Frontiers in Business, Economics and Management 
ISSN: 2766-824X | Vol. 11, No. 2, 2023 

 

288 

Pricing	Strategy	of	Convertible	Bond	with	Memory	and	
Jumps	
Lichao Tao1, Run Wen2, Biao Xu2 and Zheng Zhou3, * 

1 Zhejiang College of Construction, Zhejiang, China 
2 Department of Applied Statistics, Zhejiang University of Science and Technology, Zhejiang, China 
3 School of Accounting and Finance, Zhejiang Business College, Zhejiang, China 
* Corresponding author: Zheng Zhou (Email: 00817@zjbc.edu.cn) 

 

Abstract: In this paper, a novel pricing formula for convertible bond is developed under Chinese Regulator’s framework.  
This formula generalizes existing results by considering the long-memory property of underling assets and rare market events.  
An empirical analysis is conducted by the end of this paper. 

Keywords: Convertible Bond, Sub-Fractional Brownian Motion, Jump Diffusion, Genetic Algorithm. 
 

1. Introduction 
The idea of using mathematical tools to predict the future 

trend of financial asset is always a hot topic in the field of 
mathematical finance. The history of such idea can be traced 
back to the beginning of 20th Century, Bachelier [3] modeled 
financial asset price dynamics by using Brownian motion. 
The breakthrough came in 1973, when Black and Scholes [4], 
Merton [16] developed an explicit expression for European 
call and put options, which is the so-called BSM model. In 
the following years, extensive results are developed, see 
[8,9,11,19] and reference therein.  Under Black-Scholes-
Merton framework, the financial asset is modelled by a 
continuous-time Markov process, i.e. The geometric 
Brownian motion, but such process has limitation in explain 
the long-memory properties and rare market events. 

To overcome such difficulties, In [5], Bojdecki introduced 
a novel way of describing the asset price, which employed a 
stochastic differential equation driven by sub-fractional 
Brownian motion(sfBm). In [18], Tudor found that the sfBm 
has self-similarity and long- range dependence properties, the 
sub-fractional Brownian motion degenerated faster than 
fractional Brownian motion, which shows that the long-
memory property of asset pricing. More literature about sub-
fractional Brownian motion can be found in [10, 20, 21] and 
references therein. Moreover, Some rare market events, for 
example outbreak or war, will have huge impact on the asset 
price. In terms of mathematics, these sudden changes to the 
asset prices can be formulated by using a jump diffusion. For 
instance, Rambeerich and his co-authors [17] developed a 
numerical approximation of European and American options. 
In [14] Marina etc, derived the explicit formula for Asian 
options, where the underlying asset is driven by jump-
diffusion. Ji et al. [13] investigated the pricing of barrier 
option when the underlying asset is driven by a Sub-mixed 
fractional Brownian motion. 

Recently, a class of financial assets, called convertible bond, 
attracts the eyes of many practitioner in the Mainland, China. 
The history of convertible bond can be traced back to the 
middle of 19th century when Jacob Little and Daniel Drew 
[15] used this instrument to counter volatility risk. There are 
extensive literature about convertible bonds. In [12], Under 

BSM framework, Ingersoll developed an explicit formula for 
pricing the convertible bonds in terms of company’s stock 
prices; Bernnan et al. [6] further extended the previous 
convertible bond pricing model by introducing the stochastic 
interest rate, in which the pricing process is driven by a two 
factor model. More details can be found [1, 2] and reference 
therein. 

The assumptions of BSM framework are restrictive and 
over-ideal, if all parameters of BSM model rely on the 
historical data, then there will be a huge gap between 
theoretical and practical results. To tackle this difficulty, a 
widely used machine learning algorithm called the genetic 
algorithm(GA) will be employed in this paper. GA is proved 
to be efficient in estimating the parameters, and also it 
provides global optimal estimation of parameters. For 
instance, Chen et al. [7] employed the GA algorithm to 
optimize the investment portfolios, while Zelenkov [22] used 
GA algorithm to reorganize the historical data, which improve 
the performance in quantitative trading. 

To author’s best knowledge, there is little literature 
regrading to the pricing strategy of convertible bond 
considering both sub-fractional Brownian motion and jump 
diffusion. The main aim of this paper is to construct the 
theoretical pricing formula of convertible bond and perform 
an empirical study by estimating the corresponding 
parameters via genetic algorithm. 

The rest of this paper can be organized as follows, in 
section 2, some preliminary results will be given. In Section 
3, the pricing formula of convertible bonds will be illustrated 
under classic BSM framework with specified modification. 
The empirical research involving parameter estimation via 
GA approach combined with will conduct in the last Section. 

2. Preliminaries 
Throughout this paper, let (Ω, ℱ, P) be a complete 

probability space with a filtration {ℱt} t≥0 satisfying the 
usual condition (i.e. it is right continuous and ℱ0 contains all 
P-null sets). Let Bt be a standard m-dimensional Brownian 
motion. Recall that sfBm in short with Hurst index H ∈ (0, 
1) is a Gaussian process },{ H ）（  tH  with a mean zero 
and covariance 



 

289 

0,),|||(|
2

1
][ 2222  tsstststE HHHHHs

H
t   

For H = 1/2, the sfBm degenerate to a standard Brownian 
motion. In this paper, the underlying asset price will follow 
the the stochastic differential equation driven by sfBm and a 
jump process. 

 
 ;0,)(d TtdNdtSdSS t

H
tttt     (2.1) 

 
where µ is the average rate of return of the stock price; σ is 

the volatility of sfBm; dNt is the Poisson compensation proc- 
cess; γ is the jump rate of stock price. The initial value is given 
by S0 = S. 

Lemma 2.1 Assume that asset price follows (2.1) with 
initial value S0 = S, then the it has an explicit solution: 

 
2 2 2 2 2

t

1
exp( (1 2 ) ).

2
H H H

t tS S t t t N     
      (2.2) 

 
Proof: Let ( , )tQ S t be a

2 ,1C function, then apply the gen- 

Eralized �Ito fomula, we have 
2 2

2 2 2 1 2 2 2
2 2

1
( (2 2 ) )

2

     

H H

H
t t

Q Q Q Q
dQ S S Ht S dt

t S S t
Q Q

Sd SdN
S S

   

  

       
   

 
 
 

  

Now, by letting ,lnSQ  then 

t
H
t

HH
t NtttSS  

 ))21((lnln 22
12222

Then, the desired result can be obtained by taking expone- 
ntial of both sides. 

Remark 2.1 According to the Chinese Regulator and 
Shanghai Stock Exchange(SEC),convertible corporate bonds 
are a kind of corporate bonds which can be converted into a 

predetermined amount of the underlying company’s equity at 
certain times. Convertible bonds can be seen as a combination 
of bonds and stock options. As a result of that, 
together with the common practise in real financial fields, the 
price of convertible bond can be divided into four parts: 

• The price of bond;  

• The price of stock options;  

• Put provision;  

• Redemption. 

Or, equivalently, we have the following equations: 

Convertible Bond = Bond + Options + Put provision + 
Redemption. 

3. Asset Pricing Model 
In this section, The main result of this paper, the pricing 

formula of convertible bond will be illustrated. 

Theorem 3.1 A company issues a convertible bond at time t 
= 0, then the convertible bond price V can be formulated as: 

1 3 50 (1 ) (1 )

( )
2 4 6

{ [ ( ) ( ) ( )]}

      [ ( ) 0.7 ( ) 1.3 ( )]

t n

n I M
tt r r

r T t M
K

V S N d N d N d

e K N d N d N d

  

 

     

   

 (3.1) 

where 

2

1

2 2 2 2 2
2

2 1 2 2 2 2

ln( / ) (1 2 )( ) ( )( )

(2 2 )( ) ( )

H H H
t

H H H

S K T t r T t
d

T t T t



  





     


   
2 1 2 2 2 2

2 1 ( 2 2 )( ) ( )
H H Hd d T t T t       

22 2 2 2 2
2

3 2 1 2 2 2 2

ln(10 / 7 ) (1 2 )( ) ( )( )

(2 2 )( ) ( )

H H H
t back

H H H

S X T t r T t
d

T t T t



  





     


   
2 1 2 2 2 2

4 3 ( 2 2 )( ) ( )
H H Hd d T t T t         

22 2 2 2 2
2

5 2 1 2 2 2 2

ln(10 /13 ) (1 2 )( ) ( )( )

(2 2 )( ) ( )

H H H
t redeem

H H H

S X T t r T t
d

T t T t



  





     


   
2 1 2 2 2 2

6 5 (2 2 )( ) ( )
H H Hd d T t T t         

Remark 3.1 The proof can be divided into four parts, as 
mentioned in the last section. 

To prove this theorem, we will introduce following four 
lemmas. 

First, we consider the bond part of the convertible bond. 
Lemma 3.1 Firstly, we consider the bond case. Denote B 

be the present value of bond, it can be valued in the following 
form 

           
n

n

t
t

r

M

r

I
B

)1()1(0t 



 



        (3.2) 

where tI  is the cashflow of t-th year while M is the 
principal of the convertible bond. r be the risk-free rate, while 
n is the maturity of this bond. 

Now, we consider the stock option part of convertible 
bonds. 

Lemma 3.2 Suppose that the underlying asset’s price 
follows (2.2), denote C = C(St, t) be the value of European 
call option on time t for this asset. Then 

2
2 1 2 1 2 2 2

2
[ (2 2 ) ] 0

2
H H

t t

C C C
rS Ht S rC

t S S


  

  
     

  
 

with boundary condition ( , ) m ax( , 0).T TC T S S K   

Given the strike price K, maturity T, the value of European 
call option can be expressed explicitly, 

( )
1 2( , ) ( ) ( )

r T t
T tC T S S N d Ke N d

    

where 
2

1

2 2 2 2 2
2

2 1 2 2 2 2

ln( / ) (1 2 )( ) ( )( )

(2 2 )( ) ( )

H H H
t

H H H

S K T t r T t
d

T t T t



  





     


   
2 1 2 2 2 2

2 1 ( 2 2 ) ( ) ( )
H H Hd d T t T t         

Proof: Here, we use the idea of Delta hedge to construct a 
portfolio, which longs one unit of call option and short ∆ unit 
of underlying assets S. At time t, the value of this portfolio is

.t tC S     Within the time interval ),( dttt   , 

t td dC dS     which implies that 

( )t t t td r d r C S d       
Recall the fact that sfBm processes the following property 

2 2 2 2 1 2 1( ) 2 (2 2 ) ,  dt*dB 0H H Ht t tdS S Ht dt
     

An application of �'Ito s lemma yields 
2

2 1 2 1 2 2 2
20

0

( , ) (0,0) { ( , ) [ (2 2 ) ] ( , )}
2

            ( , )                                                                                            

t
H H

t x x

t

x

f f
f t S f x S Hx S x S dx

x S
f
x S dS

S


  

 
    

 









 

Therefor, 



 

290 

2
2 1 2 1 2 2 2

2
{ [ (2 2 ) ] }

2
H HC C CdC Ht S dt dS

t S S


  

  
    

  
 

(3.4)  Combining with the portfolio, we have 

2
2 1 2 1 2 2 2

2
     { [ (2 2 ) ] } ( )

2
H H

d dC dS

C C C
Hx S dt dS

t S S


  

  

  
     

  

  (3.5) 

Then, 
2

2 1 2 1 2 2 2
2

( ) { [ (2 2 ) ] }
2

H HC C Cd r C S dt Hx S dt
S t S


  

  
      

  
 (3.6) 

by letting ,
C

S


 


 

2
2 1 2 1 2 2 2

2
[ (2 2 ) ] 0

2
H H

t

C C C
rS Hx S rC

t S S


  

  
     

  
(3.7) 

With the Black-Scholes PDE in hand, we can solve this 
PDE with the boundary condition by using transformation 
method. 

Now let ln , , ( , ) ( , ),t tx S T t W x C t S      then we 
have 

2 2

2 2 2 2

1

1 1

t t t

t t t

C W t W

t
C W x W

S x S x S

C W W

S x S x S

  
   

   
   
   

   
   

  
   

  

 

which implies 

2 1 2 1 2 2

2
2 1 2 1 2 2

2

( ( 2 2 ) )
2

[ ( 2 2 ) ] 0 .
2

H H

H H

W W
r Ht

x
W

Ht rW
x


 




 

 

 

 
    

 


    


 

By letting 

2 2 1 2 1 2 2[ ( 2 2 ) ],
2

H HHt


       

and ( )
1 1 1 1( ), ( ), ( , ) ( , ).

Dx x A B U x e W x         

we have  

( )

1 1
2 2

( ) ( )
2 2

1 1

[ '( ) '( ) '( ) ] ,

,

D

D D

W U U
e A B UD

x x
W U W U

e e
x x x x



 

  




 

  
  

  
   

 
   

 

which implies 
2

' ' ' 2 2
2

1 1 1 1

( ) ( ) ( ) ( ) 0
U U U U

A B UD r rU
x x x

    


   
       

   
(3.8) 

Assume that 
2 2'( ) , '( ) , '( )A r B D r         

Then we have 

22 1 2 1 2 2 2 2 2 2
20

2
2 1 2 1 2 2 2 2 2 2

0

( ) [ (2 2 ) ] ( ) (1 2 )
2

( ) [ (2 2 ) ] (1 2 )           
2 2

( )                                                                    

H H H H

H H H H

A r Hs ds r

B Hs ds

D r

 




     

 
     

 

  

  

       

     






                                   









which yields 

                   
2

2
1 1

U U

x
 


 

               (3.9)             

Then, the payoff of this option at maturity t is given be 
1

1,( 0) max( , 0).
xU x e K   By applying the heat equation 

method, 
2( )1

4 1

2 2( ) ( )1 1
4 41 1

1 1 ln
1

ln ln
1 1

1 2

1
( , ) ( )

2
1

              
2 2

              

z x

z x z x

z

K

z

K K

U x e K e dz

K
e e dz e dz

I I



 




 



 

 

  

 

 

 



 

(3.10) 
where 

2( ) 21
4 11 1 2

1 1

1

1 1

ln 21
ln

21

1 1

1

1 1

2 2
2 ln

    =e ( )
2

z x
txz

K x
K

x

I e e dz e e

x K
N

 






 



   
 



 

 

 
(3.11) 

by setting 1 1
1

2

2

z x
t




 
  and 

2( ) 2
4 1 2

ln 1
2 1

2 ln
1

1

1

2 2
ln

    = ( )
2

z xt h

K x
K

K K
I e dz e dh

x K
KN



 







    




  (3.12) 

by setting 1
12

z x
h




 .Hence, 

1 1 1 1 1
1 1

1 1

2 ln ln
( , ) ( ) ( )

2 2
x x K x KU x e N KN




 
     (3.13) 

and 

1 1 1 1 1

1 1

2 ln ln
( , ) ( ) ( )

2 2
x r rx K x KW x e N Ke N  




 
      (3.14) 

Substituting 
1x  and 1  back, which yields 

      * *
1 2( , ) ( ) ( )

x rW x e N d Ke N d    (3.15) 
where 

  

2

2

2 2 2 2
* 2
1 2 2 1 2 2

2 2 2 2
* 2
2 2 2 1 2 2

ln (1 2 ) ( )

( 2 2 )
ln (1 2 ) ( )

( 2 2 )

H H

H H

H H

H H

x K r
d

x K r
d





  

   
  

   









    


 
    


 

 

By taking tTS t  ,lnx and ).,(),( tStCxW   
Remark 3.2 When H=1/2, the explicit formula for 

European call option with underlying assets driven by 
standard BM and jump diffusion is recovered. when λ = 0, the 
formula for European call option with underlying assets 
driven by sfBm is recovered. When H=1/2, λ=0 the classic BS 
model has been recovered. 

To finalize the pricing formula of a convertible bond, we 
need to consider some critical conditions in the real trading 
contract. When the stock price is lower then 70% of the strike 
price K for a period, then the Put provision condition is active, 
and the holder of convertible bond can sell the bond back to 
the company who issues such convertible bond. Then the 
following lemma is introduced: 

Lemma 3.3 According to the agreement, the sell back price 
is

backX and option price is backR satisfy the following equation: 
( )

3 4( , ) ( ) ( )
T t

back t t backR t S S N d X e N d
       (3.16) 

where 
22 2 2 2 2

2
3 2 1 2 2 2 2

ln( / ) (1 2 )( ) ( )( )

(2 2 )( ) ( )

H H H
t back

H H H

S X T t r T t
d

T t T t



  





     


   



 

291 

2 1 2 2 2 2
4 3 ( 2 2 )( ) ( )

H H Hd d T t T t         
When the KST 3.1 , then the redeem condition is active. 

Then we have the following lemma: 
Lemma 3.4 The redeem price and the redeem option price 

can be calculated as 
( )

5 6( , ) ( ) ( )
r T t

redeem t t redeemR t S S N d X e N d
    (3.17) 

where 
22 2 2 2 2

2
5 2 1 2 2 2 2

ln( / ) (1 2 )( ) ( )( )

(2 2 )( ) ( )

H H H
t redeeem

H H H

S X T t r T t
d

T t T t



  





     


   
2 1 2 2 2 2

6 5 ( 2 2 )( ) ( )
H H Hd d T t T t         

Proof of Theorem 3.1: The proof is to formulated above 
results from all lemmas above, which could be done 
accordingly. The details can be omit here. 

4. Empirical Research 
The key problem to solve in this section is to estimate the 

parameter used in Theorem 3.1. The classic MLE method has 
its own limitation, hence we adopt the GA algorithm, which 
is a well-developed algorithm? A typical GA algorithm can be 
divided into following three parts: 

Reproduction It corresponds to natural selection. Random 
ergodic sampling is used in the search procedure, and the 
probability of each individual appearing in offspring is 
calculated according to the fitness value of the individual, and 
individuals are randomly selected according to this 
probability to form the offspring population. Real value 
coding is adopted for each parameter to simplify coding and 
decoding. For example, each dimension in the four-
dimensional vector [A, B, C, D] represents a parameter, and 
different four-dimensional vectors represent different coding 
strings, and each coding string is a group of ideal parameters. 

Crossover Crossover operation of genetic operation is the 
core of genetic algorithm. The so-called interchange refers to 
the two parent individuals to replace part of the structure of 
the recombination to generate a new individual operation. 
Through crossover, the search ability of genetic algorithms 
can be greatly improved. Crossover computing, by randomly 
swapping genes between two individuals in a population 
based on crossover rates, can produce new combinations of 
genes in the hope of putting beneficial genes together. The 
design of crossover operation mainly consists of three aspects: 
the determination of crossover location, the way of 
exchanging a certain part of genes with each other and the 
determination of crossover probability. Based on the 
consideration of the length of coding string, the genetic 
algorithm in this paper adopts the parent-twin method. In this 
method, after parents are identified, all the genes after that bit 
are swapped with a random bit, and two offspring are formed 
after swapping. 

Mutation The basic content of mutation operation is to 
change the gene value on some gene locations of individual 
strings in the population. For example, a mathematical  
expression of mutation can be as follows: if there are only two 
numbers of 0 and 1 in the domain, then mutation changes 0 to 
1 and 1 to 0, thus generating a new solution.  

4.1.  Data  
According to the Wind Database, there are more than 300 

tradable convertible bond available in the Chinese market. 
Then according to the characteristics of whether the issuing 
terms of the convertible bonds are standardized and complete 

and the issuing quantity, whether the trading of the 
convertible bonds and the underlying stocks is suspended in 
the sample interval, whether there is sufficient market price 
data, financial situation, credit rating, bond maturity and so 
on. We choose one of the convertible bonds, Hope convertible 
bond (code 127015), the contents are shown in the table 
below: 

 
Table 1. Hope convertible bond main contents of relevant 

clauses 

Code 
Cred
it 

Rating 

Maturi
ty Date 

Conversi
on Price 

Redempti
on 

Price 

12701
5 

AA
A 

2026.1
.2 

19.75 25.68 

 

The compounded return rate of the 
thi  day 

1lnlnr  iii SS  , and the volatility is based on filtered 
historical data, which covers January 3rd 2020 till January 3rd 
2021, in total 236 observations,so the corresponding standard 
deviation is 

2

1
2

( )
0 .0 1 2 7 8,

1

T
i

i

r

T







 


  

which can be regarded as the daily volatility, with 

2

0 .0 0 0 8 .
1

T
i

i

r

T




  


  

As a result, the annual volatility 0.1962361   and 
the convertible counterpart is 19.63%. Compared to the 
classic BSM model, the approximated volatility is 45.3. The 
value of lambda is set to be 22, as the number that the daily 
volatility is larger than 5%. For the convenience of calculation, 
the pure bond discount rate in this paper adopts the five-year 
government’s bond benchmark interest rate of the People’s 
Bank of China in 2020, which is r = 5.32%; The risk-free rate 
is 1.35% for one-year time deposit. 

The table below summarizes all the related information: 
 

Table 2. Correlation parameter estimation results 

 H       r  
B-S Model 0.5 0 0.45 0 1.35% 
GA Model 0.77 3 0.22 0.06 1.35% 
New Model 0.7 22 0.19 0.05 1.35% 

4.2.  Simulation Results 
In this paper, the value of the Hope convertible bonds is 

collected on fortnight basis of listing on February 4, 2020, and 
a total of 12 samples over six months. The estimated 
parameters of the convertible bond pricing model under the 
sub-fractional jump diffusion environment were calculated 
via a Python program, and then the first day convertible bond 
price was simulated by using the parameters. At the same time, 
the algorithm has been tested for nearly one year, and the 
results are good. 

The value of the pure bond part of the convertible bond can 
be calculated from Equation (3.3). Take the Hope convertible 
bond as an example: the face value is 100 yuan, the duration 
is 6 years, the coupon interest rate is 0.2% in the first year, 
and then from 0.4% in the second year, it increases by 0.4% 
year by year to 2% in the sixth year. The value of the pure 
bond part of the expected convertible bond can be obtained 



 

292 

by discount as follows: 

2 3
0

4 5 6 6

100 0.2% 100 0.4% 100 0.8%

(1 ) (1 ) 1 5.32% (1 5.32%) (1 5.32%)
100 1.2% 100 1.6% 100 2% 100

   + =78.1822
(1 5.32%) (1 5.32%) (1 5.32%) (1 5.32%)

n
t
t n

t

I M
B

r r

  
    

    
  

  
   



for the option part, the number of evolution was set as 1000, the 
crossover probability was 0.7, the mutation probability was 0.02, 
and the initial population size was 50 on the genetic algorithm, 
and the parameters to be estimated were obtained as follows: 
[H,λ,σ,γ] = [0.77, 3, 0.32, 0.06]. In this case, the value of 
fitness function, namely the sum of squares of residuals, is 
82.56. Figure 1 shows the variation trend of the average 
adaptive value of the population and the variation trend of the 
best individual adaptive value of each generation in the 
evolutionary process. 

 

 

 
Figure 1: Convergence of individuals and populations 

 
As can be seen from the Figure 1, the average adaptive 

value of the population fluctuated sharply before the 180 
times of genetic algorithm operation, but with the increase of 
the number of algorithms, the fluctuation of the average 
adaptive value of the population became smaller and smaller, 
and showed a decreasing trend. The average adaptive value of 
the parameters obtained by the genetic algorithm was about 
280 at the initial stage, and the average adaptive value 
stabilized at 80 after 390 generations with the increase of the 
number of evolutionary times, showing convergence from the 
population level. The adaptive value of the best individual in 
the genetic algorithm converges very quickly, indicating that 
evolution has made the gene significantly improved. 

In order to demonstrate the effectiveness of the algorithm, 
the conclusions obtained should be applied in practice. Firstly, 
the price march on the first day of listing is compared, and the 
value calculated by fractional jump-diffusion model is 

= 7 8 .1 8 2 2 8 .6 8 1 0 0 /1 9 .7 5 = 1 2 2 .1 1V C B     
Factors in terms of value of 

+B=C+R=78.1822+7.47+5.78(7.48) 100/19.75=113.46V   
According to the BSM model, the value 

7 8 .1 8 2 2 6 .7 6 1 0 0 / 1 9 .7 5 1 1 3 .4 6V C B       
is all around the expected CB listing price of 115.04on the 

first day of listing. 
The theoretical value of each trading day is compared to 

the actual price: 

 
Figure 2: Hope convertible bond pricing results 

 
Through the intuitive analysis of the above Figure 2, it can 

be seen that the theoretical results of the three models are very 
close to the actual market price, and the four curves 

fit well, that is to say, the theoretical value can well fit the 
direction of market changes. Through comparison, it is found 
that the result of the sub-fractional jump-diffusion model is 
better than that of the B-S model. Due to the existence of the 
jump, the overall volatility of the underlying asset is shared, 
so that the pricing result is more stable than that of the B-S 
model. Then, comparing the sub-fractional jump-diffusion 
model with clauses, it can be found that the model with 
clauses added is smoother, which is more consistent with the 
actual price. Therefore, this paper believes that the sub-
fractional jump-diffusion model with clauses added is 
effective. 

Next, this paper introduces the mean square error (MSE), 
which reflects the degree of deviation between theoretical 
price and actual price, to compare the advantages and 
disadvantages of each model quantitatively. The expression 
of MSE is as follows: 

2m od

1

1
( )

n
el actual

i actual

C C
MSE

n C


   

 
Table 3. Mean square error of theoretical value of hope 

convertible bond 

 Include Clause Exclude Clause 

 
B-S 

Model 
New 

Model 
B-S 

Model 
New 

Model 
Historical 0.0611 0.0134 0.0874 0.0468 

GA 0.0579 0.0056 0.0468 0.0408 

 
The table summarizes the previous results. 
As can be seen from the table, the model in this paper is 

generally superior to the traditional B-S model, the model 
considering terms factors is superior to the model without 
considering terms factors, and the parameter effect estimated 
by genetic algorithm is superior to the parameter estimated by 
historical data, which further verifies the analysis obtained 
from Figure 2. 

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