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Journal of Probability and Statistics
Volume 2015, Article ID 513137, 20 pages
http://dx.doi.org/10.1155/2015/513137
Research Article

Approximating Explicitly the Mean-Reverting CEV Process

Department of Mathematics, University of the Aegean, Karlovassi, 83 200 Samos, Greece

Received 2 September 2015; Revised 13 October 2015; Accepted 15 October 2015

Academic Editor: Steve Su

Copyright © 2015 N. Halidias and I. S. Stamatiou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We are interested in the numerical solution of mean-reverting CEV processes that appear in financial mathematics models and are described as nonnegative solutions of certain stochastic differential equations with sublinear diffusion coefficients of the form where . Our goal is to construct explicit numerical schemes that preserve positivity. We prove convergence of the proposed SD scheme with rate depending on the parameter . Furthermore, we verify our findings through numerical experiments and compare with other positivity preserving schemes. Finally, we show how to treat the two-dimensional stochastic volatility model with instantaneous variance process given by the above mean-reverting CEV process.

1. Introduction

Consider the following stochastic models in Itô form: where represents the underlying financially observable variable, is the instantaneous volatility when or the instantaneous variance when , and the Wiener processes , have correlation .

We assume that is a mean-reverting CEV process of the above form, with the coefficients for and , since the process has to be nonnegative. To be more precise the above restriction on implies that is positive; that is, is unattainable, as well as nonexplosive; that is, is unattainable, as can be verified by Feller’s classification of boundaries [17, Proposition 5.22] (see also Appendix A). The steady-state level of is and the rate of mean-reversion is .

System (1) for is the Heston model. When we get the Brennan-Schwartz model [2, Section II] that despite its simple form cannot provide analytical expressions for

Process for , also known as the CIR process [6, (13)], by the initials of the authors that proposed it for the term structure of interest rates, has received a lot of attention and we just mention the latest two contributions to the study of such processes (see [4, 5] and references therein).

Process for has been also considered for the dynamics of the short-term interest rate [3, (1)]. The stationary distribution of the process has also been derived in [7, Proposition 2.2].

We aim for a positivity preserving scheme for the process The scheme that we propose and denote as semidiscrete (SD) preserves the analytical property of staying positive. The idea of the semidiscrete method is that we discretize a part of the original SDE and then apply Itô’s formula (cf. [8] where the method originally appeared and [5, 9, 23]). The explicit Euler scheme fails to preserve positivity, as well as the standard Milstein scheme. We intend to apply the semidiscrete method for the numerical approximation of in model (1) with and compare with other positivity preserving methods such as the balanced implicit method (BIM) (introduced by [11, (3.2)] with the positivity preserving property [25, Section 5] and its stability properties [13]; see also [14] for an extended balanced method with better stability behavior) and the balanced Milstein method (BMM) [25, Theorem 5.9] (we give in Appendix B the form of all the above schemes for the approximation of ). Finally, we approximate the stochastic volatility model (1) with In [15] a thorough treatment can be found, where also another stochastic volatility model is suggested.

Section 2 provides the setting and the main results, Theorems 1 and 2, concerning the -convergence of the proposed semidiscrete method to the true solution of mean-reverting CEV processes of the form of the stochastic volatility in (1). The rate of mean-square convergence in Theorem 1 is logarithmic and in Theorem 2 is polynomial with magnitude The main ingredient of the approach we adopt, inspired by [16], is a change of the initial Brownian motion to another Brownian motion justified by Lévy’s martingale characterization of Brownian motion.

Section 3 is devoted to the logarithmic rate of convergence of the proposed semidiscrete scheme, while Section 4 concerns the proof of the polynomial rate of convergence. In Section 5 we briefly discuss the case where we do not alter the initial Brownian motion This approach produces reduced convergence rate. Finally, Section 6 presents illustrative figures where the behavior of the proposed scheme, regarding the order of convergence, is shown and a comparison with BIM and BMM schemes is given. In Section 7 we treat the full model (1) for a special case. Concluding remarks are in Section 8 and in Appendix B we briefly present numerical schemes for the integration of the variance-volatility process

2. The Setting and the Main Results

We consider the following SDE: where are positive and Then, Feller’s test implies that there is a unique strong solution such that a.s. when a.s. Let where and

Let the partition with and consider the following process: with a.s. or more explicitly for , where represents the level of implicitness and with

Process (6) is well defined when and this is true when and Furthermore, (6) has jumps at nodes Solving for , we end up with the following explicit scheme: with solution in each step given by [1, (4.39), page 123] which has the pleasant feature

Inspired by [16] we remove the term from (6) by considering the process which is a martingale with quadratic variation and thus a standard Brownian motion with respect to its own filtration, justified by Lévy’s theorem [17, Theorem 3.16, page 157]. Therefore, the compact form of (6) becomes for where Consider also the process The process of (2) and the process of (14) have the same distribution. We show in the following that as ; thus the same holds for the unique solution of (2); that is, as To simplify notation we write as We end up with the following explicit scheme: where is as in (8).

Assumption A. Let the parameters be positive such that and consider such that , for Moreover assume a.s. and for some

Theorem 1 (logarithmic rate of convergence). Let Assumption A hold. The semidiscrete scheme (15) converges to the true solution of (2) in the mean-square sense with rate given by where is independent of and given by where

Assumption B. Let Assumption A hold where now and .

Theorem 2 (polynomial rate of convergence). Let Assumption B hold. Then the semidiscrete scheme (15) converges to the true solution of (2) in the mean-square sense with rate given by where and is the constant described in (83) and is an appropriately chosen positive parameter which satisfies (84) and always exists, , and

In the following sections we write for simplicity or for .

3. Logarithmic Rate of Convergence

3.1. Moment Bounds

Lemma 3 (moment bound for SD approximation). It holds that for any , where .

Proof of Lemma 3. We first observe that is bounded in the following way: a.s., where the lower bound comes from the construction of and the upper bound follows from a comparison theorem. We will bound and therefore , since a.s. Set the stopping time , for with the convention Application of Itô’s formula on implies where in the second step we have used the fact that , in the third step the inequality , valid for and with , and in the final step the fact that and Taking expectations in the above inequality and using that is a local martingale vanishing at , we get where we have applied the Gronwall inequality [18, (7)]. We have that Thus taking expectations in the above inequality and using the estimated upper bound for we arrive at and, taking the limit as , we get Let us fix The sequence of stopping times is increasing in and as , and thus the sequence is nondecreasing in and as Application of the monotone convergence theorem implies for any Using again Itô’s formula on , taking the supremum, and then using Doob’s martingale inequality on the diffusion term we bound and thus

Lemma 4 (error bound for SD scheme). Let be an integer such that Then for any , where the positive quantities do not depend on

Proof of Lemma 4. First we take a We get that where we have used the Cauchy-Schwarz inequality. Taking expectations in the above inequality and using Lemma 3 and Doob’s martingale inequality on the diffusion term we conclude where the positive quantity , except on , depends also on the parameters , but not on Now, for , we get where we have used Jensen’s inequality for the concave function Following the same lines, we can show that for any , where the positive quantity , except on , depends also on the parameters , but not on

For the rest of this section we rewrite again the compact form of (12) in the following way: where is given by (3) and the auxiliary process is close to as shown in the next result.

Lemma 5 (moment bounds involving the auxiliary process). For any it holds that and for one has that for any , where the positive quantities do not depend on

Proof of Lemma 5. We have that for any , where we have used (33). Using Lemma 3 we get the left part of (34). Now for and noting that we get the right part of (34), where we have used Lemma 3. The case follows by Jensen’s inequality as in Lemma 4.
Furthermore, for and , we derive that where we have used (30) and in the same manner The case follows by Jensen’s inequality.

3.2. Convergence of the Auxiliary Process to in

We will use the representation (33) and write

Proposition 6. Let Assumption A hold. Then one has for any , where and

Proof of Proposition 6. Let the nonincreasing sequence with and We introduce the following sequence of smooth approximations of (method of Yamada and Watanabe [19]): where the existence of the continuous function with and support in is justified by The following relations hold for with ,We have thatMoreover we find that where we have used properties of Hölder continuous functions and, namely, the fact that is -Hölder continuous for , that is, , and that is -Hölder continuous since Application of Itô’s formula to the sequence implies where in the second step we have used (46) and (47) and the properties of and Taking expectations in the above inequality yields where we have used Lemma 5 in the second step and the Hölder inequality and Lemmas 3 and 4 in the third step and the fact that (the function belongs to the space of real-valued measurable -adapted processes such that ; thus [20, Theorem 1.5.8] implies ). Thus (45) becomes where in the second step we have used the asymptotic relations as for any as for any as ; in the last step we have used the Gronwall inequality and is as defined in Proposition 6 while Taking the supremum over all gives (41).

3.3. Convergence of the Auxiliary Process to in

Proposition 7. Let Assumption A hold. Then one has where is independent of and given by , where

Proof of Proposition 7. We estimate the difference It holds that where in the second step we have used the Cauchy-Schwarz inequality and (46) and Taking the supremum over all and then expectations we have where in the second step we have used Lemma 4 and Doob’s martingale inequality with , since is an -valued martingale that belongs to We find that where we have used (47). Now, Lemmas 3, 4, and 5 imply where we have used the asymptotic relations for all as and the quantity is given by
Relation (56) becomes where we have used Proposition 6 in the second step with the sequence as defined there and Gronwall’s inequality in the last step and the asymptotic relation as , for any , and is independent of and given by
We take , with , to be specified soon and note that as , since as Moreover we have that Now, since there is an small enough such that We take and conclude that as which in turn implies the asymptotic relation as , with the logarithmic rate stated before. In the same way we can show as , by taking We finally arrive at by taking , which implies (53).

3.4. Proof of Theorem 1

In order to finish the proof of Theorem 1 we just use the triangle inequality, Lemma 5, and Proposition 7 to getwhere , given in the statement of Theorem 1.

4. Polynomial Rate of Convergence

We work with the stochastic time change inspired by [21]. We define the process and the stopping time The process is well defined since a.s. and (see Section 2).

The difference is estimated as in Section 3 and we get, as in (56), that where is a stopping time and independent of is as in the proof of Proposition 7. The main difference here will be the estimation of the last term in (66). The approach in Section 3 resulted in the estimation where we used the Yamada-Watanabe approach. Now, we use the Berkaoui approach. Relation (47) becomes where we have used the inequality valid for all , and Consequently, we get the upper bound