On the Parameter Estimation in the Schwartz-Smith’s Two-Factor Model

Abstract

The two unobservable state variables representing the short and long term factors introduced by Schwartz and Smith in [16] for risk-neutral pricing of futures contracts are modelled as two correlated Ornstein-Uhlenbeck processes. The Kalman Filter (KF) method has been implemented to estimate the “short” and “long” term factors jointly with unknown model parameters. The parameter identification problem arising within the likelihood function in the KF has been addressed by introducing an additional constraint. The obtained model parameter estimates are the Maximum Likelihood Estimators (MLEs) evaluated within the KF. Consistency of the MLEs is studied. The methodology has been tested on simulated data.

A Derivations of (1) and (2)

In Sect. 2.1, we define the components of a bivariate Ornstein-Uhlenbeck process as

$$\begin{aligned} d\chi _{t} = -\kappa \chi _{t}dt + \sigma _{\chi }dZ_{t}^{\chi } \end{aligned}$$

(8)

$$\begin{aligned} d\xi _{t} = (\mu _{\xi } - \gamma \xi _{t})dt + \sigma _{\xi }dZ_{t}^{\xi }, \end{aligned}$$

(9)

where $dZ_t^{\chi }, dZ_t^{\xi } \sim N(0, \sqrt{\varDelta t})$ are correlated standard Brownian motions. Here we show how to derive (1) and (2). Firstly, from (8),

$$\begin{aligned} \varDelta \chi _t = -\kappa \chi _t \varDelta t + \sigma _{\chi } \sqrt{\varDelta t} \epsilon _{\chi }. \end{aligned}$$

Therefore,

$$\begin{aligned} \chi _{t+1} = (1 - \kappa \varDelta t) \chi _t + \sigma _{\chi } \sqrt{\varDelta t} \epsilon _{\chi }. \end{aligned}$$

(10)

Similarly, from (9), we get

$$\begin{aligned} \xi _{t+1} = (1 - \gamma \varDelta t) \xi _t + \mu _{\xi }\varDelta t + \sigma _{\xi } \sqrt{\varDelta t} \epsilon _{\xi }, \end{aligned}$$

(11)

where $\epsilon _{\chi }, \epsilon _{\xi } \sim N(0,1)$. Let $Corr(\epsilon _{\chi }, \epsilon _{\xi }) = \rho$ and $w = \left( \begin{matrix} \sigma _{\chi } \sqrt{\varDelta t} \epsilon _{\chi } \\ \sigma _{\xi } \sqrt{\varDelta t} \epsilon _{\xi }\end{matrix} \right)$, then

$$\begin{aligned} W=Var(w) = \left( \begin{matrix} \sigma _{\chi }^2 \varDelta t &{} \rho \sigma _{\chi } \sigma _{\xi } \varDelta t \\ \rho \sigma _{\chi } \sigma _{\xi } \varDelta t &{} \sigma _{\xi }^2 \varDelta t \end{matrix} \right) . \end{aligned}$$

(12)

Let $X_t = \left( \begin{matrix} \chi _t \\ \xi _t \end{matrix} \right)$, $c = \left( \begin{matrix} 0 \\ \mu _{\xi }\varDelta t \end{matrix} \right)$ and $G = \left( \begin{matrix} 1 - \kappa \varDelta t &{} 0 \\ 0 &{} 1 - \gamma \varDelta t \end{matrix} \right)$. Then from (10) and (11) we get

$$\begin{aligned} X_{t+1} = c + GX_t + w_{t+1}. \end{aligned}$$

Let $\phi = 1 - \kappa \varDelta t$, $\psi = 1 - \gamma \varDelta t$. Then

$$E(X_t) = \left( \begin{matrix} (1-\kappa \varDelta t) \chi _{t-1} \\ (1 - \gamma \varDelta t) \xi _{t-1} + \mu _{\xi } \varDelta t \end{matrix} \right) $$

$$ = \left( \begin{matrix} (1-\kappa \varDelta t)^n \chi _0 \\ (1 - \gamma \varDelta t)^n \xi _0 + (1-\gamma \varDelta t)^{n-1} \mu _{\xi } \varDelta t + ... + (1-\gamma \varDelta t)^0 \mu _\xi \varDelta t \end{matrix} \right) $$

$$ = \left( \begin{matrix} \phi ^n \chi _0 \\ \psi ^n \xi _0 + \mu _{\xi } \varDelta t \frac{1-(1-\gamma \varDelta t)^n}{\gamma \varDelta t} \end{matrix} \right) $$

$$\begin{aligned} = \left( \begin{matrix} \phi ^n \chi _0 \\ \psi ^n \xi _0 + \frac{\mu _{\xi }}{\gamma }(1-\psi ^n) \end{matrix} \right) , \end{aligned}$$

(13)

and

$$Var(X_t) = G\cdot Var(X_{t-1}) \cdot G' + W = G^{n}Var(X_0)(G')^{n} + G^{n-1}W(G')^{n-1} + ... + G^0W(G')^0.$$

If we assume $Var(X_0) = 0$, we can get

$$Var(X_t) = G^{n-1}W(G')^{n-1} + ... + G^0W(G')^0$$

$$= \left( \begin{matrix} \sigma _{\chi }^2 \varDelta t \sum \nolimits _{i=0}^{n-1}\phi ^{2i} &{} \rho \sigma _{\chi } \sigma _{\xi } \varDelta t \sum \nolimits _{i=0}^{n-1}(\phi \psi )^{i} \\ \rho \sigma _{\chi } \sigma _{\xi } \varDelta t \sum \nolimits _{i=0}^{n-1}(\phi \psi )^{i} &{} \sigma _{\xi }^2 \varDelta t \sum \nolimits _{i=0}^{n-1}\psi ^{2i} \end{matrix} \right) $$

$$\begin{aligned} =\left( \begin{matrix} \sigma _{\chi }^2 \varDelta t \frac{1-\phi ^{2n}}{1-\phi ^2} &{} \rho \sigma _{\chi } \sigma _{\xi } \varDelta t \frac{1-(\phi \psi )^n}{1-\phi \psi } \\ \rho \sigma _{\chi } \sigma _{\xi } \varDelta t \frac{1-(\phi \psi )^n}{1-\phi \psi } &{} \sigma _{\xi }^2 \varDelta t \frac{1-\psi ^{2n}}{1-\psi ^2} \end{matrix} \right) . \end{aligned}$$

(14)

When $n \rightarrow \infty$, $\varDelta t = t/n \rightarrow 0$, $\varDelta t^2 = 0$, then

$$\phi ^n = (1-\frac{\kappa t}{n})^n \rightarrow e^{-\kappa t}, $$

$$\psi ^n = (1-\frac{\gamma t}{n})^n \rightarrow e^{-\gamma t}, $$

$$(\phi \psi ) ^ n = (1-(\kappa + \gamma ) t/n)^n \rightarrow e^{-(\kappa + \gamma ) t}, $$

$$ 1-\phi ^2 = 2\kappa \varDelta t, 1-\psi ^2 = 2\gamma \varDelta t, 1 - \phi \psi = (\kappa + \gamma )\varDelta t. $$

From Eqs. (13) and (14), we have

$$\begin{aligned} E(X_t) = \left( \begin{matrix} e^{-\kappa t} \chi _0 \\ e^{-\gamma t} \xi _0 + \frac{\mu _{\xi }}{\gamma }(1 - e^{-\gamma t}) \end{matrix} \right) \end{aligned}$$

and

$$\begin{aligned} Var(X_t) = \left( \begin{array}{cc} \frac{\sigma _{\chi }^2}{2\kappa }(1-e^{-2\kappa t}) &{} \frac{\rho \sigma _{\chi } \sigma _{\xi }}{\kappa + \gamma } (1-e^{-(\kappa + \gamma ) t}) \\ \frac{\rho \sigma _{\chi } \sigma _{\xi }}{\kappa + \gamma } (1-e^{-(\kappa + \gamma ) t}) &{} \frac{\sigma _{\xi }^2}{2\gamma }(1-e^{-2\gamma t}) \end{array} \right) , \end{aligned}$$

in the linearised form $Var(X_{{\varDelta } t})\approx W$ from (12).