New Bounds on American Option Prices
In Joon Kim
1
Graduate School of Management
Korea Advanced Institute Science and Technology
phone: 82-2-958-3719. fax: 82-2-958-3618.
e-mail: [email protected]
Geun Hyuk Chang
Graduate School of Management
Korea Advanced Institute Science and Technology
phone: 82-2-958-3968. fax: 82-2-958-3618.
e-mail: [email protected]
Suk Joon Byun
Graduate School of Management
Korea Advanced Institute Science and Technology
phone: 82-2-958-3352. fax: 82-2-958-3618.
e-mail: [email protected]
February 2003
1
Professor, Graduate School of Management, Korea Advanced Institute Science and Technology, 207-43
Cheongryangri-dong, Dongdaemoon-gu, Seoul, 130-012, KOREA. phone: 82-2-958-3719. fax: 82-2-9583618. e-mail: [email protected]
New Bounds on American Option Prices
Abstract
In this article, we develop new upper and lower bounds on American option prices which
improve the bounds by Broadie and Detemple. The main idea is the consideration of
doubly capped call options which have two cap prices. We present a new option price
approximation based on the two upper bounds. On average, our upper bound
extrapolation (named UBE) has an average accuracy better than a 1,000 time-step
binomial tree with a computation speed comparable to a 100 time-step binomial tree. We
also provide a new method of approximating the optimal exercise boundaries of
American options.
JEL Classification: G13
Key Words: American option, Optimal exercise boundary, Approximation, Bound, Cap
2
Introduction
In their seminal paper, Black and Scholes (1973) introduced a closed-form solution for European
options. However, American options have the early exercise feature which creates difficulties in the
valuation problem. Since a wide variety of traded options are American options, the problem of valuing
American options is an important topic in financial economics. Many papers have focused on this topic
and introduced numerical approximation methods. Broadie and Detemple (1996), Huang, Subrahmanyam,
and Yu (1996) and Ju (1998) provide a good overview of the literature.
Broadie and Detemple (1996) and Ju (1998) tested a wide array of methods of approximating American
option prices in terms of speed and accuracy. The lower and upper bound approximation (LUBA) in
Broadie and Detemple (1996), the piecewise exponential boundary approximation in Ju (1998) and the
randomization of maturity in Carr (1998) dominate other methods. Of these three methods, the LUBA has
the special advantage of providing lower and upper bounds on the option price and a lower bound on the
optimal exercise boundary as well as an accurate price approximation. The bounds were developed based
on the fact that any capped call option value with one constant cap is bounded above by the American call
option value. However, a weak point of the LUBA is that it needs regression coefficients. In order to
estimate the regression coefficients, accurate prices for a large number of options must be computed.
Because it depends on the regression technique, it cannot improve the accuracy and it is not convergent.
Extending the work by Broadie and Detemple (1996), this article provides improved lower and upper
bounds on the theoretical American call option price and also proposes a tighter lower bound on the
optimal exercise boundary of the American call option.1 Our work uses doubly capped call options which
have different cap prices in consecutive time intervals (see Section 2). Though doubly capped call options
are not traded in the real world, we imagine them just hypothetically. With controlling two constant cap
prices in different time intervals, we can get a higher option value than with one cap. However it is not
simple to manage two variables simultaneously. We will fix one cap price suitably and compute a lower
bound on the optimal exercise boundary of the American call option. It improves over the bound by
Broadie and Detemple (1996). From this lower bound, an improved upper bound on the theoretical option
price can be obtained. Based on the two upper bounds (Broadie and Detemple’ s and ours) on the option
price, we provide a new option price approximation, termed UBE (upper bound extrapolation). Though
1
the UBE is more time consuming than the LUBA, it can improve accuracy and it does not need any
regression coefficient. It is convergent in the following sense. As more and more cap prices are
considered, its value becomes more and more accurate. However, we apply only two cap prices because
we have to compute so many multivariate cumulative normal distribution functions with three or more
cap prices. For an optimal exercise boundary approximation, the extrapolation of the two lower bounds
also generates accurate results. Since our approach uses the optimal exercise boundary, hedge parameters
can be calculated analytically as in Huang, Subrahmanyam, and Yu (1996).
This article is organized as follows. Section 1 reviews the work by Broadie and Detemple (1996) on the
bounds on the American call option price. In Section 2, we introduce doubly capped call options and
present improved bounds on the American call option price. Section 3 proposes some option price
approximations based on the upper bounds, and compares these methods with several existing methods in
terms of speed and accuracy. Section 4, applies doubly capped call options to the approximation for
American capped call option prices. Concluding remarks are given in Section 5. Proofs and explicit
formulas are collected in the Appendix.
1. Review of Broadie and Detemple (1996)
This section defines notations and summarizes bounds on an American option value suggested by
Broadie and Detemple (1996). For the development of our new bounds in Section 2, more investigations
on their work are contained in the remarks. Consider an American call option with maturity T and
exercise price K . We assume that the underlying asset price S follows the stochastic differential
equation:
dSt  (r   )S t dt  S t dWt ,
where
(1)
Wt is a standard Brownian motion under the risk neutral measure. Here r ,  and 
represent risk free rate, dividend rate and volatility of the asset price respectively and they are all constant.
2
In this article, we assume   0 . Let C ( St ) denote the value of the American call option and Bt*
represent the optimal exercise boundary at time t . Bt* is a nonnegative decreasing continuous function
of time.
Consider a capped call option written on the same underlying asset with constant cap price L . When
the underlying asset price reaches the cap L before the maturity, then the option is exercised
automatically with payoff L  K . At maturity, the option payoff is max min[ ST , L]  K ,0 . If
L  max[ S t , K ] , the value of the capped call option is given by
CC(St , L)  E[(L  K )e r ( t ) 1 T  max[(ST  K ),0]e r (T t ) 1 T ] ,
(2)
where   inf{u : S u  L} . Explicit representation for Equation (2) is given in Proposition A1 in the
Appendix.2 If L  max[ S t , K ] , then define CC (S t , L)  max min[ S t , L]  K ,0 .
For any L , CC ( S t , L) is bounded above by the American call option value C ( St ) and
max L CC (S t , L)  C (S t ) . Let Lˆ (S t ) and C1l (St ) be the optimal solution of L and the optimal value,
i.e.
Lˆ (St )  arg max L CC(St , L) ,
(3)
C1l (St )  max L CC(St , L) .
(4)
This lower bound C1l (St ) in Equation (4) improves over the European call option value and the
immediate exercise value.
To compute an upper bound on C ( St ) , the following integral formula in Kim (1990) (see also Carr,
Jarrow, and Myneni (1992) and Jacka (1991)) was used:
C ( S t )  c( S t )  tT S t e  (u t ) N (d1 ( S t , Bu* , u  t )) du  tT rKe  r (u t ) N (d 2 ( S t , Bu* , u  t )) du ,
3
(5)
where c(S t ) is the value at time t of the European call option on S with strike price K and
maturity T introduced by Black and Scholes (1973):
c( St )  St e  (T t ) N (d1 ( St , K , T  t ))  Ke  r (T t ) N (d 2 ( St , K , T  t )) ,
(6)
where N () is the cumulative standard normal distribution function and
d 1 ( x, y , u ) 
ln( x / y )  (r     2 / 2)u
 u
d 2 ( x, y , u )  d 1 ( x, y , u )   u .
Broadie and Detemple (1996) proved that an upper bound on the American call option value can be
calculated from Equation (5) with a lower bound on the optimal exercise boundary. Their lower bound
L*t on Bt* is the solution of the following equation:
D(L, t )  0 ,
(7)
where D( L, t ) is the derivative of the capped call option value with respect to the constant cap L
evaluated as S t approaches L from below:
CC ( S t , L)
.
L
St  L
D( L, t )  lim
(8)
D( L, t ) is represented explicitly in Proposition A1 in the Appendix. The numerical solution of Equation
(7) can be easily obtained by using the Newton-Raphson method. Their upper bound C1u (S t ) on the
American call option value is computed by using L*t instead of Bt* in Equation (5). They described the
idea behind the bound L*t . We present another interpretation of L*t in the remarks.
4
The LUBA for the American option value that is based on the lower bound C1l (St ) and upper bound
C1u (S t ) , is computed by
 C1l  (1   )C1u ,
(9)
where the coefficient 0    1 determined by the weighted regression approach described in the
Appendix B in Broadie and Detemple (1996).
In the next section, a procedure to compute an improved lower bound on Bt* will be suggested. For
this, we need the following remarks on Lˆ ( S t ) and L*t .
Remarks on Lˆ (St ) and L*t .
From Equation (8), for L  K , we can get another representation
D( L, t )  lim

h0
CC ( L, L  h)  ( L  K )
,
h
(10)
where note that CC ( L, L)  L  K . If D( L, t )  0 (or D( L, t )  0 ), then infinitesimally, CC( L, L  h)
increases (or decreases) as h increases from 0. From the relation between Lˆ ( S t ) and L*t in Broadie
and Detemple (1996), if L  L* (or L  L* ), then D( L, t )  0 (or D( L, t )  0 ).
Bt* is the minimum of the stock price at t such that the immediate exercise value is not less than the
American call option value. Similarly, L*t is the minimum of the stock price at t such that the
immediate exercise value is not less than any constant caped call option value, i.e. if St  L*t , then
CC (St , St   )  St  K for any   0 and in this case, D( S t , t )  0 and Lˆ (S t )  S t .3 Since for each
S t , the American call option is more valuable than any capped call option, we can see that Bt*  L*t . The
above statement can be applied to other lower bounds on Bt* . For example, comparing European call
5
option value with the immediate exercise value, we can get the critical level LEt . In this case,
c( LEt )  LEt  K and LEt is a lower bound on Bt* , where c( LEt ) is the European call option price.
Since the constant capped exercise policy can make the option value higher than the European option
value for   0 (this means that c( LEt )  C1l ( LEt ) ), it is clear that L*t  LEt . In the next section, we will
find an exercise policy which makes the option value higher than the option value with the capped
exercise policy. Thus a lower bound on Bt* which is tighter than L*t can be computed.
Suppose St  L*t . Then D(S t , t )  0 and this implies that some capped call option value is higher
than the immediate exercise value, i.e. CC (S t , S t   )  max( S t  K ,0) for a suitable   0 . CC ( S t , L)
increases as the cap price L increases from S t to Lˆ ( S t ) . In this case, Lˆ ( S t ) can be obtained from
the following first order condition on the maximizing problem in Equation (4):
CC ( S t , L)
 0,
L
and S t  Lˆ (S t )  L*t . When L increases from Lˆ (S t ) , then CC ( S t , L) decreases. We can see that
CC(St , L)  CC(St , L*t ) , for L  L*t .
It can be summarized as follows: under the condition L  L*t , CC(S t , L)  S t  K if St  L*t and
CC(St , L)  CC(St , L*t ) if S t  L*t .
2. Improved bounds
Consider a doubly capped call option written on the same underlying asset with maturity T . For
L1  L2  K , L1 is the cap price by time T1 , and L2 is the cap price from T1 to maturity T .4
Consult Figure 1. The option is automatically exercised when S reaches L1 before T1 , or S T1  L2 at
time T1 , or S reaches L2 between the time interval [ T1 , T ]. If the option is not exercised before T1
6
and if ST1  L2 , then the doubly capped call option value at time T1 is the same as the capped call
option value with cap price L2 and remaining time to maturity T  T1 .
The capped call option in the
previous section is a special case of doubly capped call options with L  L1  L2 . For S  L1 , the value
of the doubly capped call option is given by
DCC ( St , L1, L2 )  E[( L1  K )e r (1 t ) 1 1 T1  ( ST1  K )er (T1 t ) 1 1 T1 , ST
1
 CC ( ST1 , L2 )e
 r (T1 t )
 L2
(11)
1 1 T1 , ST
 L2 ] ,
1
where  1  inf{u : S u  L1} and CC ( S T , L2 ) represents the capped call option value with remaining
1
time to maturity T  T1 . If S t  L1 , then we define DCC (S t , L1 , L2 )  L1  K . The explicit formula of
Equation (11) is given in Proposition A2 in the Appendix. From now, for our future work, we will fix
T1  (t  T ) / 2 the mid point of t and T .
[Insert Figure 1]
Since the above doubly capped exercise policy is an admissible policy for the American call option,
clearly DCC (S t , L1 , L2 )  C (S t ) . As in Section 1, we can obtain an improved lower bound on C (S t )
by maximizing DCC (S t , L1 , L2 ) with respect to two cap prices L1 and L2 . The lower bound C 2l ( S t )
is defined as follows:
C2l (S t )  max L , L DCC (S t , L1 , L2 ) .
1
2
(12)
However, the above maximizing problem is not simple to compute because it contains two variables. We
will focus on the upper bound explained in the next paragraphs.
To compute an improved upper bound on the theoretical American call option value, we try to find a
lower bound on Bt* which is tighter than L*t . From the remarks in Section 1, L*t is determined by
7
comparing the constant exercise policy with immediate exercise. In order to get a more accurate lower
bound on Bt* , we will fix L2 such that the value of the option with the doubly capped exercise policy is
higher than the value of the option with the constant capped exercise policy. Comparing the doubly
capped exercise policy with immediate exercise, we can get a critical level higher than L*t . The following
proposition suggests how to find a suitable L2 .
Proposition 1) Let L2  L*T1 and suppose L1  L2 . Then DCC (St , L1 , L2 )  CC(St , L1 ) .
Proof) See the Appendix.
We will fix L2  L*T1 . For L1  L2 we define the following equations analogous to Equation (8):
DCC ( S t , L1 , L2 )
,
L1
(13)
DCC ( L, L  h, L2 )  ( L  K )
,
h
(14)
DD ( L1 , t )  lim
St  L1
or to Equation (10):
DD ( L, t )  lim

h 0
*
and let LD
denote the solution of the following equation:
t
DD ( L1 , t )  0 .
(15)
The solution of the above equation can be easily obtained from the Newton-Raphson method with initial
*
value L*t . The expression for DD ( L1 , t ) is given in Proposition A2 in the Appendix. This LD
is the
t
critical level of the stock prices higher than L2  L*T1 such that the immediate exercise value is not less
*
than any doubly capped call option value with fixed L2  L*T1 and T1  (t  T ) / 2 . LD
is also a lower
t
8
bound on Bt* as L*t is, and by Proposition 1 and the remarks in Section 1, the following can be
obtained.
Proposition 2) LDt*  L*t . Here note that L*t  L2  L*T .
1
*
Hence LD
is a tighter lower bound on Bt* and, in addition, Bt* can be approximated by the
t
*
*
*
D*
extrapolation L*t ext  2LD
on Bt* and the
t  Lt . Table 1 reports the two lower bounds Lt and Lt
extrapolated value L*t ext . To test accuracies of these values, we use the recursive numerical integration
method in Kim (1990, 1994) with n  200 time steps per year for true values. As Table 1 shows, the
*
error from LD
is about one-half of the error from L*t . The results of the extrapolation are very
t
successful. The root mean squared relative error (RMSRE) is only about 0.03%. Our approximation L*t ext
can be obtained without recursive procedure.
[Insert Table 1]
As C1u is calculated with L* , an improved upper bound C 2u on the theoretical American call option
value is calculated from the Equation (5) with the lower bound LD* instead of B * . We can get the
approximate price of the American call option by the extrapolation of C1u and C 2u . We refer to this
method as UBE (Upper Bound Extrapolation). Another option price approximation method is to compute
the integral in Equation (5) with L*ext . We refer to this method as BEI (Boundary Extrapolation and
Integration). To test the accuracies of the above two approximation methods, Table 2 and 3 report the
results for 40 options in Table 1 and 2 in Broadie and Detemple (1996). Two tables also contain the two
lower bounds and their extrapolation. As we can see from the tables, UBE or BEI represent very accurate
results. In Table 1, L*ext tends to be higher than B * . So the UBE and BEI values are lower than the true
values. These characteristics are observed for the options which we test in Section 3 and we guess that
these may be common for other parameters. Since DD ( L1 , t ) contains bivariate cumulative normal
9
*
distribution functions, the computation of LD
needs much more time relative to the computation of L*t .
t
*
When we implement our methods, 4~10 points of LD
will be used in order to reduce computing time.
t
Detailed procedures and the numerical results of our approximations will be discussed in the next section.
[Insert Table 2]
[Insert Table 3]
3. Implementation and Computational Results
To increase the computational efficiency, we have to find some methods that need a small number of
LD* . For example, UBE6-60 method uses 6 points of LD* and 60 points of L* . In the first step, we
compute numerical integrations C1u,6 (with 6 points of L* ) and C 2u,6 (with 6 points of LD* ) and
calculate the extrapolated value 2C 2u,6  C1u,6 .5 Next we adjust the error from the numerical integration
with a small number of points, by using the control variate technique. 6 In this step, C1u,60  C1u,6 is added
to the above extrapolated value, where C1u,60 is computed with 60 points of L* . Through the above two
steps, the resulting approximation is given by
UBE
u
u
u
.
C6,60
 2C2,6
 2C1,6
 C1,60
(16)
In the case of BEI6-60 method, the first step is different. Computing C6B with 6 points of L*ext and
using the control variate technique, we can get the following approximation:
BEI
u
u
.
C6,60
 C6B  C1,6
 C1,60
(17)
Suitable couples of numbers, for example, (6,60), (8,64) or (10,80) can be chosen. For the numerical test,
10
we chose the above couples of numbers.
In order to examine the accuracies of our methods, we tested 2400 options. We compare our methods
with the binomial methods (denoted BN), the binomial Black-Scholes Richardson extrapolation of
Broadie and Detemple (1996) (denoted BBSR), the recursive integration methods of Huang,
Subrahmanyam, and Yu (1996) (denoted HSY) and the LUBA. We fix K =100 and vary other parameters
as follows: r =0, 0.025, 0.05, 0.075, 0.1,  =0.025, 0.05, 0.075, 0.1,  =0.1, 0.2, 0.4, 0.6, T =0.1, 0.3,
0.5, 1.0, 2.0, 3.0, S =80, 90, 100, 110, 120. These parameters are in the sample set that was used to
estimate the regression coefficients of the LUBA method in Broadie and Detemple (1996). To take
advantage of the computation of critical stock prices, we priced 5 options of different stock prices for a
given set of other parameters. We use the binomial method with 40,000 time steps as our benchmark for
the true values.
Based on the results of testing 2400 options, the speed-accuracy trade-off of the above American option
pricing methods are given in Figure 2 and 3. Two accuracy measures in the Figures are RMSE (Root
Mean Squared Error) and RMSRE (Root Mean Squared Relative Error). The RMSRE is calculated for the
options with true value >0.50. The UBE and the BEI have almost the same speed-accuracy relation. The
two figures contain the results of the UBE. Our methods reduce the RMSE relative to the LUBA as shown
in Figure 2 (by about 1/3). This accuracy level is better than that of a 1,000 time-step binomial tree. For
the RMSRE in Figure 3, our methods also represent better performance than the LUBA, though the
degree of the improvement is not as high as for the RMSE. Compared with other methods, the LUBA is
better with the RMSRE than with the RMSE, because the regression coefficients are chosen such that
they minimize the RMSRE. Consequently, our methods are more accurate than the LUBA, though they
need more computing time.
[Insert Figure 2]
[Insert Figure 3]
For the hedge ratio  , we can use the following analytic formula derived from Equation (5): 7
11
C ( S t )
 e  (T t ) N (d 1 ( S t , K , T  t ))
S t


T
t
e
 (u t )

Bu*  rK 
*
*
N (d 1 ( S t , Bu , u  t ))  n(d 1 ( S t , Bu , u  t )) *
du
Bu  u  t 

.
(18)
Like the option values,  can be approximated by using the extrapolation and the control variate
technique. Table 4 represents the computational results of 20 options in Table 3. Integration with L*ext
and extrapolation after integrations with L* and LD* generate almost the same results. Table 4 contains
the values by extrapolation after integrations (the UBE style). We use the extended tree method described
in Pelsser and Vorst (1994) when compute  using the binomial method. Though Ju (1998) pointed out
large errors of  from the LUBA method, it performs well with the parameters in Table 4. Maybe it is
because the LUBA uses some regression coefficients. Our methods have an RMSE smaller than a 1,000
time-step extended binomial tree (by about 1/2~1/3).
[Insert Table 4]
4. Application to American Capped Call options
In this section we consider an American capped call option on the underlying asset S with constant cap
L , maturity T and exercise price K . The American capped call option is a special case of barrier
options, and it is described as an up-and-out call option with rebate L  K . As introduced by Gao, Huang,
and Subrahmanyam (2000), some barrier options have analytic pricing formulas similar to Equation (5)
and the recursive integration method of Huang, Subrahmanyam, and Yu (1996), or the multi-piece
exponential boundary approximation method of Ju (1998) can be applied to approximations of their
formulas. The analytic pricing formulas of American capped call options are introduced in Broadie and
Detemple (1995, 1997) and Gao, Huang, and Subrahmanyam (2000). However, it is not simple to
compute the option values from their formulas. Lattice pricing methods for barrier options (containing
capped options) were developed by Boyle and Lau (1994), Ritchken (1995) and Figlewski and Gao
12
(1999). A drawback of the lattice methods is that they have the computing problem when the barrier (or
cap) is close to the current price of the underlying asset (the near barrier problem). The following
paragraphs introduce two lower bounds on the American capped call option price and suggest that their
extrapolated value can be an approximate option price.
The optimal exercise boundary B of the American capped call option is described as follows (see
Broadie and Detemple (1995)):
Bu  min(L, Bu* ) ,
(19)
where Bu* denotes the optimal exercise boundary of the American call option at time u . In some
special cases, the American capped call option can be priced simply. If L  BT* , then Bu  L for all
u  [t , T ] and the option value is calculated by Proposition A1. If L  Bt* , then Bu  Bu* for all
u  [t , T ] and the American capped call option and the American call option without cap have the same
value. In the other case BT*  L  Bt* , it is not simple to compute the value of the American capped call
option. We focus on this case from now.
Let t * be defined by the solution to the equation Bu*  L for u  [t , T ] . To get an approximation for
t * , consider t 0 which satisfies L*t  L . Since L*u  Bu* and Bu* is decreasing, for 0    1
0
T  t *  (T  t 0 ) .
(20)
One lower bound AC1l on the American capped call option value is just the capped call option value
CC ( S t , L) . For another lower bound, we use doubly capped call options with L1  L and T1  t * .
Since DCC (S t , L, L2 ) is bounded above by the American capped call option value for any L2  [ K , L] ,
a lower bound AC2l which improves over AC1l , can be obtained by the following optimization:
AC 2l  max K  L  L DCC (S t , L, L2 ) .
2
13
(21)
Their extrapolation 2 AC2l  AC1l can be used as an approximate price of the American capped call
option.8 To implement the above method, it is necessary to determine t * . In Equation (20) one can use
  0.85 ~ 0.9 to approximate t * from t 0 . Table 5 tests the accuracy of the above approximation with
  0.85 . We use the trinomial tree method in Ritchken (1995) with 20,000 time steps as our benchmark
for the true values. Our method does not have the near barrier problem and it is better than at least 200
time-step trinomial method by Ritchken (1995) in both speed and accuracy. The extrapolation coefficients
and  can be adjusted in favor of high accuracy, but that may be another study.
[Insert Table 5]
5. Conclusion
The optimal exercise boundary of an American call option is the critical level such that the immediate
exercise value is not less than the value of the American call option which has the optimal exercise policy.
However, the American call option value is not simple to compute. If the value of an option with a certain
admissible exercise policy can be calculated easily, then we may find the critical level such that the
immediate exercise value is not less than the value of the option with the admissible exercise policy. It is
a lower bound on the optimal exercise boundary. The higher the option price is, the tighter becomes the
bound.
In this article, we have introduced doubly capped call options and computed an improved lower bound
on the optimal exercise boundary of the American call option. The optimal exercise boundary can be
approximated with an RMSRE of 0.03% in Table 1 without recursive computations. We have presented
improved bounds on the price of the American call option and introduced the UBE and BEI methods for
the option price approximation. When we implement the UBE and the BEI, we note that L* can be
computed very quickly. So the control variate technique using L* is an efficient tool to compute early
exercise premium. If accurate optimal exercise boundary could be calculated efficiently at several points,
14
then the American option could be priced by the control variate technique. Our UBE does not need any
regression coefficients and it generates successful results. On average, it is more accurate than a 1,000
time-step binomial tree with a computation speed comparable to a 100 time-step binomial tree. It also
represents good performance in approximating hedge ratios. Consequently, by extending the work by
Broadie and and Detemple (1996), we could get better results in both bounds and approximations.
Two lower bounds on an American capped call option value have been introduced and their
extrapolation provided a good approximation for the American capped call option value. The
extrapolation coefficients may be adjusted in order to give high accuracy.
15
Appendix
Proof of Proposition 1)
Assume S  L1 . From Equation (2),
CC ( S t , L1 )  E[( L1  K )e  r ( t ) 11 T1  CC ( ST1 , L1 )e  r (T1 t ) 11 T1 ]
where CC(ST1 , L1 ) represents the capped call option price with remaining time to maturity T  T1 . The
first term in the above expectation is the same as the first term in Equation (11). For the remaining terms,
note
the
assumption
L1  L2  L*T1 .
From
the
remarks
in
section
1,
we
know
that
CC ( S T , L2 )  CC ( S T , L1 ) for ST1  L2 and that S T  K  CC ( S T , L1 ) for L2  ST1  L1 . These
1
1
1
1
imply
E[(ST1  K )e r (T1 t ) 11 T1 ,ST L2  CC(ST1 , L2 )e r (T1 t ) 11 T1 ,ST L2 ]  E[CC(ST1 , L1 )e r (T1 t ) 11 T1 ]
1
1
and DCC (S , L1 , L2 )  CC (S , L1 ) .
Q.E.D.
Proposition A1) (Broadie and Detemple (1996)) Suppose
L  max( S t , K ).
Let   T  t ,
1
1
1
b    r   2 , f  b 2  2r 2 ,   (b  f ) and   (b  f ) . Then the value of a capped call
2
2
2
option (automatically exercised at the cap price) with cap price L , maturity T and exercise price K
is given by
16
 S

CC ( S t , L)  ( L  K )  t
 L





2 /  2
S
N (d 0 )   t
 L
 



 S t e  N (d 1 ( L)     N (d 1 ( K )   
S
  t
 L
 Ke



2 (  r ) / 
 r

2
2 /  2
N (d 0 

 
Le  N (d 1 ( L)     N (d 1 ( K )   
2f  
)
 


2b /  2

S




t
 N ( d ( L)  N d ( K )   
N (d 1 ( L)  N d 1 ( K )
 L
1
1

 


 


 



and D( L, t ) can be written as
D ( L, t ) 
CC
L

St  L
 (L  K )
  f 
(2 /  2 )  N  
1 
L


 
 e 
2(b   2 )

where d 0 
2
N d

1 ( L)  
ln( S t / L)  f
 
  (L  K )
  f 
  1 
(2 /  2 )  N 
 
L
  

 

  N d 1 ( K )     e  r1
and d1 ( x) 
N d
L
2bK

2





1 ( L)
 ln( S t / L)  ln( L)  ln( x)  b
 
Proposition A2) Suppose L1  L2  max( S t , K ) . Let b ,
 N d

1 (K )

.
f ,  and  be as before and let
1  T1  t ,  2  T  t and b'  b   2 . Then the value of a doubly capped call option with exercise
price K , maturity T and two cap prices L1 (from t to T1 ) and L2 (from T1 to T ) is given by
17
DCC ( S t , L1 , L 2 ) 
2 /  2
2 /  2


 St 
 St 

( L1  K )  
N (d ( f ))   
N (d ( f )) 
 L1 

 L1 




 S t e  1 N  d (b' )   N g 1 (b' )
S 
  t 
 L1 
 Ke
2 (  r ) / 
 r1
2



L1 e  1 N d (b' )   N ( g 1 (b' )

2b /  2

 N  d (b)   N g  (b)   S t 
N (d (b)   N g 1 (b)
L 
1

 1








2 f / 2




S


 BN g  ( f ),  g  ( f ),     t 

BN
g
(
f
),

g
(
f
),


1
2
1
2
L 


 1


2 /  2 
2 f /  2
S 
 BN g  ( f ),  g  ( f ),     S t 
 ( L 2  K ) t 
BN g 1 ( f ),  g 2 ( f ),  
1
2
L 

L
 2
 1

 BN g 1 (b' ), g 2 (b' ),   BN g 1 (b' ), h  (b' ), 



2
 S t e  2    S  2b '/ 




  t 

BN
g
(
b
'
),
g
(
b
'
),


BN
g
(
b
'
),
h
(
b
'
),

1
2
1


 
  L1 

S
 ( L 2  K ) t
 L2



2 /  2











 










 BN g 1 (b' ),  g 2 (b' ),    BN g 1 (b' ), h  (b' ),  

S 
2
 S t e  2  t 
   S   2b ' / 
  t 
 L2 
BN g 1 (b' ),  g 2 (b' ),    BN g 1 (b' ), h  (b' ),  
  L1 
 BN g 1 (b), g 2 (b),   BN g 1 (b), h  (b), 



2
 Ke  r 2    S  2b / 




  t 

BN
g
(
b
),
g
(
b
),


BN
g
(
b
),
h
(
b
),

1
2
1


  L1 

2b ' /  2
 






2b /  2


 
S
 Ke  2  t
 L2








 BN g 1 (b),  g 2 (b),    BN g 1 (b), h  (b),  

2
   S   2b / 
  t 
BN g 1 (b),  g 2 (b),    BN g 1 (b), h  (b),  
  L1 
 

and DD ( L1 , t ) (in this case,  2  21 and L2  L*T1 ) can be written as
18













DD ( L1 , t ) 
DCC
L1

S t  L1
 ( L1  K )
  f 1
1 
(2 /  2 )  N  
L1


 
 e  1
 e  r1
L 
  1 
 L2 
 e  r 2
2b'
2
N d (b' )  N g
2bK
 L1
2

1 (b' )
N d (b)  N g


1 (b)
  (L  K )

2   f 1
  1  1

(
2

/

)
N
  
 
L1
 

 e
 r1
2( L2  K )
 L1  1
2




n( g 1 (b))
2 /  2
 ( 2  1 ) ln( L2 / L1 ) 
2( L2  K )  1
2


n( g1 ( f )) 
n( g 2 ( f )) N 


L1
2



 1
1
2




( 2  1 ) ln( L2 / L1 ) 
4( L2  K )
n( g 2 (b)) N 


  2 L1
 1 2


2 /  2
2 /  2

 L1 
2 f ( L2  K )  L1 


 

BN g1 ( f ),  g 2 ( f ),     
BN g1 ( f ),  g 2 ( f ),  
2



L
L
 L1
 2
 2

2b'
 e  2 2 BN g1 (b' ), g 2 (b' ),   BN g1 (b' ), h  (b' ), 


 

2b ' /  2
e
 2
 e  r 2
e
 r 2
where   1
and h  ( x) 




 


2b'  L1 
 
BN g1 (b' ),  g 2 (b' ),    BN g1 (b' ), h  (b' ),  
2 
  L2 
2bK
BN g1 (b), g 2 (b),   BN g1 (b), h  (b), 
 2 L1
 
2bK  L1 
 
 2 L1  L2 
2

2b /  2
, d ( x) 
BN g


1 ( b),  g 2 ( b),  
 1
, g i ( x) 
x 2  ln( K / L2 )  ln( L2 / L1 )  ln( S / L1 )
 2

 BN g

1 ( b), h  (b),  

x i  ln( L2 / L1 )  ln( S / L1 )
 i
,
and BN x, y, p  is the bivariate cumulative
normal distribution function defined by 9
BN ( x, y, p) 
x
1
2 1  p
2
 
y
 



ln( S / L1 )  x1




exp  (u 2  v 2  2 puv) / 2(1  p 2 ) dudv .
19
Foot notes
1.
By put-call symmetry, this can be applied to put options. For the optimal exercise boundary of any
American put option, an upper bound can be obtained.
2.
It was introduced by Broadie and Detemple (1996).
3.
D( S t , t )  0 represent
CC ( St , L)
 0 for L  S t . This is the first order condition on the
L
maximizing problem in Equation (4) such that Lˆ (St )  St .
4.
For L1  L2 , they can be also defined.
5.
When we compute numerical integrations, we use Simpson’s rule.
6.
Hull and White (1988) used control variate technique in American option pricing.
7.
It is analogous to the formula of the delta of a put option represented in Huang, Subrahmanyam, and
Yu (1996).
8.
The extrapolation coefficients can be adjusted. According to Table 5, if we determine them relating to
the ratio L / S t , we may get more accurate approximation for option values.
9.
For the approximation of the bivariate cumulative normal distribution function, see Hull (2000) p.272.
20
References
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Economy, 81, 637-654.
Boyle, P. P., and S. H. Lau, 1994, “Bumping Up Against the Barrier with the Binomial Method,” Journal
of Derivatives, 1, 6-14.
Brennan, M., E. Schwartz, 1977, “The Valuation of American Put Options,” Journal of Finance, 32, 449462.
Broadie, M., and J. B. Detemple, 1995, “American Capped Call Options on Dividend Paying Assets,”
Review of Financial Studies, 8, 161-191.
Broadie, M., and J. B. Detemple, 1996, “American Option Valuation: New Bounds, Approximations, and
Comparison of Existing Methods,” Review of Financial Studies, 9, 1211-1250.
Broadie, M., and J. B. Detemple, 1997, “The Valuation of American Options on Multiple Assets,”
Mathematical Finance, 7, 241-286.
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Decomposition Technique,” Journal of Economic Dynamics and Control, 24, 1783-1827.
Harrison, J., M., 1990, Brownian Motions and Stochastic Flow Systems, Krieger, FL, USA.
21
Huang, J., M. Subrahmanyam, and G. Yu, 1996, “Pricing and Hedging American Options: A Recursive
Integration Method,” Review of Financial Studies, 9, 277-300.
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Ritchken, P., 1995, “On Pricing Barrier Options,” Journal of Derivatives, 3, 19-28.
22
Table 1
Selected values of optimal exercise boundaries of American call options. ( K  100 )
Option
parameter
r  0 .03
  0 . 07
  0 . 40
r  0 . 03
  0 . 07
  0 . 20
r  0 . 07
  0 .03
  0 . 40
  0 .03
  0 . 20
r  0 .07
Maturity
L*
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
3.0
143.010
156.710
166.168
173.482
179.449
184.477
188.803
192.584
195.928
198.910
116.525
120.857
123.656
125.723
127.351
128.683
129.801
130.757
131.587
132.316
266.711
282.397
297.550
312.306
326.160
338.987
350.826
361.765
371.896
381.303
249.063
255.352
260.060
263.941
267.296
270.282
272.999
275.506
277.846
280.044
LD*
143.858
157.714
167.231
174.560
180.519
185.524
189.820
193.565
196.869
199.812
116.801
121.153
123.949
126.006
127.619
128.936
130.040
130.981
131.797
132.512
267.054
282.932
298.327
313.347
327.446
340.482
352.494
363.569
373.804
383.287
249.214
255.558
260.304
264.209
267.576
270.574
273.300
275.816
278.163
280.368
L*t ext
144.706
158.717
168.293
175.637
181.588
186.571
190.836
194.545
197.811
200.713
117.078
121.449
124.242
126.288
127.888
129.190
130.278
131.205
132.007
132.708
267.398
283.468
299.104
314.388
328.731
341.977
354.161
365.373
375.712
385.272
249.365
255.763
260.548
264.477
267.857
270.865
273.601
276.126
278.480
280.691
True value
144.614
158.607
168.181
175.530
181.490
186.483
190.760
194.481
197.759
200.674
117.049
121.419
124.216
126.267
127.872
129.178
130.271
131.202
132.007
132.712
267.342
283.414
299.093
314.387
328.705
341.907
354.042
365.209
375.511
385.044
249.336
255.722
260.493
264.419
267.807
270.821
273.559
276.086
278.442
280.654
The ‘True value’ column is based on the recursive numerical integration method in Kim (1990, 1994)
with n  200 time steps per year. L* is the solution of Equation (7) based on Broadie and Detemple
(1996). LD* is the solution of Equation (15) based on the doubly capped call option. L*t ext is the
extrapolated value 2LD*  L* .
23
Table 2
Bounds of American call option values and their extrapolations (maturity T  0.5 years)
Option
parameter
r  0 . 03
  0 . 07
  0 . 20
r  0 .03
  0 . 07
  0 . 40
r  0 .00
  0 .07
  0 .30
r  0 .07
  0 .03
  0 .30
Asst
Price
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
LB1
LB2
LBE
UB1
UB2
UBE
BEI
LUBA
True value
0.2178
1.3759
4.7501
11.0488
20.0000
2.6759
5.6942
10.1901
16.1101
23.2712
1.0287
3.0981
6.9845
12.8818
20.6501
1.6644
4.4947
9.2506
15.7975
23.7062
0.2179
1.3782
4.7622
11.0773
20.0000
2.6782
5.7019
10.2091
16.1438
23.3205
1.0301
3.1054
7.0065
12.9219
20.6912
1.6644
4.4947
9.2506
15.7975
23.7062
0.2180
1.3806
4.7743
11.1057
20.0000
2.6804
5.7097
10.2282
16.1775
23.3697
1.0315
3.1128
7.0285
12.9620
20.7324
1.6644
4.4947
9.2506
15.7975
23.7062
0.2196
1.3885
4.7919
11.1253
20.0620
2.6908
5.7272
10.2494
16.2006
23.3918
1.0389
3.1290
7.0509
12.9883
20.7787
1.6644
4.4947
9.2506
15.7975
23.7062
0.2195
1.3874
4.7869
11.1106
20.0312
2.6898
5.7245
10.2437
16.1903
23.3747
1.0380
3.1260
7.0427
12.9705
20.7459
1.6644
4.4947
9.2506
15.7975
23.7062
0.2194
1.3863
4.7820
11.0959
20.0004
2.6887
5.7218
10.2380
16.1800
23.3576
1.0372
3.1230
7.0345
12.9527
20.7131
1.6644
4.4947
9.2506
15.7975
23.7062
0.2194
1.3863
4.7821
11.0961
20.0002
2.6887
5.7219
10.2382
16.1802
23.3581
1.0372
3.1230
7.0347
12.9532
20.7139
1.6644
4.4947
9.2506
15.7975
23.7062
0.2195
1.3862
4.7821
11.0976
20.0000
2.6893
5.7231
10.2402
16.1817
23.3574
1.0373
3.1232
7.0355
12.9531
20.7208
1.6644
4.4947
9.2506
15.7975
23.7062
0.2194
1.3864
4.7826
11.0977
20.0004
2.6888
5.7221
10.2386
16.1812
23.3598
1.0373
3.1233
7.0355
12.9551
20.7173
1.6644
4.4947
9.2506
15.7975
23.7062
All options have K  100 . LB1 and UB1 are based on the method in Broadie and Detemple (1996). LB2, UB2 are based on the method in Section 2. The LBE and
the UBE are the extrapolations of the lower bounds and the upper bounds respectively. The BEI is the boundary extrapolation and integration method. UB1, UB2
and BEI are calculated with 200 time steps. The “true value” column is based on the binomial method with 40,000 time steps.
24
Table 3
Bounds of American call option values and their extrapolations (maturity T  3 years)
Option
parameter
r  0 . 03
  0 . 07
  0 . 20
r  0 .03
  0 . 07
  0 . 40
r  0 .00
  0 .07
  0 .30
r  0 .07
  0 .03
  0 .30
Asst
Price
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
LB1
LB2
LBE
UB1
UB2
UBE
BEI
LUBA
True value
2.5529
5.1207
9.0302
14.3710
21.3540
11.2378
15.6088
20.6562
26.3366
32.6074
5.4631
8.7658
13.0478
18.3474
24.6849
12.1447
17.3674
23.3467
29.9608
37.0993
2.5572
5.1397
9.0333
14.4111
21.3907
11.2712
15.6572
20.7203
26.4159
32.6998
5.4838
8.7993
13.0944
18.4043
24.7461
12.1448
17.3676
23.3473
29.9618
37.1010
2.5614
5.1587
9.0364
14.4512
21.4274
11.3045
15.7057
20.7844
26.4952
32.7922
5.5045
8.8329
13.1411
18.4613
24.8072
12.1449
17.3678
23.3478
29.9627
37.1028
2.5891
5.1865
9.1023
14.5037
21.5060
11.3537
15.7628
20.8496
26.5687
32.8758
5.5397
8.8783
13.1985
18.5344
24.9021
12.1453
17.3684
23.3486
29.9639
37.1040
2.5844
5.1764
9.0835
14.4726
21.4586
11.3390
15.7413
20.8200
26.5296
32.8260
5.5283
8.8593
13.1694
18.4926
24.8449
12.1452
17.3684
23.3485
29.9637
37.1037
2.5796
5.1662
9.0647
14.4415
21.4112
11.3243
15.7198
20.7904
26.4905
32.7762
5.5169
8.8403
13.1403
18.4508
24.7877
12.1451
17.3684
23.3484
29.9635
37.1034
2.5797
5.1664
9.0649
14.4417
21.4116
11.3245
15.7202
20.7908
26.4911
32.7768
5.5171
8.8406
13.1406
18.4511
24.7882
12.1452
17.3683
23.3484
29.9635
37.1033
2.5804
5.1677
9.0651
14.4443
21.4120
11.3271
15.7236
20.7926
26.4893
32.7723
5.5199
8.8435
13.1415
18.4530
24.7974
12.1453
17.3684
23.3486
29.9639
37.1040
2.5800
5.1670
9.0660
14.4434
21.4139
11.3257
15.7220
20.7933
26.4945
32.7811
5.5176
8.8416
13.1420
18.4531
24.7907
12.1452
17.3684
23.3484
29.9636
37.1034
All options have K  100 . LB1 and UB1 are based on the method in Broadie and Detemple (1996). LB2, UB2 are based on the method in Section 2. The LBE and
the UBE are the extrapolations of the lower bounds and the upper bounds respectively. The BEI is the boundary extrapolation and integration method. UB1, UB2
and BEI are calculated with 200 time steps. The “true value” column is based on the binomial method with 40,000 time steps.
25
Table 4
Deltas of American call options (maturity T  3 years)
Option
parameter
r  0 . 03
  0 . 07
  0 . 20
r  0 .03
  0 . 07
  0 . 40
r  0 .00
  0 .07
  0 .30
r  0 .07
  0 .03
  0 .30
RMSE
Asst
Price
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
80
90
100
110
120
LUBA
UBE6-60
UBE8-64
UBE10-80
HSY4
HSY8
HSY12
BN1000
True value
0.2004
0.3209
0.4614
0.6163
0.7776
0.4045
0.4740
0.5390
0.5996
0.6564
0.2854
0.3803
0.4800
0.5827
0.6861
0.4803
0.5623
0.6317
0.6895
0.7370
0.2004
0.3210
0.4615
0.6158
0.7809
0.4044
0.4741
0.5393
0.6000
0.6563
0.2854
0.3805
0.4801
0.5822
0.6856
0.4803
0.5623
0.6317
0.6895
0.7369
0.2004
0.3210
0.4615
0.6157
0.7806
0.4044
0.4741
0.5393
0.6000
0.6564
0.2854
0.3805
0.4801
0.5822
0.6854
0.4803
0.5623
0.6317
0.6895
0.7369
0.2004
0.3210
0.4615
0.6157
0.7802
0.4044
0.4741
0.5393
0.6000
0.6564
0.2853
0.3805
0.4802
0.5822
0.6853
0.4803
0.5623
0.6317
0.6895
0.7369
0.2034
0.3095
0.4473
0.6272
0.8077
0.4012
0.4641
0.5262
0.5895
0.6535
0.2795
0.3645
0.4664
0.5842
0.7054
0.4804
0.5625
0.6322
0.6903
0.7379
0.2015
0.3184
0.4658
0.6112
0.7826
0.4038
0.4711
0.5385
0.6034
0.6605
0.2835
0.3799
0.4845
0.5809
0.6790
0.4803
0.5623
0.6316
0.6894
0.7371
0.2005
0.3214
0.4596
0.6193
0.7753
0.4036
0.4741
0.5411
0.5996
0.6542
0.2846
0.3822
0.4778
0.5839
0.6863
0.4803
0.5623
0.6317
0.6894
0.7369
0.2007
0.3214
0.4620
0.6162
0.7803
0.4049
0.4746
0.5398
0.6005
0.6569
0.2858
0.3810
0.4808
0.5829
0.6859
0.4806
0.5625
0.6319
0.6895
0.7369
0.2004
0.3210
0.4616
0.6158
0.7798
0.4044
0.4741
0.5394
0.6001
0.6565
0.2853
0.3805
0.4802
0.5823
0.6854
0.4803
0.5623
0.6317
0.6895
0.7369
5.63E-04
2.60E-04
1.90E-04
1.07E-04
1.13E-02
2.80E-03
1.71E-03
4.15E-04
All options have K  100 . LUBA is based on the method in Broadie and Detemple (1996). UBE columns are based on the methods in this article. The HSY
columns are based on the recursive integration method of Huang, Subrahmanyam and Yu (1996) with discretizations of 4, 8 and 12 point. The BN 1000 represents
the extended binomial method with 1,000 time steps. The “true value” column is based on the extended binomial method with 40,000 time steps.
26
Table 5
American capped call option values.
r  0 . 07 ,
  0 . 07
r  0 . 03 ,
LBE
Ritchken
(0.2,120,0.5,90)
(0.2,120,0.5,100)
(0.2,120,0.5,110)
(0.2,120,0.5,119)
(0.2,120,0.75,90)
(0.2,120,0.75,100)
(0.2,120,0.75,110)
(0.2,120,0.75,119)
(0.2,120,1.0,90)
(0.2,120,1.0,100)
(0.2,120,1.0,110)
(0.2,120,1.0,119)
TrueValue
1.7186
5.4757
11.8606
19.1434
2.6022
6.5714
12.5972
19.2233
3.3545
7.4021
13.1327
19.2810
1.7182
5.4750
11.8602
19.1434
2.6018
6.5709
12.5967
19.2234
3.3541
7.4017
13.1324
19.2810
1.7196
5.4763
11.8626
19.1436
2.6049
6.5722
12.6005
19.2234
3.3577
7.4032
13.1365
19.2811
(0.3,145,0.5,90)
(0.3,145,0.5,100)
(0.3,145,0.5,110)
(0.3,145,0.5,120)
(0.3,145,0.5,130)
(0.3,145,0.5,140)
(0.3,145,0.5,144)
(0.3,145,0.75,90)
(0.3,145,0.75,100)
(0.3,145,0.75,110)
(0.3,145,0.75,120)
(0.3,145,0.75,130)
(0.3,145,0.75,140)
(0.3,145,0.75,144)
(0.3,145,1.0,90)
(0.3,145,1.0,100)
(0.3,145,1.0,110)
(0.3,145,1.0,120)
(0.3,145,1.0,130)
(0.3,145,1.0,140)
(0.3,145,1.0,144)
3.8725
8.2163
14.3840
22.0315
30.7320
40.1264
44.0179
5.3708
9.9241
15.9665
23.2513
31.4810
40.3769
44.0680
6.6231
11.2981
17.2494
24.2724
32.1307
40.5989
44.1125
3.8726
8.2172
14.3860
22.0350
30.7344
40.1286
44.0183
5.3697
9.9233
15.9650
23.2512
31.4812
40.3777
44.0684
6.6213
11.2968
17.2483
24.2716
32.1305
40.5992
44.1128
3.8741
8.2166
14.3873
22.0341
30.7321
40.1262
44.0179
5.3757
9.9250
15.9715
23.2556
31.4799
40.3772
44.0680
6.6261
11.2991
17.2568
24.2770
32.1315
40.5995
44.1125
Parameters
  0 . 07
LBE
Ritchken
(0.2,115,0.5,90)
(0.2,115,0.5,100)
(0.2,115,0.5,110)
(0.2,115,0.5,114)
(0.2,115,0.75,90)
(0.2,115,0.75,100)
(0.2,115,0.75,110)
(0.2,115,0.75,114)
(0.2,115,1.0,90)
(0.2,115,1.0,100)
(0.2,115,1.0,110)
(0.2,115,1.0,114)
TrueValue
1.3860
4.7716
10.9875
14.1595
2.0414
5.5661
11.3617
14.2391
2.5716
6.1304
11.6145
14.2927
1.3850
4.7697
10.9859
14.1610
2.0402
5.5647
11.3610
14.2402
2.5704
6.1294
11.6148
14.2931
1.3875
4.7726
10.9893
14.1596
2.0444
5.5678
11.3634
14.2393
2.5743
6.1315
11.6165
14.2927
(0.3,130,0.5,90)
(0.3,130,0.5,100)
(0.3,130,0.5,110)
(0.3,130,0.5,120)
(0.3,130,0.5,129)
(0.3,130,0.75,90)
(0.3,130,0.75,100)
(0.3,130,0.75,110)
(0.3,130,0.75,120)
(0.3,130,0.75,129)
(0.3,130,1.0,90)
(0.3,130,1.0,100)
(0.3,130,1.0,110)
(0.3,130,1.0,120)
(0.3,130,1.0,129)
3.4237
7.5087
13.5093
21.1515
29.0739
4.6670
8.9044
14.6985
21.8345
29.1465
5.6639
9.9588
15.5753
22.3343
29.1995
3.4216
7.5056
13.5060
21.1496
29.0738
4.6636
8.9012
14.6962
21.8331
29.1472
5.6613
9.9561
15.5735
22.3333
29.2002
3.4259
7.5104
13.5138
21.1541
29.0739
4.6701
8.9069
14.7029
21.8365
29.1466
5.6678
9.9615
15.5764
22.3353
29.1996
1.51E-03
2.36E-04
2.20E-01
2.45E-03
4.20E-04
1.05E+00
Parameters
RMSE
RMSRE
Computing Time
All options have K  100 . Parameters column represents (  , L, T  t , S t ). The “true value” column is
based on the trinomial method in Ritchken (1995) with 20,000 time-steps.
“LBE” and “Ritchken”
columns represent the extrapolation of lower bounds described in Section 4 and the trinomial method in
Ritchken (1995) with at least 200 time steps respectively.
27
L1
L2
K
Capped with L1
Capped with L2
T1
t
Figure 1
Exercise boundary of the doubly capped call option.
28
T2
lo g S p eed
4
3
2
1
0
-4
-3
B inom ial
-2
BBSR
-1
H SY
0
lo g R M S E
LU B A
UBE
Figure 2
Speed-accuracy trade-off for all calculated options (RMSE).
The RMSE (root of mean squared errors) is defined by

1 m ˆ
 Ci  Ci
m i 1

2
, where Ci is the ‘true’
option value (estimated by the binomial method with 40,000 time steps), Ĉ i is the approximate option
value estimated by the corresponding numerical method, and m =2,400 is the number of all calculated
options. Speed is measured in option prices calculated per seconds (on a 2000-MHz Pentium-Ⅳ PC).
Preferred methods are in the upper-left corner.
The binomial method results are based on the 25, 50, 100, 150, 200, 500 and 1000 time steps. The BBSR
(binomial Black and Scholes method with Richardson extrapolation of Broadie and Detemple (1996))
results are based on the 12, 24, 50, 100 and 200 time steps. The HSY (recursive integration method of
Huang, Subrahmanyam and Yu (1996)) results are based on the discretizations of 4, 6, 8, and 10 points.
The LUBA represents the lower and upper bound approximation of Broadie and Detemple (1996). The
UBE (upper bound extrapolation) of this article results are based on the discretizations of 6-60, 8-64, and
10-80 points.
29
log S peed
4
3
2
1
0
-5
-4
B inom ial
-3
BBSR
-2
H SY
-1
log R M S R E
LU B A
UBE
Figure 3
Speed-accuracy trade-off for all calculated options with true price  0.5 (RMSRE).
The RMSRE (root of mean squared relative errors) is defined by
1
m
 (Cˆ
i
 Ci ) / Ci

2
, where Ci
is the ‘true’ option value (estimated by the binomial method with 40,000 time steps), Ĉ i is the
approximate option value estimated by the corresponding numerical method, and m =2,121 is the
number of all calculated options satisfying Ci  0.5 . Speed is measured in option prices calculated per
seconds (on a 2000-MHz Pentium-Ⅳ PC). Preferred methods are in the upper-left corner.
The binomial method results are based on the 25, 50, 100, 150, 200, 500 and 1000 time steps. The BBSR
(binomial Black and Scholes method with Richardson extrapolation of Broadie and Detemple (1996))
results are based on the 12, 24, 50, 100 and 200 time steps. The HSY (recursive integration method of
Huang, Subrahmanyam and Yu (1996)) results are based on the discretizations of 4, 6, 8, and 10 points.
The LUBA represents the lower and upper bound approximation of Broadie and Detemple (1996). The
UBE (upper bound extrapolation) of this article results are based on the discretizations of 6-60, 8-64, and
10-80 points.
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