On reselling of European option
On Black and Scholes market investor buys a European call option. At each moment of time till the maturity, he is allowed to resell the option for the quoted market price. A model is proposed, under which there is
 no arbitrage possibility. It is shown that the optimal reselling problem is e...
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| Cite this: | On reselling of European option / A.G. Kukush, Yu.S. Mishura, G.M. Shevchenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 75–87. — Бібліогр.: 12 назв.— англ. |
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| author | Kukush, A.G. Mishura, Yu.S. Shevchenko, G.M. |
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| citation_txt | On reselling of European option / A.G. Kukush, Yu.S. Mishura, G.M. Shevchenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 75–87. — Бібліогр.: 12 назв.— англ. |
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| description | On Black and Scholes market investor buys a European call option. At each moment of time till the maturity, he is allowed to resell the option for the quoted market price. A model is proposed, under which there is
no arbitrage possibility. It is shown that the optimal reselling problem is equivalent to constructing nonrandom two dimensional stopping domains.
For a modified model of the market price, it is shown that the
stopping domains have a threshold structure.
|
| first_indexed | 2025-12-07T16:38:16Z |
| format | Article |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 75–87
A. G. KUKUSH, YU. S. MISHURA, AND G. M. SHEVCHENKO
ON RESELLING OF EUROPEAN OPTION
On Black and Scholes market investor buys a European call option. At
each moment of time till the maturity, he is allowed to resell the option
for the quoted market price. A model is proposed, under which there is
no arbitrage possibility. It is shown that the optimal reselling problem
is equivalent to constructing nonrandom two dimensional stopping do-
mains. For a modified model of the market price, it is shown that the
stopping domains have a threshold structure.
1. Introduction
Optimal strategies for Investor in American option were studied in the
papers [2, 4, 5, 7, 9, 12]. Construction of these strategies leads to the con-
struction of a one-dimensional stopping domain Gt for each moment t up
to maturity T . For the European call option, Investor is not entitled to
exercise the option before the time T and should wait until the maturity.
However, it is known that on real financial markets he has an opportunity to
resell the option before the maturity. Thus an investigation of the reselling
problem is essential, while, to the authors’ knowledge, there is no paper
dealing with this problem.
In this paper, we treat the following model. On the Black–Scholes security
market with an interest rate r, at the moment t0 = 0, Investor buys a
European call option with the strike price K and the maturity T , on the
stock with initial value S0, for the price CBS(S0, T ) = CBS(S0, T ; σ, K, r)
computed by the Black–Scholes formula. At any moment t ∈ (0, T ) he can
resell the option for a certain market price Cm
t , which may differ from the
“fair” price CBS(St, T − t).
The paper proposes a stochastic model of the market price Cm(t), which
does not lead to an arbitrage opportunity. It is shown that such an option,
with reselling possibility, is equivalent to certain American type derivative.
This allows to describe the optimal reselling time for the option in terms
of nonrandom stopping sets Gt, which are subsets of the two-dimensional
phase space R
+ × R
+ � (St, C
m
t ) (hereafter R
+ denotes the set of all non-
negative real numbers).
In the paper [9], analytic structure of boundary of stopping sets for the
optimal exercise of an American option is studied. For more general models
2000 Mathematics Subject Classification. Primary 62P05; Secondary 91B28, 65C50.
Key words and phrases. European option reselling, arbitrage, option market price,
implied volatility, stopping region.
75
76 A. G. KUKUSH, YU. S. MISHURA, AND G. M. SHEVCHENKO
a threshold structure of those sets is shown in [4, 5, 6], and the algorithm
is proposed for constructing the sets. This algorithm is based on dynam-
ical programming and Monte Carlo technique and relies on the threshold
structure of stopping domains.
Thus it is natural to establish a threshold structure for the stopping
sets in the problem of reselling of a European option, in particular, that
Gt is a set of points lying above a certain curve. We establish similar
threshold structure for stopping sets in the simplified model of market price
for European option, where stochastic volatility process has no memory.
The paper is organized as follows. In Section 2, it is proved that an as-
sumption about equality of a market option price and the Black–Scholes
price makes the problem of reselling lose any sense. The model for a market
price of an option in terms of implied volatility is introduced. Section 3
contains the main result about absence of arbitrage possibility in the pro-
posed model. Section 4 focuses on consistent estimates of parameters of
the option market price model. It is shown in Section 5 that the optimal
Investor’s strategy in the reselling problem is determined by nonrandom
stopping sets Gt. A modified non-arbitrage model for the option price is
proposed, in which the optimal sets have a threshold structure. A numer-
ical algorithm for construction of the optimal stopping set is given. And
Section 6 concludes.
2. Models for option market price
Consider the classical Black and Scholes market
(1)
St = S0 exp
{(
μ − σ2
2
)
t + σWt
}
,
Bt = B0e
rt, t ≥ 0.
Here St and Bt are the stock and the bond prices at the moment t, Wt
is Wiener process on the filtered probability space (Ω,F , (Ft)t≥0, P ); μ, σ,
and r are positive parameters, which we assume to be known. Positive
initial values S0 and B0 are nonrandom. Throughout the paper E[·] denotes
expectation w.r.t. P .
Assume that at the moment t0 = 0 Investor buys a European call option
with strike price K and maturity T . We suppose that the Investor buys the
option for “fair” price C0 given by the Black–Scholes formula
C0 = CBS(S0, T, σ; K, r) := S0Φ
( log S̃0
v0
+
v0
2
)
− Ke−rT Φ
( log S̃0
v0
− v0
2
)
,
where S̃0 := erT S0/K, v0 := σ
√
T , and Φ is the standard normal distribu-
tion function.
Now suppose that Investor can resell the option at any moment t for a
certain random market price Cm
t . Naturally, we will assume that Cm
0 = C0,
and Cm
T = g(ST ) = (ST − K)+ = max(ST − K, 0).
ON RESELLING OF EUROPEAN OPTION 77
The problem of optimal reselling of the option is optimization problem
(2) Ψ(τ) = E[ e−rτCm
τ ] → max
in the class of all (Markov) stopping times τ ∈ [0, T ]. The maximizing
stopping time is called optimal reselling time, and we denote it by τopt.
Later on, we will specify the filtration, under which the Markovian property
is considered.
Remark 2.1. It is natural to ask what happens in a dynamical setting, i.e.,
when Investor is allowed to dynamically trade either a stock or the option.
There are three possible cases how one can understand this dynamic trading.
We assume that dynamic trading of the bond is always allowed, that is
“selling the stock” “selling the option” means also immediate investment
into the bond.
1. Investor is not allowed to trade the stock, and is allowed to sell his
option in parts. The latter can seem meaningless, but we can understand
it as ability to sell a part of a large option holding. Thus in this sense
Investor is allowed to use some strategies of the form {γt, t ∈ [0, T ]}, where
γt ∈ [0, 1] is a decreasing adapted process indicating part of the option,
which Investor owns at the moment t. In this case Investor’s gain will be
F (γ) = −E
[ ∫ T
0
e−rtCm
t dγt
]
and it should be maximized in the set Γ of all decreasing predictable strate-
gies. Now we make two observations. The first is that F (γ) is a linear
functional, and it is continuous in the supremum norm on under some nat-
ural assumptions on Cm
t .
The second is that the set Γ is closed (in sup-norm) convex hull of the
set Ξ of processes of the form It≤τ , where τ is a stopping time. Hence
the functional F attains its maximum on Ξ. Therefore in this setting the
optimal reselling problem is reduced to (2).
2. Investor is allowed both to sell and buy either the option, or the stock
and the option. In this case we can consider the option as a new stock,
and this way we are lead to portfolio optimization in a rather standard
semimartingale setting.
3. Investor is allowed to buy and sell the stock and is allowed to sell the
option (possibly, in parts). This seems to be the most interesting setting.
It is worth to note that, in contrast to the first case, trading of the option
is truly dynamic now, because Investor can invest more to the stock after
he has resold the option, and the set of strategies is not convex anymore.
2.1. The case where market price coincides with “fair price”.
Assume that for all moments t the market price Cm
t is equal to the Black–
Scholes price Ct = CBS(St, T − t; σ, K, r),
(3) CBS(St, T − t) = StΦ
( log S̃t
vt
+
vt
2
)
− Ke−r(T−t)Φ
( log S̃t
vt
− vt
2
)
,
78 A. G. KUKUSH, YU. S. MISHURA, AND G. M. SHEVCHENKO
where
(4) S̃t := er(T−t)St/K, vt := σ
√
T − t.
It is known that there is a measure P ∗ such that Ste
−rt is a martingale w.r.t
the filtration Ft. This is equivalent to the fact that
W̃t = Wt +
μ − r
σ
t
is Wiener process under P ∗. Then
(5) Cm
t = CBS(St, T − t) = E∗[ e−r(T−t)g(ST ) | Ft ].
Here E∗[·] denotes (conditional) expectation with respect to the measure
P ∗. We will consider Markov property w.r.t. the filtration Ft. Then
(6) Ψ(τ) = e−rT E[ Yτ ],
where Yt = E∗[ g(ST ) | Ft ], 0 ≤ t ≤ T .
Lemma 2.2.The process Yt is a
a) P -supermartingale for μ ≤ r,
b) P -submartingale for μ ≥ r.
Consequently, Yt is a P -martingale for μ = r.
Proof. a) Suppose μ ≤ r. First note that
St ≥ S ′
t := S0 exp
{(
μ − σ2
2
)
t + σW̃t
}
,
hence g(ST ) ≥ g(S ′
T ). Since the distribution of S ′
t w.r.t. P ∗ is the same as
of St w.r.t. P , we can write for t ≥ s
E[ Yt | Fs ] = E
[
E∗[ g(ST ) | Ft ] | Fs
]
≥ E
[
E∗[ g(S ′
T ) | Ft ] | Fs
]
= E
[
E[ g(ST ) | Ft ] | Fs
]
= Ys,
which proves the statement a).
Statement b) is proved similarly.
Corollary 2.3. If an option market price coincides with the Black–Scholes
price, then
a) τopt = 0 for μ < r,
b) τopt = T for μ > r,
c) any stopping time is optimal for μ = r.
Proof. This follows immediately from formula (6) and Lemma 2.2, and from
the observation that Yt is a strict P -sub- or P -supermartingale in case of
strict inequalities in Lemma 2.2.
In other words, if the market price is given by the “fair” Black–Scholes
price, then
b) it makes no difference when to resell it for μ = r,
b) it should be resold immediately for μ < r,
c) it should be held till the maturity for μ > r.
ON RESELLING OF EUROPEAN OPTION 79
Thus under conditions of Corollary 2.3 the problem (2) has no practical
sense.
2.2. Stochastic model for the market price. For arbitrary moment
t ∈ [0, T ), an implied volatility σt is defined as a solution to the equation
(7) CBS(St, T − t; σt, K, r) = Cm
t , σt > 0.
If the right-hand side satisfies
(8) (St − Ke−r(T−t))+ < Cm
t < St
then this equation has a unique solution, since the left-hand side is contin-
uous, increasing in σt, tends to (St − Ke−r(T−t))+, as σt → 0, and tends to
St, as σt → +∞. It is natural to assume that inequalities (8), being univer-
sal bounds for the option price, are always valid. Thus we can construct a
model for the market price in terms of the implied volatility σt. Note that
Cm
0 = C0 implies σ0 = σ.
We model this implied volatility σt as a stochastic volatility. Here we
assume that it satisfies a linear stochastic differential equation
(9)
dσt
σt
= α dt + β dW 1
t ,
where W 1
t is Wiener process on (Ω, F̃t, P ) with a new filtration (F̃t)t≥0. The
model (9) is a standard model of stochastic volatility, see, e.g., [1].
Also we will assume the following:
Wiener processes Wt and W 1
t
(10) are jointly Gaussian and positively correlated.
This condition can be understood as follows. If St grows, then so does σt,
which makes Cm
t go beyond the “fair” price Ct = CBS(St, T − t, σ). This
corresponds to Investor’s aim to hold an option if the stock price is growing.
On the other hand, when the stock price drops, Investor is willing to get
rid of an option, which makes Cm
t go below its “fair” price.
Set Gt = Ft∨F̃t. In the reselling model (1), (7), (9) the optimal stopping
time τopt is defined as maximizer of Ψ(τ) in a class of all stoping times w.r.t.
the filtration Gt.
3. Absence of arbitrage
We start with standard definition of arbitrage.
Definition 3.1. In the model (1), (7), (9) a stopping time τ is said to
provide an arbitrage possibility, if
a) P (e−rτCm
τ ≥ C0) = 1,
b) P (e−rτCm
τ > C0) > 0.
80 A. G. KUKUSH, YU. S. MISHURA, AND G. M. SHEVCHENKO
If there is no such stopping time, then the model is called arbitrage-free.
Before stating the main result, it is convenient to introduce dimension-
free variables. Denote
C̃t = er(T−t)Ct/K, C̃m
t = er(T−t)Cm
t /K,
vm
t = σt
√
T − t.
Recall that S̃t and vt were introduced in (4). Then the Black–Scholes for-
mula is reduced to
(11) C̃t = fBS(S̃t, vt),
where
fBS(s, v) = S̃tΦ
( log S̃t
vt
+
vt
2
)
− Φ
( log S̃t
vt
− vt
2
)
,
while vm
t and C̃m
t are related by an equality
C̃m
t = fBS(S̃t, v
m
t ).
Now, the universal bounds (8) take the form
(S̃t − 1)+ < C̃m
t < S̃t.
We can assume that the the observed process (S̃t, C̃
m
t ) takes values in the
phase space
V := {(s, c) : (s − 1)+ ≤ c ≤ s}.
Theorem 3.2.There is no arbitrage possibility in the model (1), (7), (9).
Proof. Step 1. First we study the properties of the function f = fBS . We
are particularly interested in the properties of an (implicit) function s(v)
such that f(s(v), v) = f(s0, v0), s0 = S̃0 and v0 is volatility at the moment
0. We have
f ′
s(s, v) = Φ
( log s
v
+
v
2
)
,
f ′
v(s, v) =
s1/2
√
2π
exp
{
−
(v2
8
+
log2 s
2v2
)}
.
This implies that the function s(v) is decreasing. Moreover, we claim that it
is infinitely differentiable due to such property of f and the implicit function
theorem. Define for α > 0 gα(u) = s(uα), u0 = v
1/α
0 . Now we are going to
show that for some α ∈ (0, 1)
(12) gα(u) > lα(u) := g′
α(u0)(u − u0) + s0, for all u > 0, u
= u0.
First note that s(0+) = f(s0, v0) + 1 > s0, thus for some δ > 0 we have
s(v) > s0 on (0, δ], and gα(u) > s0 on (0, δ1/α]. Further,
lα(0) = g′
α(u0)(−u0)+ s0 = s0−αs′(uα
0 )uα
0 = s0−αs′(v0)v0 → s0, α → 0+ .
Thus for some α0 > 0 we have gα(u) > lα(0) ≥ lα(u) for all α ∈ (0, α0) and
u ∈ (0, δ1/α].
ON RESELLING OF EUROPEAN OPTION 81
Next,
g′′
α(u) = αuα−2
(
αs′′(uα)uα + (α − 1)s′(uα)
)
=
= −αuα−2s′(uα)
(
1 − α − α
uαs′′(uα)
s′(uα)
)
and the last term here can be evaluated by implicit differentiation
a(v) :=
vs′′(v)
s′(v)
= − v2
4
+
log2 s
v2
+
exp
{
− log2 s
2v2 − v2
8
}
Φ
(
log s
v
+ v
2
)
√
2
πs
( log s
v
− v
2
)
+
exp
{
− log2 s
v2 − v2
4
}
πsΦ2
(
log s
v
+ v
2
) ,
where s is an abbreviation for s(v). Clearly, a(v) → −∞ as v → +∞.
This means that it is bounded from above on [δ, +∞) by some number
M . Then for α ∈ (0, 1
M+1
) and u ∈ [δ1/α, +∞) it holds g′′
α(u) < 0, and
the function gα is strictly convex on [δ1/α, +∞), therefore gα(u) > lα(u),
u ∈ [δ1/α, +∞) \ {u0}.
Thus we have (12) for all α ∈ (0, min(α0,
1
M+1
)).
Step 2. Take α > 0 such that (12) holds. Recall that S̃t and σt are
geometric Brownian motions, therefore σ
1/α
t is also a geometric Brownian
motion, thus there exists an equivalent measure P α, such that S̃t and σ
1/α
t
are martingales w.r.t. P α. Consequently, (vm
t )1/α is a supermartingale w.r.t.
P α:
d(vm
t )1/α = d(σ
1/α
t (T − t)1/2α) = (T − t)1/2αd(σ
1/α
t )− 1
2α
σ
1/α
t (T − t)1/2α−1dt.
Now assume that τ is such that it satisfies Definition . In terms of C̃m
t ,
this is equivalent to the following conditions:
a′) P (C̃m
τ ≥ C̃0) = 1,
b′) P (C̃m
τ > C̃0) > 0.
Condition a′) means that f(S̃τ , v
m
τ ) ≥ f(s0, v0). But f(s, v) is increasing in
s, hence
S̃τ ≥ s(vm
τ ) = gα((vm
τ )1/α).
By (12), we can write further
S̃τ ≥ lα((vm
τ )1/α).
Taking expectations w.r.t. P α, we get by martingale property of S̃t and
supermartingale property of (vm
t )1/α, that
s0 ≥ s0 + g′
α(u0)E
α[ (vm
τ )1/α − u0 ] ≥ s0,
with equality possible only if S̃τ = lα((vm
τ )1/α) (mod P α). The last means
in particular that gα((vm
τ )1/α) = lα((vm
τ )1/α) (mod P α). But then from (12)
82 A. G. KUKUSH, YU. S. MISHURA, AND G. M. SHEVCHENKO
it follows that v
1/α
τ = u0 (mod P α), i.e., vτ = v0 (mod P α). On the other
hand,
S̃τ = gα((vm
τ )1/α) = s(v0) = s0 (mod P α).
This, however, contradicts b′), as P α is equivalent to P .
Remark 3.3. It is easy to see that Theorem 3.2 remains valid if only
the following is assumed: for each α small enough there exists probability
measure P α such that St and σ
1/α
t are martingales w.r.t. P α. This allows
to consider a wide class of semimartingale models of the stock price and
volatility, which satisfy the Novikov condition.
4. Estimation of model parameters
Estimation of α, β, and correlation coefficient ρ. According to the
condition (10), the processes Wt and W 1
t are positively correlated and we
denote the correlation coefficient by ρ. Then W 1
t can be decomposed as
(13) W 1
t = ρWt + γW 2
t , γ =
√
1 − ρ2,
where W 2
t is a Wiener process independent of Wt. Hence the processes S̃t
and W 2
t are independent as well, moreover,
(14) log
σt
σ
= x log
S̃t
S̃0
+ yt + zW 2
t ,
where
(15) x :=
βρ
σ
, y := α − β2
2
− (μ − r − σ2
2
)
βρ
σ
, z := βγ.
Suppose that the processes S̃t and σm
t are observed at the moments t0 <
t1 < t2 < · · · < tn. Then by (14) we have for k = 0, . . . , n − 1
(16)
1
bk
log
σm
tk+1
σm
tk
= x
1
bk
log
S̃tk+1
S̃tk
+ ybk + zεk,
where Δtk := tk+1 − tk, bk :=
√
Δtk, εk := (W 2
tk+1 − W 2
tk
)/bk. Random
variables {εk} are independent and have standard Gaussian distribution.
Denote
Uk =
1
bk
log
σtk+1
σtk
, ak =
1
bk
log
S̃tk+1
S̃tk
.
Then
(17) uk = xak + ybk + zεk, k = 0, . . . , n − 1.
This is a linear multiple regression model with a random regressor ak and
a nonrandom regressor bk; zεk are the observation errors with variance z2.
ON RESELLING OF EUROPEAN OPTION 83
Define the design matrix and the response vector
A =
⎡
⎣ a0 b0
...
...
an−1 bn−1
⎤
⎦ , u = (u0, . . . , un−1)
�.
Then, see e.g. [8], the maximum likelihood estimator for x and y in the
model (17) coincides with a least squares estimator and is given by the
formula
(18) (x̂, ŷ)� = (A�A)−1A�u.
An unbiased estimator for parameter z is
(19) ẑ =
⎛
⎝
∥∥∥u − A
(
x̂
ŷ
)∥∥∥
n − 2
⎞
⎠
1/2
.
Under fairly mild conditions estimates (18), (19) are strongly consistent
and asymptotically normal, as n → ∞. We substitute x̂, ŷ instead of x, y
into (15), solve the system for α, β, and ρ, and get consistent estimators
of unknown parameters. Remember that the parameters μ, σ, and r are
assumed to be known.
Thus the proposed model (1), (7) is identifiable, i.e., additional parame-
ters α, β, ρ are uniquely determined by the processes St and Cm
t .
5. Investor’s strategy in the reselling problem
5.1. Stopping sets. In dimension-free variables, the problem (2) is
equivalent to the optimization problem
(20) Ψ̃(τ) := E[ C̃m
τ ] → max
in the class of all Gt-stopping times. This problem is a problem of optimal
realization of American type option with pay-off function
(21) g̃(S̃t, C̃
m
t ) := C̃m
t ,
this is an option with maturity T on (correlated) stocks S̃t and C̃m
t . As far
as there is no discounting factor in (20), the interest rate for such an option
is r̃ = 0. Then (see e.g. [11]) the optimal reselling (or option exercise) time
τopt is given by the formula
(22) τopt = inf{t ∈ [0, T ] | C̃m
t ∈ Gt},
where the nonrandom stopping sets are given by
(23) Gt = {(s, c) ∈ V | c = ft(s, c)},
the function ft(s, c) is the reward function,
(24) ft(s, c) := sup
τ∈[t,T ]
E[ C̃m
τ | S̃t = s, C̃m
t = c ],
84 A. G. KUKUSH, YU. S. MISHURA, AND G. M. SHEVCHENKO
the supremum is taken over all Gt-stopping times, which belong to [0, T ].
Since ft is jointly continuous, Gt is a closed subset of V .
Unfortunately, we failed to prove that Gt has a threshold structure, i.e.,
that it consists of points lying beyond a certain curve. We thus propose a
modification of the model (7), which has the stopping sets with the required
property.
5.2. Modified model for option market price. For the model (1),
(7) we can rewrite (14) in a form
(25)
σt
σ
=
( S̃t
S̃0
)x
eyt+zW 2
t , t ∈ [0, T ].
We assume that a transaction can be made only at one of a finite number
of moments
(26) t ∈ ΠN := {t0 := 0 < t1 < · · · < tN := T},
and that the stock price St and the option market price Cm
t are observed
only at these moments. Instead of the relation (25), we adopt the following:
(27)
σt
σ
=
( S̃t
S̃0
)x
eyt+z
√
tεt , t ∈ ΠN ,
where εtk , k = 0, . . . , N are i.i.d. variables with standard Gaussian distrib-
ution, which are independent of St. The conditional distribution L(σt | S̃t)
in the models (25) and (27) is the same. The relation (27) means that
the additional randomness on the reselling market has no memory and is
renewed at each new moment, while in the model (25) the randomness is
accumulated from the previous trading periods.
It can be shown by reasoning similar to the proof of Theorem 3.2 that
in the discrete time model (1), (27) there is no arbitrage possibility as well
(with slightly modified definition of an arbitrage, which involves ΠN -valued
stopping times). Then the optimal reselling problem (2) is formulated in
this class of stopping times.
Remark 5.1. In the model (1), (27) the parameters can be estimated
similarly to the discussion of subsection 4.2, if we consider linear regression
of the response variable 1√
t
log
σm
t
σ
to the covariates 1√
t
log St
S0
and
√
t.
The optimal reselling problem in the model (1), (27) can be reduced to
the problem of optimal exercise of the corresponding American option with
discrete time, similarly to subsection 5.1. Then the optimal reselling time
equals
(28) τopt = min{tk : (S̃tk , C̃
m
tk
) ∈ Fk},
ON RESELLING OF EUROPEAN OPTION 85
where Fk, k = 0, 1, . . . , N are some nonrandom stopping sets. For k = N
we have FN = V , and for k = 0, . . . , N
Fk := {(s, c) ∈ V : c ≥ fk(s)},(29)
fk(s) := sup
τ∈[tk+1,T ]∩ΠN
E[ C̃m
t | S̃tk = s ],(30)
where the upper bound is taken over all Gt-stopping times.
Note that the reward function (30), in contrast to (24), depends only on
s. The reason is independence of εt in (27): indeed, for (s, c) ∈ V one has
E[ C̃m
τ | S̃tk = s, C̃m
tk
= c ] = E[ C̃m
τ | S̃tk = s ]
almost surely for τ ≥ tk+1.
We see that in the model (1), (27) the stopping sets have threshold struc-
ture. If in the model (27) the parameter x > 0, then the premium function
(30) is increasing and monotone. Moreover, at the moment tN−1 preceding
the maturity,
fN−1(s) = E[ C̃m
T | S̃tN−1
= s ] = E[ (S̃T − 1)+ | S̃tN−1
= s ]
= fBS(s, σ
√
T − tN−1),
where the function fBS is defined by (11). Thus for the moment tN−1 the
stopping set is known:
(31) FN−1 = {(s, c) ∈ V | c ≥ fBS(s, σ
√
T − tN−1)}.
The function fBS is strictly convex in s, therefore the threshold curve for
FN−1 is strictly convex.
5.3. Construction of the stopping sets in the modified model.
Following [3] and [10], we apply the dynamical programming and Monte
Carlo technique to construct stopping sets Fk in the model (1), (27). The
set FN−1 is already constructed in (31). The stopping set FN−2 is then built
in the following manner.
Fix the vertical lines grid s = si, i ≥ 1, in the phase space V . On the
line s = si we have to find a threshold point fN−2(si). In order to do that,
we first take a point M1 = (si, c1) ∈ V , and we should decide, whether or
not M1 ∈ V . First we simulate a path (S̃t, C̃
m
t ), t = tN−2, tN−1, tN , which
starts at M1. If it gets to FN−1 at the moment tN1 , we put τ(1) = tN−1
and stop, otherwise we put τ(1) = tN , and we calculate C̃m
τ(1) for this path.
Repeating this procedure for M paths, we obtain values C̃m
τ(k), k = 1, . . . , M .
If c1 ≥ 1
M
∑M
k=1 C̃m
τ(k), then we make a decision that M1 ∈ FN−2, otherwise
we decide that M1 /∈ FN−2. The threshold point M(si, fN−2(si)) is found
by the dichotomy procedure.
The similar operations are made for all vertical lines of the fixed grid,
and we use the fact that the function fN−2 is increasing. This way we have
obtained a discrete approximation of the threshold curve. Then we make a
86 A. G. KUKUSH, YU. S. MISHURA, AND G. M. SHEVCHENKO
linear interpolation. The points lying on this curve and above form a set
F̂N−2, which is an approximation to FN−2.
Next we construct an approximation F̂N−3 of FN−3 similarly. Again, we
fix a grid of vertical lines in the phase space V . Then a trial point M1 =
(si, c1) ∈ V is taken, and we generate a path (S̃t, C̃
M
t ), t = tN−3, . . . , tN ,
starting at M1. If it gets to F̂N−2 at the moment tN−2, we set τ(1) =
tN−2 and stop, if it gets to FN−1 at tN−1, we set τ(1) = tN−1 and stop,
otherwise we set τ(1) = tN . Repeating this procedure M times, we get
corresponding values C̃m
τ(k), k = 1, . . . , M . If c1 ≥ 1
M
∑M
k=1 C̃m
τ(k), we decide
that M1 ∈ FN−3, otherwise M1 /∈ FN−3. Then we again use dichotomy,
get approximation of the threshold curve and corresponding approximation
F̂N−3 of the stopping set FN−3. Next we construct F̂N−4, . . . , F̂0. Thus we
utilize the dynamic programming to construct stopping sets backwards in
time, using Monte Carlo method to calculate the reward function (24).
6. Conclusion
We have considered the problem of European option reselling and pro-
posed a stochastic model for option market price. For a wide class of models,
which includes the proposed one, absence arbitrage opportunities is shown.
Optimal strategy for Investor in this model is described by nonrandom
stopping sets in the phase space of possible stock prices and option market
prices. For the modified model the threshold structure is established.
Acknowledgements
The first author was supported by Royal Swedish Academy of Sciences
within the project “Nonlinear stochastic dynamic models of price processes”
and by Catholic University of Leuven grant. The third author was sup-
ported by INTAS foundation grant YSF 03-55-2447.
Authors expess their gratitude to Prof. W. Schoutens (Belgium), and
Prof. D. Silvestrov and Dr. A. Malyarenko (Sweden) for profitable discus-
sions.
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Department of Probability Theory and Mathematical Statistics, Kyiv
National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: alexander kukush@univ.kiev.ua
E-mail address: myus@univ.kiev.ua
E-mail address: zhora@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4459 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T16:38:16Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kukush, A.G. Mishura, Yu.S. Shevchenko, G.M. 2009-11-11T15:21:39Z 2009-11-11T15:21:39Z 2006 On reselling of European option / A.G. Kukush, Yu.S. Mishura, G.M. Shevchenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 75–87. — Бібліогр.: 12 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4459 On Black and Scholes market investor buys a European call option. At each moment of time till the maturity, he is allowed to resell the option for the quoted market price. A model is proposed, under which there is
 no arbitrage possibility. It is shown that the optimal reselling problem is equivalent to constructing nonrandom two dimensional stopping domains.
 For a modified model of the market price, it is shown that the
 stopping domains have a threshold structure. en Інститут математики НАН України On reselling of European option Article published earlier |
| spellingShingle | On reselling of European option Kukush, A.G. Mishura, Yu.S. Shevchenko, G.M. |
| title | On reselling of European option |
| title_full | On reselling of European option |
| title_fullStr | On reselling of European option |
| title_full_unstemmed | On reselling of European option |
| title_short | On reselling of European option |
| title_sort | on reselling of european option |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4459 |
| work_keys_str_mv | AT kukushag onresellingofeuropeanoption AT mishurayus onresellingofeuropeanoption AT shevchenkogm onresellingofeuropeanoption |