A Positivity-Preserving Numerical Scheme for Nonlinear Option Pricing Models
A positivity-preserving numerical method for nonlinear Black-Scholes models is developed in this paper. The numerical method is based on a nonstandard approximation of the second partial derivative. The scheme is not only unconditionally stable and positive, but also allows us to solve the discrete equation explicitly. Monotone properties are studied in order to avoid unwanted oscillations of the numerical solution. The numerical results for European put option and European butterfly spread are compared to the standard finite difference scheme. It turns out that the proposed scheme is efficient and reliable.
It is widely recognized that the value of a European option can be obtained by solving the linear Black-Scholes equation under quite restrictive assumptions (such as liquid, frictionless, and complete markets) [1, 2]. However, these restrictive assumptions are never fulfiled in reality. In order to conform the actual situation, many modified Black-Scholes models have been proposed in recent years, such as transaction costs (Leland , Palmer , Hoggard et al. , Barles and Soner , and Jandačka and Ševčovič ), illiquid market (Frey and Patie , Sircar and Papanicolaou , Liu and Yong , etc.), and volatility uncertainty (Avellaneda et al. ). These models result in quasilinear or fully nonlinear Black-Scholes equations.
In this paper, we are interested in the option pricing model with transaction costs proposed by Barles and Soner  that are motivated by Hodges and Neuberger . In their model, the value of the option satisfies the following partial differential equation: with the nonlinear volatility that reads and the terminal condition where is the price of the underlying asset, is the maturity date, is the risk-free interest rate, is the asset volatility, and , with the proportional transaction cost , the risk aversion factor , and the number of options to be sold. is the solution of the following ordinary differential equation:
The existence and uniqueness of the solution of (1.1)–(1.3) have been shown by the theory of stochastic optimal control in . However, analytical solutions cannot be found because of fully nonlinear properties of (1.1); thus, we need to compute the option values numerically.
There have been rich achievements for the numerical method of linear Black-Scholes equations (e.g., see [13–17]). As for the nonlinear situation, only a few results can be found. There is a stable numerical scheme developed in  (see also  for application to a general class of nonlinear Black-Scholes equations) for the so-called gamma equation (a quasilinear parabolic equation for the ). Recently Kútik and Mikula  did some progress in showing its stability and accuracy for nonlinear Black-Scholes equation. Company et al. [20–22] construct explicit finite difference schemes for (1.1)–(1.3), and consistency and stability are studied. However, they have the disadvantage that strictly restrictive conditions on the discretization parameters are needed to guarantee stability and positivity. The implicit schemes do not have this disadvantage, but they are quite time-consuming. Yousuf et al.  develop a new second-order exponential time differencing (ETD) scheme to avoid unwanted oscillations near the non-smooth nodes for the Hoggard-Whalley-Wilmott (HWW) model  based on the Cox and Matthews approach  and partial fraction version of the matrix exponential, but the theoretical analysis of stability and convergence are not studied. Some authors (for instance, Düring et al.  for European options, Dremkova and Ehrhardt  for American options) construct high-order compact difference schemes with frozen values of the nonlinear coefficient of the nonlinear Black-Scholes equation to make the scheme linear and show that the resulted linearized problem is stable.
On the other hand, since the value of option is nonnegative, it is very important to make numerical schemes preserve the positivity of solution. Several authors have developed some schemes that guarantee the positivity of solutions for ordinary differential equations [27, 28] and parabolic equations . In , Chen-Charpentier and Kojouharov propose an unconditionally positivity-preserving scheme for linear advection-diffusion reaction equations. They construct a nontraditional discretization of the advection and diffusion terms by the approximation of the spatial derivatives using values at different time levels. Motivated by this work, we will develop the method to a nonlinear Black-Scholes equation, and some properties (such as stability, monotonicity, and consistency,) of numerical scheme are studied in this paper. The new numerical method unconditionally preserves the positivity of the solutions, the stability, and monotonicity of the scheme. In addition, the designed numerical approximations allow us to solve the discrete equation explicitly, which reduces the time of calculation and increases the efficiency of the methods.
The rest of the paper is organized as follows. In the next section the original problem (1.1)–(1.3) is transformed into a nonlinear diffusion problem by an appropriate change of variables and some properties of the function are given. In Section 3, the discretization method is constructed. In Section 4, we prove the boundedness of coefficients, positivity, and monotonicity of the numerical scheme. Stability and consistency are studied in Section 5. In Section 6, numerical experiments for European put option and a European butterfly spread are presented to support these theoretical results. Finally, some conclusions are drawn in Section 7.
2. The Transformed Problem
For the convenience in the numerical processing and the study of the numerical analysis, we are going to transform the problem (1.1)–(1.3) into a nonlinear diffusion equation. Taking the variable transformation the original problem (1.1)–(1.3) is transformed into with the initial condition and the boundary condition
The following two lemmas give the properties of the function appearing in (1.2), which will play an important role in the numerical analysis and numerical calculation.
Lemma 2.1 (see ). The solution of ordinary differential equation (1.4) exists and is unique, and it satisfies,(i) is an increasing function mapping the real line onto the interval .(ii) is implicitly defined by (iii)if , then the function is bounded and satisfies where
Lemma 2.2 (see ). Let , then is continuously differentiable at and satisfies where and are given by (2.7), and
3. The Unconditionally Positivity-Preserving Scheme
Since the value of an option is nonnegative, it is important that numerical scheme is positivity preserving.
We see that the problem (2.2) is described in an infinite domain , which makes it difficult to construct scheme effectively. Let us consider the truncated numerical domain and discretize it in the following form. We introduce a grid of mesh points , where , and the spatial step size given by , and the time step size is . Let us denote the approximation of by and define the following finite difference approximations of derivatives:
Clearly, the approximation used to calculate the second partial derivative of with respect to is nonstandard. From (2.2), we can obtain the positivity-preserving finite-difference numerical scheme as follows:
Scheme (3.2) is equivalent to where
Remark 3.1. From property (i) of Lemma 2.1, takes values in the interval , so the coefficients are nonnegative, that is, for any .
Obviously, numerical scheme (3.3) is unconditionally positive for a nonnegative payoff . However, we cannot obtain the numerical solution explicitly since the nonlinear term involves the value at the time level , which makes it quite difficult to prove the stability of the scheme presented previously. In fact, the numerical scheme (3.3) can only be solved by a nonlinear iteration in each time step which is quite time-consuming.
In order to obtain an efficient scheme, we correct the approximation of the nonlinear coefficients in (2.2) by using the standard second-order central difference. Thus the corrected numerical scheme is as follows: where
Since the calculation of (3.5) for and requires us to know the fictitious values and , we obtain them using the linear extrapolation as follows:
Thus the values of boundary points turn out
The left boundary condition is not needed and in fact must not be prescribed in the case of a parabolic equation with degenerating diffusion term at . This is known in the literature as the Fichera condition  (see  for application in Black-Scholes equations, Chapter 8, (8.25)). The Fichera condition is just a comparison of the speed of degeneration versus. speed of advection at the boundary . Fortunately, the left boundary condition in (3.8) is a consequence of (3.5) with . The only boundary condition that can be prescribed is the right boundary condition at . For the convenience in the numerical calculation, let us denote the vectors , then numerical scheme (3.5), (3.8) can be written in matrix form where
In order to study the related properties of numerical scheme (such as monotonicity and stability), we need to know the behaviour of appearing in the nonlinear team .
Lemma 3.2. With the previous notation, appearing in the nonlinear team satisfies the scheme
Proof. From (3.6) and (3.7) we can know for and that
Putting (3.5) into the expression (3.6) of , and after a simple calculation we can obtain (3.12).
On the other hand, the most common option price sensitivities are the first and second derivatives with respect to the price of the underlying asset, that is, “delta” and “gamma”, respectively. These are important features in risk management and are challenging to compute numerically. From the transformation (2.1) we can obtain the approximations of Greeks of the option as follows:
4. Properties of the Numerical Scheme
4.1. Boundedness of the Coefficients
For the sake of convenience, we introduce the following definition.
Definition 4.1 (see ). If is a vector in , then its 1-norm is denoted by , and maximum-norm is denoted by .
The following theorem shows that the nonlinear team appearing in (3.5) is bounded.
Theorem 4.2. Let , then the following properties hold true.(i) is nonincreasing.(ii)The nonlinear team appearing in (3.5) satisfies where with , , and given by (2.7).
Proof. Property (i) is proved using the induction principle over index .
For , from Lemma 3.2 and by Remark 4.7, it follows that Taking into account (3.13) and (4.3), we have Thus property (i) is proved for .
Now, let us assume that property (i) holds true up , that is, For , from Lemma 3.2, it follows that Taking into account , we have Hence property (i) is proved completely.
On the other hand, from (3.6) and the monotonic increasing property of (see Lemma 2.1), it follows that
Since and from the property (iii) and property (i) of Lemma 2.2, we have Thus the proof of theorem is complete.
Since the value of option is nonnegative, a nice property of the numerical scheme for the pricing equation is positivity-preserving.
Clearly, all the coefficients of the numerical scheme (3.5) are nonnegative (see Theorem 4.2). Hence, for a nonnegative payoff , the following result has been established.
Proposition 4.3. The numerical scheme (3.5), (3.8) is unconditionally positive.
Remark 4.4. is also unconditionally positive, that is, if for , then for .
For the sake of convenience in the presentation, we introduce the following definition of a monotonicity preserving numerical scheme.
Definition 4.5 (see ). In numerical scheme , we say that it is monotonicity-preserving. If each time that or for all , then it occurs that or for all .
The next result shows the monotonicity of the numerical scheme.
Theorem 4.6. The numerical scheme (3.5), (3.8) is unconditionally monotonicity-preserving with .
Proof. Let us write
Assuming that for , then from (3.5), it follows that
that substituting by one gets
From (4.10), (4.11), and (4.13), it follows that
Taking into account (4.1) and (4.14) for and , we have
The monotonicity of the scheme in the internal mesh points has been proved.
In an analogous way, we can verify that
Similarly, we can prove that if , then for .
Thus the proof of theorem is complete.
Remark 4.7. If the payoff is nondecreasing with (e.g., ), then for a fixed with .
5. Stability and Consistency
Theorem 5.1. The difference scheme (3.5), (3.8) is unconditionally -stable.
Proof. From (3.5), let us write Since all the coefficients of (5.1) are nonnegative, then using triangle inequality, it follows that so that According to the definition of the maximum norm, it follows that Thus the result is established.
The proposed new difference scheme (3.5) is explicit, unconditionally positive, and unconditionally stable; however, it is not unconditionally consistent. There are extra truncation error terms since the approximations to second derivatives with respect to are at different time levels.
The following theorem gives the consistency condition of the difference scheme (3.5).
Theorem 5.2. With the previous notation, suppose that the exact solution of (1.1)–(1.3) satisfies . Then the local truncation error is given by
Proof. Let us write the scheme (3.5) in the form Using Taylor's expansion about and , it follows that where From (5.5)–(5.8), it follows that the local truncation error takes the form Let us introduce the notation Using function introduced in Lemma 2.2, it follows that From Lemma 2.2, (5.11), and , it is easy to know that Thus the result has been proved.
From Theorem 5.2, we can see that the meshes should satisfy as and go to zero in order to ensure the consistency. Therefore, the key to the convergence of the scheme is the consistency rather than the stability. In actual calculation, we can choose the time step depending on the spatial size so that inconsistent terms go to zero.
6. Numerical Experiments
In this section, we implement the positivity-preserving scheme (3.5), (3.8) on European put option and European butterfly spread. We analyse the effectiveness of the method and compare it with the numerical scheme (SFD1) given in  and a standard forward Euler finite difference scheme (SFD2) which is analysed in . The function is calculated with (2.5) using the “fsolve” function in Matlab. Both the experiments are performed for large values of to visualize the sensitivity of the methods towards high volatility.
6.1. European Put Option
A European put option is a contract where the owner of the option has the right to sell an underlying asset for a fixed amount, known as the strike price , at the expiry date . The payoff function is given by We choose the parameters as
Equations (1.1)–(1.3) give analytical solution for only (see ). Figure 1 (the left one: , the right one: ) gives the option value using scheme (3.5) and schemes (SFD1 and SFD2) in [20, 21], which shows that our scheme stable, monotonous, and is able to produce solution that is close to the exact solution, but numerical solution of scheme (6.1) appears as spurious oscillation for and .
Moreover, Figure 2 presents the related hedging parameters of European put option using the two schemes. We can see that our proposed scheme produces smoother solutions than standard scheme for the delta and gamma. In addition, the gamma is positivity preserving and is maximal as it closes the strike price .
(a) Standard scheme for delta
(b) New scheme for delta
(c) Standard scheme for gamma
(d) New scheme for gamma
Next we consider the influence of transaction costs (parameter ) on the value of European put option (Figure 3). The left one shows the change of the option value with the parameter , and the right one presents an evolution profile of the difference between our proposed scheme with transaction costs and without transaction costs. We can see that the difference is not symmetric, but decreases towards the expiry date, and is maximal close to the strike price , where the nonlinear value is significantly higher than the linear value.
(a) Linear and nonlinear cases
|(b) Nonlinear-linear ()|
6.2. European Butterfly Spread
A butterfly spread is a combination of three-call options with three-strike prices, in which one contract is purchased with two outside strike prices and two contracts are sold at the middle strike price. The payoff function is given by where , and are the strike prices that satisfy and .
We choose the following parameters: the time step , and the spatial step .
Figure 4 shows the different solutions of two schemes at for . We can see that our proposed scheme is smooth and stable at the same step sizes. Moreover, Figure 5 shows the Greeks of the numerical solution calculated with scheme (3.14), which is different from the vanilla option. The standard scheme (the left one) is unstable and produces unwanted oscillations which are not present while using our proposed scheme (the right one).
(a) Standard scheme for delta
(b) New scheme for delta
(c) Standard scheme for gamma
(d) New scheme for gamma
The last two figures show the value of European butterfly spread with different transaction costs (parameter ). It is evident from Figure 6 that butterfly spread becomes more expensive in the presence of transaction cost and the difference is maximal as it closes the strike price , which is similar with the vanilla option.
(a) Linear and nonlinear cases
|(b) Nonlinear-linear ()|
7. Discussions and Conclusions
In this paper, we have extended the numerical method in  to nonlinear situation and presented the numerical scheme for a nonlinear Black-Scholes equation in the presence of transaction costs. The numerical method is based on a nonstandard approximation of the second partial derivative by , and the nonlinear team is treated explicitly, which guarantees to solve the original problem without iteration. The scheme is unconditionally positive and stable, but it is conditionally consistent. In fact, as , the scheme effectively solves the nonlinear parabolic equation where It means that the new scheme converges to a solution of (2.2) if . Otherwise, (if is fixed) it converges to a solution of (2.2) in a different time scale. In fact, it can be seen from Theorem 5.2, where the truncation error really depends on the ratio , which is also the reason that we consider the smaller ratio in the experiment. The numerical results show that our method produces better numerical solutions than the schemes in [20, 21] with the same step sizes. In the future work, we will extend the method to the problem of American option pricing.
The authors would like to thank the anonymous referees for several suggestions for the improvement of this paper. This work is supported by the Fundamental Research Funds for the Central Universities of China (Grant no. JGK101677).
F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, pp. 637–659, 1973.View at: Google Scholar
R. C. Merton, “Theory of rational option pricing,” The Rand Journal of Economics, vol. 4, pp. 141–183, 1973.View at: Google Scholar
H. E. Leland, “Option pricing and replication with transaction costs,” Journal of Finance, vol. 40, pp. 1283–1301, 1985.View at: Google Scholar
K. Palmer, “A note on the Boyle—vorst discrete-time option pricing model with transactions costs,” Mathematical Finance, vol. 11, no. 3, pp. 357–363, 2001.View at: Google Scholar
T. Hoggard, A. E. Whalley, and P. Wilmott, “Hedging option portfolios in the presence of transaction costs,” Advanced Futures and Options Research, vol. 7, pp. 21–35, 1994.View at: Google Scholar
G. Barles and H. M. Soner, “Option pricing with transaction costs and a nonlinear Black-Scholes equation,” Finance and Stochastics, vol. 2, no. 4, pp. 369–397, 1998.View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Jandačka and D. Ševčovič, “On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile,” Journal of Applied Mathematics, no. 3, pp. 235–258, 2005.View at: Publisher Site | Google Scholar | Zentralblatt MATH
R. Frey and P. Patie, “Risk management for derivatives in illiquid markets: a simulation study,” in Advances in Finance and Stochastics, pp. 137–159, Springer, Berlin, Germany, 2002.View at: Google Scholar | Zentralblatt MATH
K. R. Sircar and G. Papanicolaou, “General Black-Scholes models accounting for increased market volatility from hedging strategies,” Applied Mathematical Finance, vol. 5, pp. 45–82, 1998.View at: Google Scholar
H. Liu and J. Yong, “Option pricing with an illiquid underlying asset market,” Journal of Economic Dynamics & Control, vol. 29, no. 12, pp. 2125–2156, 2005.View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Avellaneda, A. Levy, and A. Paras, “Pricing and hedging derivative securities in markets with uncertain volatilities,” Applied Mathematical Finance, vol. 2, pp. 73–88, 1995.View at: Google Scholar
S. D. Hodges and A. Neuberger, “Optimal replication of contingent claims under transaction costs,” Review of Future Markets, vol. 8, pp. 222–239, 1989.View at: Google Scholar
P. P. Boyle and Y. Tian, “An explicit finite difference approach to the pricing of barrier options,” Applied Mathematical Finance, vol. 5, pp. 19–43, 1998.View at: Google Scholar
R. Zvan, P. A. Forsyth, and K. R. Vetzal, A General Finite Element Approach for PDE Option Pricing Models, University of Waterloo, Ontario, Canada, 1998.
C. Vázquez, “An upwind numerical approach for an American and European option pricing model,” Applied Mathematics and Computation, vol. 97, no. 2-3, pp. 273–286, 1998.View at: Publisher Site | Google Scholar
D. Y. Tangman, A. Gopaul, and M. Bhuruth, “Exponential time integration and Chebychev discretisation schemes for fast pricing of options,” Applied Numerical Mathematics, vol. 58, no. 9, pp. 1309–1319, 2008.View at: Publisher Site | Google Scholar | Zentralblatt MATH
B. A. Wade, A. Q. M. Khaliq, M. Yousuf, J. Vigo-Aguiar, and R. Deininger, “On smoothing of the Crank-Nicolson scheme and higher order schemes for pricing barrier options,” Journal of Computational and Applied Mathematics, vol. 204, no. 1, pp. 144–158, 2007.View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. Sevcovic, B. Stehlikova, and K. Mikula, Analytical and Numerical Methods for Pricing Financial Derivatives, Nova Science, Hauppauge, NY, USA, 2011.
P. Kútik and K. Mikula, “Finite volume schemes for solving nonlinear partial differential equations in financial mathematics,” in Finite Volumes for Complex Applications. VI. Problems & Perspectives, vol. 4 of Springer Proceedings in Mathematics., pp. 643–651, Springer, Heidelbergm, Germnay, 2011.View at: Publisher Site | Google Scholar | Zentralblatt MATH
R. Company, L. Jódar, and J.-R. Pintos, “A numerical method for European option pricing with transaction costs nonlinear equation,” Mathematical and Computer Modelling, vol. 50, no. 5-6, pp. 910–920, 2009.View at: Publisher Site | Google Scholar | Zentralblatt MATH
R. Company, L. Jódar, and J.-R. Pintos, “Consistent stable difference schemes for nonlinear Black-Scholes equations modelling option pricing with transaction costs,” Mathematical Modelling and Numerical Analysis, vol. 43, no. 6, pp. 1045–1061, 2009.View at: Publisher Site | Google Scholar | Zentralblatt MATH
R. Company, L. Jódar, J.-R. Pintos, and M.-D. Roselló, “Computing option pricing models under transaction costs,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 651–662, 2010.View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Yousuf, A. Q. M. Khaliq, and B. Kleefeld, “The numerical approximation of nonlinear Black-Scholes model for exotic path-dependent American options with transaction cost,” International Journal of Computer Mathematics, vol. 89, no. 9, pp. 1239–1254, 2012.View at: Publisher Site | Google Scholar
S. M. Cox and P. C. Matthews, “Exponential time differencing for stiff systems,” Journal of Computational Physics, vol. 176, no. 2, pp. 430–455, 2002.View at: Publisher Site | Google Scholar | Zentralblatt MATH
B. Düring, M. Fournié, and A. Jüngel, “High order compact finite difference schemes for a nonlinear Black-Scholes equation,” International Journal of Theoretical and Applied Finance, vol. 6, no. 7, pp. 767–789, 2003.View at: Publisher Site | Google Scholar | Zentralblatt MATH
E. Dremkova and M. Ehrhardt, “A high-order compact method for nonlinear Black-Scholes option pricing equations of American options,” International Journal of Computer Mathematics, vol. 88, no. 13, pp. 2782–2797, 2011.View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. Burchard, E. Deleersnijder, and A. Meister, “A high-order conservative Patankar-type discretisation for stiff systems of production-destruction equations,” Applied Numerical Mathematics, vol. 47, no. 1, pp. 1–30, 2003.View at: Publisher Site | Google Scholar | Zentralblatt MATH
D. T. Dimitrov and H. V. Kojouharov, “Positive and elementary stable nonstandard numerical methods with applications to predator-prey models,” Journal of Computational and Applied Mathematics, vol. 189, no. 1-2, pp. 98–108, 2006.View at: Publisher Site | Google Scholar | Zentralblatt MATH
Q. A. Dang and M. Ehrhardt, “Adequate numerical solution of air pollution problems by positive difference schemes on unbounded domains,” Mathematical and Computer Modelling, vol. 44, no. 9-10, pp. 834–856, 2006.View at: Publisher Site | Google Scholar | Zentralblatt MATH
B. M. Chen-Charpentier and H. V. Kojouharov, “An unconditionally positivity preserving scheme for advection-diffusion reaction equations,” Mathematical and Computer Modelling.View at: Publisher Site | Google Scholar
G. Fichera, “Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,” Atti della Accademia Nazionale dei Lincei, vol. 5, pp. 1–30, 1956.View at: Google Scholar | Zentralblatt MATH
G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 3rd edition, 1996.