δ = 1.0 and a barrier given by Hn = {x : x ≥ 85}. We use M = 107
simulations for an underlying process with a yearly drift of μ = 0.0 and
yearly volatility of σ = 0.24. It is seen in Table 1 that when sN−1,j = 97 the
measure L̃N−1,j is first time negative. From that we can conclude that the
strategy to exercise the option at that moment is more profitable than to
continue to hold the option until moment N . We also see that the variance
increases as the underlying price approaches the barrier. Note also that
Ẽcont
N−1,j for sN−1,j = 86 is less than for sN−1,j = 87, this is because of the
large probability of crossing the barrier.
In Table 2 we show the probability of making classification errors for
different number of simulations. It is seen that the probability of making
classification error has the greatest value around sN−1,j = 97, and from Ta-
ble 1 it is known that it is the boundary of the stopping domain. Note also
that the probability of making such errors are relatively large close to the
boundary even when the number of simulations are as large as M = 1 · 106.
So the number of simulations should be large close to the boundary.
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pN−1,j
sN−1,j M= 5 · 104 M= 1 · 105 M= 1 · 106 M= 1 · 107 M= 1 · 108
99.00 0.0 0.0 0.0 0.0 0.0
98.00 1.11 · 10−16 0.0 0.0 0.0 0.0
97.00 0.3292 0.2929 0.1451 0.0468 7.074 · 10−12
96.00 0.3249 0.2618 4.758 · 10−10 0.0 0.0
95.00 0.2766 0.0739 3.597 · 10−13 0.0 0.0
94.00 0.1662 0.0305 1.188 · 10−14 0.0 0.0
93.00 0.0672 0.0128 3.186 · 10−14 0.0 0.0
92.00 0.008088 0.0344 1.155 · 10−14 0.0 0.0
91.00 0.0379 0.001539 6.661 · 10−16 0.0 0.0
90.00 0.001421 0.003090 1.110 · 10−16 0.0 0.0
89.00 5.026 · 10−05 7.286 · 10−09 0.0 0.0 0.0
88.00 0.0 0.0 0.0 0.0 0.0
Table 2: The probability of classification error pN−1,j, note that the proba-
bility of making classification errors at the boundary of the stopping domain
is large and has preferred size first for M = 108 simulations.
Lets now move back in time to the moment n = N − 2 with all other
parameters the same. First it is noted that parameter L̃N−2,j become neg-
sN−2,j g(sN−2,j) HN−1 Ẽcont
N−2,j σ̃2
N−2,j L̃N−2,j d̃2
N−2,j
99.00 1.00 85.00 1.42556 2.18098 0.42556 2.18098
98.00 2.00 85.00 2.17862 2.65320 0.08931 0.66330
97.00 3.00 85.00 3.05612 2.69609 0.01871 0.29957
96.00 4.00 85.00 4.00748 2.45158 0.00187 0.15322
95.00 5.00 85.00 4.99254 2.19546 -0.00149 0.08782
94.00 6.00 85.00 5.98865 2.05237 -0.00189 0.05701
93.00 7.00 85.00 6.98787 1.98238 -0.00173 0.04046
92.00 8.00 85.00 7.98657 1.94025 -0.00168 0.03032
91.00 9.00 85.00 8.97966 1.94184 -0.00226 0.02397
90.00 10.00 85.00 9.93983 2.19356 -0.00602 0.02194
89.00 11.00 85.00 10.7710 3.60404 -0.02082 0.02979
88.00 12.00 85.00 11.1982 8.96621 -0.06681 0.06227
87.00 13.00 85.00 10.6576 22.6948 -0.18019 0.13429
86.00 14.00 85.00 8.54428 41.6160 -0.38969 0.21233
Table 3: Estimated values of the expected continuation profit Ẽcont
N−2,j, the vari-
ance σ̃2
N−2,j and for the measures L̃N−2,j, d̃2
N−2,j. Note that the measure L̃N−2,j
changes sign when entering the optimal stopping domain.
ative first when sN−2,j = 95 compared with sN−1,j = 97 in Table 1. So
the strategy to continue is more profitable compare to continue first when
the underlying asset starts at 95 for moment N − 2 compared with 97 for
moment N − 1. This effect can be explained by that the probability of the
underlying assets to go down is greater with two days to maturity than with
one day left. The same effects that was seen in Table 1 for Ẽcont
N−1,j is also
seen here for Ẽcont
N−2,j, the estimate decreases close to the barrier, but in this
case for sN−2,j = 87 instead of sN−1,j = 86 as in Table 1. The estimated
variance also increases rapidly close to the barrier, it is five times higher for
OPTIMAL STOPPING FOR AMERICAN OPTIONS 109
sN−2,j = 86 than sN−2,j = 88.
Finally in Table 4 the probabilities of making classification errors are
given for moment n = N − 2. It is again seen that the greatest probability
of making such errors are close to the boundary of the optimal stopping
domain. We also see that in order to have a good accuracy the number of
simulations is less than when moment n = N − 1 is considered. At this
moment to get a good estimate M = 1 · 106 number of simulations will give
a good estimate. However the biasses discussed earlier will at smaller n be
greater and make the total probability of classification errors greater than
for moment n = N − 1.
As mentioned above, the estimator T̂
(M)
n g(sn,j) has biasses, due to that
it uses already estimated structure of the stopping domain will make that
the probability of classification errors increases when n becomes smaller. To
pN−2,j
sN−1,j M= 5 · 104 M= 1 · 105 M= 1 · 106 M= 1 · 107 M= 1 · 108
99.00 0.0 0.0 0.0 0.0 0.0
98.00 0.0 0.0 0.0 0.0 0.0
97.00 4.59 · 10−14 0.0 0.0 0.0 0.0
96.00 0.042724555 0.023568059 3.14 · 10−08 0.0 0.0
95.00 0.412963812 0.20353611 9.92 · 10−09 0.0 0.0
94.00 0.44102954 0.030851588 8.83 · 10−12 0.0 0.0
93.00 0.125290687 3.03 · 10−04 9.61 · 10−12 0.0 0.0
92.00 0.088964356 6.66 · 10−04 0.0 0.0 0.0
91.00 2.33 · 10−04 5.65 · 10−09 0.0 0.0 0.0
90.00 0.0 0.0 0.0 0.0 0.0
89.00 0.0 0.0 0.0 0.0 0.0
88.00 0.0 0.0 0.0 0.0 0.0
Table 4: The probability of classification error pN−2,j, note that the barrier
gets estimated at sN−2,j = 95 for this setting. Also note that the greatest
probability is at the boundary of the stopping domain.
minimize this biasses Δ should be chosen small enough, and the number of
simulations M close to the boundary should be chosen large to make the
structure used accurate.
The choice of Δ will also generate classification errors, due to the con-
struction with intervals In,j. To avoid this type of error the Δ should be
chosen to be small enough. From a practical point of view the Δ should
be chosen with respect to the accuracy in comparison with the preferred
runtime of the algorithm.
As mentioned the probability of making classification errors are closest
to the boundary of the stopping domain. When knowing the structure of
the stopping domain improvements can be made. To get quicker result and
better accuracy, an interval halving method is suggested. So instead of cre-
ating a static mesh, it is quicker to, for each n first determine if the point
in the middle of the interval [sl, su] belongs to the stopping domain or the
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Figure 1: The stopping domain for a standard put payoff
continuation domain. Then by knowing the structure of the stopping do-
main to the corresponding payoff function the interval can be made smaller
to in the end being minimized around the boundary of the stopping domain
until the interval In,j is as small to have a preferred accuracy.
6. Examples of optimal stopping domains
In all examples it is assumed that the price process is following a geomet-
rical random walk, i.e given by (1) where Yn = eμ+σWn , having the daily
parameters μ = 0.0, σ = 0.05, and Wn, n = 1, 2, . . . is a sequence of i.i.d
normally distributed random variables.
Further we consider a constant yearly interest rate of r = 1%, a matu-
rity N = 50 and we use M = 100000 simulations on each point in the mesh
to evaluate if the point is included into the stopping domain or not. The
reason why such unrealistic parameters for μ and σ are studied is that the
interesting effects of the stopping domains will get more visible then.
In this section examples of stopping domains for the specified payoff
functions presented in Section 3, together with knock out domains will be
considered.
We consider the knock out domain of the following form
Hn = {x ≥ 0 : x ≤ α}, (15)
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Figure 2: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
where α ∈ R+, or
Hn =
{ ∅ n /∈ [α, β],
x c ≤ x ≤ d, n ∈ [α, β].
(16)
For each payoff function defined in section 3, investigation is made about
the structure of the stopping domain in combination with differentkinds of
knock out domains. This section is finished with a general discussion about
how the structure is generated.
Domains generated from standard put payoff function. We will
start our investigation from the simplest case, that is the payoff function
that corresponds to the standard put option. The standard put option has
the following payoff function (9). Through out all simulations in this section
we have used a payoff function having parameters K = 35 and a = 1.0.
The first example given in Figure 1, there is no knock out domain at all
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Figure 3: Upper left: The considered payoff function Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
considered. In this case it is seen that the stopping domain is monotonically
increasing in time, that because it becomes less profitable to own the option
when time decreases. This result will we use as a benchmark for standard
payoff function when investigating the influence given by different knock
out domains.
The first knock out domain considered is constant in time and is given by
equation (15) with parameter α = 20. The result is given in Figure 2, and
we see that due to the introduced knock out domain the stopping domain is
no longer monotonic in time. So the greatest payoff we can receive for this
contract is if the underlying process would hit the stopping domain close to
day 40 and we would get a payoff of 5.50. The reason of this shape is because
of that the probability of hitting the knock out domain decreasing in time
which make the continuation strategy more competitive, and whe we will
come closer to maturity N the stopping strategy is again more competitive.
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Figure 4: The stopping domain generated by a piecewise linear payoff func-
tion having parameters K1 = 35 K2 = 15 and a1 = 1.0 a2 = 3.5
Also note that if our model would be continuous then the probability
of crossing the barrier and entering the knock out domain would be zero.
Then the stopping domain would be as in Figure 1.
In the third example given in Figure 3, we consider the standard put
payoff together with a small knock out domain as a strip described by
equation (16) having parameters c = 25, d = 35, α = 33.5 and β = 32.5.
This corresponds to a small box. Here we see that the small box influence the
stopping domain all the time until day 35 when there is no more possibility
to enter the knock out domain. From this moment the stopping domain
has the same shape as for ordinary put payoff function with no knock out
domain at all, as in Figure 1. We also note that if the knock out domain are
defined as above but a option contract with maturity at day 50 and starting
at day 34 with S34 = 30 is considered. Then we may have to exercise the
option if the price goes up and if the price goes down. So the stopping
domain for this contract starting from day 34 are the same as for a contract
starting at day 0 and both are valid until day 50.
By adding more boxes the stopping domain will have more waves like in
Figure 3. If the
boxes are placed in a larger distance from the boundary of the stopping
domain given in Figure 1 the stopping domain will not grow together with
the stopping domain generated by the knock out box, and a small island
shaped stopping domain will appear instead. Similar result are presented
for quadratic payoff function in Figure 15.
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Figure 5: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
Domains generated from piecewise linear put payoff function.
We now consider a more complex stopping domain. The piecewise linear
payoff function given by equation (10). In this investigation we let the payoff
function have parameters K1 = 35 K2 = 15 and slopes a1 = 1.0 a2 = 3.5.
This contract corresponds to a portfolio of one standard put option with
strike price 35, and 2.5 options with strike price 15, all options having the
same maturity fifty days into the future. We again start with a benchmark
example having no knock out domain at all. As seen in Figure 4 the result
has a multi threshold structure and form a harbor. This structure appear
because that when the underlying process is close to K2 but have not yet
crossed it, the probability of crossing it in the future is large which makes
the continuation strategy more competitive than the stopping strategy, see
Jönsson, Kukush and Silvestrov (2004, 2005) and Kukush and Silvestrov
(2004) for more intensive studies of this payoff function and the effects
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Figure 6: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
generated by it.
Now a knock out domain is introduced, in first example we consider the
piecewise payoff function described above together with a standard barrier
of down and out type. This knock out domain is given by equation (15)
having parameter α = 10. The structure of the stopping domain generated
by this combination is presented in Figure 5. By comparing the results in
Figure 4 with Figure 5, it is seen that the introduced barrier makes the
disjoint stopping domain in Figure 4 grow together and form an inner lake.
The reason why it grows together is due to the probability of getting knocked
out is large in this area so the stopping strategy becomes more competitive,
i.e the expected reward for a price process starting in that region will be
less than the reward received if the contract it exercised at that moment. It
is again worth noting that if a contract is defined between day 25 and 50,
this closed lake is now open depending on which S0 that contract has.
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Figure 7: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
In the next example we consider the piecewise linear payoff function
together with a small knock out domain in shape of a box. The knock out
box is placed underneath the harbor in Figure 4. The box is described by
equation (16) having parameters c = 15, d = 25, α = 16 and β = 17 and
the result is presented in Figure 6. Again we compare with the result in
Figure 4 and first thing we note is that the harbor in Figure 4 is extended
from day 17 to day 0. Secondly, it is seen that this small knock out domain
generate an inner lake as in Figure 5.
But a difference from Figure 5 is that a contract starting at S0 has an
opportunity to get the higher payoff underneath the harbor.
In the last example of this section we consider the piecewise payoff func-
tion together with two knock out boxes. The setting is similar with Fig-
ure 6 but we now have two knock out boxes underneath the harbor. The
boxes are placed after each other with some space in between the boxes.
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Figure 8: The stopping domain generated by a stepwise payoff function
having parameters K1 = 40 K2 = 25 and B1 = 10 B2 = 20
The knockout domains are defined by equation (16) and having parameters
c1 = 15, c2 = 35, d1 = 25, d2 = 45, α = 16 and β = 17. We see that intro-
ducing one more box, it will generate one more inner lake in the stopping
domain. The result is presented in Figure 7.
So to summarize this section: By introducing one more box on top of the
stopping domain as in Figure 3 it will generate one wave shaped stopping
domain on top. By introducing more boxes on top close to the boundary,
the stopping domain will have more waves. By moving the boxes further
away from the boundary of the stopping domain small islands will appear
as in Figure 15, that will prevent the price process to enter the knock out
domain. Finally, by introducing more boxes underneath the harbor, more
inner lakes will appear.
Domains generated from stepwise payoff function. Now we con-
sider stopping domains generated from the two-step stepwise payoff function
defined in equation (11). The considered payoff function have parameters as
B1 = 10, B2 = 20, K1 = 40, K2 = 25. This option contract corresponds to
a portfolio of ten digital options with strike price 40, and ten digital options
with strike price 25, all having the same maturity fifty days into the future.
The first considered case is the stepwise payoff function without any
knock out domain at all as a benchmark. The result is presented in Figure
8. When looking at the figure it is firstly seen that the stopping domain
again form a harbor shaped stopping domain at the upper strike price, but
now it starts at day 0 and continues all the time up to maturity. The harbor
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Figure 9: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
shaped stopping domain starts at strike price K1 and continues until the
probability of crossing second strike price K2 is large enough. Secondly it is
seen that because of the constant reward you receive for this contract you
should exercise the option as soon as you cross a strike price K1 or K2. If
however S0 of this option contract is below K1 but above K2 the strategy
can be to continue to hold on to the option because of the high probability
of crossing also K2 and receive a greater reward. If more steps is introduced
in the payoff function, then for each step, a new harbor will appear in the
stopping domain, see Jönsson (2001) for further studies of stopping domains
for this type of payoff function.
First considered combination of the stepwise payoff function with a
knock out domain, is when the knock out domain corresponds to a bar-
rier option of down and out type with a barrier at 20. This is given by
equation (15) having parameter α = 20. The result are presented in Figure
OPTIMAL STOPPING FOR AMERICAN OPTIONS 119
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Figure 10: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
9 and it is seen that due to the large probability of hitting the knock out
domain, starting from an early moment, the harbor in Figure 8 have grown
together with the lower part of the stopping domain and, as for the piece-
wise payoff function, formed an inner lake. Again it is worth noting that if
the contract starts at day 30 the stopping domain that are giving the inner
lake structure, is then in the past for the process, and instead we again have
a harbor structure. as in Figure 8.
Second considered combination is the stepwise payoff function together
with a knock out domain having the shape of a small box given by equation
(16), having parameters c = 25, d = 35, α = 31 and β = 29. The result
is presented in Figure 10. It is seen that the stopping domain generated
by this combination almost create an inner lake similar with Figure 9 or
Figure 6 for example.
The structure in Figure 10 is due to the fact that the small knock out
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Figure 11: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
box are placed in a small continuation region, thats why the possibility for
the price process to get past is almost 0. As in previous sections if a con-
tract between day 35 and day 50 is constructed, then the starting point for
this contract will be after the knock out domain in time, and the stopping
domain for this contract will be as in Figure 10 from day 35 and forward.
Next combination is the stepwise payoff function together with two
knock out boxes.
Both knock out boxes are placed in the small continuation region un-
derneath the harbor giving the price process a small possibility to get
past them without either hit the knock out domain or the stopping do-
main. The boxes are described by equation (16) and having parameters
c1 = 25, c2 = 35, d1 = 30, d2 = 40, α1 = 29, α2 = 26 and β1 = 31, β2 = 28.
The result is presented in Figure 11. Due to the different levels in prices of
the two boxes, which gives the price process a small probability of passing
OPTIMAL STOPPING FOR AMERICAN OPTIONS 121
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Figure 12: The stopping domain generated by a quadratic put payoff func-
tion having parameters K = 35 and a = 1.0
the boxes, an inner lake are again generated. If however the distance be-
tween the harbor and the stopping domain should be greater there would be
a small opening to get pass the knock out domains and to enter the inner
lake structure in Figure 11.
The same reasoning as earlier can be applied here too. That if the
process starts at moment 41 the stopping domain creating the inner lake
are in the past and in this case the stopping domain would be similar with
the stopping domain in Figure 8 from moment 41 and forward. By adding
more small boxes as in Figure 11 small openings or inner lakes can be
constructed. There should however be noted that for this contract the pos-
sibility of changing the structure is limited to the small continuation regions
under the harbors.
Domains generated from quadratic payoff function. In this sec-
tion several combinations of knock out domains together with the quadratic
payoff function are considered. The quadratic payoff function is given by
equation (12), and we will consider following parameters, K = 35 a = 1.0.
Again the first considered case is the quadratic payoff function with no
knock out domain at all, as a benchmark. The result are presented in Figure
12.
The structure generated by the quadratic payoff function is monotonic
and looks similar with the structure generated by the standard put option
given in Figure 1, the difference is that the quadratic put payoff makes the
continuation strategy more competitive because if the underlying make a
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Figure 13: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
small change in price down, the payoff will have a big change.
Now we consider the quadratic payoff function together with a knock
out domain. The knock out domain corresponds to a straight barrier and
it is given by equation (15) with parameter α = 10. This corresponds to
a barrier put option of down and out type having a barrier at 10, with a
quadratic payoff function. The result is given in Figure 13 and it is seen that
the stopping domain is no longer monotonic. That because of the probabil-
ity of entering the knock out domain is greater when t is far from T . This
will give the effect of the boundary going down and giving the opportunity
to stop at a larger payoff at a later date. But at day 45 due to so short time
left to maturity the stopping strategy will become more profitable. These
two factors will give the structure of the stopping domain. It is also noted
that this stopping domain has similar structure as the standard put option
with a straight barrier given in Figure 2. But with the quadratic put the
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Figure 14: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
continuation strategy is more competitive which will make the continuation
region larger than for the standard put payoff.
Now instead of considering a knock out domain that is homogeneous in
time, we consider a knock out domain that has the shape of a small box
together with the quadratic payoff function. The box is given by equation
(16) with parameters c = 25, d = 35, α = 26 and β = 24. The result is
given in Figure 14. Since the box lies close enough to the stopping domain
generated by the quadratic payoff function, the stopping domain generated
by the small knock out domain has grown together with the stopping do-
main given by the quadratic put payoff function, and generated one joint
set of the stopping domain. The result in Figure 14 is similar with the result
for the standard put payoff with similar knock out domain given in Figure
3, but with the difference coming from the quadratic payoff function.
In the next case all the settings are the same apart from that the knock
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Figure 15: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
out domain has been moved further away from the stopping domain given
in Figure 12 and the box is now given by equation (16) but with parameters
c = 25, d = 35, α = 31 and β = 29. The result is given in Figure 15. Now
when the distance is greater the stopping domain is now two disjoint sets
one similar with what the quadratic put payoff would generate. Compare
with Figure 12, and we see that we first get a stopping domain similar with
Figure 12 that corresponds to the stopping domain given by the payoff func-
tion. We also have one set that is covering the knock out domain and some
small area around the knock out domain to make sure that the option will
be exercised before entering the knock out domain. The effect would be
similar if a knock out domain is introduced far from the stopping domain
generated by other payoff functions too.
Domains generated from logarithmic payoff functions. In this
section several combinations of knock out domains together with the loga-
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Figure 16: The stopping domain generated by a logarithmic put payoff
function having parameters K = 35 and a = 1.0
rithmic payoff function are considered. The logarithmic payoff function is
given by equation (13) with parameters K = 35 and a = 1.0. The first
considered example is again the logarithmic payoff function with no knock
out domain at all, as a benchmark, the result is presented in Figure 16. The
stopping domain is again similar with Figure 1 but here the stopping region
is greater, that due to the logarithmic payoff function, and the fact that a
small change in price will give an even smaller change in payoff.
Secondly, we consider a case with the logarithmic payoff function in com-
bination with a constant barrier at 20. The constant barrier are described
by equation (15) having parameter α = 20. The result here is similar as
for the standard put payoff with barrier given in Figure 2, but here the
stopping region again is greater.
Now we consider the logarithmic payoff function in combination with a
knock out domain having the shape of a small box. The box is given by
equation (16) with parameters c = 25, d = 35, α = 32 and β = 30. The
result is presented in Figure 18. We note that the knock out domain give
some effects in the stopping domain but not so visible as in previous sec-
tions.
More generally, we see that for stopping domains generated by the log-
arithmic payoff function, the effect coming from the knock out domain is
small. That effect can be explained due to the fact that a small change in
price of underlying will give an even smaller change in payoff.
Discussion. It is seen that the combination of different payoff func-
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Figure 17: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
tions with different kind of knock out domains can yield complex stopping
domains. If the stepwise payoff is excluded similar effects can be observed
for the same type of knock out domains. If Figure 2, Figure 13 and Figure
17 are compared many similarities of there structure is found. The pay-
off function will only determine the size of the stopping domain. For the
logarithmic payoff the continuation strategy is less competitive due to the
small payoff you gain for the risk you take of keeping the contract. For the
quadratic payoff it is the opposite, you will gain a large payoff for keep-
ing the contract which makes the continuation strategy more competitive.
Similar effects can also be seen in Figure 5, but it is not as visual since the
barrier is far from the upper boundary of the stopping domain. By consid-
ering the inner lake this effect is also found for the piecewise linear payoff
function.
If instead Figure 3, Figure 14 and Figure 18 are compared similarities
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Figure 18: Upper left: The considered payoff function. Upper right: The
structure of knock out domain. Down: The corresponding optimal stopping
domain.
in their structure will be found for them too, again the payoff function will
determine the size of the stopping domain. The effect seen in Figure 15
can be seen for the other payoff functions too, that if a box is placed far
away from the stopping domain, but still in-the-money, a small island will
be created around that box.
By comparing Figure 6, Figure 7, Figure 10 and Figure 11 similar ef-
fects again can be found, the difference in the structure depends upon the
discrete steps in reward for the stepwise payoff function.
To summarize: A knock out domain of barrier type will due to the prob-
ability of entering it give a convex stopping domain for all payoff functions.
The stopping domain will also suggest exercise at a lower profit than for
an ordinary contract if a barrier type option of down and out is consid-
ered. That because the probability of entering the knock out domain will
128 ROBIN LUNDGREN
affect the optimal strategy. A box shaped knock out domain close above
the boundary of the stopping domain for a payoff function with no knock
out domain, will make the combined stopping domain to cover the knock
out domain and form a wave shaped stopping domain. If the box is further
away the stopping domain will form a small island around the box and
some small surroundings. If the box is placed in between a harbor and the
stopping domain the stopping domain can grow together and form an inner
lake. By adding more boxes more waves and inner lakes will appear.
7. Conclusions
This paper presents the results of experimental studying of the structure
of optimal stopping domains for American knock out options. The optimal
stopping time is given by first hitting moment of the optimal stopping do-
main (7), the structure of optimal stopping domain is given by the recursion
(8).
From Table 1 and Table 3 it is seen that the volatility increases rapidly
and the expectation decreases close to the knock out domain. From Table
2 and Table 4, it is seen that the probability of making classification errors
has the greatest value near the boundary of the optimal stopping domain
and maturity.
Stopping domains can poses complex multi-threshold structure deter-
mined by the combinations of different payoff functions and different knock
out domains. Several combinations is presented in section 6. By knowing
the structure of the stopping domain the interesting points for investiga-
tion can be chosen, hence speed of generating the stopping domains can be
increased.
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Department of Mathematics and physics, Mälardalen University,
Box 883, SE-721 23 Väster̊as, Sweden.
E-mail: robin.lundgren@mdh.se
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