In our paper we build a reccurence from generalized Garman equation and discretization of 3-dimensional domain. From reccurence we build an algorithm for computing values of an option based on time, momentan volatility of support and value of support on afinancial derivatives, Black-Scholes PDE, Garman PDE, reccurence, algorithm

Danubianu Mirela

Maxim Ioan

Naaji Antoanela

Socaciu Tiberiu

English

Research Papers in Economics

1044 
ALGORITHM FOR GENERALIZED GARMAN EQUATION IN OPTION PRICING OF A FINANCIAL 
DERIVATIVES WITH STOCHASTIC VOLATILITY MODELS 
 
Socaciu Tiberiu 
University Stefan cel Mare of Suceava Faculty of Economics and Public Administration, Computer Science Chair 
Suceava, str. Universitatii nr. 13, Corp H 
and 
West University Vasile Goldis of Arad Computer Science Faculty, Computer Science Chair Arad, Bd. Revolutiei nr. 
81 e-mail: socaciu@seap.usv.ro tel: 0744 820 827 
Danubianu Mirela 
University Stefan cel Mare of Suceava Faculty of Electrical Engineer and Computer Science, Computers Chair 
Suceava, str. Universitatii nr. 13, Corp C e-mail: mdanub@eed.usv.ro tel: 0744 820 827 
Maxim Ioan 
University Stefan cel Mare of Suceava Teachers’ Training Department Suceava, str. Universitatii nr. 13, Corp A e-
mail: maximioan@yahoo.com tel: 0744 820 827 
Naaji Antoanela 
West University Vasile Goldis of Arad Computer Science Faculty, Computer Science Chair Arad, Bd. Revolutiei nr. 
81 e-mail: anaaji@uvvg.ro tel: 0744 820 827 
 
In our paper we build a reccurence from generalized Garman equation and discretization of 3-dimensional domain. From 
reccurence we build an algorithm for computing values of an option based on time, momentan volatility of support and value 
of support on a spot market of an exchange. 
 
Keywords: financial derivatives, Black-Scholes PDE, Garman PDE, reccurence, algorithm 
 
JEL classification: G13, C69 
 
1. Garman functions attached on generalized model with stochastic volatility 
In Black-Scholes model
623 we supose that volatility is a constant value. Unfortunately, volatility is not constant, and 
more, can not be predicted for periods greater than several months or even can not be observed directly. For these 
reasons, it seems natural to present itself as the volatility of a random variable that satisfies a stochastic process. 
Assume that the variation rate asset support satisfy a stochastic differential equation (SDE) of the form: 
 
dS(t) = A(t,S(t),V(t))dt+B(t,S(t),V(t))dW1(t)      (1) 
 
but in addition we assume that the square of volatility (v), follows the stochastic process: 
 
dv(t) = C(t,S(t),V(t))dt+D(t,S(t),V(t))dW2(t),      (2) 
 
where the two Wiener processes W1(t)t≥0 şi W2(t)t≥0 are correlated: 
 
    (dW1(t))(dW2(t)) = ρdt. 
 
In this case the value of an option is a function of three variables, V (t, S, v) and because we have two sources of 
risk need to build a portfolio of coverage to include the option in question and two other components. Assume the 
existence of a derivative financial unca on the same support, so that the value of this option is a function of three 
variables, V1 (t, s, v) to check with a law counterparts of V (t, S, v). After applying extended lemma of Ito
624, we 
obtain the existance of a special value β named as the market price of volatility risk
625, where: 
 
β = (–rV+VSrS+Vt+AVS+CVv+½B
2VSS+½D
2Vvv+ρBDVSv–AVS)/(DVv)     (3) 
 
If β is known, (3) rewrite as: 
 
Vt+rSVS+(C–Dβ)Vv+½B
2VSS+½D
2Vvv+ρBDVSv–rV = 0   (4) 
 
                                                       
623 see [Black1973]. 
624 see [Socaciu2009a]. 
625 see [Moodley2005], p. 8. 1045 
is known as Garman equation. Note that (4) is a partial differential equation (PDE), like Black-Scholes equation, 
obtained from SDE of model. 
 
2. Particular cases of Garman functions 
 
1. Black-Scholes; in conditions: 
 
B
2 = vS
2 
C = D = 0 ó dv = 0 (constant volatility) 
v = σ
2 
 
we re-obtain Black-Scholes equation
626: 
 
    Vt+rSVS+½σ
2S
2VSS–rV = 0 
 
2. Heston; in conditions: 
 
A = µS 
B = v
½S 
C = k(θ–v) 
D = ξv
½  
    β = λv
½  
 
we re-obtain
627: 
 
Vt+rSVS+(k(θ–v)–ξλv)Vv+½vS
2VSS+½ξ
2vVvv+ρSξvVSv–rV = 0  (5) 
 
3. other cases for particular models can be obtained via particular A, B, C, D and β. An incomplete list of models
628 
can be retrieved from [Sundaresan2000]
629. 
 
3. Garman equation on domains’ frontier 
 
Equation (5) has on frontiers
630:  
 
V(t, 0, v) = 0, with 0 ≤ t ≤ T, all v.        (6) 
 
and:  
 
V(T, S, v) = payoff(S), with 0 ≤ S, all v.       (7) 
 
We need add a new condition on v axis. Simplest condition is based on zero-volatility. In this case, (2) is vanished, 
and (1) will be rewrite as a regullar BS equation with a specific solutions like
631: 
 
V(t, S, 0) = BS (t, S)            (8) 
 
4. Domain’s discretization and reccurence obtained from Garman equation 
 
We will apply a Finite Element Method (FEM
632) for resolve (5). We know that FEMs are main method for resolve 
PDE
633. First step is to reduce initial domain D
i:  
 
D
i = [0,T] x R
+ x R
+  
                                                       
626 see [Black1973]. 
627 see [Lazăr2006], [Mikhailov2003]. 
628 with specification of A and B. 
629 an other incomplete list can be retrieved in [Lazăr2006]). 
630 like in [Socaciu2009b]. 
631 Numerical solutions will be presented in [Socaciu2009a], [Socaciu2009b] and [Socaciu2009c]. 
632 see [Gârbea1990]. 
633 see [Coman1994]. 1046 
 
to a limited domain D:  
 
D = [0,T] x [0,MaxS] x [0,MaxV], 
 
where MaxS is a big number that cover all values of potential quotations, and MaxV is an big number that cover all 
potential volatilities. After discretization with  t,  S and  v steps we will obtain a grid of points. We denote:  
 
V[i,j,k] = V(i( t),j( S),k( v)), with 0 ≤ i ≤ M, 0 ≤ j ≤ N, 0 ≤ k ≤ P 
 
where number of points on axis:  
 
M := MaxS/( S) 
N := T/( t) 
P := MaxV/( v) 
 
than (4) will be rewrite as: 
 
(V[i,j,k]-V[i-1,j,k])/( t)+rj( S)(V[i,j+1,k]-V[i,j-1,k])/(2( S)) 
+(C–Dβ) (V[i,j,k+1]-V[i,j,k-1])/(2( v)) 
+½B
2(V[i,j+1,k]-2V[i,j,k]+V[i,j-1,k])/( S)
2 
+½D
2(V[i,j,k+1]-2V[i,j,k]+V[i,j,k-1])/( v)
2 
+ρBD(V[i,j+1,k+1]-V[i,j+1,k-1]-V[i,j-1,k+1]+V[i,j+1,k+1])/(( S)( v)) 
–rV[i,j,k] = 0 
 
That can be rewrite as: 
 
a(V[i,j,k]-V[i-1,j,k])+b(V[i,j+1,k]-V[i,j-1,k])+c(V[i,j,k+1]-V[i,j,k-1]) 
+d(V[i,j+1,k]-2V[i,j,k]+V[i,j-1,k])+e(V[i,j,k+1]-2V[i,j,k]+V[i,j,k-1])
 
+f(V[i,j+1,k+1]-V[i,j+1,k-1]-V[i,j-1,k+1]+V[i,j+1,k+1])–gV[i,j,k] = 0 
                    (9) 
 
where: 
 
    a = ( t)( S)
2( v)
2 
    b = ½rj( t)( S)
2( v)
2 
    c = ½(C–Dβ)( t)( S)
2( v) 
    d = ½B
2( t)( v)
2 
    e = ½B
2( t)( v)
2 
    f = ρBD( t)( v)( S) 
    g = r( t)( v)
2( S)
2 
 
But (9) can be rewrite in explicit form: 
 
V[i–1,j,k] = pV[i,j,k]+qV[i,j+1,k]+rV[i,j–1,k]+sV[i,j,k+1]+tV[i,j,k–1] 
+uV[i,j+1,k+1]–uV[i,j+1,k–1]–uV[i,j–1,k+1]+uV[i,j+1,k+1]  (10) 
 
where: 
 
    p = (a–2d–2e–g)/a 
    q = (b+d)/a 
    r = (d–b)/a 
    s = (c+e) 
    t = (e–c) 
    u = f/a 
 
Note that coefficients p-u are based on t, S and v, and we must rewrite all of them in spirit of (i,j,k). 
 
5. Numerical algorithm based for Garman equation based on triple reccurence 
 1047 
Numerical algorithm is based on (10) reccurence with supplementally (6)-(8) frontier condition is: 
 
For i:=MaxM downto 0 do 
  For j:=0 to MaxN+MaxM do 
    For k:=0 to MaxP+MaxM do 
      If i=MaxM then 
        V[i,j,k]:=payoff(j( S)) 
      ElseIf j=0 then 
        V[i,j,k]:=0 
      ElseIf k=0 then 
        V[i,j,k]:=BS(i,j) 
      Else 
        V[i,j,k]:=pV[i+1,j,k]+qV[i+1,j+1,k] 
        +rV[i+1,j–1,k]+sV[i+1,j,k+1]+tV[i+1,j,k–1] 
        +uV[i+1,j+1,k+1]–uV[i+1,j+1,k–1]–uV[i+1,j–1,k+1] 
        +uV[i+1,j+1,k+1] 
      End If 
    End For 
  End For 
End For 
 
where BS is a function that call a numerical solution of Black-Scholes like equation on zero-volatility condition. 
Note that iteration will be computed for j to MaxN+MaxM and for k to MaxP+MaxM because we have some “out 
of range” problems if we compute V[0,MaxN,MaxP] and cheap resolvation is to compute a supradomain of D. 
 
6. Further works 
We work on obtaining parallel version of this algorithm with slicing of D like in [Socaciu2009a], [Socaciu2009b] 
and [Socaciu2009c]. We try to obtain similar formulae for a multi-SDE model, like Chen
634. We want to start a 
parallel project for obtain on symbolic way an extendended Ito type lemma
635 and from this lemma to build an 
Garman-type equation. 
 
Refferences 
1. [Black1973] Fischer Black, Myron Scholes, The Pricing of Options and Corporate Liabilities, in Journal of 
Political Economy, 81 (3), pp. 637–654. 
2. [Chen1996] Lin Chen, Stochastic Mean and Stochastic Volatility -- A Three-Factor Model of the Term Structure 
of Interest Rates and Its Application to the Pricing of Interest Rate Derivatives, Blackwell Publishers, 1996. 
3. [Coman1994] Gheorghe Coman, Analiza numerică, Ed. Libris, Cluj, 1994. 
4. [Gârbea1990] Dan Gârbea, Analiza cu elemente finite, Editura Tehnica, Bucuresti, 1990, 248 pagini. 
5. [Iftimie2008] Bogdan Iftimie, Financial models in continous time, Editura MatrixRom, Bucuresti, 2008, ISBN 
978-973-755-350-8, 126 pagini. 
6. [Lazăr2006] Lazăr Vasile Lucian, Modele de evaluare a op iunilor, in Studia universitatis Stiinte Economice, Nr. 
(5) an 2006, volumul II, online at  
7. http://www.uvvg.ro/studia/economice/index.php?categoryid=10&p2_articleid=74&p142_dis=3& 
p142_template=Default. 
8. [Mikhailov2003] Mikhailov, S., Nogel, U., Heston’s stochastic volatility model. Implementation, calibration and 
some extensions, in Wilmott, 2003, 74–94, online at  
9. http://www.wilmott.com/pdfs/051111_mikh.pdf. 
10. [Moodley2005] Nimalin Moodley, The Heston Model: A Practical Approach with Matlab Code, University of 
the  Witwatersrand,  Johannesburg,  South  Africa,  online  at 
http://www.math.nyu.edu/~atm262/fall06/compmethods/a1/nimalinmoodley.pdf. 
10.  [Sundaresan2000]  Suresh  M.  Sundaresan,  Continuous-Time  Methods  in  Finance:  A  Review  and  an 
Assessment,  in  The  Journal  of  Finance,  Volume  55,  Issue  4,  august  2000,  pp.  1569-(5)22,  online  at 
http://www.kelley.iu.edu/clundbla/sund.pdf.  
11. [Socaciu2009a] Tiberiu Socaciu, Paralelizarea ecua iilor de tip Black-Scholes si Garman (Parallelization of 
Black-Scholes and Garman equations, in Romanian), Editura InfoData, will be appear. 
12. [Socaciu2009b] Tiberiu Socaciu, Ioan Maxim, Algorithm for a Message-based Architecture for Black-Scholes 
Equation, in IECS 2009, (5)th International Economic Conference "Industrial Revolutions, from the Globalization 
                                                       
634 See [Socaciu2009a], [Chen1996]. 
635 See [Socaciu2009a], [Iftimie2008]. 1048 
and Post-Globalization Perspective", 7-8 mai 2009, Sibiu, Romania, section 5, 2009, pp. 236-241, ISBN 978-973-
739-775-1. 
13. [Socaciu2009c] Tiberiu Socaciu, O rezolvare paralela a ecuatiei Black-Scholes, at Zilele Academice Aradene, 
8-10 mai 2009. 
   

ALGORITHM FOR GENERALIZED GARMAN EQUATION IN OPTION PRICING OF A FINANCIAL DERIVATIVES WITH STOCHASTIC VOLATILITY MODELS

http://steconomice.uoradea.ro/anale/volume/2009/v4-management-and-marketing/214.pdf

ALGORITHM FOR GENERALIZED GARMAN EQUATION IN OPTION PRICING OF A FINANCIAL DERIVATIVES WITH STOCHASTIC VOLATILITY MODELS

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