On the Rate of Convergence of Binomial Greeks
San-Lin Chung, Weifeng Hung and Pai-Ta Shih***
Abstract
The recent literature indicates that the most efficient binomial models, including
binomial Black-Scholes (BBS) model of Broadie and Detemple (1996), the flexible
binomial model (FB) of Tian (1999), the smoothed payoff (SPF) approach of Heston
and Zhou (2000), and the generalized Cox-Ross-Rubinstein (GCRR) model of Chung
and Shih (2007), generally yield binomial option prices with monotonic and smooth
convergence. However, their efficiency for the calculation of delta and gamma are not
known. To fill the gap of the literature, we investigate the convergence patterns and
the rates of convergence of these binomial models for calculating delta and gamma.
The method proposed by Hull (1993, hereafter Hull method) and extended binomial
tree proposed by Pelsser and Vorst (1994, hereafter extended method) are applied to
these binomial models. The numerical results indicate that by using the Hull method
or the extended method, these binomial models can generate monotonic convergence
with the rate of convergence O(1/ n) for calculating delta and gamma. Moreover, the
GCRR-XPC model with an extended tree is the most efficient model to compute
deltas and gammas for options when a two-point extrapolation formula is used.
Keywords: binomial option pricing model, Greeks, monotonic convergence, rate of
convergence
JEL Classification: G13

Corresponding author, Department of Finance, National Taiwan University, 85, Section 4, Roosevelt
Road, Taipei 106, Taiwan, R.O.C.. Tel: 886-2-3366-1084. Email: chungs@management.ntu.edu.tw.

Department of Finance, Feng-Chia University. 100, Wenhwa Rd., Seatwen, Taichung, Taiwan 40724,
Taiwan, R.O.C.. Tel: 886-04-24517250#4173. Email: wfhung@fcu.edu.tw.
***
Department of Finance, National Taiwan University, 85, Section 4, Roosevelt Road, Taipei 106,
Taiwan, R.O.C.. Tel: 886-2-3366-1093. Email: ptshih@management.ntu.edu.tw.
Introduction
Binomial methods, developed first by Cox, Ross and Rubinstein (1979, CRR
thereafter), are well-known for their flexibility and efficiency in calculating option
price. One stream of the literature modifies the lattice or tree type to improve the
accuracy and computational efficiency of binomial option prices. The pricing errors in
the discrete-time models are mainly due to “distribution error” and “nonlinearity
error” (see Figlewski and Gao (1999) for thorough discussions). Within the literature,
there are many proposed solutions that reduce the distribution error and/or
nonlinearity error. For example, Broadie and Detemple (1996) and Heston and Zhou
(2000) suggested modified binomial models by replacing the binomial prices prior to
the end of the tree by the Black-Scholes values, or by smoothing payoff functions at
maturity, and computing the rest of binomial prices as usual. The advantage of their
approach is that the resulting binomial option prices converge to the Black-Scholes
prices monotonically and smoothly, and thus the standard Richardson extrapolation
can be applied to obtain solutions with high accuracy. The other improved lattice
approaches including Omberg (1988), Leisen and Reimer (1996), Figlewski and Gao
(1999), Widdicks, Andricopoulos, and Newton, and Duck (2002).1
In this paper, we focus on recent binomial models whose binomial option prices
converge to the “true” value monotonically and smoothly. In other words, the pricing
1
Omberg (1988) developed a family of efficient multinomial models by applying the highly efficient
Gauss-Hermite quadrature to the integration problem (e.g. N (d1 ) in the Black-Scholes formula)
presented in the option pricing formulae. Leisen and Reimer (1996) modified the sizes of up- and
down-movements by applying various normal approximations (e.g. the Camp-Paulson inversion
formula) to the binomial distribution derived in the mathematical literature. Figlewski and Gao (1999)
proposed the so-called adaptive mesh method which sharply reduces nonlinearity error by adding one
or more small sections of fine high-resolution lattice onto a tree with coarser time and price steps.
1
errors of these binomial option prices are of the same sign and decrease at a known
rate as the number of time steps increases.2 These models include the binomial
Black-Scholes (BBS) model of Broadie and Detemple (1996), the flexible binomial
model (FB) of Tian (1999), the smoothed payoff (SPF) approach of Heston and Zhou
(2000), and the GCRR-XPC model of Chung and Shih (2007).3
Although the above models have been widely applied to price options, their
convergent patterns and rate of convergence for calculating the hedge ratios are not
well known. To fill this gap, we apply the method proposed by Hull (1993) and
extended method proposed by Pelsser and Vorst (1994) to calculate Greeks under
these models. We find that by using Hull method and extended method, these
binomial models can generate monotonic convergence with O  1n  rate for
calculating deltas and gammas.4 Thus one can apply the extrapolation formula to
enhance the computation of the hedge ratios. Among all the binomial models
considered in this paper, the GCRR-XPC model is the most efficient one for the
evaluation of deltas and gammas for European options when a two-point extrapolation
is used.
The rest of the paper is organized as follows. Section 2 briefly reviews the
binomial models considered in this paper. The rates of convergence of applying
2
Binomial models with the smooth and monotonic convergence property are the most accurate ones in
the recent literature because their accuracy can be substantially improved by applying the Richardson
extrapolation technique (Chang, Chung, and Stapleton, 2007).
3
In the GCRR-XPC model, a stretch parameter (  ) is used to adjust the lattice so that the strike price
is located in the center of the final nodes. See Section 2.1 for details of the GCRR-XPC model.
4
Numerical differentiation formulae can also be applied to these binomial models for the calculations
of Greeks. However, by using numerical differentiation formulae, it needs to construct two binomial
trees to calculate delta and to construct three binomial trees to calculate gamma. Thus, it takes more
time to calculate Greeks. Besides, the numerical results show that the convergent patterns of binomial
Greeks using numerical differentiation formulae are not smooth and monotonic. Thus, we do not apply
numerical differentiation formulae in this paper.
2
extended binomial tree to CRR model, SPF model and GCRR-XPC model are proved
and the extrapolation formula is discussed in Section 3. Section 4 presents the
numerical results of various binomial models for the evaluations of deltas and
gammas of options. Section 5 concludes the paper.
2. Reviews of the binomial option pricing models
We assume that the Black-Scholes economy holds and thus options can be
valued as if the investor is risk-neutral. In other words, the options can be priced
under the risk-neutral measure where stock price follows a geometric Brownian
motion given by
dS
 (r  q ) dt   dZ ,
S
(1)
where S is the stock price, r is the risk-free rate, q is the dividend yield, and 
is the instantaneous volatility of S .
The binomial option pricing model was first developed by Cox, Ross, and
Rubinstein (1979) and Rendleman and Bartter (1979). Consider the pricing of an
option maturing at time T . In an n-period binomial model, the time to maturity
[0, T ] is partitioned into n equal time steps t  T / n . If the stock price is S in this
period, then it is assumed to jump either upward to uS with probability p or downward
to dS with probability 1  p in the next period, where 0  d  e( r  q ) t  u and
0  p  1 . The binomial model is completely determined by the jump sizes u and d
and the risk-neutral probability p.
2.1. The binomial models with monotonic and smooth convergence property
In the traditional CRR model, the following three conditions are utilized to
determine u, d, and p.
3
pSu  (1  p) Sd  Se( r  q ) t ,
pu 2  (1  p)d 2   pu  (1  p)d    2 t ,
2
(2)
ud  1.
The first two conditions are used to match the mean and variance of the stock price in
the next period, and the third condition is imposed arbitrarily by CRR. With these
three conditions, one can easily determine the binomial parameters as follows:
u  e
t
, d  e
t
, and p  (e( r q ) t  d ) /(u  d ) .
According to the Central Limit Theorem,5 the discrete distribution of the asset
price under the CRR model will converge to its continuous-time limit (i.e.
Black-Scholes model). In other words, as t  0 , the price distribution of the CRR
model converges to a lognormal distribution,


2
ln ST  N  ln S0  (r  q  )T ,  T  .
2


d
(3)
As a result, the option prices calculated by the binomial model will also converge to
the Black-Scholes price.
Although the CRR model is a well known and widely used model for valuing
options, the option prices calculated from it usually converge to the Black-Scholes
price in a wavy erratic fashion. In order to enhance the accuracy of the binomial
option prices, the recent literature has developed several ways to overcome the wavy
erratic problem embedded in the CRR model. Four important articles are discussed
and applied in this paper. These papers developed binomial models with monotonic
and smooth convergence property. Monotonic convergence is desirable because more
time steps do guarantee more accurate prices. Moreover, smooth convergence is also
5
The following condition should be satisfied to apply the Central Limit Theorem:
p u  e( r  q ) t  1  p  d  e( r  q ) t
3
3
[ p(u  e( r  q ) t )2  1  p  (d  e( r  q ) t )2 ]1.5 n
 0 as t  T / n  0 .
4
advantageous because the standard Richardson extrapolation can be used to enhance
the accuracy.
Broadie and Detemple (1996) proposed a binomial model termed Binomial
Black and Scholes (BBS) model to price options. The BBS model is identical to the
CRR model except that at one time step before option maturity the Black-Scholes
formula replaces the usual “continuation value.” Broadie and Detemple (1996)
showed that the option prices obtained from the BBS model converge to the
Black-Scholes formula smoothly and monotonically. Thus, they suggested using the
Richardson extrapolation to enhance the accuracy.
Tian (1999) developed a flexible binomial (FB) model with a “tilt” parameter (  )
that alters the shape and span of the binomial tree. The parameters of the FB binomial
model are as follows:
 t   2 t
ue
, d e
 t   2 t
e( r  q ) t  d
.
, p
ud
(4)
With a positive tilt parameter,  > 0, the up movement is larger than the
corresponding up move in a standard CRR tree. Consequently, the central nodes
depict an upward sloping line. The resulting tree span is thus shifted upward. The
exact opposite is true for binomial trees with a negative tilt parameter,  < 0. Fig. 1
illustrates the effect of the “tilt” parameter on binomial trees.
<Insert Fig. 1. Here>
Tian (1999) first proved that his FB model converges to its continuous-time
counterparts (i.e. the Black-Scholes model) for any value of the tilt parameter.6 Tian
6
However, the tilt parameter must satisfy the following inequality in order to have “nonnegative
probability”:

( r  q)
2

1
 t
.
5
(1999) then suggested the way to select a particular tilt parameter that enhances the
rate of convergence of binomial prices to their continuous-time limit. This is done by
selecting a tilt parameter such that a node in the tree coincides exactly with the strike
price at the maturity of the option. This idea was inspired by Leisen and Reimer
(1996). 7 As a result, one can derive the formula for the tilt parameter as the
following:

ln( SX0 )  (2 j0  n) t
n 2 t
,
(5)
 ln( SX )  n ln(d0 ) 
0
where j0  
 , [.] denotes the closest integer to its argument, X is the
ln( ud00 )



strike price, u0  e
t

, and d 0  e
t
. Using the FB model with the above tilt
parameter, Tian (1999) showed that monotonic and smooth convergence is possible
for pricing standard European and American options, and extrapolation methods are
used to enhance the accuracy.
Heston and Zhou (2000) showed that the accuracy or rate of convergence of the
binomial model depend crucially on the smoothness of the payoff function. They have
developed an approach to smooth the payoff function. Intuitively, if the payoff
function at singular points can be smoothed, the binomial recursion might be more
accurate. They defined the smoothed payoff function G  x  as follows:
G  x 
1
2 x

x
x
g  x  y  dy ,
(6)
where g  x  is the payoff function. This is a rectangular smoothing of g  x  . The
7
Leisen (1996) independently proposed a binomial model that has some similar features as the flexible
binomial model. See his SMO model. However, his choice of u and d is rather ad hoc. A
systematic way of determining u and d is proposed here through the choice λ.
6
smoothed function, G  x  , can be easily computed analytically for most payoff
functions used in practice. For example, the smoothed payoff function of a European
put option with a strike price X can be derived as follows:
0,



e t  e t

,
G ( ST )=  X  ST
2


t

 X  t  ln ST
X
ST  


e

2 t
2 t 


ln ST  ln X   t
ln ST  ln X   t
t

(7)
X
 ,   t  ln ST  ln X   t
ST 
Applying the smoothed payoff function G  x  , instead of g  x  , to the binomial
model yields a rather surprising and interesting result. The associated SPF model
converge at the O  1n  rate to its continuous-time limit and this convergence is
smooth and monotonic as the number of time steps increases.
Chung and Shih (2007) proposed a generalized CRR (GCRR) model by adding
a stretch parameter into the model specification. In the GCRR model the jump sizes
and probability of going up are as follows:
u  e
t
, d e
1
  t

, and p 
e( r  q ) t  d
,
ud
(8)
where   0 is a stretch parameter which determines the shape (or spanning) of the
binomial tree. The CRR model is obviously a special case of our GCRR model with
  1 . When t  0 , i.e., the number of time steps n grows to infinity, the GCRR
binomial option prices also converge to the Black-Scholes formulae for plain vanilla
European options.
Generally speaking, the rate of convergence of the GCRR model for pricing
7
options is of order O
 
1
n
when   1 .8 This rate of convergence is inferior to other
binomial models such as the BBS model and the FB model. Thus, to best apply the
GCRR model, Chung and Shih (2007) suggested using the so-called GCRR-XPC
model, where the lattice is set up in a way that the strike price is in the center of the
final nodes, to price options. In particular, the binomial option prices of the
GCRR-XPC model not only converge at a high order (of order O  1n  ) but also
converge smoothly and monotonically to the Black-Scholes formulae. Thus in the
numerical analysis of this paper, we adopt the GCRR-XPC model to calculate hedge
ratios.
2.2. Calculating hedge ratios with binomial models
Greeks (or hedge ratios) are the sensitivity of the option price with respect to the
change of the underlying risk factors. For the Hull method, the estimated delta is
calculated at time t  T / n and the estimated gamma is calculated at time 2t in
the standard tree where the current underlying asset price and option price are S0,0
and C0,0 , respectively. In Fig. 2, the estimates of delta and gamma can now be
obtained as follows:
C  C1,0
ˆ  1,1
,
S1,1  S1,0
 C2,2 C2,1     C2,1 C2,0  
2

C
  S2,2  S2,1     S2,1  S2,0  
ˆ  2  
.
1
S
S

S
 2,2 2,0 
2
(9)
(10)
<Insert Fig. 2. Here>
Instead of using the Hull method, we also apply the extended method to
calculate delta and gamma. In the extended binomial tree, we build the binomial tree
8
Please refer to Theorem 2 of Chung and Shih (2007).
8
starting from two time steps back from today. Fig. 3 illustrates the extended binomial
tree, where the current underlying asset price and option price are S0,1 and C0,1 ,
respectively. The estimates of delta and gamma can now be obtained as the following:
C  C0,0
ˆ  0,2
,
S0,2  S0,0
(11)
 C0,2 C0,1     C0,1 C0,0  
2

C
  S0,2  S0,1     S0,1  S0,0  
.
ˆ  2  
1
S
 S0,2  S0,0 
2
(12)
<Insert Fig. 3. Here>
3. The rate of convergence of the binomial models and the
extrapolation formula
When the convergence pattern of the binomial option prices is monotonic and
smooth, one can apply the extrapolation technique to enhance its accuracy. However,
the extrapolation formula depends on the rate of convergence of the binomial model.
Thus we first discuss the rate of convergence of the binomial models discussed in this
paper.
Leisen and Reimer (1996) proved that the convergence is of order O  1n  for
the CRR model. Since the FB model of Tian (1999) is a fine tune of the CRR model,
it is straightforward to verify that its convergence is also of order O  1n  . Moreover,
Heston and Zhou (2000) showed that the rate of convergence can be improved by
smoothing the option payoff functions and cannot exceed
1
n
at some nodes of the
tree.9 They showed that the rate of convergence of the binomial model with smoothed
payoff functions (e.g. the BBS model and the SPF model) is also of order O  1n  .
Finally, Corollary 1 of Chung and Shih (2007) proved that the rate of convergence of
9
Please see p. 57 of Heston and Zhou (2000).
9
the GCRR-XPC model is of order O  1n  . Thus, all the binomial models considered in
this paper have the same rate of convergence, i.e. of order O  1n  for calculating option
prices.
Let e(n) be the pricing error of the n-step binomial model, i.e.:
e(n)  Cn  CBS ,
(13)
where Cn is the n-step binomial price and C BS is the Black-Scholes price. Define
the error ratio as:
 (n)  e(n / 2) e(n).
(14)
The error ratio is a measure of the improvements in accuracy as the number of time
steps doubles. Given the fact that all the binomial models considered in this paper
have the convergence of order O  1n  , we obtain that the error ratio will converge to 2
as the number of time steps increases. Thus the Black-Scholes price can be
approximated as follows:
CBS (n)  2Cn  Cn 2 .
(15)
It is not difficult to show that the pricing error after applying the extrapolation can be
written as:
e(n)  CBS (n)  CBS  (2   (n))e(n) .
(16)
Thus the pricing error after applying the extrapolation critically depends on the
difference between 2 and the error ratio. If the error ratio  ( n ) is within the range of
(1, 3) , the absolute pricing error with extrapolation is smaller than the absolute
pricing error without extrapolation. Otherwise, applying the extrapolation method to
the binomial prices will increase rather than decrease the pricing errors.
Similarly, to derive the extrapolation formula for hedge ratios, one needs to
10
derive the rate of convergence of the Greeks calculated from the binomial models. In
the following, the theoretical proofs of the convergent rates of applying the extended
method to CRR, SPF, and GCRR-XPC for calculating deltas are offered.10 Similarly,
calculating gamma also has the same rate of convergence and the detailed proofs are
available on request.
Theorem 1. Let ˆ n,CRR be the n-period CRR binomial delta of a standard European
put option with extended tree and  BS is the true delta. Therefore,

 n ,CRR =  BS 
d12
2
e
f   (n)   O  1n  ,
n 2
(17)
where f  (n)  = 2d 2 ( (n)   (n) 2 ) , N(．) is the cumulative density function of the
standard
normal
distributions,
d1 
ln XS  (r   2 / 2)T
 T
and  (n) 
ln SXx
ln du
,
S x  Su m d nm , and m is the largest integer which satisfies Su m d n  m  X .
Proof: It is assumed that Su m d n m  X and m is the largest integer. In other words,
m
ln XS
n
n

  ( n) =  z ,
2
2 2 t
(18)
ln X  ln Su m d nm
  (n) and 0   (n)  1 . Since
where z 
  (n) , we have
2 t
2 t
ln
X
S
Su m d nm  Xe2
t
.
(19)
On the other hand,
10
Generally, BBS and SPF can be classified as the models with the smoothed payoff function. On the
other hand, FB and GCRR-XPC can be classified as the models with the adjusted tree so that the strike
price is allocated at one of the final nodes. Therefore, in addition to the convergent rate of applying
extended binomial tree method to CRR, only the convergent rates of applying extended binomial tree
method to SPF and GCRR-XPC for calculating deltas are offered for simplicity.
11
ert  e
p
where a 
r
 t
e
e
t
 t

1
 a t  O  1n  ,
2
(20)
2
2 .
2
In the extended tree, the option value at the node with stock price equal Su 2 at
present time is
m 1


e rT  C nj p j (1  p)n j X  Su 2u j d n j ,
j 0
(21)
and the option value at the node with stock price equal Sd 2 at present time is
m 1


e rT  C nj p j (1  p)n  j X  Sud 2u j d n  j .
j 0
The estimated delta is  n,CRR 
(22)
(21)  (22)
. We denote (21)-(22) =A+B+C, where
S (u 2  d 2 )
n m
n  m 1
,
A   Xe rT Cmn p m 1  p   Cmn 1 p m1 1  p 


B=
nm
n  m1
e rT Cmn p m 1  p  Su 2u md nm  Cmn 1 p m1 1  p 
Sd 2u m1d nm1  ,


m
C= e rT  C nj p j 1  p 
j 0
n j
 Su
2
(23)

 Sd 2 u j d n j .
According to Chung and Shih (2007),
m
C
n j
 rT


e
C nj p j 1  p  u j d n j

2
2
S (u  d )
j 0

0.5   (n) 
  N  d1 
 O  1n  .


np 1  p  

Because Cmn 1  Cmn 
nm
and Su m d nm  Xe2
m 1
12
t
by (19), we have
(24)
A+B
 Xe rT Cmn p m 1  p 
= Xe  rT Cmn p m 1  p 
where D = u 2e2
nm
nm
t
t 

n  m p 2
e
m 1 1 p
t 
1
nm p 
m  1 1  p 
(25)
D,

1  2nz
nm
Since

m  1 1  2( zn1)
 2 2
u e

n  m p 2
e
m 1 1 p
t
1 
nm
nm p
. First, let’s check
.
m 1
m 1 1  p
2

 2 z   2( z  1) 4  z  1
 1   1 


...
 , by (18),
n 
n
n2



2 ln XS
nm
 1
O
m 1
n t

.
e1  ( n )
n
(26)
Besides, we have
u 2e2
t
 1  2 t 1   (n)  2 2 t 1   (n)  O
2

e2  ( n )

n15
,
(27)
and by (20)
p
 1  4a t  O  1n  .
1 p
(28)
Therefore, D defined in (25) is
2ln XS

D  1 
O
 n t

e1  ( n )
n

 1  4a
× 2 t  (n)  2 2 t 2 (n)  O
t  O  1n 

e3  ( n )

n15

+ 2 t 1   (n)  2 2 t 1   (n)  O
2

e2  ( n )

n15

 4 t 0.5   (n)  2 2 t 2 (n)  8at (n)

4ln XS
2
 (n)  2 2 t 1   (n)  O
n

e4  ( n )
n1.5
On the other hand, according to Feller’s (1971, P.53~P.54),
13

(29)
n!  2 n
n
1
2 n
e
1  O   .
1
n
Therefore,
Cmn 
Since
1
1
1  O  1n   .
1
1 
2 n 1  m nm 2 m m 2
 n
n
m 1 z
  according to (18),
n 2 n
1  O  1n  

1
.
C 

n 1
1
1
2 n  1 
2 z nm 2
2 z m 2
1  n 
  1  n 
2
n
m
(30)
On the other hand, by (20),
p
m
1  p 
nm
n
m
n m
1
   1  x  1  x  ,
2
(31)
where x  2a t  O  1n  . Therefore, by (30) and (31),
C p 1  p 
n
m
m
nm
1
1  1 x 



2 n  1   1  2nz 
 
2

nm
 1 x 

2z 
 1 n 
n
m
1
1
2z 2
n
1
2z 2
n
1   1  
m
 1  x   1  x 1  2nz 


 1  O  1n  
2z  
2z 
n  1 n   1 x 1 n 
2
4
1
.
(32)


y 2 y3
According to the Taylor’s series of ln(1  y )    y 
  ...  and the
2
3


definitions of z and x in (18) and (31),
n
m
 1  x   1  x 1  2nz 
nx 2
2z 2
ln  2 z  
 2z  
 2 xz 
O
2
n
 1 n   1 x 1 n 
 
2
1
n
z 

= 2  a t  n 
 O
n

14
 .
1
n
(33)
 2 
r 

2 
ln XS
2 (n) 
z
 ( n) 1 
 d2
Since a t n 

T

,
2
n 
2
n
2 T
n
d
(33)  
2
 2 (nn )

2
2
O
 
1
n
d22 2d2 (n)

O
2
n

 .
1
n
Therefore,
d 22 2 d 2 ( n )

2
n

Xe  rT Cmn p m 1  p 
nm
e
 Xe rT
n
2
4
1
S
 1  O

2
n
4
e

d12
2
e

2 d 2 ( n )
n
  1  O 
1
n
1
n
 .
1  O

  1  O 
1
n
 .
1  O

  1  O 
1
n

1
n
(34)
By (29) and (34), A+B in (25) is
( A  B)  S
1
2
n
4
e

d12
2
e

2 d 2 ( n )
n
1
n

 4 t  0.5   (n)   2 2 t 2 (n)  8at (n)

4ln XS
n
 (n)  2 2 t 1   (n)   O
2

e4  ( n )
n1.5

(35)
Therefore, after tedious calculations, we have
 n,CRR 
(21)  (22) A  B  C
(35)


 (24) .
2
2
2
2
S (u  d ) S (u  d ) S (u 2  d 2 )

  N  d1  
d12
2
e
f   (n)   O  1n  ,
n 2
where f  (n)  = 2d 2   (n)   2 (n)  .
(36)
Q.E.D.
According to Theorem 1, although using CRR with an extended tree to calculate
delta has the rate of convergence O  1n  , the convergent pattern is oscillatory due to the
15
relative allocation of the strike price between two most adjacent nodes. However, SPF
and GCRR-XPC models demonstrated that the CRR’s binomial prices monotonically
converge to the Black-Scholes formula at the rate of O( 1n ). The rates of convergence
of applying extended tree to SPF and GCRR-XPC models in calculating delta are
proved in the following theorems.
Theorem 2. Let ˆ n, SPF be the n-period SPF binomial delta of a standard European
put option with extended tree and  BS is the true delta,
 n , SPF =  BS  O  1n  .
(37)
Proof: For simplicity, it is assumed that  (n)  0.5 . The proofs for the other values of
 (n) are quite similar, and they are available on request. In the extended tree, the
option value at the node with stock price equal Su 2 at present time is
m 1

e t  e
e  rT  C nj p j (1  p) n  j  X  Su 2u j d n  j

2 t
j 0

t
  rT n m
n m
  e Cm p (1  p )

 X  t  ln( Su m1d nm1 ) 

X
e t
X 
 

m 1 n  m 1
×

Su d

,
2 t
2 t
2 t 



(38)
and the option value at the node with stock price equal Sd 2 at present time is
m 1
e  rT  C nj p j (1  p ) n  j ( X  Sd 2u j d n  j
j 0
e

 rT
n
m 2
C
p
m 2
(1  p)
n m2
e
 e 
2 t
t

t
)

 X  t  ln Su m1d nm1 
X
 


2 t


e  t
X 
Su m 1d n  m 1 
.
2 t
2 t 
The estimated delta is denoted by  n, SPF 
(39)
(38)  (39)
. For the convenience of proof,
S (u 2  d 2 )
16
We define


  t  ln Su m1d nm1 

 t
X
Su m1d nm1
1 

 e
A=  

.
X
2


t
2


t
2


t




Xe rT A[Cmn p m (1  p)nm  Cmn  2 p m 2 (1  p)n m2 ]
(38)  (39)
Let’s first calculate
in
.
2
2
S (u 2  d 2 )
S (u  d )
Because ln
Su m1d nm1
 2 t 1   (n)  ,
X
 t (1 2 ( n ))
 1]
2
1
 [e
A=   (1   (n))  
  t 0.5   (n)   O
2 t
2


E1  ( n )
n
.
(40)
In addition,
Xe rT [Cmn p m (1  p) n  m  Cmn  2 p m  2 (1  p) n m  2 ]
 (n  m  1)(n  m) p 2 
 Xe rT Cmn p m (1  p)nm 1 
.
(m  1)(m  2) (1  p) 2 

Since p 
a
r
(41)
[1  2( zn1) ][1  2nz ]
1
(n  m  1)(n  m)
 a t  O  1n  and
where

2
(m  1)(m  2)
[1  2( zn1) ][1  2( zn 2) ]
2
2
2
and z 
ln
X
S
2 t
  ( n) ,
 (n  m  1)(n  m) p 2 
4ln XS
(r  2 )
1



4
t  O


(m  1)(m  2) (1  p) 2  n t


2

4d 2
O
n

E1  ( n )
n

E2  ( n )
n
.

(42)
By (34) and (42), we have
(41)= S
1
2
n
4
e

d12
2

e
2 d 2 ( n )
n
(1  O
 ) 1  O    4nd
1
n
1
n
Therefore, by (40) and (43),
17
2
O

E2  ( n )
n
 .
(43)
Xe rT A[Cmn p m (1  p)n m  Cmn  2 p m  2 (1  p )n m  2 ]
S (u 2  d 2 )

1 2
e
n 2

d12
2
(d 2 ) 0.5   (n)   O
2

E3  ( n )
n1.5

.
2
1 2  d21

e (d 2 )   (n)   2 (n)   O  1n 
n 2
(44)
From the above theorem, we know

d12
2
U D
2e
  N  d1  
d 2   (n)   2 (n)   O  1n  ,
2
2
Su  Sd
n 2
(45)
where
m 1

e t  e
U = e  rT  C nj p j (1  p )n  j  X  Su 2u j d n  j

2 t
j 0

t

 ,

m 1

e t  e 
D = e  rT  C nj p j (1  p )n  j  X  Sd 2u j d n  j

2 t
j 0

t

 .

Therefore by (44) and (45), the estimated delta
 n, SPF 
(40)  (41)
= (44)+(45) =  N (d1 )  O  1n  .
2
2
S (u  d )
Q.E.D.
Theorem 3. Let ˆ n,GCRR  XPC be the n-period GCRR-XPC binomial delta of a
standard European put option with extended tree and  BS is the true delta,
 n ,GCRR  XPC =  BS  O  1n  .
(46)
Proof: Please refer to Appendix 1.
To sum up, in this section, we discuss the rate of convergence of the binomial
models and offer the theoretical proofs of the rates of convergence of applying
extended method to CRR, SPF and GCRR-XPC to calculate deltas. It is found that
applying extended tree to SPF and GCRR-XPC can generate monotonic convergence
with O  1n  rate for calculating Greeks.
18
4. Numerical results and analysis
In the following numerical analysis, we will extensively illustrate the
convergent patterns, rate of convergence and numerical efficiency of various binomial
models of calculating delta and gamma. First, to show the error ratios of various
binomial models defined in (14), the following parameters are chosen: stock price=40,
strike price = 45, volatility = 0.2, time to maturity= 6 months, riskless interest
rate=6% and dividend yield= 0 so that we can make sure if the extrapolation
technique can be applied to enhance the accuracy.
For calculating deltas, from Tables 1 and 2, it shows that applying Hull method
or applying the extended method to these binomial models result in quite stable error
ratios around 2. It means that they converge to the true delta monotonically with
O(1/n) rate. Therefore, the extrapolation technique can be applied to enhance the
accuracy. We first choose a large set of options (243 options). The 243 parameter sets
are drawn from the combinations of X  {35, 40, 45}, T  {1/12,4/12,7/12}, r  {3%,
5%, 7%}, q  {2%, 5%, 8%},   {0.2, 0.3, 0.4}, and S  40 . We report the results
for different numbers of time steps n  {20, 40, 60, 80, 100, 500, 1000} in order to
examine the convergence property. The accuracy measure used in this paper is the
root mean squared (RMS) absolute error. RMS absolute error is defined as
1 m 2
RMS 
 ei ,
m i 1
(47)
where ei  cˆi  ci is the absolute error, ci is the “true” value (estimated from the
Black-Scholes formula), cˆi is the estimated value from the binomial model.
<Insert Tables 1 and 2 Here>
19
To clarify the role played by Richardson extrapolation, we first compare the
numerical efficiency of each model on a raw basis without any extrapolation
procedure. From Table 3, the extended method on average underperforms Hull
method without extrapolation.11 Overall, the SPF with Hull method is the best.
However, when Richardson extrapolation is applied, Table 4 shows that the extended
method on average outperforms the Hull method in calculating Greeks. 12 The
improvement of using Richardson extrapolation for the extended method is because
the error ratios of the extended method are more like to be 2 for these four binomial
models compared to the Hull method as shown in Table 1 and Table 2. Overall,
applying extended method to GCRR-XPC is the best.
For calculating gammas, from Table 5 and Table 6, applying Hull method or
extended method to these binomial models result in quite stable error ratios around 2,
so the extrapolation technique can be applied to enhance the accuracy. Again, to
clarify the role played by Richardson extrapolation, we first compare the numerical
efficiency of each model on a raw basis without any extrapolation procedure. In Table
7, it shows that the extended method is on average superior to the Hull method.13
Overall, the performance of FB with the extended tree is the best. However, when
Richardson extrapolation is used in Table 8, FB performs worst and this is consistent
with Table 5 and Table 6 in which the error ratios of using FB are most unstable
11
This is true for all these four binomial models except for GCRR-XPC model.
12
The numerical results of using extended method are better than those of using Hull method for
GCRR-XPC model and FB model. Besides, the numerical result of using extended method is almost as
good as those of using Hull method for BBS model.
13
The numerical results of using extended method are better than those of using Hull method for CRR
model, GCRR-XPC model, and FB model. However, the numerical result of using Hull method is
better than those of using extended method only for BBS model and SPF model.
20
around 2. Overall, applying extended tree to GCRR-XPC is the best. To sum up,
applying extended tree to GCRR-XPC with extrapolation results in the most efficient
estimates of deltas and gammas.
<Insert Tables 3 to 8 Here>
5. Conclusion
This paper uses various binomial models to calculate Greeks and discuss the
convergent pattern, convergent rate, and the numerical efficiency among these models.
We contribute to the literature at least three aspects. First, although BBS model of
Brodie and Detemple (1996) , SPF of Heston and Zhou (2000), FB of Tian (1999),
and GCRR-XPC of Chung and Shih (2007) have been widely applied to price options,
their convergent patterns and rate of convergence for calculating the hedge ratios are
not well known. To fill this gap, we apply the Hull method and the extended method
to these models in order to compute deltas and gammas.
Second, we apply the Hull method and extended method to these models and
compare numerical efficiency of calculating Greeks. It is found that applying the
extended method to GCRR-XPC is the most efficient binomial model in calculating
Greeks when a two-point extrapolation formula is used.
Finally, we derive the convergence rate of the binomial model in calculating
Greeks. We prove that applying the extended method to GCRR-XPC model and SPF
model converge to the true delta monotonically with the rate of convergence O(1/n).
21
Appendix 1 (Proof of Theorem 3)
At the node with stock price equal to Su1u2 , the option price is
n
1
2
e rT  C nj p j (1  p)n j ( X  Su1u2  u1j d1n j ) ,
(A1)
j 0
and at the node with stock price equal to Sd1d 2 , the option price is
n
1
2
e rT  C nj p j (1  p)n j ( X  Sd1d 2  u1j d1n j ) ,
(A2)
j 0
where u1  e
 ( n ) t
, u2  e
 t
 (n)
, d1  e
 t
 (n)
and d 2  e   ( n )
t
. The estimated delta is
(A1)  (A2)
defined as ˆ n ,GCRR  XPC 
and
S (u1u2  d1d 2 )
n
1
 n2 1

2
n j
n j
n j
n j 

(A1)  (A2)  X  C j p (1  p)   C j p (1  p)
 j 0

j 0


n
1
 n2 1

2
j n j n j
n j
j n j n j
n  j   rT

+  Su1u2  u1 d1 C j p (1  p)   Sd1d 2  u1 d1 C j p (1  p )
e
 j 0

j 0


Because C(nn / 2) 1 
n!
n
n/2
n


 C(nn / 2) 
,
n2
 n   n   n   n  n 1
  1 !  1 !   !  ! 2
2  2  2 2
n
1
 n2 1

2
n j
n j
n j
n  j   rT

X  C j p (1  p)   C j p (1  p)
e
 j 0

j 0


n
n
n
n
1
1 

  X Cnn/2 p 2 (1  p) 2  C(nn /2) 1 p 2 (1  p) 2  e  rT



n
p   rT
  XC(nn /2) p n /2 (1  p)n /2  1 

e .
 n  2 1 p 
22
(A3)
1
 a t  O  1n  ，where
2
According Chung and Shih (2007), p 
 2 
ln   r 
T
2 
1  r ln XS   1
1
1

.
a  
 
 d2
2   T 2  2
T
2
T
X
S
(A4)
Therefore,
n
 1
 1
 2
p (1  p)    a t  O  1n    a t  O  1n   
 2

 2
n
2
n
2
n
 1 2
   1  4a 2 t  O
 4  
  .
n
2
1
n
3
(A5)
2
On the other hand,
n
1
 n2 1

2
 Sd d  u j d n  j C n p j (1  p ) n  j  Su u  u j d n  j C n p j (1  p )n  j  e  rT

1 2
1 1
j
1 2
1 1
j


j 0
j 0


n
2
  ( Sd1d 2  Su1u2 )  C nj p j (1  p) n  j u1j d1n  j  e  rT
j 0
n
n
1
1
2
2
1 2 1
1
 Sd d u
d
n
( n /2) 1
C
n n
2 2
1 1
p
n
1
2
Since Su d  X , u1  e
(1  p )
 ( n ) t
n
1
 rT
2
e
, u2  e
n
2
1 2 1
n
2
1
 Su u u d C
 t
 (n)
, d1  e
n
n /2
 t
 (n)
n
2
n
2
p (1  p )  e  rT .
and d 2  e   ( n )
t
(A6)
,
n
n
n
n
n n
n
n
1
1
1
1

  rT
n
2
2
2
2
2 2 n
2
2
Sd
d
u
d
C
p
(1

p
)

Su
u
u
d
C
p
(1

p
)
 1 2 1 1 ( n /2) 1
 e
1 2 1 1
n /2


n
n
n n

ddu
n
p   rT
 Cnn/2 p 2 (1  p ) 2 Su12 d12 u1u2  1 2 1 

e
d1
n  2 1 p 

C
n
n /2

p
n   rT
p (1  p)  X  u1u2 

e
1  p n  2 

n
2
n
2
By (A3), (A6) and (A7),
23
(A7)
n
2
n
2
XC p (1  p) [u1u2  1]  rT
(A1)  (A2)
  C nj p j (1  p)n j u1j d1n j  e rT 
e .
S (u1u2  d1d2 )
S (u1u2  d1d 2 )
j 0
n
n /2
n /2
(A8)
Because
2

1
p(1  p)     a t  O  1n 
2

2
1
 [1  x ]
4
where x  4a 2 t  O
n
(A9)
 ,
1
n1.5
n
X Cnn/2 p 2 (1  p) 2 [u1u2  1]  rT X n  1 
 e  Cn /2  
S
u1u2  d1d 2
S
4
Since (1  x)
e

d22
2
e

n /2
e
( d1  a T )2
2
ln(1 x )
e

n
2
d12
2
e
( x
x2
n
...)
2
2
n /2
 e2 a tn e
2
O
1  x 
n /2
u1u2  1
 e rT
u1u2  d1d 2
(A10)
  , by (A4) and
1
n
S rT
e ，we have
X
(1  x)
n /2
e

d22
2
   S e d2  erT  eO  .
e
O
2
1
1
n
1
n
X
and
n
d
X
 1  2 S rT  21  rT O 1n  (u1u2  1)
e e e e
(A10)  Cnn/2  
S
u1u2  d1d 2
4 X
n
2
d
 1  2  1 O 1  (u1u2  1)
 Cnn/2    e 2  e n
.
u1u2  d1d 2
4
2
(A11)
According to Feller’s (1971, p.53 ~ p.54),
n!  2  n
n
1
2 n
e
If we set n  2k ，we have
24
1  O   .
1
n
C
n
n /2
n
2
(2k )!  1 
1
  
 
k !k !  2 
4

2k

2k 
1
2
e 2 k 1  O  21k   1
 2k
2
2
2 k 2 k 1e2 k 1  O  1k  
2 (2k )
1
1  O  1k   .
k
Therefore,
n
n
n /2
C
1
1
 1 2
1  O  1n  
1  O  1n  ,
  
k
n / 2 
4
(A12)
and
(A11)  e

e

d12
2

O 1 
u u 1
1
1  O  1n    e n  1 2
u1u2  d1d 2
2  (n / 4)
d12
2
1  d21
e
2
1
1  O  1n    1  O

2  (n / 4)
    12  O  
1
n
1
 O  1n  .
2  (n / 4)
1
n
(A13)
In (A8), according to Chung and Shih (2007),
n /2
n /2
J 0
J 0
 C nj p j (1  p) n  j u1j d1n  j e  rt   C nj p* j (1  p* )n  j


0.5
  O  1n  .
  N  d1 
*
* 

np (1  p ) 

where p* 
(A14)
pu1
.
e r t


0.5
(A14)    N (d1 )  N (d 1 )
 O  1n 

np* (1  p* )

1  d1
  N (d1 )   e 2
2
1
 O  1n  .
2  (n / 4)
25
(A15)
Finally, we have
(A1)  (A2)
ˆ n ,GCRR  XPC 
 (A13)  (A15)   N ( d1 )  O  1n  .
Su1u2  Sd1d 2
(A16)
Q.E.D.
26
References
Broadie, M., Detemple, J., 1996. American option valuation: New bounds,
approximations, and a comparison of existing methods. Review of Financial
Studies 9, 1211-1250.
Chang, C.C., Chung, S.L., Stapleton, R., 2007. Richardson extrapolation techniques
for the pricing of American-style options. Journal of Futures Markets 27,
791-817.
Chung, S.L., Shih, P.T., 2007. Generalized Cox-Ross-Rubinstein binomial model.
Management Science, 53, 508-520.
Cox, J.C., Ross, A., Rubinstein, M., 1979. Option pricing: A simplified approach.
Journal of Financial Economics 7, 229-263.
Feller, W., 1971. An introduction to probability theory and its applications. NY: John
Wiley&Sons, Inc.
Figlewski, S., Gao, B., 1999. The adaptive mesh model: A new approach to efficient
option pricing. Journal of Financial Economics 53, 313-351.
Heston, S., Zhou, G., 2000. On the rate of convergence of discrete-time contingent
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Hull, J.C., 1993. Options, futures and other derivatives. NJ: Prentice-Hall
International, Inc.
Leisen, D., Reimer, M., 1996. Binomial models for option valuation-examining and
improving convergence. Applied Mathematical Finance 3, 319-346.
Omberg, E., 1988. Efficient discrete time jump process models in option pricing.
Journal of Financial and Quantitative Analysis 23, 161-174.
Pelsser, A., Vorst, T., 1994. The binomial model and the Greeks. Journal of
Derivatives 1, 45-49.
27
Rendleman, R., Batter, B., 1979. Two-state option pricing. Journal of Finance 34,
1093-1110.
Tian, Y.S., 1999. A flexible binomial option pricing model. The Journal of Futures
Markets 19, 817-843.
Widdicks, M., Andricopoulos, A.D., Newton, D.P., Duck, P.W., 2002. On the
enhanced convergence of standard lattice methods for option pricing. Journal of
Futures Markets 22, 315-338.
28
 0
 0
 0
Fig. 1. The flexible binomial tree for different λ values
S 2,2
S1,1
C2,2
C1,1
S 0,0
S 2,1
S1,0
C0,0
S 2,0
C 2,1
C1,0
C2,0
Fig. 2. The Hull method
C0,2
S 0,2
S 1,1
C 1,1
S 2,0
S 0,1
C 0,1
C2,0
S 1,0
C1,0
S 0,0
C0,0
-2 t
- t
0
-2 t
- t
Fig. 3. The extended binomial tree
29
0
TABLE 1
Delta and error ratios for European put options by using Hull method
GCRR-XPC
FB
Delta
Error
Error ratio
Delta
Error
Error ratio
20
-0.719297
-0.010454
—
-0.716577
-0.007734
—
40
-0.714714
-0.005871
1.780563
-0.712432
-0.003589
2.154864
80
-0.712009
-0.003166
1.854442
-0.710782
-0.001939
1.850665
160
-0.710508
-0.001665
1.901807
-0.709920
-0.001077
1.800574
320
-0.709704
-0.000861
1.932918
-0.709386
-0.000543
1.983631
640
-0.709284
-0.000441
1.953739
-0.709094
-0.000251
2.165007
1280
-0.709067
-0.000224
1.967875
-0.708973
-0.000130
1.936146
2560
-0.708956
-0.000113
1.977577
-0.708908
-0.000065
1.983873
5120
-0.708900
-0.000057
1.984291
-0.708875
-0.000032
2.016248
Time steps n
BBS
SPF
Delta
Error
Error ratio
Delta
Error
Error ratio
20
-0.713379
-0.004536
—
-0.711156
-0.002313
—
40
-0.711087
-0.002244
2.021622
-0.709940
-0.001097
2.107662
80
-0.709950
-0.001107
2.026989
-0.709392
-0.000549
2.000170
160
-0.709394
-0.000551
2.007205
-0.709103
-0.000260
2.109401
320
-0.709119
-0.000276
1.999329
-0.708974
-0.000131
1.977799
640
-0.708981
-0.000138
2.003059
-0.708909
-0.000066
1.977977
1280
-0.708911
-0.000068
2.015835
-0.708876
-0.000033
1.996055
2560
-0.708877
-0.000034
2.000920
-0.708860
-0.000017
2.012228
5120
-0.708860
-0.000017
1.995844
-0.708851
-0.000008
1.990182
Black-Scholes
-0.708843
Time steps n
The number of time steps starts at 20 and doubles each time subsequently. The left side of the
table reports delta, delta errors, and error ratios.
30
TABLE 2
Delta and error ratios for European put options by using extended method
GCRR-XPC
Time steps n
FB
Delta
Error
Error ratio
Delta
Error
Error ratio
20
-0.701162
0.007682
—
-0.700546
0.008297
—
40
-0.704986
0.003857
1.991408
-0.704675
0.004169
1.990367
80
-0.706911
0.001933
1.995742
-0.706751
0.002093
1.992035
160
-0.707876
0.000967
1.997881
-0.707804
0.001039
2.013715
320
-0.708359
0.000484
1.998943
-0.708324
0.000519
2.000775
640
-0.708601
0.000242
1.999472
-0.708581
0.000262
1.981833
1280
-0.708722
0.000121
1.999736
-0.708713
0.000131
2.003349
2560
-0.708783
0.000061
1.999868
-0.708778
0.000065
2.001280
5120
-0.708813
0.000030
1.999934
-0.708810
0.000033
1.998430
BBS
Time steps n
SPF
Delta
Error
Error ratio
Delta
Error
Error ratio
20
-0.697113
0.011731
—
-0.695143
0.013701
—
40
-0.702848
0.005996
1.956450
-0.701771
0.007073
1.937062
80
-0.705803
0.003040
1.972039
-0.705262
0.003582
1.974643
160
-0.707314
0.001529
1.987975
-0.707027
0.001817
1.971769
320
-0.708077
0.000766
1.995427
-0.707934
0.000910
1.996845
640
-0.708460
0.000384
1.996501
-0.708389
0.000455
2.000010
1280
-0.708651
0.000193
1.993193
-0.708616
0.000228
1.998972
2560
-0.708747
0.000096
1.999067
-0.708730
0.000114
1.997423
5120
-0.708795
0.000048
2.001171
-0.708786
0.000057
2.001033
Black-Scholes
-0.708843
The number of time steps starts at 20 and doubles each time subsequently. The left side of the
table reports delta, delta errors, and error ratios.
31
TABLE 3
RMS errors of delta estimates (without extrapolation)
Time steps n
20
40
60
80
100
500
1000
Hull method
CRR
GCRR-XPC
FB
BBS
SPF
1.97E-03
7.49E-04
9.90E-04
5.24E-04
4.30E-04
6.82E-05
4.04E-05
(0.08)
(0.14)
(0.20)
(0.28)
(0.35)
(4.13)
(15.34)
6.23E-03
3.40E-03
2.35E-03
1.80E-03
1.47E-03
3.17E-04
1.62E-04
(0.13)
(0.20)
(0.29)
(0.38)
(0.50)
(5.70)
(21.16)
4.05E-03
2.07E-03
1.48E-03
1.07E-03
8.43E-04
1.70E-04
8.68E-05
(0.08)
(0.12)
(0.16)
(0.21)
(0.28)
(4.20)
(16.15)
1.77E-03
8.66E-04
5.71E-04
4.28E-04
3.41E-04
6.79E-05
3.38E-05
(3.88)
(7.64)
(11.40)
(15.16)
(18.93)
(97.36)
(203.27)
8.78E-04
4.41E-04
2.91E-04
2.20E-04
1.74E-04
3.50E-05
1.75E-05
(0.13)
(0.19)
(0.27)
(0.36)
(0.47)
(5.27)
(18.56)
Extended method
CRR
GCRR-XPC
FB
BBS
SPF
7.47E-03
3.94E-03
2.29E-03
1.93E-03
1.49E-03
3.24E-04
1.45E-04
(0.10)
(0.15)
(0.21)
(0.29)
(0.36)
(4.16)
(15.40)
5.09E-03
2.55E-03
1.70E-03
1.28E-03
1.02E-03
2.05E-04
1.02E-04
(0.15)
(0.22)
(0.31)
(0.40)
(0.52)
(5.75)
(21.24)
4.95E-03
2.50E-03
1.66E-03
1.24E-03
1.01E-03
2.02E-04
1.00E-04
(0.10)
(0.13)
(0.17)
(0.22)
(0.29)
(4.23)
(16.21)
7.31E-03
3.73E-03
2.51E-03
1.89E-03
1.51E-03
3.05E-04
1.53E-04
(4.27)
(8.02)
(11.78)
(15.54)
(19.31)
(97.75)
(203.68)
8.71E-03
4.45E-03
2.99E-03
2.25E-03
1.80E-03
3.63E-04
1.82E-04
(0.15)
(0.21)
(0.29)
(0.38)
(0.49)
(5.31)
(18.63)
The CPU time (in seconds) required to value 243 deltas is given in parentheses.
32
TABLE 4
RMS errors of delta estimates (with extrapolation)
Time steps n
20
40
60
80
100
500
1000
Hull method
GCRR-XPC
FB
BBS
SPF
2.48E-04
6.74E-05
3.08E-05
1.76E-05
1.14E-05
4.68E-07
1.18E-07
(0.16)
(0.26)
(0.38)
(0.51)
(0.66)
(7.38)
(26.75)
2.89E-03
1.25E-03
6.36E-04
3.49E-04
2.18E-04
2.85E-05
9.44E-06
(0.10)
(0.16)
(0.21)
(0.28)
(0.37)
(5.43)
(20.42)
6.13E-04
1.70E-04
8.17E-05
4.53E-05
3.05E-05
1.01E-06
7.68E-07
(5.93)
(11.56)
(17.22)
(22.88)
(29.26)
(145.38)
(300.24)
4.01E-04
1.27E-04
6.41E-05
3.75E-05
2.26E-05
2.04E-06
6.96E-07
(0.16)
(0.25)
(0.36)
(0.49)
(0.62)
(6.82)
(23.46)
Extended method
GCRR-XPC
FB
BBS
SPF
1.55E-04
4.29E-05
1.98E-05
1.14E-05
7.38E-06
3.08E-07
7.73E-08
(0.19)
(0.29)
(0.41)
(0.54)
(0.69)
(7.44)
(26.85)
9.31E-04
4.36E-04
2.43E-04
1.40E-04
9.08E-05
9.56E-06
2.94E-06
(0.13)
(0.17)
(0.22)
(0.30)
(0.38)
(5.47)
(20.49)
6.17E-04
1.71E-04
8.20E-05
4.56E-05
3.06E-05
1.01E-06
7.68E-07
(6.53)
(12.14)
(17.79)
(23.45)
(29.85)
(145.96)
(300.85)
9.22E-04
2.41E-04
1.06E-04
6.70E-05
4.05E-05
2.19E-06
9.22E-07
(0.19)
(0.28)
(0.39)
(0.52)
(0.66)
(6.93)
(23.82)
Note: The CPU time (in seconds) required to value 243 deltas is given in parentheses.
33
TABLE 5
Gamma and error ratios for European put options by using Hull method
GCRR-XPC
Time steps n
FB
Gamma
Error
Error ratio
Gamma
Error
Error ratio
20
-0.001827
-0.001827
—
0.001526
0.001526
—
40
-0.000883
-0.000883
2.068289
0.000841
0.000841
1.814899
80
-0.000425
-0.000425
2.080183
0.000383
0.000383
2.197365
160
-0.000205
-0.000205
2.076239
0.000149
0.000149
2.562954
320
-0.000099
-0.000099
2.065368
0.000073
0.000073
2.044971
640
-0.000048
-0.000048
2.052693
0.000046
0.000046
1.600160
1280
-0.000024
-0.000024
2.040803
0.000021
0.000021
2.147565
2560
-0.000012
-0.000012
2.030747
0.000010
0.000010
2.041972
5120
-0.000006
-0.000006
2.022736
0.000005
0.000005
1.959054
BBS
Time steps n
SPF
Gamma
Error
Error ratio
Gamma
Error
Error ratio
20
0.061341
0.000717
—
0.061022
0.000398
—
40
0.060984
0.000360
1.993806
0.060824
0.000200
1.991723
80
0.060803
0.000179
2.003719
0.060727
0.000103
1.935925
160
0.060713
0.000089
2.016477
0.060684
0.000060
1.732304
320
0.060669
0.000045
1.987626
0.060653
0.000029
2.078587
640
0.060646
0.000022
1.997408
0.060638
0.000014
2.054581
1280
0.060635
0.000011
2.012954
0.060631
0.000007
2.000481
2560
0.060630
0.000006
2.002770
0.060628
0.000004
1.969727
5120
0.060627
0.000003
1.993477
0.060626
0.000002
2.028263
Black-Scholes
0.060624
The number of time steps starts at 20 and doubles each time subsequently. The left side of the
table reports gamma, gamma errors, and error ratios.
34
TABLE 6
Gamma and error ratios for European put options by using extended method
GCRR-XPC
Time steps n
FB
Gamma
Error
Error ratio
20
0.059092
-0.001532
—
40
0.059851
-0.000773
80
0.060236
160
Gamma
Error
Error ratio
—
0.060874
0.000249
1.981377
0.060814
0.000190
1.315248
-0.000388
1.990661
0.060690
0.000066
2.876399
0.060429
-0.000195
1.995324
0.060631
0.000007
10.114306
320
0.060527
-0.000097
1.997660
0.060627
0.000002
2.762609
640
0.060575
-0.000049
1.998830
0.060631
0.000007
0.351503
1280
0.060600
-0.000024
1.999415
0.060627
0.000002
2.857169
2560
0.060612
-0.000012
1.999707
0.060625
0.000001
2.256359
5120
0.060618
-0.000006
1.999854
0.060625
0.000001
1.770236
BBS
Time steps n
Gamma
Error
20
0.060102
-0.000522
40
0.060373
-0.000251
80
0.060501
160
SPF
Error ratio
Gamma
Error
Error ratio
0.059794
-0.000830
—
2.076224
0.060215
-0.000409
2.028231
-0.000124
2.033228
0.060425
-0.000200
2.051288
0.060562
-0.000062
1.998139
0.060533
-0.000091
2.181807
320
0.060594
-0.000030
2.028958
0.060578
-0.000047
1.960983
640
0.060609
-0.000015
2.009468
0.060601
-0.000024
1.972058
1280
0.060617
-0.000008
1.983999
0.060612
-0.000012
2.001634
2560
0.060620
-0.000004
1.997418
0.060618
-0.000006
2.019246
5120
0.060622
-0.000002
2.010265
0.060621
-0.000003
1.983790
Black-Scholes
0.060624
The number of time steps starts at 20 and doubles each time subsequently. The left side of the
table reports gamma, gamma errors, and error ratios.
35
TABLE 7
RMS errors of gamma estimates (without extrapolation)
Time steps n
20
40
60
80
100
500
1000
Hull method
CRR
GCRR-XPC
FB
BBS
SPF
2.18E-03
1.05E-03
7.25E-04
5.29E-04
4.16E-04
8.27E-05
4.13E-5
(0.08)
(0.14)
(0.20)
(0.28)
(0.35)
(4.13)
(15.34)
2.25E-03
1.11E-03
7.41E-04
5.56E-04
4.45E-04
8.98E-05
4.50E-05
(0.13)
(0.20)
(0.29)
(0.38)
(0.50)
(5.70)
(21.16)
2.34E-03
1.13E-03
7.58E-04
5.62E-04
4.51E-04
8.96E-05
4.46E-05
(0.08)
(0.12)
(0.16)
(0.21)
(0.28)
(4.20)
(16.15)
1.39E-03
6.72E-04
4.43E-04
3.30E-04
2.64E-04
5.23E-05
2.61E-05
(3.88)
(7.64)
(11.40)
(15.16)
(18.93)
(97.36)
(203.27)
9.23E-04
4.48E-04
2.94E-04
2.20E-04
1.75E-04
3.46E-05
1.74E-05
(0.13)
(0.19)
(0.27)
(0.36)
(0.47)
(5.27)
(18.56)
Extended method
CRR
GCRR-XPC
FB
BBS
SPF
9.90E-04
5.14E-04
2.99E-04
2.30E-04
1.96E-04
3.91E-05
2.02E-05
(0.10)
(0.15)
(0.21)
(0.29)
(0.36)
(4.16)
(15.40)
1.11E-03
5.66E-04
3.80E-04
2.86E-04
2.29E-04
4.62E-05
2.31E-05
(0.15)
(0.22)
(0.31)
(0.40)
(0.52)
(5.75)
(21.24)
7.38E-04
3.69E-04
2.42E-04
1.86E-04
1.48E-04
2.93E-05
1.47E-05
(0.10)
(0.13)
(0.17)
(0.22)
(0.29)
(4.23)
(16.21)
1.56E-03
8.03E-04
5.41E-04
4.08E-04
3.27E-04
6.61E-05
3.31E-05
(4.27)
(8.02)
(11.78)
(15.54)
(19.31)
(97.75)
(203.68)
1.98E-03
1.02E-03
6.90E-04
5.20E-04
4.17E-04
8.44E-05
4.23E-05
(0.15)
(0.21)
(0.29)
(0.38)
(0.49)
(5.31)
(18.63)
The CPU time (in seconds) required to value 243 gammas is given in parentheses.
36
TABLE 8
RMS errors of gamma estimates (with extrapolation)
Time steps n
20
40
60
80
100
500
1000
Hull method
GCRR-XPC
FB
BBS
SPF
3.51E-04
1.27E-04
7.48E-05
5.19E-05
3.91E-05
4.66E-06
1.77E-06
(0.16)
(0.26)
(0.38)
(0.51)
(0.66)
(7.38)
(26.75)
6.83E-04
2.30E-04
9.97E-05
9.15E-05
4.78E-05
4.87E-06
1.76E-06
(0.10)
(0.16)
(0.21)
(0.28)
(0.37)
(5.43)
(20.42)
1.99E-04
4.63E-05
2.03E-05
1.16E-05
6.97E-06
2.79E-07
9.94E-08
(5.93)
(11.56)
(17.22)
(22.88)
(29.26)
(145.38)
(300.24)
2.23E-04
5.15E-05
2.39E-05
1.70E-05
9.63E-06
9.33E-07
2.95E-07
(0.16)
(0.25)
(0.36)
(0.49)
(0.62)
(6.82)
(23.46)
Extended method
GCRR-XPC
FB
BBS
SPF
1.30E-04
3.67E-05
1.70E-05
9.76E-06
6.32E-06
2.63E-07
6.62E-08
(0.19)
(0.29)
(0.41)
(0.54)
(0.69)
(7.44)
(26.85)
5.38E-04
1.63E-04
6.49E-05
5.89E-05
3.43E-05
3.31E-05
1.26E-06
(0.13)
(0.17)
(0.22)
(0.30)
(0.38)
(5.47)
(20.49)
1.80E-04
5.12E-05
2.35E-05
1.40E-05
8.92E-06
3.99E-07
1.17E-07
(6.53)
(12.14)
(17.79)
(23.45)
(29.85)
(145.96)
(300.85)
2.93E-04
8.07E-05
3.94E-05
2.28E-05
1.48E-05
9.93E-07
3.32E-07
(0.19)
(0.28)
(0.39)
(0.52)
(0.66)
(6.93)
(23.82)
The CPU time (in seconds) required to value 243 gammas is given in parentheses.
37