Recursive Estimation for Continuous Time Stochastic Volatility Models Using the Milstein Approximation ()
1. Introduction
In the last three decades, semimartingales have received considerable attention with the emphasis being placed on state space models. From an econometric standpoint, time-varying volatility models have been widely developed, recognizing that the volatility and the correlation of assets change over time (see for example [4]). State space models in which the conditional mean of the observed process is modeled as a stochastic process are useful in parameter estimation. For example, stochastic volatility models are widely employed to estimate volatility parameters [3,5].
In [2], the estimating function approach was used for the recursive parameter estimation in models with semimartingales. In [1,6,7], the estimating function method was used for the estimation of state space models in the Bayesian setup. Parameter estimates obtained in [2] involve the evaluation of the stochastic integrals based on the observation of the complete path of the observed process. However, for continuous time models, it is more appropriate to study parameter estimates based on discretely observed data. In order to study the inference for diffusion processes based on discretely observed data, one has to approximate the continuous time diffusion by a discrete process. For some interest rate models (e.g. Vasicek, Cox-Ingersoll-Ross), discrete time approximation has been used to study parameter estimation (see [8,9] and the references therein).
Recursive estimation expresses the estimate of the parameter at time 
in terms of the parameter at time 
and an adjustment based on the observation at time
. Continuous time volatility models have been studied in [10]. However, the recursive parameter estimation based on discrete approximation have not been studied in the literature.
In most realistic situations, the diffusion cannot be observed continuously, so discrete time approximations to stochastic integrals or a direct approach using discrete time observations is required. For extended versions of the Cox-Ingersoll-Ross (CIR) model (see [11]), closed form expressions for the first four conditional moments cannot be obtained easily by using Ito’s formula, as was done for the non-extended CIR model (see [9]). Recently, [11] uses the Milstein approximation [12] to obtain the first two conditional moments of a diffusion. For diffusion models with a finite number of parameters, [9] uses the Milstein approximation to obtain the first four conditional moments and to construct the optimal estimating functions for the Vasicek model of the form
with
, 
, and
. One of the drawbacks of this one-factor model is that it is not in general possible to calibrate it so that it fits the presently observed term structure. For example, [13, p. 171] points out that for the above Vasicek model, which depends on three parameters, 
, 
, and
, it is not possible to choose values of those parameters so that the entire observed term structure of interest rates is fitted exactly by the model. To solve the problem, Kennedy proposes to allow time-varying parameters in the drift term of the Vasicek model.
Consider a diffusion process given by the time-homogeneous stochastic differential equation of the form
(1)
where 
and 
are the drift and diffusion functions, respectively, and 
is the standard Brownian motion. A special case of (1) is the Brownian motion with constant drift and diffusion coefficients:
where
. In this case, the conditional distribution of 
given 
is a normal with mean 
and variance
. If we consider the geometric Brownian motion given by
with
, then 
becomes a Brownian motion with drift with 
and
. In this case, the conditional distribution of 
given 
is also normal. The CIR process can be reparameterized to the following form:
Extended versions of the CIR process model have been proposed for modelling interest rate processes. For example, some consider the constant elasticity of variance process of the form
or the nonlinear drift diffusion process (see [14]) given by
For more general extended models, the diffusion is a function of the observation 
and hence, closed form expressions of the conditional distributions, as well as closed form expressions for the conditional moments cannot be easily obtained by solving differential equations obtained by repeated application of Itô’s formula. However, the Milstein approximation can be used to obtain the first four conditional moments.
If we consider a discretisation in small intervals of time
, then the Milstein approximation applied to (1) produces
(2)
where 
and
, i.i.d.
Unlike the Euler approximation for diffusion processes, the Milstein method in (2) gives a non-Gaussian time series model for
. The distribution implied by the Milstein approximation is a mixture of a normal and chi-square distribution. Moreover, for the extended CIR model and for more general diffusion processes, Ito’s approximation cannot be used to obtain closed form expressions for the first four conditional moments. In this paper, first we use the Milstein approximation to discretise the continuous time diffusion processes and then study the recursive estimates of latent state variables. We also show how the proposed method can be used to derive zero coupon bond prices in the incomplete information environment. In this case, the valuation exercise and the recursive estimation (learning) of the unobserved state variable are performed simultaneously by market participants.
2. State Space Models
In order to construct an optimal recursive estimate for non-normal stochastic volatility models, we start with the following discrete time example.Let the discrete-time state space model of the observed process 
and the state process 
be given by:
(3)
where 
and 
are positive constants, and possibly measurable with respect to the 
-field 
generated by the observations of 
up to and including time
. In addition, 
and 
are two standard Gaussian sequences of identically distributed random variables with
. The following lemma will be used to prove our main Theorem.
Lemma 1 Assume that 
and 
with
. Then
.
Proof 1 It follows from the theorem on Normal correlation that the conditional expectation and conditional variance of 
given 
are give by
and
.
Using the law of total expectation, we also have
Hence, the correlation between 
and 
is given as
The following theorem establishes the recursive estimation for the state space model (3).
Theorem 1 Given the state space model (3), and the class of all estimators of the form:
the
, which minimizes the mean-square error,
is given by
Moreover, the mean-square error is given as
Proof 2 The difference 
is given by
Squaring the above expression, taking expectations, and using the results of Lemma 1 it follows that the conditional mean-square error at 
is given by
Differentiating 
with respect to 
and setting the first derivative to zero, we have
Solving for
, we obtain
Corollary 1 Let the state space model be of the form
where 
and 
are two sequences of independent and identically distributed random variables having mean zero and variance 
and
, respectively. In the class of estimates of the form:
the 
which minimizes the mean-square error
is given by
In addition, the mean-square error is given as
Proof 3 The result follows from Theorem 1 by setting
, 
, 
, and
.
3. General Model
In the continuous-time setting, consider the general state space model of the form
where 
and 
are two uncorrelated standard Brownian motions. If we consider a discretisation in small intervals of time
, 
, then the Milstein approximation gives a non-Gaussian discrete state-space model of the form:
(4)
where 
and
, and 
and 
are two independent standard Gaussian sequences of independent and identically distributed random variables.
We relate the discretised model (4) to the discrete-time model (3) by letting
, 
, 
, and
. In addition, we have
, 
, 
,
, 
,
, and
. It now follows from Theorem 1 that the recursive estimator is of the form
where
and the mean-square error is given as
Example 1 (Klebaner’s Model) [15] considers a state space model in which the conditional mean of the observed diffusion process is modeled by the Black-Scholes process (see [16]) and given by:
where 
and 
are two independent standard Brownian motions. In this case, the Milstein approximation leads to
(0.5)
We relate (5) to the discrete-time model (3) by letting
, 
, 
, and
.
Also, we put
, 
, 
,
, 
and
. It now follows from Theorem 1 that the recursive estimator is of the form
where
and the mean-square error is given as
Example 2 (Hull and White Model) [17] proposed a stochastic volatility model in which the conditional variance of the observed diffusion process is modeled by a Black-Scholes process and given by:
where 
and 
are two correlated standard Brownian motions with
. We use Ito’s formula to obtain
:
To simplify the Milstein approximation, we treat the coefficient on 
as a function of only
. In this case, the Milstein approximation leads to
(6)
We relate (6) to the discrete-time model (3) by letting
, 
, 
,
.
Also, we put
, 
, 
, 
, 
and
. It now follows from Theorem 1 that the recursive estimator is of the form
where
and the mean-square error is given as
When correlation
, the model simplifies to
Example 3 (CIR Model) Consider the CIR model for observed process 
given by
and the state process 
follows a diffusion process of the form
In this case, the Milstein approximation for 
and 
leads to
(7)
respectively.
We relate (7) to the discrete-time model (3) by letting
, 
, 
, 
and
. Also, we put
, 
,
, 
, 
and
. It now follows from Theorem 1 that the recursive estimator is of the form
where
and the mean-square error is given as
4. Bond Valuation with Recursive Learning under Milstein Approximation
We now present the computation of a zero coupon bond price in the setting of a two-factor CIR model. In twofactor models, in general, bond yields are deterministic (and usually affine) functions of two factors. There are at least two reasons for why two-factor (or even multi-factor) models are more preferable to single-factor models. First, the empirical difficulties of fitting the shape of the term structure of zero rates and their volatilities and the variation of interest rate spreads in single-factor models are well known. Second, there are institutional restrictions on the behavior of interest rates that mandate more factors than one. Central banks tend to target certain levels (or ranges) of interest rates. These levels themselves may change over time as economic conditions change. As an example we consider a variant of the two-factor CIR model presented in [18]. The model defines the short rate as a CIR process with long-run mean (also known as central tendency) being itself a CIR process:
where
. The Milstein approximation is readily available:
(8)
and
. Note that the new state variable processes are no longer normal. Rather, they are a mixture of normal and chi-squared random variables.
Because investors do not observe
, the task of pricing a zero coupon bond is a two-stage exercise. First, investors estimate the latent central tendency process,
. For that purpose, we assume they use the rule described in Theorem 1, so that
and
This last term simplifies to
Second, investors value the bond conditional on the pair
. Thus, investors’ problem is the joint problem of estimation of the latent state process and simultaneous valuation of the bond.
The fundamental valuation principle in asset pricing states that if there is no arbitrage, then there exists a positive pricing kernel (also called stochastic discount factor (SDF)) such that the following condition is satisfied by any 
-period return on any asset at any time:
(9)
In our example we are interested in an 
-period return on a zero coupon default-free bond, 
where 
is the time 
price of a zero coupon bond with 
periods remaining until maturity. The complete information version of this model is affine, and the solution for a bond price in the complete information case is available in continuous time. Here, we can start with discrete-time SDF
(10)
Finding SDF parameter restrictions requires the knowledge of the following integral of an exponential-quadratic function of a standard normal variable,
:
(11)
with transversality condition
. The condition that the expectation of an 
-period SDF has to give us the 
-period short rate allows us to find SDF coefficient restrictions:
Using the fundamental pricing Equation (9), the SDF expression (10), and the expression for the expectation of the exponential-quadratic function of the standard normal variable in (11), we have
(12)
For SDF (10) to be consistent with restriction (12), we must have
Inserting SDF (10) into the pricing Equation (9), we obtain the following expression for the price of a zerocoupon bond maturing at time 
(let
):
By definition, the yield on this bond is given by
Unfortunately, the learning implications of the model render the final bond expression non-affine in the state variables. The expectation above, however, can be easily computed using Monte Carlo integration.
When constructing the term structure of interest rates we make maturities, 
, range from one year to 10 years. The discretisation time step, 
, is kept constant at 
of a year. As a base case for our simulations we take the following parameter values. We choose the speed of mean reversion in both the short rate and the central tendency to be
, so that they are consistent with high persistence of the state variables. E.g., for
, the persistence of the non-Gaussian AR(1) short rate process in (8) is equal to 
. Both 
and 
have virtually identical impact on the term structure of zero yields1. This influence, however, is strong as we might expect. Intuitively, larger speed of mean reversion pulls the state variables faster to the long run mean,
. The result is that all yields are larger with the intermediate yields being affected the most, which increases the concavity of the term structure as represented in Figure 1.
The shape of the term structure strongly depends on the relative position of the current short rate with respect to the long run mean of the central tendency, 
2. Our model produces rich patterns of the term structure similar to non-discretised CIR models. If the short rate is below the mean, the term structure is upward-sloping, otherwise, it is inverted. For our numerical results we set the long run mean of the central tendency at 
in the base case. The level of 
has a strong effect on both the levels and the curvature of the term structure, with the latter being affected the most by 
than any other parameter of the model (see Figure 2).
Our numerical simulations show that, interestingly, the instantaneous volatilities of both the short rate and the central tendency are largely irrelevant for the shape and level of the term structure. We start with the base case values of the volatilities given by
. As an example, the yields on a 
-year and 
-year zeros in the base case are 
and
, respectively. If we increase 
substantially to, say, 0.1, the corresponding new yields are identical to those obtained with base case parameters. Likewise, if we increase 
from 0.01 to 0.1, we do not see any change in any of the yields3.
The base case risk premiums are 
and
. Zero yields are largely insensitive to the value of
. However, the second risk premium, which is the loading on the non-Gaussian component in the SDF, has strong influence on the term structure. This non-Gaussian risk premium affects zero rates of all maturities in the same way leading to parallel shifts in the yield curve. Even though the shape of the term structure is largely not affected, the yields are very sensitive to the level of the second risk premium. E.g., a change in 
from the base case level of 0.001 to 0.05 adds about 980 basis points to yields of all maturities as shown in Figure 3.
5. Conclusion
Recently, it has been demonstrated (see [19]) that the diffusion process can be well approximated by the Milstein approximation rather than the Euler approximation.
Figure 1. Term structure as a function of the speed of mean reverison in the short rate. We use base parameters presented in the text to generate the term structure of zero rates. The underlying model is the discretized version of a continuous-time Cox-Ingersoll-Ross (CIR) model with central tendency of the short rate also following a CIR process. We use the Milstein discretization scheme. The curves represent the mean yields over 10,000 Monte Carlo iterations. The time step in the Milstein scheme is 1/500 of a year. The speed of mean reversion parameter, 
ranges from 0.5 to 2. In our simulations, we assume that both the short rate and the central tendency start at 0.01. We also assume that the posterior variance of the central tendency estimate, λt, starts at the level of two instantaneous standard deviations of the central tendency, 
per year.
Figure 2. Term structure as a function of the central tendency of the short rate. We use base parameters presented in the text to generate the term structure of zero rates. The underlying model is the discretized version of a continuous-time Cox-Ingersoll-Ross (CIR) model with central tendency of the short rate also following a CIR process. We use the Milstein discretization scheme. The curves represent the mean yields over 10,000 Monte Carlo iterations. The time step in the Milstein scheme is 1/500 of a year. The long run mean of the central tendency, θ, ranges from 0.1 to 0.4. In our simulations, we assume that both the short rate and the central tendency start at 0.01. We also assume that the posterior variance of the central tendency estimate, λt, starts at the level of two instantaneous standard deviations of the central tendency, 
per year.
Figure 3. Term structure as a function of the second rsik premium, λ2. We use base parameters presented in the text to generate the term structure of zero rates. The underlying model is the discretized version of a continuous-time Cox-IngersollRoss (CIR) model with central tendency of the short rate also following a CIR process. We use the Milstein discretization scheme. The curves represent the mean yields over 10,000 Monte Carlo iterations. The time step in the Milstein scheme is 1/500 of a year. The non-Gaussian risk premium, λ2, ranges from 0.5 to 2. In our simulations, we assume that both the short rate and the central tendency start at 0.01. We also assume that the posterior variance of the central tendency estimate, λt, starts at the level of two instantaneous standard deviations of the central tendency, 
per year.
In this paper, we study the recursive estimates for various classes of discretely sampled continuous time stochastic volatility models using the Milstein approximation. We also provide an example of joint valuation of a zerocoupon bond and learning about an underlying state variable under incomplete information environment.
NOTES
2In our simulations, we assume that both the short rate and the central tendency start at 0.01. We also assume that the posterior variance of the central tendency estimate, 
, starts at the level of two instantaneous standard deviations of the central tendency, 
, i.e., 
per year.
3Only if we increase these volatilities to unrealistic levels by a factor of 1000, do the yields decline. The decline, however, is minuscule, half a basis point or less.