INTERNATIONAL JOURNAL OF ENHANCED RESEARCH IN SCIENCE TECHNOLOGY AND
ENGINEERING (IJERSTE), ISSN NO: 2319-7463
Vol. 3 Issue 2, February-2014, pp: (…-..), Available online at: www.erpublications.com
Modified Black-Scholes Model based on Monte
Carlo Simulation
Dilip Senapati1, Nikhil Rajput2, Binay Kumar Singh3, Amit Kumar Singh4
1,4
School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi-110067, India
2
Department of Computer Science, Ramanujan College, Delhi-110019, India
3
Department of Computer Science, Indian School of Mines, Dhanbad-826004, India
Abstract: Understanding stock market dynamics is a challenging problem. We have proposed a novel approach for
Black-Scholes model with periodic interest rate using Monte Carlo simulation. In this paper, we have also analyzed
the First Passage Time (FPT) distribution under some given strike price. We have applied Monte Carlo simulation
to the underlying Stochastic Differential Equation (SDE), which produces nonlinear dynamics of stock price.
Computer simulation and comparison with standard example is also provided to support our approach.
Keywords: Black-Scholes, First Passage Time, No-Arbitrage, Periodic Rates, Monte Carlo Simulation.
Introduction
From last few decades, understanding the dynamics of stock market is a challenging as well exciting problem. Several
mathematical and computational techniques have been developed to contemplate the characteristics of stock market. But
mathematical formulation is intractable in nature, so clearly, to capture the variations of stock market one has to apply the
well-known Monte Carlo simulation. It is a common practice to model stock market dynamics by the well-known BlackScholes model 1973[1]. Several researchers [2,3,4,5,6] have proposed various models for better understanding of the nature
of stock price movement. Researchers also developed some computational schemes for simulation of stock prices and
related quantities of interest. The limitations of Black-Scholes model are that it assumes the interest rate as time
independent; however, in real world, markets interest rates are known to be functions of time. In this paper, we have
considered the time dependent interest rate Black-Scholes model. The basis for interest rate modeling depends on effects of
business cycles on interest rate. Using this modified Black-Scholes model, we are interested in examining the effects of
seasonal periodicities in the temporal evolution of stock price dynamics. This model is using analytical results and Monte
Carlo simulation of stock price dynamics for the corresponding SDEs. During the analysis of the model, we have obtained
conditions of no-arbitrage.
Black-Scholes model and Monte Carlo Simulation
The Black -Scholes model has been the most important mathematical model to describe the stock price dynamics. This
model especially calculates the fair option pricing in stock market, which is resembled as observed pricing. The
generalized Black-Scholes model has been suggested in the literature[2]. Black-Scholes model is represented by the SDE
as given below:
(1)
dS   Sdt   SdW (t ), S (t  0)  S
0
here, W(t) is the Weiner process. The solution of this equation (1) is given by
1 2

    t 
2


S (t )  S0e
tZ
The corresponding probability density function S (t ) for is
2

 
1 2  


   log(S / So )       t  
2   

 exp  
2



2 t
P( S , S0 , t )  f (S , t )  




,S  0

S 2 t

0, S  0
(2)
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INTERNATIONAL JOURNAL OF ENHANCED RESEARCH IN SCIENCE TECHNOLOGY AND
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1.2
1
 =0.4
 =0.6
 =0.8
f(S)
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
S
3
3.5
4
4.5
5
Figure 1: Lognormal density function at time for t=2: µ=0.005, S 0 =1
Monte Carlo method is a simulation technique, which uses random numbers and these numbers are independent and
identically distributed. This method is generally used in optimization, numerical integration, and simulating financial
market. The analytical solution of a vast class of SDEs is not possible; therefore we resort to Monte Carlo simulation of
SDEs.
Referencing [6.7,8,9], we have used the following discretized scheme for simulation
S (ti 1 )  S (ti )   (ti ) S (ti )h   (ti ) S (ti ) h Zi+1 ,
(3)
Where Zi+1N (0,1) and Z1, Z2… are independent normal variates and h is the step size.
180
160
140
S(t)
120
100
80
60
40
0
1
2
3
4
5
Time
6
7
8
9
10
Figure 2: Sample paths of S(t) with µ=.05, σ=0.1, h=.001
Modified Black –Scholes Model
Our proposed modified Black-Scholes model considers time dependent interest rates which is oscillatory. The underlying
stock S(t) under time dependent interest rate is modeled as given below:
dS (t )   (t ) S (t )dt   (t ) S (t )dW (t ), S (t  0)  S0
(4)
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INTERNATIONAL JOURNAL OF ENHANCED RESEARCH IN SCIENCE TECHNOLOGY AND
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d  (t )  r (t )  (t ) dt ,
(5)
where, μ(t) is the time dependent drift, W(t) represents Wiener process σ is the market volatility, and r(t) is the
corresponding sinusoidal interest rate represented as:
r (t )  r0  r cos(t ), r0  r
(6)
where, r0 is the mean interest rate and r cos(t ) is the interest rate which is periodic with frequency ω. For further
calculations, we will use the time-average interest rate given below:
ravg (t )  r0 
r

sin(t ),
(7)
Interestingly, with the deterministic time structure in the parameters, under the no-arbitrage condition the proposed
model is found to follow the same pde as originally proposed by Black-Scholes[3]. The pde is as under:
2
2
f (t , S (t ))
f (t , S (t ))    S (t )    2 f (t , S (t ))
 r (t ) f (t , S (t ))   r (t ) S (t ) 




t
x
2
2 x


(8)
Thus, under no-arbitrage condition in the Black-Scholes model, the time-dependent drift of the underlying stock is equal
to the interest rate of the bond, we have
 (t )  r (t ).
Therefore, it is easy to see that the stochastic evolution of the underlying stock price follows lognormal process with
oscillatory mean. The pdf of the process is:

 
 S

   log 

 
1
 S0
exp 

p ( S , t )   S 2 t





0, S  0

 
2
   ravg (t ) 
2
 
2 2t
2
  
t
  
  
,S  0



Computer Simulation and Comparisons
As it is interesting to note that we have explicit expression for the time dependent density function for stock prices. The
density function represents lognormal process with periodic drift. The stochastic evolution of the stock price has been
adopted as a first passage time (FPT) problem [10]. When the price of the underlying asset S(t) with initial price as S0
reaches the strike price (K), the option is exercised.
The FPT of stock price is accordingly defined as:
FPT  inf t  0 : S (t )  K , S0  K
7000
6000
5000
f(t)
4000
3000
2000
1000
0
30
35
40
45
Time
50
55
60
Figure 3: Histogram of FPT distribution for model driven by periodic interest rate.
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INTERNATIONAL JOURNAL OF ENHANCED RESEARCH IN SCIENCE TECHNOLOGY AND
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S0  10, 0  0.05,   0.2, r0  0.1, r  .1,   0.05
0.07
0.06
0.05
f(t)
0.04
0.03
0.02
0.01
0
0
5
10
15
20
25
Time
30
35
40
45
50
Figure 4: FPT distribution for model driven by periodic interest rate.
8000
7000
6000
f(t)
5000
4000
3000
2000
1000
0
35
40
45
50
55
60
Time
Figure 5: Histogram of FPT distribution without periodic interest rate.
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INTERNATIONAL JOURNAL OF ENHANCED RESEARCH IN SCIENCE TECHNOLOGY AND
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0.08
0.07
0.06
f(t)
0.05
0.04
0.03
0.02
0.01
0
0
5
10
15
20
25
Time
30
35
40
45
50
Figure 6: Probability Density of FPT without periodic interest rate.
Conclusion
In this paper, we have presented a novel approach for the dynamic evaluation of stock prices. The proposed model is
formulated as a SDE having periodic interest rate. It provides a greater insight, because it recovers the existing lognormal
process of stock market behavior in the absence of oscillating drift in our model. Also, it is interesting to find out that the
strike price can be derived from a first pass passage time nature of stock market under the rigorous Monte Carlos
simulation. Although, some future work is still possible and one can find option price based on our model may be helpful
for portfolio manager, hedge funder, and investors.
References
1.
2.
3.
4.
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E. Barucci, L. Landi and U. Cherubini, “Computational Finance”, IEEE Computational Science & Engineering,
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