This paper shows how can be estimated the value of an option if we assume the double- Heston model on a message-based architecture. For path trace simulation we will discretize continous model with an Euler division of time.Monte Carlo; algorithms; computational financial engineering; derivatives evaluation; Blackï¿½Scholesï¿½Merton model; Heston model; double-Heston model; generalized double-Heston model.

Bogdan Patrut

Tiberiu Socaciu

English

Research Papers in Economics

BRAND. Broad Research in Accounting, Negotiation, and Distribution 
ISSN 2067-8177, Volume 1, Issue 1, 2010 
 
 1  
Algorithm for Financial Derivatives Evaluation  
in Generalized Double-Heston Model 
 
Tiberiu Socaciu 
“Ştefan cel Mare” University of Suceava, Faculty of Economics, Romania, 
“Vasile Goldis” West University of Arad, Faculty of Informatics, Romania 
“Babes-Bolyai” University of Cluj-Napoca, Faculty of Economics, Romania 
socaciu@inf.ro 
 
Bogdan Pătruţ 
“Vasile Alecsandri” University of Bacău, Faculty of Sciences, Romania 
bogdan@edusoft.ro 
 
Abstract 
This paper shows how can be estimated the value of an option if we assume the double-
Heston model on a message-based architecture. For path trace simulation we will discretize 
continous model with an Euler division of time. 
Keywords: Monte Carlo; algorithms; computational financial engineering; derivatives 
evaluation; Black–Scholes–Merton model; Heston model; double-Heston model; 
generalized double-Heston model. 
 
1. BSM model, Heston model, Double-Heston model 
From physical models, the following situation has reached acceptance: a financial asset 
interest rate follows a normal law, where the mean is the drift rate and the deviation is the volatility. 
This leads to a model that is currently accepted in finance:, the model of geometric Brownian 
motion. This model (known as Black–Scholes–Merton model in finance and financial engineering, 
see [1]) is a stochastic differential equation (1):  
 
dS(t) = m S(t) dt + s S(t) dB(t),  (1) 
 
where: 
a) (S(t), t≥0) is a stochastic process for the value of stock;  
b) m is a static parameter for the drift rate of return;  
c) s
2 is a static parameter for the volatility of stock (s≥0);  
d) (B(t), t≥0) is a standard Wiener process.  
 
Another model is assumed by Heston (see [2]) and it consists from two stochastic 
differential equations. The Heston model corrects some inconsistency of the Black–Scholes–Merton 
model, for example:  
a) in reality, volatility is not a static parameter; it can be used as static value only on short periods 
(this value will obtain on calibration process, usual with a statistical estimator);  
b) on long periods, it is possible that interest rate series did not verify a normal law.  
The Heston model is described by the following coupled stochastic differential equations 
(2), (3):  
 
dS(t) = A(S(t), v(t), t) dt + B(S(t), v(t), t) dB1(t)      (2) 
   dv(t) = C(S(t), v(t), t) dt + D(S(t), v(t), t) dB2(t)     (3) 
 
where:  
a) (S(t), t≥0) is a stochastic process for value of stock;  
b) (v(t), t≥0) is a stochastic process for volatility of value of stock;  BRAND. Broad Research in Accounting, Negotiation, and Distribution 
ISSN 2067-8177, Volume 1, Issue 1, 2010 
 
 2  
c) A(S, v, t), B(S, v, t), C(S, v, t), D(S, v, t) are three parametric algebraic functions;  
d) (B1(t), t≥0) and (B2(t), t≥0) are two r-correlated standard Wiener processes, i.e. (4):  
 
dB1(t) dB2(t) = r dt    (4) 
 
For Wiener processes, more details can be found in [3]. For the basic Heston model we have 
(5):  
a) A = S (t) m      
b) B = S (t) v(t)           (5) 
c) C = K (θ – v(t))     
d) D = ξ v(t)       
 
where:  
a) m is a drift of rate;  
b) θ  is long run average price volatility; as t tends to infinity, the expected value of v(t) tends to θ;  
c) K is the rate at which v(t) reverts to θ;  
d) ξ is the volatility of the volatility; as the name suggests, this determines the variance of v(t).  
 
Note that for C = D = 0 we obtain a static volatility model (Black–Scholes–Merton) (6) 
 
dv(t) = 0.  
 
The Double-Heston model (see [4]) is described by the following coupled stochastic 
differential equations (7), (8), (9): 
 
   dS(t) = M(S(t), v1(t), v2(t), t) dt + S1(S(t), v1(t), t) dB1(t) + S2(S(t), v2(t), t) dB2(t)   (7) 
dv1(t) = C1(S(t), v1(t), t) dt + D1(S(t), v1(t), t) dB3(t)       (8) 
dv2(t) = C2(S(t), v2(t), t) dt + D2(S(t), v2(t), t) dB4(t)      (9) 
 
where: 
 
a) (S(t), t≥0) is a stochastic process for value of stock;  
b) (v1(t), t≥0) is a stochastic process for half-volatility of value of stock;  
c) (v2(t), t≥0) is a stochastic process for half-volatility of value of stock;  
d) M(S, v1, v2, t), S1(S, v1, t), S2(S, v2, t), C1(S, v1, t), D1(S, v1, t), C2(S, v2, t), D2(S, v2, t) are 
three/four parametric algebraic functions;  
e) (B1(t), t≥0) and (B3(t), t≥0) are two r1-correlated standard Wiener processes, i.e. (10):  
 
dB1(t) dB3(t) = r1 dt    (10) 
 
f) (B2(t), t≥0) and (B4(t), t≥0) are two r2-correlated standard Wiener processes, i.e. (11):  
 
dB2(t) dB4(t) = r2 dt    (11) 
 
g) (B1(t), t≥0) and (B2(t), t≥0) are two independent standard Wiener processes.  
 
For the basic Double-Heston model we have (12): 
 BRAND. Broad Research in Accounting, Negotiation, and Distribution 
ISSN 2067-8177, Volume 1, Issue 1, 2010 
 
 3  
 
a) M = S(t) m         
b) S1 = S(t) v1(t)      
c) S2 = S(t) v2(t)      
d) C1 = K1 (θ1-v1(t))        (12) 
e) C2 = K2 (θ2-v2(t))      
f) D1 = ξ1 v1(t)      
g) D2 = ξ2 v2(t)      
 
where: 
a) m is a drift of rate;  
b) θ1 is long run average price volatility; as t tends to infinity, the expected value of v1(t) tends to 
θ1;  
c) θ2 is long run average price volatility; as t tends to infinity, the expected value of v2(t) tends to 
θ2;  
d) K1 is the rate at which v1(t) reverts to θ1;  
e) K2 is the rate at which v2(t) reverts to θ2;  
f) ξ1 is the volatility of the volatility; as the name suggests, this determines the variance of v1(t); 
g) ξ2 is the volatility of the volatility; as the name suggests, this determines the variance of v2(t).  
 
Any financial derivative based on support with price S(t) at time t, with quotation at time t 
and a value S of support as V(S,v1, v2, t), where (12): 
 
+ + → × × × R T T T R V ] , 0 [ ] , 0 [ ] , 0 [ :  (12) 
 
and at maturity time T will generate an generate an payoff (13): 
 
payoff: R+ → R+ 
 
For example, European options CALL and PUT has payoff functions (14): 
 
        payoff (x) = max{0, x - E}      (14) 
    payoff  (x)  =  max{0,  E - x}   
 
where E is excercise price of option. 
 
2. Path trace simulation for option's pricing in generalized Double-Heston model 
First, we discretize continuous dimension of time. Let us denote (15): 
 
t[k] = t[0] + k∆, 0≤k≤N   (15) 
 
where: 
a) ∆ = (T-t[0])/N 
b) T is the maturity time of option; 
c) N is a number of time units (like days, hours, minutes, etc); note that sometimes is used 
transaction days - in this case, discretization hasn't a constant step. 
 BRAND. Broad Research in Accounting, Negotiation, and Distribution 
ISSN 2067-8177, Volume 1, Issue 1, 2010 
 
 4  
Because for a standard Wiener process (B(t), t≥0) we can obtain a standard normal random 
variable series (X[B(t)], t≥0) with (16): 
dB(t) = X(dt)½     (16) 
 
we can build a simulation step as (17): 
 
M ← M(S[k], v1[k], v2[k], t[k])      
S1 ← S1(S[k], v1[k], t[k])      
S2 ← S2(S[k], v2[k], t[k])             
C1 ← C1(S[k], v1[k], t[k])      
D1 ← D1(S[k], v1[k], t[k])      
C2 ← C2(S[k], v2[k], t[k])     (17) 
D2 ← D2(S[k], v2[k], t[k]) 
Δ + Δ + Δ + ← + 2 2 1 1 ] [ ] 1 [ X S X S M k S k S  
v1[k+1] ← v1[k] + C1 ∆ + D1  Δ 3 X  
 v2[k+1] ← v2[k] + C2 ∆ + D2  Δ 4 X  
 
where X1 and X3 are r1-correlated, X2 and X4 are r2-correlated. A simple method to generate two r 
correlated normal values is (18): 
 
X ←  NormRand()       
Z ← NormRand()          ( 1 8 )  
Y ← r X + 
2 1 r −  Z    
 
where NormRand is a function that produces independent real random numbers between 0 and 1, 
with normal distribution. 
A complete simulation for interval [t0, T] in N steps with evaluation of payoff is function 
Simulation, described below: 
 
FUNCTION Simulation() 
S ← S0 
v1 ← v10 
v2 ← v20 
t ← t0 
∆ ← (T – t0) / N 
FOR k ← 1, N 
t ← t + ∆ 
X1 ← NormRand() 
X2 ← NormRand() 
Y3 ← NormRand() 
Y4 ← NormRand() 
X3 ← r1 X1 + 
2
1 1 r − Y3 
X4 ← r2 X2 + 
2
2 1 r − Y4 
M ← M(S, v1, v2, t) 
S1 ← S1(S, v1, t) BRAND. Broad Research in Accounting, Negotiation, and Distribution 
ISSN 2067-8177, Volume 1, Issue 1, 2010 
 
 5  
S2 ← S2(S, v2, t) 
C1 ← C1(S, v1, t) 
D1 ← D1(S, v1, t) 
C2 ← C2(S, v2, t) 
D2 ← D2(S, v2, t) 
SS ← S + M ∆ + S1 X1  Δ  + S2 X2  Δ  
vv1 ← v1 + C1 ∆ + D1 X3  Δ  
vv2 ← v2 + C2 ∆ + D2 X4  Δ  
S ← SS 
v1 ← vv1 
v2 ← vv2 
END FOR 
RETURN payoff(S) 
END FUNCTION 
 
3. Monte Carlo method for Option's Pricing in Double-Heston Model 
Because for a level of acceptance α, where 0< α <1, a trust interval for E[S(T)] is [s - a, s + 
a], with (19): 
 
s = [Simulation() + Simulation() + ... + Simulation()] / N   (19) 
 
and (20): 
a = F(α/2)σ/M½       (20) 
 
where: 
a) N is number of simulations; 
b) F is the inverse function for CDF (cumulative distribution function) of standard normal 
distribution; it means that (21) or (22): 
 
Prob(s - a < E[payoff(S(T))] < s + a) = 1 – α (21) 
Prob(E[payoff(S(T))] = s + O(M½)) = 1 – α.   (22) 
 
where big–O notation is a Buchmann–Landau symbol (see [5]). Algorithm for evaluation of 
E[payoff(S(T))] is described below, in Serial_Simulation function: 
 
FUNCTION Serial_Simulation() 
LOCAL x 
x ← 0 
FOR i ← 1, M  
x ← x + Simulation()  
ENDFOR  
RETURN x/M 
END FUNCTION 
 
4. Further works 
Like in [6] we will to parallelize Monte Carlo algorithm for generalized Double-Heston 
model. Also, we want to build a Merton-Garman like PDE for option pricing like in [1] for 
generalized Double-Heston model, and build some parallelization of PDE numerical solving, like in 
[4]. BRAND. Broad Research in Accounting, Negotiation, and Distribution 
ISSN 2067-8177, Volume 1, Issue 1, 2010 
 
 6  
 
5. Acknowledgment 
This work was supported by the grant ID 2585, sponsored by NURC – Romanian National 
University Research Council (CNCSIS - Consiliul Naţional al Cercetării Ştiinţifice din 
învăţământul Superior, see [7]). 
 
6. References 
[1] Fischer Black, Myron Scholes, The Pricing of Options and Corporate Liabilities, in Journal of 
Political Economy, 1973, 81 (3), pp. 637–654. 
[2] Steven L. Heston, A closed–form solution for options with stochastic volatility with applications 
to bond and currency options, in The Review of Financial Studies, 1993, Volume 6, number 2, pp. 
327–343.  
[3] Daniel Revuz, Marc Yor, Continuous martingales and Brownian motion, second edition, 
Springer–Verlag 1994.  
[4] Peter F. Christoffersen, Steven L. Heston, Kris Jacobs, The Shape and Term Structure of the 
Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well, preprint SSRN, 
20 febr. 2009, online at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=961037, last access: 7 
mar. 2010. 
[5] Eric W. Weisstein, Landau Symbols, in MathWorld––A Wolfram Web Resource, online at 
http://mathworld.wolfram.com/LandauSymbols.html, last access: 13 jun. 2009. 
[6] Tiberiu Socaciu, Ilie Parpucea, Bazil Parv, Maria Parv, Parallel Algorithms for Financial 
Derivatives Evaluation in Generalized Heston Model, in Scientific Studies and Research 
Series Mathematics and Informatics, Vol. 19 (2009), No. 2, 479 – 492 
[7] http://www.cncsis.ro/PNCDI%20II/Idei/2008/Exploratorie/Prop_finantare/Proiecte_ 
propuse_spre_finantare_domeniul_13_3B.html, last access 1 august 2009.  
 

A closed–form solution for options with stochastic volatility with applications to bond and currency options, in The Review of Financial Studies,

Ilie Parpucea, Bazil Parv, Maria Parv, Parallel Algorithms for Financial Derivatives Evaluation in Generalized Heston Model,

Landau Symbols, in MathWorld––A Wolfram Web Resource, online at http://mathworld.wolfram.com/LandauSymbols.html, last access: 13

Marc Yor, Continuous martingales and Brownian motion, second edition,

Myron Scholes, The Pricing of Options and Corporate Liabilities,

The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well, preprint SSRN, 20 febr. 2009, online at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=961037, last access: 7

Algorithm for Financial Derivatives Evaluation in Generalized Double-Heston Model

http://www.edusoft.ro/brand/RePEc/bra/journl/brand_1_socaciu_heston_ok.pdf

Algorithm for Financial Derivatives Evaluation in Generalized Double-Heston Model

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics