On the rate of convergence of barrier option prices in binomial market to those in continuous time market
We estimate the rate of convergence of barrier option price in a discrete time binomial market to such in a continuous time market.
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| Цитувати: | On the rate of convergence of barrier option prices in binomial market to those in continuous time market / O. Soloveiko, G. Shevchenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 165-173. — Бібліогр.: 8 назв.— англ. |
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irk-123456789-45752009-12-08T12:00:37Z On the rate of convergence of barrier option prices in binomial market to those in continuous time market Soloveiko, O. Shevchenko, G. We estimate the rate of convergence of barrier option price in a discrete time binomial market to such in a continuous time market. 2008 Article On the rate of convergence of barrier option prices in binomial market to those in continuous time market / O. Soloveiko, G. Shevchenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 165-173. — Бібліогр.: 8 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4575 en Інститут математики НАН України |
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We estimate the rate of convergence of barrier option price in a discrete time binomial market to such in a continuous time market. |
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Soloveiko, O. Shevchenko, G. |
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Soloveiko, O. Shevchenko, G. On the rate of convergence of barrier option prices in binomial market to those in continuous time market |
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Soloveiko, O. Shevchenko, G. |
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Soloveiko, O. |
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On the rate of convergence of barrier option prices in binomial market to those in continuous time market |
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On the rate of convergence of barrier option prices in binomial market to those in continuous time market |
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On the rate of convergence of barrier option prices in binomial market to those in continuous time market |
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On the rate of convergence of barrier option prices in binomial market to those in continuous time market |
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On the rate of convergence of barrier option prices in binomial market to those in continuous time market |
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on the rate of convergence of barrier option prices in binomial market to those in continuous time market |
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Інститут математики НАН України |
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2008 |
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On the rate of convergence of barrier option prices in binomial market to those in continuous time market / O. Soloveiko, G. Shevchenko // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 165-173. — Бібліогр.: 8 назв.— англ. |
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Theory of Stochastic Processes
Vol.14 (30), no.3-4, 2008, pp.165-173
OLENA SOLOVEIKO AND GEORGIY SHEVCHENKO
ON THE RATE OF CONVERGENCE OF BARRIER
OPTION PRICES IN BINOMIAL MARKET TO
THOSE IN CONTINUOUS TIME MARKET
We estimate the rate of convergence of barrier option price in a dis-
crete time binomial market to such in a continuous time market.
1. Introduction
A barrier option is a derivative with a payoff that depends on the fact
whether asset price crosses certain level during certain time interval. Thus,
payment for barrier option depends on the behavior of the price asset during
all the time interval, i.e. barrier option is a particular case of exotic option.
The simplest barrier options are calls and puts that are knocked out or
knocked in by the underlying asset itself. The payoff of a knock-out option
is made if underlying asset price does not cross the barrier, such options
are of two types: if asset price does not cross the barrier below, then such
an option is called “up-and-out”, if from above – “down-and-out”. Payoff
of a knock-in option is made if underlying asset price crosses the barrier,
they also are of two types accordingly: “up-and-in” and “down-and-in”.
Altogether there are eight types of barrier options.
For example, the payoff function of up-and-in option is given by
C =
{
(ST −K)+, if max0≤t≤T St ≥ H ,
0 else,
where H is a barrier level (H > S0 and H > K), K is a strike price.
Payoffs for the rest options are determined in the same way. Barrier options
are among the most popular path-dependent option traded in exchanges
worldwide and also over-the-counter markets.
Invited lecture.
2000 Mathematics Subject Classifications. Primary: 91B28; Secondary: 60G50,
60F05
Key words and phrases. Barrier option, fair price, complete market, binomial market,
Black–Scholes market
165
166 OLENA SOLOVEIKO AND GEORGIY SHEVCHENKO
The problem of pricing and hedging barrier option in the models with
continuous time is rather complete, and analytical formulae for the prices of
such options are known only in the most easy cases. Therefore the problem
of asymptotic estimation of the prices of such options arises. The simplest
asymptotic methods is the method of time discretization, which can be
described in the following way. Time interval is divided into m equal parts
and now the asset price model with discrete time is considered. In such a
formulation we can approximately calculate option price using Monte Carlo
simulations, modelling the path of the underlying asset price. From the
other side, the opposite problem could arise: let we have analytical formula
for option price in continuous time model. Then the demand may come to
estimate the price of the option with payoff realized when the asset price
crosses the barrier level, and this price is observed only in certain time
moments (for example, daily when stock exchange is closing).
From the practical point of view, when we approximately estimate the
price of the option it is important to know the quality of such an estimation,
i.e. the order of the error.
In [1] authors introduce a simple continuity correction for approximate
pricing of discrete barrier option. Their method uses formulae for the prices
of continuous barrier options but shifts the barrier to correct for discrete
monitoring. Compared with using the unadjusted continuous price, their
formula reduces the error from O( 1√
m
) to o( 1√
m
), as the number of moni-
toring points m increases. The correction is justified both theoretically and
experimentally.
Theorem 1. [1] Let V (H) be the price of a continuously monitored knock-
in or knock-out down call or put with barrier H, and let Vm(H) be the price
of the corresponding discrete monitored barrier option. Then
Vm(H) = V (He±βσ
√
T/m) + o(
1√
m
),
where + applies if H > S0, and − applies if H < S0, β = −ζ(1/2)/
√
2π ≈
0.5826, with ζ the Riemann zeta function.
The paper [6] extends an approximation by Broadie et al. in [1] for
discretely monitored barrier options by covering more cases and giving a
simpler proof. The paper [4] also continues the work of Broadie and deter-
mine formulae to estimate the price of discrete up-and-out/in calls, down-
and-out/in puts and double barrier option. The methods used here lead to
slightly different barrier correction formulae. In [2] the rate of convergence
for lookback options and other exotic options is obtained.
The model considered in [8] investigates the rate of convergence of option
price in discrete market, but this price is not fair in the sense that it might
be not unique. Discrete market, generated by the increments of geometric
RATE OF CONVERGENCE OF BARRIER OPTION PRICES 167
Brownian motion, is not complete, so there are many “fair prices”. Thus
it would be better to have result for convergence of the unique price in
complete market. That because in our work we consider discrete binomial
market and investigate the rate of convergence of fair price of barrier option
in such market to correspondent price on continuous market. We have
proved that the rate of convergence is lnn/
√
n, where n is the number of
periods in the binomial market.
2. Main result
Let (Ω,F , P ) be a complete probability space with filtration {Ft, t ≥ 0},
{Wt, t ≥ 0} is standard Ft-Brownian motion on it. Consider Black–Scholes
financial market model, where we have two assets: riskless (bond), whose
price at the moment t equals
Bt = B0 exp
{∫ t
0
rsds
}
,
and a risky asset (stock), whose price is
St = S0 exp
{∫ t
0
μsds+ σWt
}
,
where Wt is standard Brownian motion defined before. Volatility σ > 0
is assumed to be constant. For simplicity, we assume that P itself is a
martingale measure for discounting process of risky asset price, i.e. μt =
rt−σ2/2. Besides, we demand the interest rate rt to be Lipschitz continuous,
i.e. for every t, s ∈ [0, T ]
|rt − rs| ≤ C|t− s|, (1)
where C is a constant.
In the market with continuous time the fair option price is defined as the
expectation of discounting payoff for the option given martingale measure.
Let IA denote the indicator of an event A, MT = max {St, t ∈ [0, T ]}, mt =
min {St, t ∈ [0, T ]}. Then, for instance, European up-and-out call option
price is given by
V (H) = E
(
exp
{
−
∫ T
0
rtdt
}
(ST −K)+I{MT<H}
)
,
where K > 0 is a strike price, H > S0 is a barrier, and European down-
and-in put option price is given by
V (H) = E
(
exp
{
−
∫ T
0
rtdt
}
(K − ST )+I{mT ≤H}
)
,
168 OLENA SOLOVEIKO AND GEORGIY SHEVCHENKO
where H < S0 is a barrier. In Merton’s paper [7] an explicit form for the
price of knock-out call option is established, when the risk-neutral interest
rate r is constant.
Now consider a binomial market model with discrete time, which is
constructed as follows. Divide time interval [0, T ] into n ≥ 1 parts, define
Δ = T
n
, ti = iΔ, i = 0, . . . , n. Let ξi, i = 0, . . . , n − 1 be independent
identically distributed random variables, such that P (ξi = 1) = P (ξi =
−1) = 1
2
. The risky asset price in the binomial market model is defined as
Sbti = S0 exp
{ i−1∑
j=0
(μjΔ + σξj
√
Δ)
}
, i = 1, . . . , n;
on [ti, ti+1) we put Sbt = Sbti , and set the interest rate to be equal to rti .
Instead of Brownian motion, the role of “random driver” of financial market
in the binomial model is played by a random walk {Ξi}, defined as
Ξi :=
i−1∑
j=0
ξj.
An analogue of European up-and-out call option in the binomial model has
the payoff function (SbT −K)+I{Mb
T<H}, consequently, the price is
V b
n (H) = E
(
exp
{
−
n−1∑
i=0
rtiΔ
}
(SbT −K)+I{Mb
T<H}
)
,
where M b
T = max0≤i≤n Sbti = maxt∈[0,T ] S
b
t .
The following is the main result about convergence of price in binomial
model to the one in continuous model. We need the following result.
Theorem 2. The difference of European up-and-out call options fair prices
in discrete binomial and continuous models under the assumption (1) sat-
isfies
V (H) − V b
n (H) = O
( lnn√
n
)
, n→ ∞.
Proof. In the following C will denote a generic positive constant, which may
depend only on σ, the Lipschitz continuity parameter of rt, H , K, S0, i.e.
the inputs of our problem.
In order to prove our result, we will use an approximation result from
[8]. In the discrete time market define a discretized version of S:
Sdti = S0 exp
{ i−1∑
j=0
(μjΔ + σZj
√
Δ)
}
, i = 1, . . . , n,
RATE OF CONVERGENCE OF BARRIER OPTION PRICES 169
where Zj = (Wtj+1
− Wtj )/
√
Δ, and consider European up-and-out call
option with a payoff (SdT −K)+I{Md
T<H}, where Md
T = max0≤i≤n Sti . Its fair
price is
V d
n (H) = E
(
exp
{
−
n−1∑
i=0
rtiΔ
}
(SdT −K)+I{Md
T<H}
)
.
It is proved in [8] that V (H) − V d
n (H) = O(1/
√
n), n → ∞. Thus, it is
enough to prove that V d
n (H) − V b
n (H) = O(lnn/
√
n), n→ ∞.
It is clear that
|V d
n (H) − V b
n (H)| ≤ C
∣∣∣E((SdT −K)+I{Md
T<H}
)− E
(
(SbT −K)+I{Mb
T<H}
)∣∣∣ .
(2)
Now we apply the result of Komlós, Major and Tusnády [5]. It says that
for any given λ > 0 it is possible to construct independent random variables
ηi
d
= ξi and independent standard random variables ζi, 0 ≤ i ≤ n− 1, such
that for some positive constants K
P
(
max
0≤i≤n−1
|Si − Ti| > K lnn + x
)
≤ Ke−λx, (3)
where
Si =
i∑
j=0
ηj , Ti =
i∑
j=0
ζj.
Note that (3) implies E(max0≤i≤n−1 |Si − Ti|2) ≤ C ln2 n. Indeed, denoting
R = max0≤i≤n−1 |Si − Ti|, we have
E(R2) ≤ (2K + 2)2 ln2 n+ E(R2I{R>(2K+2) lnn})
≤ C ln2 n+
∫ ∞
0
P (R2 > (2K + 2)2 ln2 n + x)dx
≤ C ln2 n+
∫ ∞
0
P (R > (K + 1) lnn + x/2)dx
≤ C ln2 n+Kn−λ
∫ ∞
0
e−λ
√
x/2dx ≤ C ln2 n.
In the following we will assume without loss of generality Kλ > 1/2.
As long as {ξi, i = 0, . . . , n− 1} d
= {ηi, i = 0, . . . , n − 1} and {Zi, i =
0, . . . , n − 1} d
= {ζi, i = 0, . . . , n − 1}, in order to estimate the difference
|V d
n (H)− V b
n (H)| we can assume that ξi = ηi and Zi = ζi, because this will
not change the expectations in (2). Now write
|V d
n (H) − V b
n (H)| ≤ C(I1 + I2),
170 OLENA SOLOVEIKO AND GEORGIY SHEVCHENKO
where
I1 =
∣∣E([(SdT −K)+ − (SbT −K)+]I{Mb
T<H})
∣∣
≤ E(|(SdT −K)+ − (SbT −K)+|I{Mb
T<H}) ≤ E(|SdT − SbT |I{Mb
T<H}),
I2 =
∣∣E((SdT −K)+[I{Md
T<H} − I{Mb
T<H}])
∣∣
≤ CE
(|I{Md
T<H} − I{Mb
T<H}|
)
≤ C
(
P (Md
T < H,M b
T ≥ H) + P (Md
T ≥ H,M b
T < H)
)
.
Processes Sd and Sb are of the form S0e
x, hence from inequality |ex− ey| ≤
(ex + ey)|x− y| we obtain
I1 ≤ CE
(
|SbT + SdT |σ
√
Δ
∣∣∣∣ n−1∑
j=0
(Zj − ξj)
∣∣∣∣I{Mb
T<H}
)
Using the Cauchy–Bunyakovsky inequality, we get:
I1 ≤ Cσ
√
Δ
(
E(|SdT + SbT |2I{Mb
T<H})
)1/2 ×
(
E
[ n−1∑
j=0
(
Zj − ξj+l
)]2
)1/2
.
Now
E(|SdT + SbT |2I{Mb
T<H}) ≤ 2E
[(
(SdT )2 + (SbT )2
)
I{Mb
T<H}
]
≤ C
(
E(S0 exp{2CT + 2σTWT}) +H2
) ≤ C,
as exp{σTWT} is integrable, and |μt| is bounded. On the other hand, as it
was pointed above,
E
[ n−1∑
j=0
(
Zj − ξj+l
)]2
≤ C ln2 n,
thus we have
I1 ≤ C
√
Δ lnn ≤ C
lnn√
n
.
Now turn to I2. Both probabilities are estimated in a similar manner, so
we will estimate only the first one. Write
P (Md
T < H,M b
m ≥ H)
≤ P (H − δ ≤Md
T < H,M b
T ≥ H) + P (Md
T < H − δ,M b
T ≥ H)
≤ P (H − δ ≤Md
T < H) + P (Md
T < H − δ,M b
T ≥ H) =: P1 + P2.
It is easy to see that Md
T has a bounded density, so P1 ≤ Cδ.
RATE OF CONVERGENCE OF BARRIER OPTION PRICES 171
Now we observe that P (H − δ ≤ Md
T < H,M b
T ≥ H) implies that for
some i SdT < H − δ < H ≤ SbT , so, by taking logarithms, we have
√
Δ
i−1∑
j=0
(ξj − Zj) > Cδ,
which implies
i−1∑
j=0
(ξj − Zj) > Cδ
√
n.
Now take δ = 2K lnn/
√
n. With this choice we have from (3) P2 ≤
Cn−λK ≤ C lnn/
√
n. Summing up, we have I2 ≤ C lnn/
√
n, and the
assertion of the theorem follows. �
3. Modelling
As in [8], we give an example showing how fast the price in discrete
binomial model converges to correspondent price in continuous model.
Consider the drift function of the form:
μt =
{
μ1, 0 ≤ t < T/2,
μ2, T/2 ≤ t ≤ T.
This function (and corresponding interest rate rt) does not satisfy the con-
dition of continuity (1), which we have impose on it. But, if we look into
the proof of Theorem 2, it is not difficult to see that it is enough to have
the condition (1) fulfilled only for t = ti, s ∈ [ti, ti+1), which is true for such
a function.
According to [3] we have that for Brownian motion Xt with initial value
x and constant drift coefficient μ simultaneous density of the distribution
of maximum Mt on interval [0, t], of the points Tt of the maximum and of
the values Xt is given by
P (Xt ∈ dz,Mt ∈ dy, Tt ∈ ds) =
(y − x)(y − z)
π
√
s3(t− s)3
×
exp
(
− (y − x)2
2s
− (y − z)2
2(t− s)
− μ(x− z) − μ2t
2
)
dz dy ds
=: ft,x,μ(z, y, s)dz dy ds
when x ≤ y and z ≤ y; when x > y or z > y it equals to zero. Noting
ν(T ) = exp
{ − ∫ T
0
rtdt
}
= exp
{
− T
2
(μ1 + μ2 + σ2)
}
and using the fact,
that Zt = 1
σ
lnSt is a Brownian motion with the drift ν1 = μ1
σ
on [0, T
2
] and
172 OLENA SOLOVEIKO AND GEORGIY SHEVCHENKO
ν2 = μ2
σ
on [T
2
, T ], we can get the European up-and-out call option fair price
as
V (H) = E
(
ν(T )(ST − K)+I{τ(H,S)>T}
)
= ν(T )E
(
(ST − K)+I{ sup
[0,T ]
St<H}
)
=
ν(T )E
(
E
(
(ST − K)+I{
sup
[ T
2 ,T ]
St<H
} ∣∣FT
2
)
I{
sup
[0, T
2 ]
St<H
}) =
ν(T )E
( ∫ T
2
0
∫ 1
σ
lnH
ZT
2
∫ y
−∞
(eσz − K)+fT
2
,ZT
2
,ν2
(z, y, s)dz dy ds I{
sup
[0, T
2 ]
St<H
}) =
ν(T )
∫ T
2
0
∫ 1
σ
lnH
Z0
∫ v
−∞
fT
2
,Z0,ν1
(x, v, u)×∫ T
2
0
∫ 1
σ
lnH
x
∫ y
−∞
(eσz − K)+fT
2
,x,ν2
(z, y, s)dz dy ds dx dv du.
The last integral is rather difficult to calculate because of its high di-
mension. Nevertheless, integrals in y and v can be evaluated in closed form,
with the use of the standard normal distribution function; we do not give
the result of this integrating — formulae are very intricate — and give only
the final estimation for the integral.
n V b
n (H) V b
n (H) − V (H) (V b
n (H) − V (H))n1/2/ lnn
10 0,5675 0,0932 0,1279
20 0,5001 0,0257 0,0383
50 0,5772 0,1028 0,1858
100 0,5438 0,0694 0,1507
200 0,4991 0,0237 0,0658
500 0,4921 0,0177 0,0638
1000 0,4906 0,0162 0,0742
2000 0,4785 0,0041 0,0241
Table 1: Price V (H) of the European up-and-out call option in continuous
model and the price V b
m(H) of the same option in discrete binomial model.
Let take the following meanings of parameters: S0 = 100, σ = 0.1,
K = 100, H = 105, T = 0,2, μ1 = 0,1, μ2 = 0,2. Then with accuracy 10−4
V (H) = 0,4744.
To estimate the order of the rate of convergence for the option prices with
discrete time, using Monte Carlo simulations for the estimation of math-
ematical expectation, we will generate 100000 trajectories of asset price
(50000 trajectories for m = 1000, 2000). The results we have got are noted
RATE OF CONVERGENCE OF BARRIER OPTION PRICES 173
in the table 1. We should note that the option prices with discrete time are
bigger and decreasing when size of partition increasing. This property is
natural, because in the case when the quantity of the points in our division
increases, the moment set in which we examine does asset price cross given
level or not also increases. There is no clear evidence however from this
data whether the estimate for the rate of convergence is sharp.
Conclusions
We have proved that barrier option fair prices in discrete binomial Black–
Scholes model with non-constant drift coefficient converges to corresponding
price in continuous model, and the rate of convergence could be estimated
as O( lnn√
n
), where n is the number of operational moments in the discrete
binomial model. Thus is a result for convergence of the unique price in a
complete market. Numerical example is presented.
References
1. Broadie, M., Glasserman, P., Kou, S. G. A continuity correction for discrete
barrier options, Math. Finance 7 (1997), 7, 325–349.
2. Broadie, M., Glasserman, P., Kou, S.G. Connecting discrete and continuous
path-dependent options, Fin. Stoch. 3 (1999), 55–82.
3. Csáki, E., Földes A., Salminen P. On the joint distribution of the maximum
and its location for a linear diffusion, Ann. Inst. Henri Poincaré, Probab.
Stat. 23 (1987), 179–194.
4. Hŏrfelt, P. Extension of the corrected barrier approximation by Broadie,
Glasserman, and Kou, Fin. Stoch. 7 (2003), 231–243.
5. Komlós, J., Major, P., Tusnády, G. An approximation of partial sums of in-
dependent RV’s and the sample DF. II, Z. Wahrscheinlichkeitstheor. Verw.
Geb. 34 (1976), 33–58.
6. Kou, S. G. On pricing of discrete barrier options, Statistika Sinica 13
(2003), 955–964.
7. Merton, R.C. Theory of rational option pricing, Bell J. Econom. Manage.
Sci. 4 (1973), 141–183.
8. Soloveiko, O., Shevchenko, G. The rate of convergence of barrier option
price with non-constant drift in discrete time to such in continuous time,
Theory Probab. Math. Stat. 79 (2008), 166–172.
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: osoloveyko@univ.kiev.ua
Department of Probability Theory and Mathematical Statistics,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: zhora@univ.kiev.ua
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