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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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A Lie Algebraic and Numerical Investigation of the Black-Scholes Equation with Heston Volatility Model

Merger J* and Borzi A

Research Assistant, Department of Mathematics, University of Würzburg, Chair for Scientific Computing Emil-Fischer-Straße 30, Room 02.013, 97074 Würzburg, Germany

Corresponding Author:
Merger J
Research Assistant, Department of Mathematics
University of Würzburg, Chair for Scientific Computing Emil-Fischer- Straße 30
Room 02.013, 97074 Würzburg, Germany
Tel: +499313180793
E-mail: [email protected]

Received date: December 19, 2015; Accepted date: January 25, 2016; Published date: January 27, 2016

Citation: Merger J, Borzi A (2016) A Lie Algebraic and Numerical Investigation of the Black-Scholes Equation with Heston Volatility Model. J Generalized Lie Theory Appl S2:006. doi: 10.4172/2469-9837.1000S2-006

Copyright: © 2016 Merger J, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

This work deals with an extension of the Black-Scholes model for rating options with the Heston volatility model. A Lie-algebraic analysis of this equation is applied to reduce its order and compute some of its solutions. As a result of this method, a five-parameter family of solutions is obtained. Though, these solutions do not match the terminal and boundary conditions, they can be used for the validation of numerical schemes.

Keywords

Lie algebra; Black-Scholes equation; Differential equations; Lie symmetries; Diffeomorphisms

Introduction

Black and Scholes [1] assumed a financial market, where a risk free bond with constant interest rate r, an asset with price S that is modelled by a geometric Brownian motion, and call and put options related to this asset can be traded. With the assumption of an arbitrage free market and in the framework of It ’s stochastic differential equations, it is possible to derive the well-known Black-Scholes partial differential equation for the fair price of an option V. As the assumptions of this first modelling attempt are in practice too restrictive, several extensions of this model were proposed. One possible direction is to discard the assumption of a constant volatility for the geometric Brownian motion of the asset price and to assume that it is itself a random variable governed by a stochastic differential equation [2]. The resulting stochastic differential problem is given by

(1a)
image (1b)

where we suppose that image are two stochastically independent Wiener processes. The model constants α, m and L are supposed to be positive. The drift term of (1b) is built in a way such that the average of vt tends to have approximately the value m. In particular, if L is zero then vt is deterministic and converges exponentially to m as t tends to infinity. In this case, the option price behaves according to the solution of the usual Black-Scholes equation with a constant volatility image

Based on (1), it is possible to derive a partial differential equation for the price of an option V [2] as it has been done for the model with constant volatility in studies of Gunther and Jungel [3]. In contrast to the standard Black-Scholes equation the PDE that arises with Heston’s volatility model involves one more argument representing the current volatility of the market. The resulting two-dimensional Black-Scholes equation is as follows.

image (2)

In application, this equation is augmented with the following terminal and boundary conditions

image (3)

In this setting, a denotes whether a call (a = 1) or a put (a = −1) option is considered. T represents the time when one is allowed to buy or sell a share of an asset for the prescribed price K, whereas λ is the parameter that models the price of volatility risk [2]. In (2), x represents the asset price S and y denotes the current volatility v. As the asset price and the current volatility are always positive, we are searching for a solution of the above differential problem (2)-(3) in the domain

image

In the following, analytical and numerical solutions of (2) together with the boundary and terminal conditions (3) are sought. In particular, a quick review of Lie symmetries of partial differential equations is given in Section 2. In Section 3, this method is applied to the 2-dimensional Black-Scholes equation (2) and we derive a five-parameter family of analytical solutions. In Section 4, convergence properties of the Chang- Cooper discretization are tested with the given analytical solutions. A section of conclusion completes the exposition of our work.

Lie Theoretical Analysis of Differential Equations

In this section, we illustrate how Lie symmetries can be used to determine analytical solutions of partial differential equations. Applications of this method can be found in literature of Bordag [4] and Naicker V, Andriopoulos K, Leach [5]. Our review is based on the book of Stephani [6].

Many partial differential equations for a function u that is dependent on n variables xi (image) can be written as follows

image (4)

with an analytic function H, where yk denotes subsequently the independent variables xi, the dependent variable u, and its derivativesimage Equation (4) defines a manifold in some multi dimensional Euclidean space and its solutions are sub-manifold. Diffeomorphisms of Rn can be used to permute the set of solution and find solutions with special properties. Therefore, we observe that a one parameter group of diffeomorphisms image can be completely determined by the first order differential operator

image (5)

which is called infinitesimal generator or symbol. In fact, the group action on a point imagecan be computed by solving the initial value problem image (6)

Moreover, the coefficients of the symbolimage with respect to a change of variables y = T(x) with a transformation image can be computed as follows

image (7)

where X(Ti) denotes the application of the first order differential operator X on the function Ti. It can be shown that there always exists a set of canonical variables in which the symbol has the normal form

image;[7].

As the symbol X acts only on the independent variables, it is prolonged to act in a higher dimensional space including also the dependent variable and its derivatives up to the order of the partial differential equation. The prolonged infinitesimal generator is defined as follows

image

where the coefficients image are given by

image (8)

Here, the total differentiation operator image is used.

A Lie symmetry of a PDE is defined as a group of transformations of the independent and dependent variables such that set of solutions is invariant under these transformations. From the fact that the image of a solution satisfies the PDE, i.e.image for all ε, it can be shown that

X(H(yk)) = 0, (9)

holds everywhere on the solution manifold H(yk) = 0, where X is the prolonged symbol of the transformation group. For a given Lie symmetry, we seek to find its canonical variables wk as the corresponding symbol is image and (9) then reads as image Hence, the resulting PDE written in the new variable wk is independent of w1 and therefore involves one independent parameter less. Computing solutions of the transformed PDE, which should be easier as less independent variables are involved, and reversing the transformation, provides solutions of the original PDE.

Lie Analysis of the Two-dimensional Black-Scholes Equation

In this section, the Lie method is used to find solutions to the 2-dimensional Black-Scholes equation (2) written as H = 0, where H is defined as follows

image

We assume that this equation admits a Lie symmetry with the infinitesimal generator

image

Then, we first apply the prolonged symbol

image

to the function H and then evaluate the resulting function X(H) on the solution manifold H = 0. The resulting expression yields zero, whenever X is the generator of a Lie symmetry. The exact expressions for the prolonged coefficients Φ1, Φ2, Φ3, Φ11 and Φ22 according to (8) are given by

image
image
image
image
image
image
image
image
image
image
image

where sub-indices x, y, and V of ξ, γ, τ and Φ denote partial derivatives with respect to the given variables. The equation H = 0 is solved for Vt and inserted into X(H) = 0. Afterwards this single equation splits up into the determining equations, since the derivative variables (Vx, Vy, Vxx, …) are linearly independent. Among the resulting equations, there are simple ones as image Hence, we solve the following remaining system of partial differential equations

image (I)
image (II)
image (III)
image (IV)
image (V)
image (VI)

for the functions image,image,image and image. Notice that equations (IV,V and VI) are similar to the original PDE we are trying to solve. Inserting the special form of Φ into (VI), it splits up into the following two equations

image (VI')
image (VI'')

due to the fact that φ, β, and τt are independent of V. The function β is independent of the other functions and equations. Furthermore, it must be a solution of the PDE (VI'') whose Lie symmetries we are looking for. So the transformation imagewith its infinitesimal generator image is a Lie symmetry, mapping solutions onto solution. As the Lie symmetry corresponding to the coefficient function β is as difficult to find as solving the PDE directly, it is not significant for our purpose and we do not take (VI'') into account any more.

Instead, let us focus on (II). Differentiating it two times with respect to y yields image whose solution is given by imageInserting this expression in (II) gives γ = τt y. Hence, γ is independent of x and since (I) holds, the function ξ is independent of y, i.e. image

The same idea works with (III), whose second derivative with respect to x is imageThe general solution to this ordinary differential equation (ODE) is image Notice that (III) can only be satisfied by image With this knowledge, Equations (III) and (IV) simplify to

image (III')
image (IV')

From (IV') one can directly derive image as φy is not dependent on y. Differentiating (III') with respect to x gives

image

Notice that the functions τ and D are independent of y. Hence

image and image hold, i.e. image (III') becomes image which means that image andimageprevious results in (VI'), we obtain

image

As the coefficient of y must equal zero, τt = 0 holds and consequently Bt is equal to zero. Hence, the most general solution of the determining equations is

image
image
image
image

where c1, c2 and c3 are real constants and β is a solution of the Black- Scholes equation.

Hence, the only Lie symmetries that (2) admits have the following infinitesimal generators

image

Apart from the last symmetry this is a three dimensional, solvable Lie algebra, i.e. the commutator image of two arbitrary symmetries X and Y equals zero.

Next, we determine the canonical variables for the Lie symmetry with fixed constants c1, c2 and c3. Therefore, we search for three functionally independent invariants image, image, and imagethat satisfy the following equation

image

The evaluation of (7) with the choice of the following new variables

image,image and image shows that the Lie symmetry with respect to the new variables has the desired normal symbol image and that v, w, u, and s are the canonical variables.

In order to rewrite (2) in the new variables, we differentiate image with respect to x, y and t and obtain

image

Hence, the 2-dimensional Black-Scholes equation in the new variables is given by

image

and by setting c1 = c2 = rc we cancel out terms with u and uv. Therefore, the reduced Black-Scholes equation is as follows

image

In order to find solutions to (10), we assume image and obtain

image

which is equivalent to

image

Since the left-hand side of the equation depends only on w and the right hand side only on v, both sides must be equal to a constant C. Hence, we obtain two decoupled ordinary differential equations

image (11)
image (12)
image

Regarding the second ordinary differential equation (12), we transform it into Kummer’s equation

image

by defining image where image
imageimage
image and imageThe general solution of Kummer’s equation is given by

image

where M and U are Kummer’s functions of the first and second kind, respectively. For further details, [8]. In case of C = 0, (12) is given by

image

and we directly see that the first derivative of Ψ is a multiple of

image and hence a general solution in the interval (0, ∞) is given by

image

Having found solutions u for (10), we obtain solutions V of (2) by applying the reverse variable transformation as follows

image (13)

To summarize, we obtain the following five parameter family of solutions to the two-dimensional Black-Scholes equation

image

where image

These functions do not satisfy the boundary conditions (3) given in Section 1. In order to check this we write down the boundary conditions in the new variables v, w, and u. They are given by

image (i)
image (ii)
image (iii)
image (iv)

So if V would be a solution to the two-dimensional Black-Scholes equation subject to the given boundary conditions and if it corresponds to a solution u of the reduced equation, then u must satisfy the boundary conditions above. Note that, while (i) imply (ii) and (iii), (i) is not consistent with (iv). As (i) determines u and therefore u is independent of w, (iv) cannot be satisfied as image which equals neither v nor image

Numerical Solution of the Two-dimensional Black- Scholes Equation

This section deals with a numerical scheme to calculate an approximation to the solution of the Black-Scholes equation (2). We work with the proposal of Chang-Cooper scheme [9] and analyzed in studies of Mohammadi and Borz [10]. This disretization scheme is often used for Fokker-Planck equations, as its solutions are probability density functions and therefore are non-negative and their integral over its domain equals 1. These two properties are preserved by the Chang-Cooper (CC) difference scheme. In the case of the Black- Scholes equation the solution is also non-negative, as it models the price of an option, which must be non-negative. Hence, the choice of the CC scheme guarantees that the numerical solution will be non negative In order to apply the Chang-Cooper discretization scheme the two dimensional Black-Scholes equation (2) must be written in flux form. This is not possible, as the coefficient of V is −r and not image However, introducing the following new variables image,image,image and image and computing the derivatives of V with respect to the new variables as follows

image

we obtain the following PDE

image (15)

We can write (15) in flux form as follows

image (16)

where

image

At this point we would like to stress three important properties of the flux functions. To begin with, they are all independent of the time variable t. Hence, the left-hand side of the resulting linear system of equations is the same for each time iteration and the corresponding matrix must be computed only once. Moreover, both B x and B y are linear functions and therefore Lipschitz continuous with the constants

image and image Finally, these functions must be positive in our domain. This is the case when the condition

image (17)

is satisfied.

The transformed Black-Scholes equation (16) must be solved subject to the following transformed initial and boundary conditions:

image (18)

There are several problems that arise during implementation:

• The domain of the problem (16) subject to (18) is imageand therefore unbounded in the space dimensions. Moreover the boundary conditions are given as a limit. For numerical purpose the domain was limited to image and it was assumed that the function attain the limit values already at the finite boundaries.

• When image corresponds to a point outside of the domain the values image with its coefficients are added to the right hand side of the linear system of equations, as they are known.

• The boundary condition for x → ∞ is given only in terms of the derivative of V with respect to x. Therefore the value imageis approximated by

image

• The derivative term is known and can be put to the right-hand side of the equation.

The values of the function on the boundary y = 0 are not given.

Fortunately, as y goes to zero, By goes to image and imagetends to zero. Assuming image,imagetends to zero, as imagegoes to infinity. Hence, the coefficient of image is zero and this function values need not to be known for the calculations [9].

In the following, the numerical scheme is applied to the function type C = 0 in (14). After the variable transformation, that is used to write (2) in flux form, the test function becomes

image

Note that this function has a singularity in y = 0, if and only if image.As this infinite value might arise problems while numerical calculations, the set of parameters is chosen such that image holds. In particular, the test function was calculated in the domain image with the following parameters

image

Unfortunately, it is not possible to chose a set of parameters such that the test function has no singularity at y = 0 and additionally there exists a domain where all flux functions are positive. That is because image imply image and therefore the necessary condition for (17) is not fulfilled. Consequently, there is no proof in this case, that the numerical solution is positive. Nevertheless, the convergence order can be observed. Figure 1 shows the difference of the numerical solution to the exact test function in terms of the norm

image

The plot data is shown in Table 1 where N, M and Q is the number of grid points in the x-, y- and t-dimension, respectively. A small time-step size is used in order to have a small error for the time discretization and to investigate the dependence of the error on the spatial-grid size.

N 26 51 76 101
M 26 51 76 101
Q 500 500 500 500
hx = hy 1/25 1/50 1/75 1/100
||f-fex|| 0.243 0.122 0.083 0.065

Table 1: Numerical error for different grid sizes.

The numerical experiments show that the discretization that is used provides only first-order convergence, i.e., doubling the grid point number in each spatial dimension and therefore halving the grid size h results in an error that is half as big as before. Notice that secondorder convergence is proven in literature of Mohammadi and Borz [10] with the assumption of zero boundary conditions. In order to validate this theoretical result the same procedure is done with a Gaussian bell function that is almost zero on the boundaries:

image

In contrast to the test function f, this function Φ does not satisfy the partial differential equation (16). Hence the deviation to the PDE

image

is added to the right hand side of the linear system of equations according to

image

Numerical experiments are performed with the grid sizes according to Table 2.

N 26 51 76 101
M 26 51 76 101
Q 500 500 500 500
hx = hy 1/25 1/50 1/75 1/100
|| F-Fex|| 3.25*10-3 0.83*10-3 0.37*10-3 0.21*10-3

Table 2: Numerical error for different grid sizes.

Φ was approximated in the domain image with the parameters being

image

Here, the flux functions are positive, as image With the logarithmic plot of the error in Figure 2 we can see that second-order convergence is obtained.

In the following, we use the Chang-Cooper numerical scheme to calculate a numerical solution. Here, this numerical solution in the case of a call option is compared to the solution of the Black-Scholes equation, where the volatility is assumed to be constant. It is given by

image (19)
image

where Φ is the cumulative distribution function of a normally distributed random variable with mean 0 and variance 1. This function is given by

image

The spatial domain of discretization is image After reversing the variable transformation the option price can be evaluated for image and image The following parameters are used

image

The model constant m represents the square of the average volatility and the stochastic process tends to this value. Hence, if one starts the process with the value y = 0.5, the stochastic process for the volatility image is likely to be almost constant to image As you can see in the lower left plot of Figure 3, the calculated price is nearly the same with both models. In contrast, regarding the case of a currently volatility lower thanimage the price of the option calculated with the extended Black-Scholes equation is higher than that of the model with constant volatility, because it takes into account that the volatility will rise. This effect can be seen in upper left and right plots. Finally, in the lower right plot the simpler model overestimates the price, when the initial volatility is higher than image , due to the fact that it is likely to fall.

In addition, the numerical solution satisfies the so-called Put- Call-Parity. The price of a call option C and the price of a put option P subject to the same asset with price x, that have the strike price K and the expiry date T in common, are related by the following formula [3]

image

We compute also the price for the put option and observe the absolute deviation for the Put-Call-Parity formula that we average along the y-dimension. Figure 4 shows the result depending on x and t. Apart from small x values the error is in the range of the numerical error of the Chang- Cooper-Scheme. The drastic increase of the error for x → 0 is due to the fact, that the boundary condition for image for the Black-Scholes equation in flux form is applied at the finite value image.This corresponds to imageas the transformation was image Consequently, the numerical solution for the put option takes the value image at image whereas the correct value is imageas C tends to zero as x goes to zero and therefore the price for the put option is image To conclude, it is evident why there is such a great error for small x, and moreover it is not relevant as x gets never so small in applications.

Conclusion

The aim of this work was to solve the partial differential Black- Scholes equation with Heston volatility model. Therefore, an analytical technique due to Sophus Lie that can be use to reduce the number of independent variables of a partial differential equation was presented and applied to the Black-Scholes equation. A five-parameter family of solutions was found. These functions do not satisfy the boundary conditions of the option price problem and henceforth numerical schemes are necessary to obtain approximate solutions. In the last part of this work the Chang-Cooper discretization scheme was used to calculate the option price function numerically. Its convergence was tested with an exact solution of the PDE, which was found by the Lie theoretical analysis. Finally, the numerical scheme was applied to compute the price of an option and good result were obtained in accordance with economic reasoning.

Acknowledgement

This work was supported in part by project “Multi - ITN STRIKE - Novel Methods in Computational Finance” by EU Grant Agreement Nr. 304617 and by BMBF project 05M2013 ‘ROENOBIO: Robust energy optimization of fermentation processes for the production of biogas and wine.

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