Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process
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. In Kukush et al. (2006) On reselling of European option, Theory Stoch. Process., 12(28), 75-87, a similar problem was investigate...
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| Cite this: | Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process / M. Pupashenko, A. Kukush // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 114-128. — Бібліогр.: 6 назв.— англ. |
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| author | Pupashenko, M. Kukush, A. |
| author_facet | Pupashenko, M. Kukush, A. |
| citation_txt | Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process / M. Pupashenko, A. Kukush // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 114-128. — Бібліогр.: 6 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. In Kukush et al. (2006) On reselling of European option, Theory Stoch. Process., 12(28), 75-87, a similar problem was investigated for another model of the market price. We propose a more realistic model based on Cox-Ingersoll-Ross process. Discrete approximation for this model is investigated, which is arbitrage–free. For this discrete model, a formula for penultimate optimal stopping domains is derived.
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Theory of Stochastic Processes
Vol.14 (30), no.3-4, 2008, pp.114-128
MYKHAILO PUPASHENKO AND ALEXANDER KUKUSH
RESELLING OF EUROPEAN OPTION IF THE
IMPLIED VOLATILITY VARIES AS
COX-INGERSOLL-ROSS PROCESS
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. In Kukush et al. (2006) On
reselling of European option, Theory Stoch. Process., 12(28), 75-87,
a similar problem was investigated for another model of the market
price. We propose a more realistic model based on Cox-Ingersoll-Ross
process. Discrete approximation for this model is investigated, which
is arbitrage–free. For this discrete model, a formula for penultimate
optimal stopping domains is derived.
1. Introduction
In this paper we consider the European call option. For this type of
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 we investigate the reselling 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 the
European call option with the strike price K and the maturity T on the
stock with initial value S0, at the price C0 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 Ct.
The paper is organized as follows. In Section 2 we propose a model for
the market price in terms of implied volatility, where the latter follows Cox-
Ingersoll-Ross process. In Section 3 we describe optimal stopping domains
in terms of implied volatility. Sections 4-6 focuse on discrete approximation
Invited lecture.
2000 Mathematics Subject Classifications: 62P05, 65C50, 91B28.
Key words and phrases. Arbitrage, Cox-Ingersoll-Ross process, European option re-
selling, implied volatility, optimal stopping domain, option market price.
114
RESELLING OF EUROPEAN OPTION 115
of the proposed model and some properties of it are derived. The formula for
penultimate stopping domain is derived in Section 7. Section 8 contains the
main result that the proposed discrete model is arbitrage–free, and Section
9 concludes.
2. Model for option market price
Consider the classical Black and Scholes market in continuous time [5]:{
St = S0e
(μ−σ2
2
)t+σWt , t ≥ 0,
Bt = B0e
rt, t ≥ 0.
(1)
Here μ, σ, and r are positive parameters, S0 and B0 are positive and nonran-
dom, Wt is Wiener process on the filtered probability space (Ω,F , (Ft)t≥0, P ).
Consider a European call option with maturity T and pay-off function
g(ST ) = (ST −K)+ = max{ST −K, 0}. We suppose that the Investor buys
the option at fair price
C0 = E∗
S0
e−rTg(ST ) =: f(S0, T ; σ, r), (2)
Here E∗ is expectation w.r.t. the martingale measure P ∗, and E∗
S0
denotes
the expectation (w.r.t. P ∗) provided S0 is the value of the stock price at
t = 0. It is well known that under P ∗, μ = r holds.
Now, suppose that the Investor can resell the option at any moment
t ∈ [0, T ] for a certain market price Cm
t . Naturally, we assume that
Cm
0 = C0, Cm
T = g(ST ). (3)
The problem of the optimal reselling of the option is an optimization
problem:
Ψ(τ) := Ee−rτCm
τ → max (4)
in the class of all (Markov) stopping times τ ∈ [0, T ]. The maximizing time
is called an optimal reselling time and we denote it by τopt.
The ”fair” market price at moment t ∈ [0, T ] equals
Ct = f(St, T − t; σ, r) := E∗[e−r(T−t)g(ST )|St]. (5)
Corollary 2.3 from [2] states the following:
Corollary 1. If an option price coincides with the Black-Scholes price,
then:
a) τopt = 0 if μ < r,
b) τopt = T if μ > r,
c) any stopping time is optimal if μ = r.
116 M. PUPASHENKO AND A. KUKUSH
If Cm
t = Ct for all t ∈ [0, T ] then Corollary 1 holds true, and the problem
(4) has no practical sense. Therefore we choose a stochastic model for the
market price. At any moment t ∈ [0, T ] an implied volatility σt is defined
as a solution to the equation
f(St, T − t; σt, r) = Cm
t , σt > 0. (6)
Under natural assumptions, see [2], the equation (6) has unique solution.
Note that Cm
0 = C0 implies σ0 = σ.
We model σt as a stochastic volatility. The model for σt as geometric
Brownian motion presented in [2] is not appropriate in practical sense, be-
cause of σt can considerably deviate from σ, as t grows. Instead we propose
the model based on Cox-Ingersoll-Ross process, see [1]:
dσ2
t = −α(σ2
t − σ2)dt+ β
√
σ2
t dW
′
t , σ0 = σ, (7)
where α, β > 0 and β2 ≤ 2ασ2, W
′
t is a Wiener process on (Ω,F , (F̃t)t≥0, P ),
with a new filtration (F̃t)t≥0. Under imposed restrictions on α, β, and σ,
the process (7) is well defined for all t ≥ 0 [1]. Also we assume the following:
Wiener processes Wt and W
′
t
are jointly Gaussian and positively correlated.
(8)
This condition can be understood as follows. If St grows, then so does σt,
which makes Cm
t go beyond the ”fair” price Ct = f(St, T − t; σ, r). 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, the Investor is willing to
get rid of an option, which makes Cm
t go below its ”fair” price.
Let Gt = Ft ∨ F̃t. In the reselling model (1), (6), and (7), the optimal
stopping time τopt is defined as a maximum point of Ψ(τ) in a class of all
stopping times w.r.t. the filtration Gt.
3. Stopping sets
Similarly to Section 5.1 from [2] we have
τopt = inf{t ∈ [0, T ] : (St, C
m
t ) ∈ Gt}, (9)
where nonrandom stopping sets are given by
Gt = {(s, c)|s ≥ 0, c = ft(s, c)}, (10)
the function ft(s, c) is a reward function,
ft(s, c) := sup
τ∈[t,T ]
E[e−r(τ−t)Cm
τ |St = s, Cm
t = c], (11)
RESELLING OF EUROPEAN OPTION 117
and the upper bound is taken over all Gt - stopping times valued in [t, T ].
Since ft is jointly continuous, Gt is a closed subset of [0,∞) × [0,∞). By
definition, ft(0, c) = 0.
It is helpful to rewrite (9)-(11) in terms of St and σt:
τopt = inf{t ∈ [0, T ] : (St, σt) ∈ Ht}, (12)
Ht := {(s, d)|s ≥ 0, d = ht(s, d)}, (13)
ht(s, d) := sup
τ∈[t,T ]
E[e−r(τ−t)Cm
τ |St = s, σt = d]. (14)
Relations (9)-(11), as well as (12)-(14), are based on the next observa-
tion. The problem (4) is a problem of optimal realization of an American
type option on two correlated assets St and Cm
t with pay-off function
g(St, C
m
t ) = Cm
t . (15)
4. Discrete approximation
In order to construct ε−optimal strategies of Investor, see [3], we deal
with discrete approximations. Divide [0, T ] into n parts, and let Δ = T/n.
We approximate the model (1) by the discrete model, which is a famous
approximation of the Black-Scholes market by the Cox-Ross-Rubinstein
market: ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
Sn(tj+1) = Sn(tj)e
σ
√
Δδj ,
Btj = Btj−1
erΔ, j = 1, n, tj = j
n
T, where
δj = δnj = ±1 - i.i.d. with distribution:
P (δj = 1) = eμΔ−e−σ
√
Δ
eσ
√
Δ−e−σ
√
Δ
=: pn,
P (δj = −1) = 1 − pn =: qn,
(16)
Consider the reselling problem for the binomial market (S(tj), Btj , j =
0, n). The fair value of the European option with the pay-off from Section
2 equals [4]:
C0n = E∗
S0
e−rTg(ST ) =: fn(S0, T ; σ, r).
Here E∗ corresponds to the martingale measure P ∗
n , for which instead of
(16) we have
P ∗(δj = 1) =
erΔ − e−σ
√
Δ
eσ
√
Δ − e−σ
√
Δ
=: p∗n.
The fair value at the moment t = k
n
T equals
Ctn = E∗[e−r(T−t)g(ST )|St] =: fn(St, T − t; σ, r).
Introduce a market option price Cm
tn with Cm
0n = C0n, C
m
Tn = CTn = g(ST ).
118 M. PUPASHENKO AND A. KUKUSH
For t ∈ { k
n
T, k = 0, n− 1}, the implied volatility σnt is defined as a
solution to the equation
Cm
tn = fn(St, T − t; σnt, r). (17)
We model the implied volatility as a stochastic volatility in such a way,
that the process σn(t), which is a linear interpolant of {σnt, t = 0, T
n
, . . . , T},
converges to {σt, t ∈ [0, T ]} from (7) in the sense of the first two moments.
The σn(t) is presented in more detail in Section 5.
5. Approximation of implied volatility
In this section we derive the discrete approximation of the implied
volatility described by model (7).
Return back to (7). Let
zt = σ2
t , t ≥ 0; z0 = σ2. (18)
Then by (7) we have
dzt = −α(zt − σ2)dt+ β
√
ztdW
′
t , z0 = σ2. (19)
The stochastic differential equation (19) describes the Cox-Ingersoll-
Ross process. Its first two moments and covariance function are as follows,
see [6]:
Ezt = σ2, (20)
V arzt =
σ2β2
2α
(1 − e−2αt), (21)
Cov(zt, zs) =
σ2β2
2α
· e−α(s+t)(e2α(s∧t) − 1), (22)
where s ∧ t := min(s, t).
Based on (19) we propose an approximation scheme for zt. From (19)
we have
zt+h − zt = −α
∫ t+h
t
(zt − σ2)dt+ β
∫ t+h
t
√
ztdW
′
t .
Then,
zt+h − zt ≈ −α(zt − σ2)h+ β
√
hztγth, γth ∼ N(0, 1). (23)
We need an approximation with Bernoulli variables. Therefore instead of
γth we use εth which equals ±1 with equal probabilities. Then,
zt+h ≈ zt(1 − αh) + αhσ2 + β
√
hztεth. (24)
RESELLING OF EUROPEAN OPTION 119
Now we use the relation (24) for a uniform partition of [0, T ] with step
Δ = T
n
. We write zj = zn(
j
n
T ), j = 0, n. Here is the approximation scheme:
zj+1 = zj(1 − αΔ) + αΔσ2 + β
√
zjΔεnj, (25)
where εnj = ±1 with equal probabilities, and {εnj, j = 0, n} is i.i.d. se-
quence. Then we set zn(t) = zj , t ∈ [ j
n
T, j+1
n
T ), j = 0, n− 1; zn(T ) = zn.
Next we find the upper and lower bounds for all zj , j = 0, n.
Lemma 1. For β2 < 2ασ2 and Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
< 1
α
we have the lower
bound for zj:
zj > Z(L) :=
(
−β√Δ +
√
β2Δ + 4α2σ2Δ2
2αΔ
)2
, for all j = 0, n. (26)
Proof. We prove by induction. For all Δ > 0, the base of induction z0 > Z(L)
holds true, indeed
z0 = σ2 =
(
−β√Δ + β
√
Δ +
√
4α2σ2Δ2
2αΔ
)2
> Z(L).
For fixed j ≤ n− 1 we assume that zj > Z(L) and want to prove that
zj+1 > Z(L). For Δ > 0 we have that Z(L) from (26) satisfies
Z(L)(1 − αΔ) + αΔσ2 − β
√
Z(L)Δ = Z(L). (27)
Now we will prove the next inequality for β2 < 2ασ2 and Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
:
zj(1 − αΔ) + αΔσ2 − β
√
Δzj > Z(L)(1 − αΔ) + αΔσ2 − β
√
ΔZ(L). (28)
We can rewrite (28) as follows:
(zj − Z(L))(1 − αΔ) > β
√
Δ(
√
zj −
√
Z(L)).
Since zj > Z(L) and Δ < 1
α
we have
√
zj +
√
Z(L) >
β
√
Δ
(1 − αΔ)
,
√
zj +
√
Z(L) > 2
√
Z(L),
and we can prove that:√
Z(L) =
−β√Δ+
√
β2Δ+4α2σ2Δ2
2αΔ
≥ β
√
Δ
2(1−αΔ)
⇔
⇔ (1 − αΔ)
√
β2Δ + 4α2σ2Δ2 ≥ β
√
Δ ⇔
⇔ 2α(2ασ2 − β2) + αΔ(β2 − 8ασ2) + 4α4σ2Δ2 ≥ 0 ⇐
⇐ 2α(2ασ2 − β2) + αΔ(β2 − 8ασ2) ≥ 0 ⇔ 0 < Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
.
120 M. PUPASHENKO AND A. KUKUSH
Then by (27) and (28) we have for β2 < 2ασ2 and Δ < 4ασ2−2β2
α(8ασ2−β2)
next
relations:
zj+1 ≥ zj(1 − αΔ) + αΔσ2 − β
√
Δzj >
> Z(L)(1 − αΔ) + αΔσ2 − β
√
ΔZ(L) = Z(L).
Lemma 1 is proven.
Note that from Lemma 1, for β2 < 2ασ2 and Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
, it follows
immediately that zj > 0 for all j = 0, n, and relation (25) is well defined.
But it is easy to show that for the positivity of zj, the condition β2 < 2ασ2
can be subsided.
Lemma 2. For Δ < 1
α
we have the upper bound for zj:
zj < Z(U) :=
(
β
√
Δ +
√
β2Δ + 4α2σ2Δ2
2αΔ
)2
, for all j = 0, n. (29)
Proof. We prove by induction. For all Δ > 0, the base of induction z0 < Z(U)
holds true, indeed
Z(U) =
(
β
√
Δ +
√
β2Δ + 4α2σ2Δ2
2αΔ
)2
> σ2 = z0.
For fixed j ≤ n− 1 we assume that zj < Z(U) and want to prove that
zj+1 < Z(U). For Δ > 0 we have that Z(U) from (29) satisfies
Z(U)(1 − αΔ) + αΔσ2 + β
√
ΔZ(U) = Z(U). (30)
Now, we have the next inequalities for Δ < 1
α
:
zj+1 ≤ zj(1 − αΔ) + αΔσ2 + β
√
Δzj <
< Z(U)(1 − αΔ) + αΔσ2 + β
√
ΔZ(U). (31)
Then (30) and (31) imply
zj+1 ≤ zj(1 − αΔ) + αΔσ2 + β
√
Δzj <
< Z(U)(1 − αΔ) + αΔσ2 + β
√
ΔZ(U) = Z(U).
Lemma 2 is proven.
In the next two lemmas we prove the convergence
zn(t) → zt, as n→ ∞ (32)
RESELLING OF EUROPEAN OPTION 121
in the sense of convergence for the first two moments of finite-dimensional
distributions. Denote uj = zj − σ2. From (25) we get
uj+1 = uj(1 − αΔ) + β
√
Δ(uj + σ2)εnj, u0 = 0. (33)
Lemma 3. Convergence (32) holds true in the sense of the first moments,
that is
lim
n→∞
Ezn(t) = Ezt.
Proof. First we consider t = T . From (33) we have
Euj+1 = Euj(1 − αΔ), Eu0 = 0,
therefore
Euj = 0, j = 0, n. (34)
Then
lim
n→∞
Ezn = lim
n→∞
E(un + σ2) = σ2 = EzT .
Next, verify (32) for all t ∈ (0, T ). We have zn(t) = uλn+σ2, λn =
[
t
Δ
]
;
λ→ t
T
, as n→ ∞. Then by (33) we have
Euλn = 0, (35)
and
lim
n→∞
Ezλn = lim
n→∞
E(uλn + σ2) = σ2 = Ezt.
Lemma 3 is proven.
Lemma 4. Convergence (32) holds true in the sense of the second moments,
that is
lim
n→∞
Cov(zn(s), zn(t)) = Cov(zs, zt).
Proof. Similarly to Lemma 1, first we consider t = T . From (33) we get
V ar(uj+1) = V ar(uj(1 − αΔ)) + β2V ar(εnj ·
√
uj + σ2)Δ.
We have
V ar(εnj
√
uj + σ2) = V ar
√
uj + σ2V arεnj + (E
√
uj + σ2)2V arεnj +
+(Eεnj)
2V ar
√
uj + σ2 = V ar
√
uj + σ2 + (E
√
uj + σ2)2 =
= E(uj + σ2) = σ2,
and
V ar(uj+1) = V ar(uj(1 − αΔ)) + β2σ2Δ, V ar(u0) = 0.
122 M. PUPASHENKO AND A. KUKUSH
Then by induction we obtain the variance of uj:
V ar(uj) = σ2β2Δ
j−1∑
i=0
(1 − αΔ)2i = σ2β21 − (1 − αΔ)2j
2α− α2Δ
, j = 1, n;
V ar(u0) = 0, (36)
moreover
V ar(un) = σ2β21 − (1 − αΔ)2n
2α− α2Δ
. (37)
Now,
(1 − αΔ)2n = (1 − αΔ)
1
αΔ
·2nαΔ = (1 − αΔ)
1
αΔ
·2αT → e−2αT , as n→ ∞.
Then from (37) we have
lim
n→∞
V ar(zn) = lim
n→∞
V ar(un) =
σ2β2
2α
(1 − e−2αT ) = V ar(zT ).
Next, similarly to Lemma 1, verify (32) for all t ∈ (0, T ). We have
zn(t) = uλn + σ2, λn =
[
t
Δ
]
; λ→ t
T
, as n→ ∞.
By (36) we get
V ar(uλn) = σ2β21 − (1 − αΔ)2λn
2α− α2Δ
. (38)
Next,
(1 − αΔ)2λn = (1 − αΔ)
1
αΔ
·2αnλΔ = (1 − αΔ)
1
αΔ
·2αTλ → e−2αt, as n→ ∞.
Then from (38) we have the desirable convergence of variances:
lim
n→∞
V ar(zλn) = lim
n→∞
V ar(uλn) =
σ2β2
2α
(1 − e−2αt) = V ar(zt).
Next we have to verify the convergence of the covariance functions. First
we show that the covariance functions of uj , ui and zj , zi are equal. We
assume j > i (the case j = i was considered earlier). We have
Cov(zi, zj) = Cov(ui, uj),
Cov(ui, uj) = E
[
ui(uj−1(1 − αΔ) + β
√
(uj−1 + σ2)Δεnj−1)
]
=
= (1 − αΔ)E(uiuj−1) + β
√
ΔE(
√
uj−1 + σ2εnj−1ui) =
= (1 − αΔ)Cov(ui, uj−1) + β
√
ΔEεnj−1 ·E(
√
uj−1 + σ2ui) =
= (1 − αΔ)Cov(ui, uj−1).
RESELLING OF EUROPEAN OPTION 123
Then by induction for j ≥ i, using (36) we have:
Cov(zi, zj) = (1−αΔ)j−iV ar(zi) = σ2β2(1−αΔ)i+j
(1 − αΔ)−2i − 1
2α− α2Δ
. (39)
Similarly to the convergence of variances, we have the convergence of co-
variance functions:
Cov(uλ1n, un) → Cov(ut, uT ),
Cov(uλ2n, uλ1n) → Cov(us, ut), as n→ ∞,
where for all s, t such that 0 < s < t < T , λ1n = [ t
Δ
], λ2n = [ s
Δ
] and
λ1 → t
T
, λ2 → s
T
, as n→ ∞.
Lemma 4 is proven.
6. Correlation
Now due to condition (8) we want to derive a correlation between εnj
and δnj . Let
EWtW
′
t = 2ρ1t, 0 < ρ1 <
1
2
. (40)
Here 2ρ1 is the correlation coefficient between the two processes. Now,
we introduce a correlation between εnj and δnj due to the table of joint
probabilities:
δn�εn -1 1
-1 (1 − pn) − 1−ρ
2
1−ρ
2
1 pn − ρ
2
ρ
2
Then Eεnj = 0, Eδnj = pn− (1− pn) = 2pn− 1, and Eδnj → 0, as n→ ∞.
Next we find the covariance between εnj and δnj.
E(εnj · δnj) =
(
ρ
2
+ (1 − pn) − 1 − ρ
2
)
−
(
pn − ρ
2
+
1 − ρ
2
)
= 2ρ− 2pn,
and we investigate its convergence:
Cov(εnj, δnj) = E(εnj · δnj) − E(εnj)E(δnj) → 2ρ− 1, as n→ ∞.
Then we set ρ = ρ1 + 1
2
which corresponds to (40). Moreover from (40) we
have that 1
2
< ρ < 1.
124 M. PUPASHENKO AND A. KUKUSH
7. Structure of stopping domain in discrete time
For fixed Δ, we have two sequences
Sj = Sn(tj), j = 0, n, n =
T
Δ
, (41)
σj = σn(tj), j = 0, n, σ0 = σ, (42)
where σ is historical volatility, which describes the behavior of the stock
price St in continuous time. In discrete time we have
τopt = τopt,n = min{tj : (σj , sj) ∈ Γj}, (43)
where Γj ⊂ [0,∞) × [0,∞).
Consider European call option with pay-off function from Section 2. If at
the moment tj the relation eσ
√
Δ(n−j) · Sj ≤ K holds true, then Sn(T ) ≤ K,
and the market option price equals zero at moments t ≥ tj . Therefore we
have the next relation:
[0,∞) × [0, Ke−σ
√
Δ(n−j)] ⊂ Γj , j ≤ n− 1.
Due to Section 3 we can write the stopping sets as follows:
Γj =
{
(v, s) ⊂ [0,∞) × [0,∞) : Cm
tj
∣∣∣
Sj=s,σj=v
≥ (44)
≥ sup
tj+1≤τ≤T
E
[
e−r(τ−tj) · Cm
τ
∣∣Sj = s, σj = v
]}
∪
∪
(
[0,∞) × [0, Ke−σ
√
Δ(n−j)]
)
, j ≤ n− 1;
Γn = [0,∞) × [0,∞).
We can simplify this formula for the case j = n− 1. Introduce a function
p∗n(v) =
erΔ − e−v
√
Δ
ev
√
Δ − e−v
√
Δ
. (45)
First, we find the LHS of inequality in (44):
Cm
tj
∣∣∣
Sj=s,σj=v
= fn(s,Δ(n− j), v) = E∗
σj=v
[
e−rΔ(n−j)g(Sn)
∣∣Sj = s
]
=
= e−rΔ(n−j)
n−j∑
l=0
(
n− j
l
)
g(sev
√
Δle−v
√
Δ(n−j−l))(p∗n(v))
l(1 − p∗n(v))
n−j−l,
and for j = n− 1 we get:
Cm
tn−1
∣∣
Sn−1=s,σn−1=v
= e−rΔ
[
g(sev
√
Δ)p∗n(v) + g(se−v
√
Δ)(1 − p∗n(v))
]
. (46)
RESELLING OF EUROPEAN OPTION 125
Then, we can rewrite the RHS of inequality in (44) for j = n− 1:
E
[
e−rΔCm
T
∣∣Sn−1 = s, σn−1 = v
]
= e−rΔE
[
g(Sn)
∣∣Sn−1 = s, σn−1 = v
]
=
= e−rΔ
[
g(seσ
√
Δ)pn + g(se−σ
√
Δ)(1 − pn)
]
, (47)
where pn is defined in (16).
Then from relations (44), (46), and (47) we can write the formula for
Γn−1:
Γn−1 =
{
(v, s) ⊂ [0,∞) × [0,∞) : g(sev
√
Δ)p∗n(v) +
+g(se−v
√
Δ)(1 − p∗n(v)) ≥ g(seσ
√
Δ)pn + g(se−σ
√
Δ)(1 − pn)
}
∪
∪
(
[0,∞) × [0, Ke−σ
√
Δ]
)
. (48)
Now, based on (48) we plot some stopping domains (all plots are in the
coordinate plane (v, s)). Parameters of our model are the following:
K = 5, n = 100, α = 2, σ = 1, β =
√
2, ρ = 0.7, r = 0.1,
and μ takes one of three values: 0.05, 0.1, or 0.2.
Stopping sets Γn−1 for three relations between μ and r on the plane (v, s):
8. Absence of arbitrage
In this section we consider Markov stopping times with discrete values
{t0, t1, . . . , tn} = {0,Δ, . . . , nΔ}. We start with a standard definition of
arbitrage.
Definition 1. In a model (16), (17), (18), and (25) a stopping time τ
provides an arbitrage possibility if:
a) P (e−rτCm
τ ≥ C0) = 1,
126 M. PUPASHENKO AND A. KUKUSH
b) P (e−rτCm
τ > C0) > 0.
Lemma 5. For β2 < 2ασ2 and Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
we have the next inequality:
σj+1∗(σj) < σj < σ∗
j (σj+1), for all j = 0, n− 1, (49)
where
σj+1∗(σj) :=
√
σ2
j (1 − αΔ) + αΔσ2 − βσj
√
Δ,
σ∗
j+1(σj) :=
√
σ2
j (1 − αΔ) + αΔσ2 + βσj
√
Δ.
Proof. First we prove that σj+1∗(σj) < σj . We have to verify that:
σ2
j (1 − αΔ) + αΔσ2 − βσj
√
Δ < σ2
j ⇔ αΔσ2
j + βσj
√
Δ − αΔσ2 > 0.
It is easy to solve the latter inequality:
σj >
−β√Δ +
√
β2Δ + 4α2σ2Δ2
2αΔ
,
but this holds true for β2 < 2ασ2 and Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
due to Lemma 1.
Next, we prove σ∗
j+1(σj) > σj . We must verify that:
σ2
j (1 − αΔ) + αΔσ2 + βσj
√
Δ > σ2
j ⇔ αΔσ2
j − βσj
√
Δ − αΔσ2 < 0.
It is easy to solve the last inequality:
0 < σj <
β
√
Δ +
√
β2Δ + 4α2σ2Δ2
2αΔ
,
which holds true for Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
< 1
α
due to Lemma 2. Lemma 5 is
proven.
Theorem 1. For β2 < 2ασ2 and Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
there is no arbitrage pos-
sibility in the discrete model (16), (17), (18), and (25).
Proof. We construct a martingale measure P ∗∗ = P ∗∗
n under which e−rtCm
t
is a martingale w.r.t. the filtration Gt (if such a measure exists then there
is no arbitrage possibility), which means:
E∗∗
(
e−rΔ(j+1)Cm
Δ(j+1)
∣∣∣Gj) = e−rΔjCm
Δj ⇔
⇔ E∗∗
(
e−rΔ(j+1)Cm
Δ(j+1)
∣∣∣Sj , σj) = e−rΔjCm
Δj ⇔
⇔ E∗∗
(
fn(Sj+1, T − Δ(j + 1); σj+1, r)
∣∣∣Sj, σj) =
= fn(Sj, T − Δj; σj , r). (50)
RESELLING OF EUROPEAN OPTION 127
Further in this section we write f j(Sj, σj) = fn(Sj , T − Δj; σj , r).
We choose a measure P ∗∗ such that εnj, j = 0, 1, ..., n have the same
distribution as under P ∗
n , and εnj and δnj are independent with
P ∗∗(εnj = 1) = h(Sj), P ∗∗(εnj = −1) = 1 − h(Sj).
Then by (25) and (49) we have
E∗∗
(
f j+1(Sj+1, σj+1)
∣∣∣Sj, σj) =
= E∗∗
(
f j+1(Sj+1, σ
∗
j+1(σj))h(Sj) + f j+1(Sj+1, σj+1∗(σj))(1 − h(Sj))
∣∣∣Sj, σj) =
= f j(Sj , σ
∗
j+1(σj))h(Sj) + f j(Sj , σj+1∗(σj))(1 − h(Sj)).
Now we can rewrite (50):
f j(Sj, σ
∗
j+1(σj))h(Sj) + f j(Sj, σj+1∗(σj))(1 − h(Sj)) = f j(Sj , σj).
And we can choose h(Sj) as follows:
h(Sj) =
f j(Sj, σj) − f j(Sj, σj+1∗(σj))
f j(Sj, σ∗
j+1(σj)) − f j(Sj, σj+1∗(σj))
, (51)
and 0 < h(Sj) < 1 for β2 < 2ασ2 and Δ ≤ 4ασ2−2β2
α(8ασ2−β2)
by the strict mono-
tonicity of f j in the second argument and by Lemma 5.
Thus P ∗∗ is equivalent to P , and P ∗∗ is a martingale measure. Theorem
1 is proven.
9. Conclusion
We considered the reselling problem for European option and proposed
a stochastic model for the market option price. For this model, we con-
structed the discrete approximation and proved the absence of arbitrage in
it. Optimal strategy of the Investor in this discrete model is described by
nonrandom stopping sets in the phase space of possible implied volatility
and stock price. We derived the formula for penultimate stopping domain.
The structure of this domain is illustrated by plots for some numerical values
of parameters.
References
1. Cox, J., Ingersoll, J., and Ross, S. (1985) A theory of the term structure of
interest rates, Econometrica, 53, 385-407.
2. Kukush, A.G., Mishura, Yu.S., and Shevchenko, G.M. (2006) On reselling
of European optoin, Theory Stoch. Process., 12(28), 75-87.
128 M. PUPASHENKO AND A. KUKUSH
3. Kukush, A.G., and Silvestrov, D.S. (1999) Optimal stopping strategies for
American type options with discrete and continuous time, Theory Stoch.
Process., 5(21), 71-79.
4. Shiryaev, A.N., Kabanov, Y.M., Kramkov, O.D., and Mel’nikov, A.V.,
(1994) Towards the theory of pricing of option of both European and Amer-
ican types. I: Discrete time, Theory Probab. Appl., 39, no.1, 23-79.
5. Shiryaev, A.N., Kabanov, Y.M., Kramkov, O.D., and Mel’nikov, A.V.,
(1994) Towards the theory of pricing of option of both European and Amer-
ican types. II: Continuous time, Theory Probab. Appl., 39, no.1, 80-129.
6. Swishchuk, A. (2004) Modeling of variance and volatility swaps for financial
markets with stochastic volatility, WILMOTT Magazine September, Issue
2, 64-72.
Department of Probability Theory and Mathematical Statistics,
National Taras Shevchenko University of Kyiv, Volodymyrska st.
64, 01033 Kyiv, Ukraine.
E-mail: myhailo.pupashenko@gmail.com
Department of Mathematical Analysis, National Taras Shevchenko
University of Kyiv, Volodymyrska st. 64, 01033 Kyiv, Ukraine.
E-mail: alexander kukush@univ.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4573 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T17:17:50Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Pupashenko, M. Kukush, A. 2009-12-07T15:37:58Z 2009-12-07T15:37:58Z 2008 Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process / M. Pupashenko, A. Kukush // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 114-128. — Бібліогр.: 6 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4573 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. In Kukush et al. (2006) On reselling of European option, Theory Stoch. Process., 12(28), 75-87, a similar problem was investigated for another model of the market price. We propose a more realistic model based on Cox-Ingersoll-Ross process. Discrete approximation for this model is investigated, which is arbitrage–free. For this discrete model, a formula for penultimate optimal stopping domains is derived. en Інститут математики НАН України Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process Article published earlier |
| spellingShingle | Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process Pupashenko, M. Kukush, A. |
| title | Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process |
| title_full | Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process |
| title_fullStr | Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process |
| title_full_unstemmed | Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process |
| title_short | Reselling of European option if the implied volatility varies as Cox-Ingersoll-Ross process |
| title_sort | reselling of european option if the implied volatility varies as cox-ingersoll-ross process |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4573 |
| work_keys_str_mv | AT pupashenkom resellingofeuropeanoptioniftheimpliedvolatilityvariesascoxingersollrossprocess AT kukusha resellingofeuropeanoptioniftheimpliedvolatilityvariesascoxingersollrossprocess |