Partial differential equation, parabolic Black-Scholes type, is used in evaluating equity options, that paying constant and continue dividends or in evaluate options in which interest rate, volatility and dividend are dependent on time.stocks, options , futures, forward, portofolio, model

Turcan Radu Olimpiu Calin

English

Research Papers in Economics

  795 
„BLACK-SCHOLES MODEL USED TO EVALUATE STOCKS OPTIONS” 
Turcan Radu 
University: University of Oradea  
Faculty: The Faculty of Social and Human Sciences 
 
 
 
Partial differential equation, parabolic Black-Scholes type: 
 
0
2
1
2
2
2 2 = −
∂
∂
+
∂
∂
+
∂
∂
rV
S
V
rS
S
V
S
t
V
σ
 
 
is used in evaluating equity options, that paying constant and continue dividends or in evaluate 
options in which interest rate, volatility and dividend are dependent on time. 
 
Keywords  :  stocks,    asset-support,options  (call  and  put),  contracts  (futures  and  forward), 
portofolio, model 
 
JEL Classification:  G12, G13 
 
 
1.1  Black-Scholes equation 
 
Further we intend to Black-Scholes equations, capitalizing above the  strategy used by Paul 
Wilmott: 
   
                Be  ￿=V(S;t) – ￿S              (1) 
 
a portfolio consists of option, with value V=V(S;t) 
and quantity: ￿S (ie: „short” position, for asset-support: S). 
 
Also, consider the value of „￿”: 
 
              
S
V
∂
∂
= ∆         (random component: ￿(„delta”))        (2) 
 
Replaced, value of „￿”:       , 0 =
∂
∂
− ∆ ⇔
∂
∂
= ∆
S
V
S
V
 
 
in his expression: d￿, we obtain: 
 
              dS
S
V
dt
S
V
S dt
t
V
d
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
∆ −
∂
∂
+ ￿ ￿
￿
￿
￿ ￿
￿
￿
∂
∂
+
∂
∂
=
￿ ￿ ￿
0
2
2
2 2
2
1
σ π       (3) 
 
or equivalent:    796
           dt
S
V
S dt
t
V
d ￿ ￿
￿
￿
￿ ￿
￿
￿
∂
∂
+
∂
∂
=
2
2
2 2
2
1
σ π                (4) 
          (ie: „increase” of ￿ portofolio, free of risk)         
           
 
But the variation of ￿, ie: d￿ can write (using the nonarbitration principle) in the form:
   
           d￿=r￿dt                             (5) 
 
Conclusion: 
 
Replaced the relations: (1), (2) and (4) ￿ (5), we obtain: 
 
           0 :
2
1
2
2
2 2 ≠ ￿
￿
￿
￿
￿
￿
∂
∂
− = ￿ ￿
￿
￿
￿ ￿
￿
￿
∂
∂
+
∂
∂
dt dt
S
V
S V r dt
S
V
S dt
t
V
σ              (6) 
 
￿        ￿
￿
￿
￿
￿
￿
∂
∂
− =
∂
∂
+
∂
∂
S
V
S V r
S
V
S
t
V
2
2
2 2
2
1
σ              (7) 
Ordering the terms in relation (29`), we obtain finally equation seeking, respectively: 
 
         0
2
1
2
2
2 2 = −
∂
∂
+
∂
∂
+
∂
∂
rV
S
V
rS
S
V
S
t
V
σ   (Black-Scholes equation)    (8) 
 
1.2 Interpretation of the notions: 
 
1 O ) Black-Scholes equation (deduced above), was written for the first time in this form, in 
1969, but published 4 years later, in 1973, when Fischer Black and Myron Scholes proved 
the correctness of this equation. 
 
2 O )  Black-Scholes  equation  models  a  very  important  financial  and  economic 
phenomenon: the assessment of stock options. 
 
3 O )  The  Black-Scholes  equation,  there  are  very  important  variables  and  parameters, 
respectively: 
 
• S=S(￿;￿) ￿ stochastic variable (is the value of the asset-holder) 
 
Where:                                                       
                 ￿= drift (or unexpected return) 
                 ￿= asset volatility                         parameters associated asset price „S” 
 
    t = time (deterministic variable) 
       r = interest rate (without risk) 
       V = V(S;t) ￿ call option value (option to purchase). 
   797 
4 O ) Terms:  , ;
2
2
S
V
and
S
V
t
V
∂
∂
∂
∂
∂
∂
 which appeared in Black-Scholes equation, is: „indicators 
of sensitivity” to change the option value „V”, respectively: 
 
          ￿￿
￿￿
￿￿ ￿ pointer „delta”, stands for: 
 
„percentage of modified (percentage of modified (￿V) option value V, depending on the 
value of asset-suport (￿S)” 
 
or even: ￿￿
￿￿
￿￿  ￿ „hedge delta” useful (theoretical and practical) to eliminate the risk of 
a portfolio; 
 
￿ ￿
￿￿
￿   ￿ pointer „theta”, is: 
„modified of option price (￿V), depending on the variation of time (￿t)” 
 
! ￿
￿"￿
￿￿" #
￿
￿￿$
￿￿
￿￿% ￿ pointer „gama”, is 
„sensitivity (excessive sensitivity)of ￿￿
￿￿
￿￿, to price variations of asset-support S” 
 
        
          5 O ) Finally, in the Black-Scholes equation, deduced previously, appear two terms: &'
￿￿
￿￿  
   and rV, where: 
 
- term:
 
&'
￿￿
￿￿ , is the convection term of equation; 
- term: rV, is the reaction term (which balances the first term of the equation  
￿￿
￿ ). 
   
   6 O ) In conclusion, bringing together the 4 (four) terms: 
    
￿￿
￿  (￿
￿
)*)') ￿"￿
￿￿" (&'
￿￿
￿￿￿￿￿￿i -rV;
   
 
       terms characterized briefly, above, we obtain Black-Scholes equation:  
 
    
  
￿￿
￿  +
￿
)*)') ￿"￿
￿￿" + &'
￿￿
￿￿ , &- ￿ .
 
 
equivalent to a special equations of fluid mechanics, type: „reaction-convection-diffusion”. 
 
1.3 Black-Scholes equation solving  
 
 General overview: 
 
      Method for solving Black-Scholes equation:     
   
            
￿￿
￿  +
￿
)*)') ￿"￿
￿￿" + &'
￿￿
￿￿ , &- ￿ .    (9) 
 
     aimed at obtaining the solution (exact) for the partial differential equation, parabolic type.   798
 
In literature, there are two broad categories of methods of solving this equation (ie: this type 
of partial differential equations). 
 
- analitic methods 
- numeric methods. 
 
The analytical methods we can mention: 
 
Method  I:  Transforming  equation  (8),  in  an  equation  with  constant  diffusion  coefficient, 
using the change of variable: 
              -/'(01 ￿ 234￿56 7 8/9(:1 
 
In finally, „finction”/solution U(x;￿) satisfy the basic equation of diffusion: 
 
              
￿;
￿6 ￿
￿";
￿4"                                              (10) 
(equation easier to solve than Black-Scholes equation (9)). 
Method II, using Green function: 
     
                    -￿/'(01 ￿
<=>/?=@1
A￿￿B)C/D￿ 12
EFGHIJ
J￿KL/>=M
"N"1/?=@1"O
"N"/?=@1                                          (11) 
 
 
Observations: 
 
1) In the limit (when t￿T, and T-t￿0 and 2￿P/D￿ 1￿2Q=1) function (derivative I of V,      
   given by (11) becomes Dirac delta function, ie:  
 
„A function that is one (zero) everywhere except one point where it is infinite, so that”: 
 
          R-￿/'(01S0 ￿ ￿                                                                         (12) 
 
     2) Function given by (11) is called Green function, which by integration, becomes: 
 
-/'(01 ￿
<=>/?=@1
AB)C/D￿ 1R 2
￿
EFGHIJ
J￿KL$>=M
"N"%/?=@1"O
"N"/?=@1 ￿
T 7 UVWXYY/'￿1
Z￿￿
￿￿       (13) 
where:  
Payoff(S)=V(S;T) ￿ ie: „profit/or loss for S  
is equal to the option V, at time t=T”                                                              (14) 
 
There are other analytical methods to assess Black-Scholes equation, for example:  
Fourier and Laplace transformations. 
 
The methods numericede solving partial differential equations, parabolic type (that is, 
and the equation: Black-Scholes), we can mention: 
 
Finite difference method with the following: 
 
- explicit finite difference method and   799 
- implicit finite difference method 
 
Very popular in the literature (to category: „numerical methods”) is the Crank-Nicolson method, 
which performs: „mean between the explicit and implicit method”. 
 
Explicit method, applied Black-Scholes equation, to implement Black-Scholes model is applied 
to options: call and put European-style (or American type). 
 
Default method, Crank-Nicolson equation applied to options: call and put European type, is more 
difficult to program, but has greater precision. 
 
Final conclusion: 
Black-Scholes Equation  0
2
1
2
2
2 2 = −
∂
∂
+
∂
∂
+
∂
∂
rV
S
V
rS
S
V
S
t
V
σ  represents  a partial differential 
equation,  parabolic  type  which  corresponds  to  the  Black-Scholes  Model  published  by  the 
American economists Fischer Black and Myron Scholes in 1973. This model may be used with 
success in evaluating equity options.  
 
References: 
1.  Black, F. si Scholes, M.,  “The pricing of options and corporate liabilities” – Journal of 
Political Economy, 81, 637-659, 1973 
2.  Crank, J.C., “Mathematics of Diffusion”, Oxford, 1989 
3.  Cox, J. si Rubinstein, M., “Options Markets”, Prentice Hall, 1985 
4.  Fama, E. “The Behavior of stock prices”, Journal of Business, 38, 34-105, 1965 
5.  Hull, J. “Options, Futures and Other Derivatives”, 6
th Ed. Prentice Hall, 2005 
6.  Merton, R.C.  “Options pricing when underlying stock returns are discontinuous”, Journal 
of Financial Economics, 3, 125-144 
7.  Neftici, S. “An Introduction to the Mathematics of Financial Derivatives”, Academic Press 
8.  Wilmott, P. (1994) “Discrete Charms”, Risk Magazine, 7, 48-51 
9.  Wilmott, P (1995) “Derivative – Inginerie financiara, Teorie si practica”, Ed. Economica, 
2002 
 
 
 
   

„BLACK-SCHOLES MODEL USED TO EVALUATE STOCKS OPTIONS”

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