Finance Stochast. 1, 239–250 (1997)
c Springer-Verlag 1997
On Leland’s strategy of option pricing
with transactions costs
Yuri M. Kabanov1,? , Mher M. Safarian2
1 Central Economics and Mathematics Institute of the Russian Academy of Sciences, and Laboratoire
de Mathématiques, Université de Franche-Comté, 16 Route de Gray, F-25030 Besançon Cedex,
France
2 Humboldt University, Unter den Linden, 6, D-10117 Berlin, Germany
Abstract. We compute the limiting hedging error of the Leland strategy for the
approximate pricing of the European call option in a market with transactions
costs. It is not equal to zero in the case when the level of transactions costs is a
constant, in contradiction with the claim in Leland (1985).
Key words: Transactions costs, asymptotic hedging, call option, Black-Scholes
formula
JEL classification: G13
Mathematics Subject Classification (1991): 90A09, 60H05
1. Introduction
In his paper devoted to the problem of option pricing in the presence of transactions costs, Heyne Leland (1985) suggested a trading strategy based on the nice
idea of a periodic revision of a hedging portfolio using modified Black–Scholes
betas. He assumed that the level k of transactions costs is a constant and claimed
that the terminal value of the portfolio approximates the payoff as the length of
a revision interval tends to zero. In a footnote remark he also mentioned that the
same holds also when the level is k0 n −1/2 , n being the number of revision intervals. Both of these results are considered very helpful for practitioners, and the
paper is widely quoted in the literature. However, Leland’s arguments were on
a heuristic level and his conclusions have to be considered only as conjectures.
? The research for this paper was carried out during a visit to Humboldt University at Berlin
supported by the Volkswagenstiftung. The authors express their thanks to Uwe Küchler for fruitful
discussions and to anonymous referees for important remarks and corrections.
Manuscript received: February 1996; final version received: October 1996
240
Yu.M. Kabanov, M.M. Safarian
Recently, Lott (1993) provided a rigorous mathematical proof of the footnote
remark (together with a study of another approximating strategy). In the present
note we show that, unfortunately, the main conjecture of Leland for the case of
constant level of the transactions costs fails, and we calculate the hedging error.
We also prove that the approximation result still holds in the case where the level
is k0 n −α , α ∈]0, 1/2[, k0 > 0.
2. Description of the model
The stock price dynamics is given by the geometric Brownian motion
St = S0 exp{(µ − σ 2 /2)t + σWt },
where W is the Wiener process. The bond price is constant over time and equal
to one (certainly, this is not a restriction). In the absence of transactions costs
the “fair” price at time t of the European call option maturing at T = 1 with the
striking price K , i.e. with the terminal payoff H = h(S1 ) = (S1 − K )+ , is given
by the Black–Scholes formula Vt = C (t, St ) where
√
(1)
C (t, x ) = C (t, x , σ) := x Φ(d ) − K Φ(d − σ 1 − t),
Φ is the standard normal distribution function with the density ϕ, and
d = d (x , σ) :=
ln(x /K ) 1 √
√
+ σ 1 − t.
σ 1−t 2
The terminal payoff is replicated by the value at maturity of the self-financing
portfolio which has initial endowment C (0, S0 ) and at time t contains φt :=
Cx (t, St ) units of the stock (and hence Vt − Cx (t, St )St units of the bond).
Assume that in the stock market the cost of a single transaction is a fixed
fraction of its trading volume and the corresponding coefficient is k = kn (our
definition corresponds to one half of Leland’s round trip coefficient). Let us
b (0, S0 ) and
consider the self-financing trading strategy with initial endowment C
n
the portfolio containing at time t a number ξt of shares of the stock given by
the formula
ξtn :=
n
X
i =1
φbti −1 I]ti −1 ,ti ] (t) =
n
X
bx (ti −1 , Sti −1 )I]ti −1 ,ti ] (t)
C
i =1
b (t, x ) := C (t, x , σ
bx (t, St ),
b), φbt := C
where ti := i /n, C
r
r
γ
2 √
2
2
2
, γ := 2
k n=2
k0 n 1/2−α
σ
b := σ 1 +
σ
π
π
(2)
(to simplify formulae we omit the dependence on n in obvious cases). The value
process now has the form
On Leland’s strategy of option pricing with transactions costs
b (0, S0 ) +
Vt (ξ ) = C
Z
t
ξun dSu − k
n
0
X
241
Sti |ξtni − ξtni −1 |.
(3)
ti ≤t
Remarks. 1) We follow here the definition adopted by Lott (1993). Leland (1985)
considered instead of self-financing an H -admissible strategy but the problem is
essentially the same.
2) A reader may have some trouble with boundary effects. There is a transaction
at time t = 0 when the investor enters the market. The last transaction at t = 1 is
also special: the contract may admit different specification for the final settlement,
e.g., to deliver or not to deliver the stock. We exclude these particular transactions
from our considerations.
Theorem 1 Assume that k = kn = k0 n −α where α ∈]0, 1/2], k0 > 0. Then
P − lim V1 (ξ n ) = H .
n→∞
(4)
Theorem 2 Let k = k0 > 0 be a constant. Then
P − lim V1 (ξ n ) = H + J1 − J2
n→∞
(5)
where
J1 := min{K , S1 },
(
2 )
Z
1 ∞ S1
v ln(S1 /K ) 1
√ G(S1 , v, k0 ) exp −
+
J2 = J2 (k0 ) :=
d v,
4 0
2
v
2
v
Z ∞
2
1
2k0 ln(S1 /K )
k0 √
+ √ e −x /2 dx
G(S1 , v, k0 ) := √
x −
2π −∞
2πv
2π
(6)
(7)
(8)
Remark.
The integral in (8) can be calculated explicitly, in particular, G(S1 , v, 0) =
√
2/ 2π. From the other hand, it is easy to check that
(
2 )
Z ∞
1
v ln(S1 /K ) 1
S1
√ exp −
J1 := min{K , S1 } = √
+
d v = J2 (0).
2
v
2
v
2 2π 0
It follows that 0 ≤ J2 − J1 ≤ Bk0 where the constant B depends on S1 and K .
Thus, the option is always underpriced in the limit though the hedging error is
small for small values of k0 (see Fig. 1).
3. Conclusion
We have shown that the limiting error in Leland’s hedging strategy for the approximate pricing of the European call is equal to zero only when the level of
transaction costs decreases to zero as the revision interval tends to zero. In the
case when the level of transaction costs is a constant the limiting hedging error
is given in Theorem 2 and, in general, is not equal to zero.
242
Yu.M. Kabanov, M.M. Safarian
0
-0.2
J(1)-J(2)
154
-0.4
152
-0.6
-4
150
S
-3.5
148
-3
lg k
-2.5
146
-2
Fig. 1. Dependence of J1 − J2 on k = k0 and S = S1 for K = 150
Appendix A. Proof of Theorem 1
We start with calculations that are common for both theorems, assuming also
without loss of generality that P is the risk-neutral measure (i.e., µ = 0).
b do not depend on n) has been
The case of α = 1/2 (when σ
b and hence C
considered in Lott (1993). So we suppose from now on that α ∈ [0, 1/2[. This
implies that σ
b2 = O(n 1/2−α ) → ∞ as n → ∞). However, some ideas from
Lott’s study work well here and we use them in several places below, e.g., in
Lemma 4.
By the Ito formula we have that
bx (0, S0 ) + Mtn + Ant
bx (t, St ) = C
C
where
Z
Mtn
:=
Ant
:=
Z t
b
bxx (u, Su )d wu ,
Cxx (u, Su )dSu =
σSu C
0
0
Z t
1 2 2b
b
Cxt (u, Su ) + σ Su Cxxx (u, Su ) du.
2
0
t
The process M n is a square integrable martingale on [0, 1] with
( 2 )
Z t
1
σ2
ln(Ss /K ) 1 √
n
√
+ σ
b 1−s
exp −
ds.
hM it =
2π 0 σ
b2 (1 − s)
σ
b 1−s 2
Following Lott we represent the difference V1 − H in the form convenient
for a further study.
On Leland’s strategy of option pricing with transactions costs
243
Lemma 1 We have V1 − H = F1 (ξ n ) + F2 (ξ n ) where
Z
F1 (ξ n )
1
:=
(ξtn − φbt )dSt ,
0
F2 (ξ n )
1
γσ
2
:=
Z
1
(9)
bxx (t, St )dt − k
St2 C
0
n
X
|ξtni − ξtni −1 |Sti .
(10)
i =1
Proof. According to the Black–Scholes theorem the claim H admits the representation
Z 1
φu dSu .
(11)
H = C (0, S0 ) +
0
Comparing (3) and (11) we get that
Z
V1 − H
1
(ξtn
=
− φbt )dSt +
Z
0
−k
1
b (0, S0 ) − C (0, S0 )
(φbt − φt )dSt + C
0
n
X
|ξtni − ξtni −1 |Sti .
i =1
We have left to check that
b (0, S0 ) − C (0, S0 ) = 1 γσ
C
2
Z
1
bxx (t, St )dt −
St2 C
0
Z
1
(φbt − φt )dSt .
0
This identity follows easily from the Ito formula and the observation that C (t, x )
is the solution of the parabolic equation (1/2)σ 2 x 2 Cxx + Ct = 0 with the boundary
bxx + C
bt = 0
b (t, x ) is the solution of (1/2)b
condition C (1, x ) = h(x ) while C
σ2 x 2 C
with the same boundary condition. Lemma 2 For any α ∈ [0, 1/2[
P − lim F1 (ξ n ) = 0.
n→∞
(12)
Proof. By the Lenglart inequality (see, e.g., Jacod and Shiryayev (1987)) it is
sufficient to show that
Z
1
P − lim
n→∞
(ξtn − φbt )2 σ 2 St2 dt = 0.
(13)
0
Here the integrand is bounded (by σ 2 supt≤1 St2 ). But for all t < 1 we have that
ξtn → 1 and φbt → 1 as n → ∞. The study of F2 (ξ n ) is more delicate.
P5
Put ∆t := 1/n. It is easily seen that F2 (ξ n ) = i =1 Lni where
244
Yu.M. Kabanov, M.M. Safarian
Ln1
σ
:=
Z
γ
2
bxx (t, St )dt
St2 C
0
−σ
Ln2
1
γ
2
Z
1
0
n
X
bxx (ti −1 , Sti −1 )I]ti −1 ,ti ] (t)dt,
St2i −1 C
i =1
n
γX 2 b
σ
Sti −1 Cxx (ti −1 , Sti −1 )∆t
2
:=
i =1
n
X
−k σ
bxx (ti −1 , Sti −1 )|wti − wti −1 |,
St2i −1 C
i =1
Ln3
kσ
:=
n
X
bxx (ti −1 , Sti −1 )|wti − wti −1 |
St2i −1 C
i =1
−k
n
X
Sti −1 |Mtni − Mtni −1 |,
i =1
Ln4
:=
k
n
X
Sti −1 |Mtni − Mtni −1 | − k
n
X
i =1
Ln5
:=
k
n
X
Sti −1 |ξtni − ξtni −1 |,
i =1
(Sti −1 − Sti )|ξtni − ξtni −1 |.
i =1
Lemma 3 For any α ∈ [0, 1/2[ we have
Z
γ 1 2b
S Cxx (t, St )dt → J1 a.s.,
σ
2 0 t
σ
and, hence,
Ln1
γ
2
Z
1
0
n
X
bxx (ti −1 , Sti −1 )I]ti −1 ,ti ] (t)dt → J1 a.s.,
St2i −1 C
i =1
→ 0 a.s. when n → ∞.
Proof. After the substitution v = σ
b2 (1 − t) the first integral can be written as

Ã
!2 
Z b
 v ln(S
σ2 S
/K ) 1 
σ γ 1
1−v/b
σ2
1−v/b
σ2
√
√
+
dv
exp −
 2
2σ
b2 2π 0
v
2 
v
and the second one as
Z b
n S
σ2 X
σ γ 1
1−vi −1 /b
σ2
√
√
2
2σ
b 2π 0
vi −1
i =1

Ã
!2 
 v
ln(S
/K
)
2
1 
1−vi −1 /b
σ
i −1
× exp −
+
I]v ,v ] (v)d v

2
vi −1
2  i i −1
b2 (1 − ti ). Clearly, both expressions tends a.s. to the integral
where vi := σ
On Leland’s strategy of option pricing with transactions costs
1
√
2 2π
Z
∞
0
245
(
2 )
v ln(S1 /K ) 1
S1
√ exp −
+
dv
2
v
2
v
(which is equal to J1 = min{K , S1 }) since γ/b
σ 2 → 1/σ and the integrands above
/ K ) are dominated by the function of the form
(when S1 (ω) =
c1 (ω)[v −1/2 e −c2 /v I[0,ε] (v) + I]ε,N ] (v) + e −c3 /v I[N ,∞[ (v)]
and we get the result (see Remark after Theorem 2).
(14)
Lemma 4 For any α ∈ [0, 1/2[ we have Ln2 → 0 in probability.
Proof. The sequence of independent random variables
p
|wti − wti −1 | − n −1/2 2/π
is a martingale difference with respect to the discrete filtration (Fti ),
2
p
E |wti − wti −1 | − n −1/2 2/π = (1 − 2/π)n −1 = (1 − 2/π)∆t.
By the Lenglart inequality we need to check only the convergence to zero in
probability of the sequence
σ 2 (1 − 2/π)k 2
n
X
bxx2 (ti −1 , Sti −1 )∆t
St4i −1 C
i =1
σ 2 = O(n −1/2−α ).
which for almost all ω is of the order k 2 /b
Lemma 5 For t ∈ [0, 1[ we have
bxx2 (t, St ) =
ESt2 C
where
√
σ t
,
a := √
σ
b 1−t
b2
1
1
√
exp
−
2πb
σ 2 (1 − t) 2a 2 + 1
2a 2 + 1
b :=
ln(S0 /K ) − σ 2 t/2 1 √
√
+ σ
b 1 − t.
2
σ
b 1−t
Proof. Let η be a standard normal random variable. Then for any a and b
b2
1
E exp{−(aη + b)2 } = √
exp − 2
.
2a + 1
2a 2 + 1
Since
1
bxx2 (t, St ) = 1
St2 C
2π σ
b2 (1 − t)
( 2 )
ln(S0 /K ) − σ 2 t/2 1 √
σwt
√
√
+
+ σ
exp −
b 1−t
2
σ
b 1−t
σ
b 1−t
the result follows.
246
Yu.M. Kabanov, M.M. Safarian
As a corollary we get that for t ∈ [1/2, 1[
bxx2 (t, St ) ≤
ESt2 C
c
.
σ
b 1−t
√
(15)
We shall use below some simple bounds for higher order derivatives of the
bx (t, x ) = Φ(db) where db := d (x , σ
b). We have:
function C
bxx (t, x )
C
=
bxxx (t, x )
C
=
bxxxx (t, x )
C
=
bxt (t, x )
C
=
bxxt (t, x )
C
=
1
√
ϕ(db),
xσ
b 1−t
Ã
!
−1
db
√
1+ √
ϕ(db),
x 2σ
b 1−t
σ
b 1−t
Ã
!
"
1
db
2
√
1+ √
x3 σ
b 1−t
σ
b 1−t
!#
Ã
db
db
+ 2
ϕ(db),
1+ √
σ
b (1 − t)
σ
b 1−t
1 ln(x /K )
1 σ
b
ϕ(db),
−
+ √
2σ
b(1 − t)3/2 4 1 − t
1
1
−
2 xσ
b(1 − t)3/2
1 ln(x /K )
1
1 σ
b
−
ϕ(db).
+ √
+ √
2σ
b(1 − t)3/2 4 1 − t
xσ
b 1−t
It follows that for t < 1 we have
1
c
1
2
bxxx
C
+
,
(t, x ) ≤
2
x4 σ
b2 (1 − t) σ
b4 (1 − t)
1
c
1
1
2
bxxxx
+
(t, x ) ≤
+
,
C
2
3
x4 σ
b2 (1 − t) σ
b4 (1 − t)
σ
b6 (1 − t)
σ
b
b
√
,
|Cxt (t, x )| = c 1 +
1−t
c
1
1
b
|Cxxt (t, x )| =
.
+
x σ
b(1 − t)3/2 1 − t
Lemma 6 For any α ∈ [0, 1/2[ the sequence Ln3 → 0 in probability.
Proof. Using the inequality ||a| − |b|| ≤ |a − b| we get that
n
Z ti
X
n
bxx (ti −1 , Sti −1 ) − St C
bxx (t, St )]d wt Sti −1 σ[Sti −1 C
|L3 | ≤ k
ti −1
i =1
and it sufficient to show that the sequence
n Z ti
X
bxx (ti −1 , Sti −1 ) − St C
bxx2 (t, St )]d wt [Sti −1 C
i =1
ti −1
(16)
(17)
(18)
(19)
On Leland’s strategy of option pricing with transactions costs
247
tends to zero in probability. But by the Burkholder and Jensen inequalities
n
Z ti
X
bxx (ti −1 , Sti −1 ) − St C
bxx (t, St )]d wt ≤
E
[Sti −1 C
ti −1
i =1
≤c
n
X
≤c
ÃZ
!1/2
ti
bxx (ti −1 , Sti −1 ) − St C
bxx (t, St )] dt
[Sti −1 C
≤
2
E
i =1
ti −1
ÃZ
n
X
ti
!1/2
bxx (ti −1 , Sti −1 ) − St C
bxx (t, St )] dt
E [Sti −1 C
2
.
ti −1
i =1
It follows from (15) that the last summand in the right-hand side of the above
inequality is finite and tends to zero. By the Ito formula
bxx (t, St )] = ft d wt + gt dt
d [St C
where
ft
:=
gt
:=
bxx (t, St ) + σ 2 St2 C
bxxx (t, St ),
σSt C
bxxt (t, St ) + 1 σ 2 St3 C
bxxxx (t, St ) + σ 2 St2 C
bxxx (t, St )
St C
2
and we can estimate all other summands as follows:
!1/2
ÃZ
ti
2
bxx (ti −1 , Sti −1 ) − St C
bxx (t, St )] dt
E [Sti −1 C
ti −1
ÃZ
≤
hZ
ti
Z
t
fu d w u +
E
ti −1
ti −1
√
≤ 2(∆t)1/2
ÃZ
!1/2
gu du dt
ti −1
Z
ti
Efu2 du
ti −1
i2
t
+ ∆t
ti
!1/2
E gu2 du
ti −1
1/2
1
1
+ 4
− ti ) σ
b (1 − ti )2
1
1
1
+c(∆t)3/2
+
+ 2
3/2
1 − ti σ
b (1 − ti )
σ
b(1 − ti )
1/2
1
1
+ 4
+ 6
σ
b (1 − ti )2 σ
b (1 − ti )3
1
1
≤ c∆t
+ 2
b (1 − ti )
σ
b(1 − ti )1/2 σ
1
+c(∆t)3/2
σ
b1/2 (1 − ti )3/4
1
1
1
1
+
+
+
+
.
b2 (1 − ti ) σ
(1 − ti )1/2 σ
b( 1 − ti )1/2 σ
b3 (1 − ti )3/2
≤ c∆t
σ
b2 (1
248
Yu.M. Kabanov, M.M. Safarian
It is clear that
n−1
X
i =1
∆t
σ
b(1 − ti )1/2
n−1
X
(∆t)
1/2
i =1
(∆t)1/2
∆t
σ
b1/2 (1 − ti )3/4
n−1
X
i =1
(∆t)1/2
i =1
(∆t)1/2
σ
b−2 ln n → 0,
−1/2 −1/2
n
Z
1
σ
b
n −1/2
Z
1
0
dt
→ 0,
(1 − t)3/4
dt
→ 0,
(1 − t)1/4
n −1/2 ln n → 0,
∆t
− ti )3/2
n −1/2 σ
b−3 n 1/2 → 0,
σ
b3 (1
and the result follows.
dt
→ 0,
(1 − t)1/2
∆t
σ
b(1 − ti )
n−1
X
i =1
1
0
∆t
(1 − ti )1/4
n−1
X
Z
0
∆t
σ
b2 (1 − ti )
i =1
n−1
X
σ
b−1
Lemma 7 For any α ∈]0, 1/2[ the sequence Ln4 → 0 in probability and bounded
in probability for α = 0.
Proof. Using again the inequality ||a| − |b|| ≤ |a − b| we get that
|Ln4 | ≤ k
n
X
Sti −1 |Ati − Ati −1 |
i =1
≤ kc(ω)
n Z
X
ti
bxt (u, Su )|du
|C
i =1 ti −1
n Z ti
X
+kc(ω)
i =1
bxxx (u, Su )|du.
σ 2 Su2 |C
It follows from (16) that
n−1 Z ti
n−1 X
X
2 2 b
σ Su |Cxxx (u, Su )|du ≤ c
i =1
ti −1
i =1
≤ c(b
σ
−1
(20)
ti −1
+σ
b
−2
∆t
∆t
+ 2
b (1 − ti )
σ
b(1 − ti )1/2 σ
ln n) → 0.
But the first sum for any α ∈ [0, 1/2[ converges to a finite limit
(
2 )
Z ∞
1 ln(S1 /K )
1 v ln(S1 /K ) 1
1
+ √ exp −
+
−
dv
c(ω)k∞ √
v
2
2
4 v
v 3/2
2π 0 2
where k∞ := lim kn (which is to zero when α > 0 and to k0 when α = 0). The
convergence to zero of the last summand in (20) follows from (15). On Leland’s strategy of option pricing with transactions costs
249
Lemma 8 For any α ∈ [0, 1/2[ the sequence Ln5 → 0 in probability.
Proof. It is sufficient to show that the sequence
n
X
k
|ξtni − ξtni −1 | = k
n
X
i =1
bx (ti , Sti ) − C
bx (ti −1 , Sti −1 )|
|C
i =1
is bounded in probability.
Pn But this fact follows easily from Lemma 7 since we
proved above that k i =1 Sti −1 |Mtni − Mtni −1 | converges in probability to J1 . Thus, we established that for α ∈]0, 1/2[ the sequence F2 (ξ n ) converges to
zero in probability and Theorem 1 is proved. Appendix B. Proof of Theorem 2
In view of Lemmas 1, 2, 8, and 3, 4, 6 it remains to show only that
k0
n
X
Sti −1 |ξtni − ξtni −1 | → J2 .
i =1
Put
Zin
:=
√
)
σ
b2 n(wti − wti −1 ) − ln(Sti −1 /K √
+ √ ,
2σ(1 − ti −1 ) n 4σ n Z̃in
:=
Zin − E (Zin | Fti −1 )
Evidently,
n
X
Sti −1 |ξtni − ξtni −1 | − k0−1 J2 = I1n + I2n + I3n
i =1
where
I1n
:=
n
X
Sti −1 |ξtni − ξtni −1 | −
i =1
n
X
σSt2i −1 Cxx (ti −1 , Sti −1 ) wti − wti −1
i =1
ln(Sti −1 /K )
1 2 ∆t +
σ
b ∆t ,
−
2σ(1 − ti −1 )
4σ
I2n
:=
n
X
bxx (ti −1 , Sti −1 )Z̃in n −1/2 ,
σSt2i −1 C
i =1
I3n
:=
n
X
bxx (ti −1 , Sti −1 )E (Zin | Fti −1 )n −1/2 − k −1 J2 .
σSt2i −1 C
0
i =1
p
Since σ
b2 / n → 2 2/πk0 σ we get using the definition (8) that
√
St
1X
p i −1
G(Sti −1 , σ
b2 (1 − ti −1 ))ϕ(db(Sti −1 , σ
b))b
σ 2 ∆t
4
σ
b 1 − ti −1
n
I3n
=
i =1
+o(1) − k0−1 J2 → 0
a.s.
250
Yu.M. Kabanov, M.M. Safarian
By the same considerations as in Lemma 4 we can show that I2n → 0 in
probability. Indeed, for any n the sequence Z̃in is a martingale difference (with
respect to the discrete filtration (Fti −1 )) and for a certain easily calculated function
R we have
n
X
bxx2 (ti −1 , Sti −1 )E ((Z̃in )2 | Fti −1 )n −1
σ 2 St4i −1 C
i =1
≤ c(ω)n
−1/2
n
X
R(Sti −1 , σ
b2 (1 − ti −1 )) → 0
a.s.
i =1
implying by the Lenglart inequality the convergence I2n → 0 in probability.
Finally,
|I1n |
≤
n
X
bxx (ti −1 , Sti −1 )(wti − wti −1 )|
Sti −1 |Mti − Mti −1 − σSti −1 C
i =1
+
n
X
bxx (ti −1 , Sti −1 )
Sti −1 Ati − Ati −1 − σSti −1 C
i =1
ln(Sti −1 /K )
1 2
× −
+
σ
b ∆t .
2σ(1 − ti −1 ) 4σ
It follows from Lemma 6 that the first sum in the right-hand side of the above
inequality converges to zero. The second sum is equivalent to the sum
Z
n
ti
X
b
b
Cxt (t, St )dt − Cxt (ti −1 , Sti −1 )∆t Sti −1 ti −1
i =1
which tends to zero as n → 0.
References
1. Jacod, J., Shiryayev A.N.: Limit Theorems for Stochastic Processes. Berlin–Heidelberg–New
York: Springer 1987
2. Leland, H.E.: Option pricing and replication with transactions costs. J. Finance 40(5), 1283–1301
(1985)
3. Lott, K.: Ein Verfahren zur Replikation von Optionen unter Transaktionkosten in stetiger Zeit,
Dissertation. Universität der Bundeswehr München. Institut für Mathematik und Datenverarbeitung, 1993