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1 Introduction
The CIR process goes back the works of Feller in the 1950s, see e.g. [32], and was used by Cox, Ingersoll and Ross [10] to model short term interest rates. It is the solution to the following stochastic differential equation (SDE)
| (1) |
where and is a Brownian motion. The parameters can be interpreted as follows: is the long run mean of the process, is its speed of mean reversion and is its volatility. We assume the initial value to be deterministic. We denote the Feller index of the CIR process by
Note that the CIR process takes positive values only. Additionally, if almost all sample paths are strictly positive. The CIR process is used in particular to model the volatility of the asset price in the Heston model [23]. Here, the SDE for the price process and its volatility are given by
| (2) |
where is deterministic, is the risk-free interest rate, determines the correlation between the two processes and , are independent Brownian motions. Usually, the log-Heston model instead of the Heston model is considered in numerical practice. This yields the SDE
| (3) |
where and . Since the right hand side of the SDE for the log-asset price does not depend on , its approximation reduces to the approximation of a Riemann integral of and an Itō integral of . So, the main difficulty is the approximation of the CIR process . Since the CIR process takes positive values only and the diffusion coefficient is a square root and thus not globally Lipschitz continuous, much effort has been devoted to this problem in the last 25 years, see Subsection 1.1.
In this manuscript, we are looking at Euler discretization schemes for the CIR process and the log-Heston model. We will work with an equidistant discretization
with . A naive Euler discretization will give negative values and is not well defined due to the square root coefficient. Therefore a ”fix” is required. A summary of the existing Euler schemes for the CIR process and a numerical comparison can be found in [27], where a general framework for Euler schemes for the CIR process is proposed as
| (4) | ||||
for with and suitable functions that are chosen from
Here we will study the Euler schemes with given by
| (5) |
or
| (6) |
The first set of conditions modifies the coefficients of the CIR process to deal with negative values, which may arise in the computation. For example, is replaced by or . After the approximation has been computed, is again applied to obtain , since may be still negative. The second set of conditions is different. Here after each Euler step or , respectively, is applied to avoid negative values. See also Subsection 2.1 and Subsection 2.2
Table 1 shows all Euler schemes that are presented in [27] in detail. The Full Truncation Euler was introduced in the same paper. The origin of the Euler with Absorption fix is unknown, the Symmetrized Euler was analyzed in [6] for example. The scheme from Higham and Mao was first analyzed in [24] and the Partial Truncation Euler was first introduced in [14].
| Scheme | |||
|---|---|---|---|
| Absorption (AE) | |||
| Symmetrized (SE) | |||
| Higham and Mao (HM) | |||
| Partial Truncation (PTE) | |||
| Full Truncation (FTE) |
Results involving a (polynomial) convergence rate for these Euler schemes are rare and usually come along with a strong restriction on the Feller index, see Subsection 1.1. In this manuscript, we will prove the -convergence rate of for all these schemes if (with arbitrarily small). Furthermore, we will show that this result carries over to the log-Heston model if the price process is discretized with the standard Euler scheme, i.e. with
| (7) | ||||
where and .
Thus, we recover (up to an arbitrarily small ) the standard convergence order of the Euler scheme for SDEs with globally Lipschitz continuous coefficients.
For the case we can obtain e.g. for the Euler schemes given by Equations (4), (5), (7) convergence order
. However, this estimate does not seem to be sharp, see our simulation study in Section
5.We conclude this section with a summary of previous results in the literature, further new results and an outline of the remainder of this manuscript.
1.1 Previous results
The strong approximation of the CIR process has been intensively studied in the last years. The first works on this topic are [14, 2, 24], which prove strong convergence (without a polynomial rate) of various explicit und implicit schemes using the Yamada-Watanabe approach.
One of the schemes of [2] is the drift-implicit square root Euler scheme which is well defined and positivity preserving for . This scheme turned out to be accessible to a more detailed error analysis, see [15, 3, 30, 21]. In particular, for the -approximation at the final time point [3] establishes convergence order 1 for , while [15] gives convergence order for and [21] yields order for .
A breakthrough for the (very challenging) case was provided by [19] and [20]. In particular, the truncated Milstein scheme of [20] attains -convergence order in this regime.
So, which rates are best possible for the (non-adaptive) -approximation of the CIR process at the final time point? This question has been answered by the works [31] and [17], which yield333We will study the optimal -approximation of stochastic volatility models in the forthcoming work [29].
where is the set of measurable functions . Thus, the convergence rate of the truncated Milstein scheme for and the rate of the drift-implicit square root Euler for are optimal.
In contrast to this, convergence rate results for the explicit Euler schemes in Table 1 have been rare. In [5], the authors prove -convergence order for the Symmetrized Euler but with a strong restriction on the Feller index. For FTE the -convergence order for and is shown in [12]. As mentioned, [27] provides a survey and numerical comparison of Euler-type schemes. Further contributions on the strong approximation of the CIR process can be found in [16, 9, 7].
We are not aware of any results concerning the strong approximation of the log-Heston model except [26, 1]. In [1] the drift-implicit square root Euler for the CIR process is combined with an Euler discretization of the log-Heston process and -convergence order is obtained for , while [26] uses a drift implicit Milstein discretization of the CIR process instead and obtains -convergence for .
The strong approximation of the full Heston model, i.e. of instead of
carries an additional burden, since the SDE for the asset price has superlinear coefficients and admits moment explosions, i.e.
for certain parameter constellations and , see e.g. [4]. The article [11], where exponential integrability results for several Euler-type methods for the CIR process have been established, is dedicated to this problem.1.2 Further results
Our analysis is taylor-made for the -approximation and based on the Tanaka-Meyer formula combined with a clever control of the arising local time of the error process. We found this approach in [13], where the approximation of SDEs with irregular drift and additive noise has been studied. For the -approximation with we could deduce the upper bound for the convergence order by a standard application of the Hölder inequality. However, this bound is unlikely to be sharp, compare e.g. [5] and [12], so we do not spell out this result in detail.
More importantly, our results can in particular be helpful for the Monte-Carlo pricing of (path-dependent) European options, since they allow the control the bias:
Proposition 1.3.
Let , and as in Theorem 1.1. Moreover, let be a measurable mapping which satisfies:
-
there exists an such that
for all measurable ;
-
.
Finally, set with .
Then we have
1.3 Notation and Outline
As already mentioned, we will work with an equidistant discretization
with and . Furthermore, we define and . Constants whose values depend only on and the choice of will be denoted in the following by , regardless of their value. Other dependencies will be denoted by subscripts, i.e. means that this constant depends additionally on the function and the parameter
. Moreover, the value of all these constants can change from line to line. Finally, we will work on a filtered probability space
where the filtration satisfies the usual conditions, and (in-)equalities between random variables or random processes are understood
-a.s. unless mentioned otherwise.The remainder of the manuscript is organized as follows. We first show and collect some preliminary results in Section 2. The proofs of Theorem 1.1 and Proposition 1.2 are carried out in Section 3 and Section 4, while the proof of Proposition 1.3 is also given in Section 4. Finally, our simulation study is presented in Section 5.
2 Time-continuous extensions of the schemes and other preliminary results
In this section, we will present the discretization schemes in detail that we are analyzing and a couple of preliminary results that are needed to prove our main theorems. The first one is a well-known result for the CIR process.
Lemma 2.1.
Let . Then we have
The next auxiliary result on the smoothness of the CIR process and the log-Heston-SDE is also well known:
Lemma 2.2.
Let . Then we have
The following lemma gives us a bound for the expected local time in zero of a semimartingale. It is taken from [13].
Lemma 2.3.
For any and any real-valued, continuous semimartingale , we have
The following statement can be verified by a simple computation.
Lemma 2.4.
For and , we have
We also will require the following well known statement on the moments of a martingale in terms of its quadratic variation, see e.g. Proposition 3.26 and Remark 3.27 in Chapter III of [25].
Proposition 2.5.
Let be a continuous martingale and . Then there exist constants such that
and
2.1 Euler schemes – Case I
For the choice (4), (5), (7) the time-continuous extensions and read as
| (8) |
with , and . Note that and are globally Lipschitz continuous with Lipschitz constant and satisfy
| (9) |
Moreover note that
| (10) |
The next lemma can be shown by some tedious but straightforward computations, since the coefficients of the Euler scheme are of linear growth.
Lemma 2.6.
Let . There exists such that
2.2 Euler schemes – Case II
For (4), (6), (7) we obtain the Symmetrized Euler (SE) and the Euler with Absorption (AE). We can write the time-continuous extension of (SE) as
on each interval and the time-continuous extension of (AE) as
respectively. Now, let . We define
and use the Tanaka-Meyer formula for and for to obtain
and
Here is the local time of in . For almost all the map is continuous and non-decreasing with . See e.g. Theorem 7.1 in chapter III of [25]. We can rewrite both schemes as
| (11) | ||||
with and .
Lemma 2.7.
Let and . Then, there exists a such that
Proof.
The next two lemmas are Propositions 3.6 and 3.9 from [28].
Lemma 2.8.
For and we have that
| (12) |
for .
Lemma 2.9.
Let , , and . Then, there exists a constant such that
The following Lemma gives a control of the non-martingale terms, which arise additionally in the expansion of SE and AE, i.e. in (11).
Lemma 2.10.
Let , and . Moreover, let be bounded and be of linear growth. Then we have
and
Proof.
(a) We start with the second assertion. Note that the integral under consideration is a pathwise Riemann-Stieltjes integral, since is positive and non-decreasing with . With we then have
It follows
and Lemma 2.9 gives
Now we have to distinct the cases and . (i) If , we can choose and such that
Since it follows that
and consequently
(ii) For , observe first that for and so choosing gives
Since
where is the Euler-Mascheroni constant, we can conclude that
Hence the assertion follows in this case by choosing sufficiently small.
2.3 The Euler scheme for the log-price process
The time-continuous extension of the Euler scheme for the log-price process in the Heston model is given by
| (13) | ||||
As , we can choose one of the previously introduced schemes for the CIR process. We have the same results concerning the moment stability and the local smoothness as before.