Estimation of the drift

Jiang and Knight (1997) apply the Kolmogorov forward equation (3.65) to develop a nonparametric estimator for the drift m. Integrating (3.66) with respect to x yields for the drift function Using (3.67) we define a nonparametric estimator of m by replacing the diffusion coefficient σ², the marginal density f and its derivative f⁰ by their nonparametric estimators.

Estimators for σ² and f are already given in the previous sections. From (3.54) and the definition of f_T⁽ⁿ⁾(x), a natural way to estimate the derivative of the marginal density is

Jiang and Knight (1997) show, that the estimatorm⁽ⁿ⁾_T (x) given by m⁽ⁿ⁾_T (x)^def= 1 is pointwise consistent for m(x). Under additional technical assumptions,Jiang and Knight (1997) show, that S_T⁽ⁿ⁾ is differentiable.

3.4.4 Fixed Sample Properties

To investigate the performance of the proposed estimators we apply them to simulated diffusion processes.

The general stochastic differential equation, that the simulated processX follows is

dX(t) = m{θ, X(t)}dt+σ{θ, X(t)}dW(t) t >0 (3.69) where θ is a parameter vector. To get discrete observations of X we use a Milstein scheme as in (3.52).

In the empirical analysis we test parametric models for the diffusion coefficient of the spot rate and various stock price processes. For this reason we will investigate the fixed sample properties of the nonparametric estimators applied to these models.

A summary of the investigated models is given in Table 3.1. The parameters are

57

Name σ(x) θ₃

constant (VC) θ₃ 0.013

square root (CIR) θ₃√

x 0.066 Chan, Karolyi, Longstaff, Sanders (CKLS) θ₃x^1.5 1.2 Table 3.1: Diffusion coefficient models used in the simulation study

chosen accordingly to the estimated values in Ahn and Gao (1999). They estimated the parameters of different models applied to the one month US treasury bill rate.

Note, that they did not estimate the parameters for all combinations of drift and diffusion coefficients that we use here. However, the parameters in our simulation study generate trajectories, that are positive and in the range of about 0.02 - 0.2 and thus might be a good choice to simulate interest rate processes.

Ahn and Gao (1999) also estimate a model introduced by Duffie and Kan (1996), where the diffusion coefficient is given by σ(x) =√

θ₃+θ₄x. Since this model is not consistent for values ofX(t) smaller than−θ3/θ4 we will not use it in our simulation study.

To study the performance of the drift function estimator given in 3.4.3 and the influence of the unknown drift m(x) on the estimates of σ², we combine each of the three diffusion coefficients with a drift function proposed by Ahn and Gao (1999), i.e. we simulate paths from three diffusion models. The function is given in Table3.2 along with the parameter values estimated by Ahn and Gao (1999). For the reason of empirical relevance we use the given parameter values in our simulation.

Name m(x) θ1 θ2

Ahn-Gao model (AG) θ₁(θ₂−x)x 3.4 0.08 Table 3.2: Ahn-Gao model used in the simulation study

For every model we simulate 1000 paths of length nT = 2500. To simulate the trajectories we apply the Milstein scheme (3.52) with δ = 1/10. This means, that we calculate 10 realizations per day but sample the data daily. Since the parameter values given in Tables 3.1 and 3.2 are annual values, we choosen = 250 (250 trading days per year), i.e. we have T = 10 years of observations.

We start with the diffusion coefficient estimation. The estimator we apply is the local time estimator given in (3.60). The mean of the estimated functions is shown

in Figure 3.6 along with the 90% empirical confidence bands (green) and the true function (red). We find in the figures that the means of the estimates for the three diffusion coefficient functions are close to the corresponding true functions and we therefore conclude, that the estimators are unbiased in this situation.

In the first figure, constant σ², we find, that the confidence bands are small for states x of the process near 0.08. The reason is, that the level of mean reversion of the first model is 0.08 and therefore realizations of the process close to 0.08 occur more frequently than realizations in other regions of the state space. The middle and lower plot show that the width of the confidence bands increases with the level of X. The reason is that σ² is increasing in x and the variance of the estimator S_T⁽ⁿ⁾ depends on the level of the true function σ² as it can be seen from Proposition 3.1.

With the knowledge about a nonparametric estimate of σ² we are now able to estimate the drift coefficient as in 3.4.3. The results for the Ahn-Gao drift function estimated from the three models described above are shown in Figure 3.7. The plots show the mean of the estimates (black), the true drift function (red) and the empirical 90% confidence band (green). As for the estimation of σ² the mean of the estimated drift is close to the true drift in all three models. However, it seems, that the estimator under-estimates the drift for values of xlarger than 0.08 (the level of mean-reversion) and over-estimates the drift for x <0.08.

Even if the bias of the drift estimator in this situation is negligible, 0 is contained in all confidence bands. Heuristically this means, that a drift function that is constantly 0 seems to be very likely in the situation here. Since the used parameter values correspond to estimated values by Ahn and Gao (1999) for an interest rate process, we doubt that for these kind of processes one can significantly distinguish between a zero drift and a mean reverting or quadratic drift. The reason is not only, that the drift is close to zero, but that the confidence bands are quite large. In particular 0 is included. The large confidence bands can be explained by the relative small sample size T, i.e. the number of years (T = 10) in which we have observations is too small to get a reliable drift estimate and to distinguish between a non zero and a zero drift.

The situation would change when we use different parameters. In particular, when we increase the speed of adjustment parameter θ₁, we increase the influence of the drift on the instantaneous behavior of the process. In the empirical analysis we come again to that point.

It also appears from Figure3.7that the diffusion coefficient does have an influence on the preciseness of the drift estimation. In particular, diffusion coefficients, that are increasing in x, like the square root and the CKLS model, produce non constant confidence bands for the drift estimation. Heuristically speaking, the drift estimate becomes more imprecise the larger the state of the process, and thus the larger the diffusion coefficient function. The reason is, that the impact of the drift on the incre-ments X_i+1 −X_i of the process becomes smaller when the diffusion coefficient gets larger. The diffusion coefficient can be interpreted as the instantaneous variance of

59

Constant diffusion coefficient

0.04 0.06 0.08 0.1

State space of X

2468

0.00012+sigma^2(X)*E-5

Square root diffusion coefficient

0.04 0.06 0.08 0.1 0.12

State space of X

12345

0.0001+sigma^2(X)*E-4

CKLS diffusion coefficient

0.05 0.1 0.15

State space of X

0246

sigma^2(X)*E-3

Figure 3.6: Nonparametric estimates ofσ² together with the empirical 90% confidence band.

these increments. And the larger this variance is, the more difficult is the estimation of the instantaneous expectation m(x). This phenomena can be compared to a uni-variate mean estimation, the higher the variance of the underlying random variable the more imprecise is the mean estimator.