Computation | Free Full-Text | Closed-Form Formula for the Conditional Moments of Log Prices under t

1. Introduction

The Heston model [ 1 ] is a diffusion model that consists of two stochastic processes, where the dynamics of the first process involve the instantaneous variance process, namely the Cox–Ingersoll–Ross (CIR) process [ 2 ]. The model can be applied to basic derivative products with various prices and allows the instantaneous variance to be the CIR process having its mean reversion. The Heston model appears in a wide variety of real-world applications in different branches of mathematical science; see [ 1 , 3 , 4 , 5 , 6 ] for more details. For example, a well-known application of the Heston model is the variance swap pricing based on log prices; see [ 7 , 8 , 9 , 10 , 11 ]. The model can be also applied to the energy industry as financial applications; see [ 12 , 13 , 14 ] for more details. In the variance swap context observed by Zhu and Lian [ 9 ], two main types of swap valuation approaches exist, numerical and analytical methods. However, the numerical method increases the computational burden. In contrast, the analytical method is poorly studied in general. Investigating the Heston model’s properties is still challenging.

Commonly, the CIR process is applied to describe the dynamics of the interest rates or instantaneous variance in the Heston model of the stochastic interest rate or stochastic volatility models. In practice, strong empirical evidence shows that the movements in practice tend to involve time; see [ 15 , 16 , 17 , 18 , 19 ]; thus, the dynamics of the processes are usually governed by time parameters. In the case of the CIR process, it becomes the extended CIR (ECIR) process [ 16 ], which has been investigated in numerous research works such as [ 16 , 17 ], who studied an extension of the CIR process that is more suitable for the data than the CIR process. In 2011, Grzelak and Oosterlee [ 20 ] discussed an extension of the Heston model by replacing the constant interest rate by a stochastic environment. However, the properties of the process, such as conditional moments, central moments, and variance, are rarely investigated. In this study, we call the Heston model with time-parameter functions the inhomogeneous Heston model. Unlike the Heston model, the conditional moments and properties of the inhomogeneous Heston model cannot be derived directly by using the transition probability density function (PDF) due to the arbitrary time-parameter functions in the process that make the transition PDFs unknown.

Under sufficient conditions, this research provides a closed-form formula for the conditional moments of log prices under the inhomogeneous Heston model. Based on a partial differential equation (PDE) according to the infinitesimal generator [ 21 ], a recursive coefficient function was obtained. The obtained formula was derived by solving the PDE without requiring any knowledge of the transition PDFs. The presented formula is more general in terms of time-dependent parameters than other approaches in the literature. Some essential properties of the inhomogeneous Heston model are observed, which can be beneficial for statistical applications, such as for calculating the variance swap valuation, which is also more general in terms of the time-dependent parameters than Zhu and Lian’s results [ 9 ].

The rest of the paper is organized as follows: Section 2 provides a brief overview of the Heston model and the inhomogeneous Heston model. Section 3 mentions the main methodology to address the relevant concepts for our proposed results, which are closed-form formulas for the conditional moments of the inhomogeneous Heston model. Section 4 provides the essential properties of the closed-form formula. Section 5 experimentally validates our proposed formulas for the inhomogeneous Heston model for time-inhomogeneous cases via Monte Carlo (MC) simulations. Section 6 concludes the paper.

2. Inhomogeneous Heston Model

The original Heston model was first introduced by Heston [ 1 ] for application to option pricing, which satisfies the dynamics of the following stochastic differential equations (SDEs):

is a stochastic instantaneous variance,

is the long-term average of variance of the price,

is the rate of reverting to

is the volatility of volatility, and

are two Winner processes. The stochastic instantaneous variance in the Heston model can be represented by the CIR process [ 2 ].

In 2011, Grzelak and Oosterlee [ 20 ] discussed an extension of the Heston model, replacing constant interest rate r by time-dependent interest rate

. In addition, References [ 16 , 17 , 18 ] studied an extension of the CIR process, which is more suitable for the data than the CIR process. Consequently, we considered an extended Heston model by using

and a general form of the ECIR that appeared in [ 18 ] to describe the stochastic instantaneous variance and having a general form as follows:

are correlated Winner processes with correlation coefficient

based on a filtered probability space

generated by an adapted the Brownian motion

, where

is a sample space, Q is a risk-neutral measure, and the family

of the

-field on

parametrized over

is a filtration. By applying Itô’s lemma [ 21 ] with

for all

to the first equation of ( 2 ), the process for the log price can be rewritten as:

, the two assumptions below proposed by Maghsoodi [ 17 ], Rogers and Williams [ 23 ], and Ekström et al. [ 24 ] are required.

Assumption 1.

The functions

, and

in the ECIR process

given in ( 4 ) are strictly positive, smooth, and continuous time-dependent functions on

. Moreover, the ECIR process

holds the inequality

According to Assumption 1, this paper defines the parameter function space of the inhomogeneous Heston model ( 4 ) as follows:

. The idea of our results relies on a solution of the PDE given in the infinitesimal generator for a two-dimensional diffusion process [ 21 ], which corresponds to the solution of ( 6 ). Roughly speaking, by expressing the solution of the PDE as a polynomial expression, we can solve its coefficients to obtain a closed-form formula directly. The motivation for the form of conditional moments, that is a solution to the PDE, is based on [ 25 , 26 , 27 , 28 , 29 ]; considering that both SDEs in the inhomogeneous Heston model ( 4 ) have linear drift and linear squared diffusion coefficients, the infinitesimal generator maps polynomials to polynomials; see more details in [ 25 , 26 , 30 , 31 ].

3. Main Results

This section presents a closed-form formula for the conditional moments corresponding to the two-factor model ( 4 ) based on the solution of the PDE according to the infinitesimal generator for the two-dimensional diffusion process [ 21 ]. The formula is provided in Theorem 1 as function

. Next, we derive a formula for the first conditional moment of log price

as a consequence of Theorem 1, which is represented in Corollary 1.

Theorem 1.

Suppose that

is an integer and

follows the dynamics described by the inhomogeneous Heston model ( 4 ), then:

can be obtained by solving the system of recursive ordinary differential equations (ODEs), for all

Proof.

From Appendix A of Rujivan and Zhu [ 32 ], we obtain the PDE associated with the Heston model ( 2 ) when

as follows: provide the system of recursive ODEs shown in ( 8 )–( 12 ), as required. □

Corollary 1.

Suppose that

follows the dynamics described by the inhomogeneous Heston model ( 4 ), then:

to complete the proof. □

Note that our proposed results in this section are more general than other results in the existing literature [ 7 , 8 , 9 , 10 , 11 , 33 , 34 ]. Unlike the methods proposed in [ 7 , 8 , 9 , 10 , 11 , 33 , 34 ], our method is based on the PDE generated from the infinitesimal generator for the two-dimensional diffusion process, which attempts to assume the solution of the infinitesimal generator in the form of a combination of polynomial expansions between

One primary concern when we work with the inhomogeneous Heston model ( 4 ) is that ( 8 )–( 12 ) and ( 25 ) may be indirectly evaluated as exact solutions. Therefore, the conditional moment ( 7 ) cannot be expressed as an exact formula. To overcome this issue, a numerical integration method is required, for example, the simple and well-known methods, such as the trapezoidal rule and Simpson’s rule, or higher accuracy methods, such as the Chebyshev integration method [ 35 , 36 , 37 ], which has been illustrated to produce a much higher accuracy than the other mentioned integration methods when using the same discretizing nodes.

4. Mathematical Properties

This section provides the benefits of all our results in Section 3 , such as the first and second conditional moments, conditional variance, conditional central moments, and conditional skewness for the log price

of the two-factor model ( 4 ) with piecewise constant parameters. Then, we can solve the system of recursive ODEs; see more details in [ 38 ].

4.1. First Conditional Moment

By applying Corollary 1, the first conditional moment of the log price based on the Heston model, ( 4 ) with

, and

, can be expressed as:

Remark 1.

The recursive coefficient functions of the first conditional moment in our result agree with the result in Proposition 2.3 of Rujivan and Zhu [ 32 ] by comparing

in their work, respectively.

4.2. Conditional Variance and Central Moments

The n th moment about the mean (the n th central moment) is expressed as:

where the zeroth conditional moment is equal to 1. The first few well-known conditional central moments have intuitive interpretations:

, known as the conditional zeroth central moment, is equal to 1; the first central conditional moment

is equal to 0; the second conditional central moment

is called conditional variance; higher orders, such as the third and fourth conditional central moments, are used to define conditional standardized moments, well known as the skewness and kurtosis, respectively.

In the case of conditional variance

, it can be calculated by the first and second conditional moments by using the following formula:

Then, we use a symbolic package, namely Dsolve , in Mathematica for solving the initial value problem given in Theorem 1 when

. The solutions can be expressed as:

Remark 2.

The recursive coefficient functions of the second conditional moment in our result agree with the results proposed by Rujivan and Zhu [ 32 ] by comparing term by term the coefficients of

4.3. Conditional Skewness and Higher Conditional Moments

Conditional skewness can be computed from the first, second, and third conditional moments by using the following formula:

Higher conditional moments using the n th conditional moments can be computed by applying Theorem 1 as well. To obtain the recursive coefficient functions,

, we solve

foreach term in the order of the diagram illustrated in Figure 1 .

The steps to solve recursive coefficient functions in Figure 1 are explained as follows. The recursive function coefficient

is solved first and is always equal to 1. Next, we compute

, and

by solving the system of ODEs of ( 9 ). After obtaining

, we can solve

by using Formula ( 10 ). For each

, when

is obtained,

can be solved using Formula ( 11 ). After terms

, and

are obtained, we can solve terms

, and

by using Formulas ( 10 ) and ( 11 ) again. Continue this process until

are obtained. Recursive function coefficient

is computed lastly by solving Formula ( 12 ).

Moreover, we give the pseudocode in Algorithm 1 from Figure 1 to obtain the n th conditional moment for

Algorithm 1 Recursive coefficient functions.

Input:

conditional moments

Output:

1:: for to 0 (step ) do
2:: for to 0 (step ) do
3:: if then
4:: if then
5:: Solve ( 8 )
6:: else
7:: Solve ( 9 )
8:: end if
9:: else if then
10:: if then
11:: Solve ( 10 )
12:: else
13:: Solve ( 11 )
14:: end if
15:: else if then
16:: Solve ( 12 )
17:: end if
18:: end for
19:: end for

5. Numerical Validation

These experimental validations discuss the inhomogeneous Heston model ( 4 ) by employing the Euler–Maruyama (EM) discretization method [ 39 ] to the model. We let

, respectively, be time-discretized approximations of

, which are generated on the time interval

steps, i.e.,

. Then, the EM approximation for ( 4 ) is performed by:

are mutually independent.

In this study, the numerical simulations for obtaining the comparison results were implemented by applying MATLAB R2021b running on a laptop computer configured with the following details: Intel(R) Core(TM) i7-5700HQ, CPU @2.70 GHz, 16.0 GB RAM, Windows 10 Pro, Version 20H2, and 64 bit operating system.

For the numerical testing, we used parameters

, and

0.001

. The comparison results between Formula ( 7 ) given in Theorem 1 and the MC simulations with 10,000 sample paths are shown in Figure 2 and Figure 3 . These figures illustrate that the results obtained from the MC simulations (colored circles) completely match with the closed-form Formula ( 7 ) (colored lines) for the first and second conditional moments, thereby validating the accuracy of the closed-form Formula ( 7 ) obtained from Theorem 1.

In addition, Table 1 and Table 2 demonstrate the mean absolute errors (MAEs) between Formula ( 7 ) and the MC simulations and the average runtimes (ARTs) of the MC simulations for different numbers of the sample paths: 20,000, 40,000, and 80,000, to validate the accuracy and efficiency of our proposed formula. These ARTs are the average of the computational times to calculate the MC simulations for fixing

at each initial value

. Table 1 and Table 2 conclude that the more the sample path numbers increase, the more the MAEs decrease. However, the ARTs also increase. Moreover, we can see the efficiency of our proposed Formula ( 7 ) from Theorem 1, which provides the exact value of

. Our proposed Formula ( 7 ) employs a small computational time around 0.1532 s. Finally, we depict the surface plots of the first and second conditional moments

by varying

at different

in Figure 4 and Figure 5 .

6. Conclusions

In this study, we derived a closed-form formula of conditional moments for the inhomogeneous Heston model ( 4 ) in the term of a polynomial expansion, as provided in Theorem 1. We further presented a formula for the first conditional moment

proposed in Corollary 1 as a consequence of Theorem 1. Additionally, some essential properties of the inhomogeneous Heston model ( 4 ), such as the first and second conditional moments, and the skewness were observed and discussed in Section 4 . The algorithm to solve the recursive coefficient functions ( 8 )–( 12 ) given in Theorem 1 was provided in Algorithm 1.

Finally, we validated our closed-form formula for the first and second conditional moments by comparing it with the MC simulations via several experimental examples in Section 5 . The experiments in each example indicated that our proposed formula and the MC simulations completely matched. Figure 2 and Figure 3 and Table 1 and Table 2 confirmed that our closed-form formula provides good accuracy and reduces the computational burden compared with the MC simulations.

This technique presented here can be also applied to the other two-factor models, such as Schwartz’s two- or three-factor model. However, one primary concern for our presented formulas is that the coefficients

in ( 7 ) cannot be obtained directly by solving the system of recursive ODEs. In such a case, a numerical method is required to manipulate those coefficients.

Author Contributions

Conceptualization, K.C. and P.S.; methodology, K.C. and P.S.; software, K.C. and P.S.; validation, K.C. and P.S.; formal analysis, K.C. and P.S.; investigation, K.C. and P.S.; writing—original draft preparation, K.C. and P.S.; writing—review and editing, K.C. and P.S.; visualization, K.C. and P.S.; supervision, K.C. and P.S.; project administration, K.C. and P.S.; funding acquisition, K.C. and P.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

We are grateful for a variety of valuable suggestions from the anonymous Referees, which substantially improved the quality and presentation of the results. All errors are the authors’ own responsibility.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

ART	Average runtime
CIR	Cox–Ingersoll–Ross
ECIR	Extended Cox–Ingersoll–Ross
EM	Euler–Maruyama
MAE	Mean absolute error
MC	Monte Carlo
ODE	Ordinary differential equation
PDE	Partial differential equation
PDF	Probability density function
SDE	Stochastic differential equation

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Figure 2. Validation tests of the first and second conditional moments for the initial values

at different

. ( a ) The first conditional moment

. ( b ) The second conditional moment

Figure 2. Validation tests of the first and second conditional moments for the initial values

at different

. ( a ) The first conditional moment

. ( b ) The second conditional moment

Figure 3. Validation tests of the first and second conditional moments for the initial values

at different

. ( a ) The first conditional moment

. ( b ) The second conditional moment

Figure 3. Validation tests of the first and second conditional moments for the initial values

at different

. ( a ) The first conditional moment

. ( b ) The second conditional moment

between our formula and the MC simulations together with the average runtimes of the MC simulations.

x No. of Paths

between our formula and the MC simulations together with the average runtimes of the MC simulations.

x No. of Paths

39.96

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Chumpong, K.; Sumritnorrapong, P. Closed-Form Formula for the Conditional Moments of Log Prices under the Inhomogeneous Heston Model. Computation 2022 , 10 , 46. https://doi.org/10.3390/computation10040046 AMA Style
Chumpong K, Sumritnorrapong P. Closed-Form Formula for the Conditional Moments of Log Prices under the Inhomogeneous Heston Model. Computation . 2022; 10(4):46. https://doi.org/10.3390/computation10040046 Chicago/Turabian Style
Chumpong, Kittisak, and Patcharee Sumritnorrapong. 2022. "Closed-Form Formula for the Conditional Moments of Log Prices under the Inhomogeneous Heston Model" Computation 10, no. 4: 46. https://doi.org/10.3390/computation10040046 Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. Terms and Conditions Privacy Policy We use cookies on our website to ensure you get the best experience.
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