Commodity derivative valuation under a factor model with time-varying market prices of risk

Abstract

It is well known that market prices of risk play an important role in commodity derivative valuation. There is an extensive literature showing that market prices of risk vary through time. Based on these results, a factor model, with two long- and short-term factors, with market prices of risk depending on these underlying asset factors is proposed and estimated, using data from crude oil, heating oil, unleaded gasoline and natural gas futures prices traded at NYMEX. The valuation results obtained with an extensive sample of commodity American options traded at NYMEX show that this model with time-varying market prices of risk outperforms standard models with constant market prices of risk.

Appendix: Kalman filtering

Detailed accounts for Kalman filtering are given in Harvey (1989) and also in Bakshi and Wu (2010) among others.

Let $Z_t=(\xi _t\chi _t\alpha _t\alpha _t^*)^{\prime }$ be the vector of all factors.¹⁰ The “risk-neutral” SDE of $Z_{t}$ can be expressed as $dZ_t =\left( {b^{\diamondsuit }+AZ_t} \right) dt+\Omega dW_{Zt}^{\diamondsuit }$, where $dW_{Zt}^{\diamondsuit }$ is a vector of independent Brownian motions, and therefore $\hbox {Var}(dZ_{t})=R= \Omega \Omega ^{T}$ ($\Omega ^{T}$ is the transpose matrix of $\Omega$), with the restriction explained above: $b^{\diamondsuit }=(\begin{array}{cccc} \mu _t-\lambda _{\xi 0}&-\lambda _{\chi 0}&0&0\end{array})$ and:

$$\begin{aligned} A=\left( {\begin{array}{cccc} -\lambda _{\xi _1}&{}\quad -\lambda _{\xi _2}&{}\quad 0 &{}\quad 0 \\ -\lambda _{\chi _1}&{}\quad -\kappa -\lambda _{\chi 2}&{}\quad 0 &{}\quad 0 \\ 0&{}\quad 0&{}\quad 0&{}\quad \varphi \\ 0&{}\quad 0&{}\quad -\varphi &{}\quad 0 \\ \end{array}} \right) \end{aligned}$$

Under this notation$X_t=cZ_t$, where $c=(\begin{array}{cccc}1&1&1&0\end{array})$.

It is easy to prove that the (unique) solution of that problem is (Oksendal 1992):

$$\begin{aligned} Z_t=e^{A t} \left[ {Z_0 +\int _0^t {e^{-A s}} b^{\diamondsuit }ds+\int _0^t {e^{-A s}} \Omega dW_{Zs}^{\diamondsuit }} \right] \end{aligned}$$

(9)

It is clear that, under the risk-neutral measure, given $Z_0$, $Z_t$ is Gaussian, with mean and variance¹¹

$$\begin{aligned} E^{*}\left[ {Z_t } \right]&= e^{A t} \left[ {Z_0 +\int _0^t {e^{-A s}} b^{\diamondsuit }ds} \right] \end{aligned}$$

(10)

$$\begin{aligned} Var^{*}\left[ {Z_t}\right]&= e^{A t} \left[ {\int _0^t e^{-A s}R(e^{-A s})^{T}ds}\right] (e^{A t})^{T} \end{aligned}$$

(11)

As $X_t=c Z_t =\xi _t +\chi _t +\alpha _t$, then under the risk-neutral measure, $X_t$ is also Gaussian with mean and variance:

$$\begin{aligned} E^{*}\left( {X_t } \right)&= cE^{*}\left( {Z_t } \right) \\ Var^{*}\left( {X_t } \right)&= cVar^{*}(Z_t )c^{T} \end{aligned}$$

This provides a valuation scheme for all sorts of commodity contingent claims as financial derivatives on commodity prices, real options, investment decisions and other more. In particular, the price of a futures contract traded at time “$t$” with maturity at time “$t+T$” is: $F_{t,T} =E^{*}\left[ {S_{t+T} \left\| {I_t } \right. } \right] =\exp \left\{ {E^{*}\left[ {X_{t+T} \left\| {I_t } \right. } \right] +\frac{1}{2}Var^{*}\left[ {X_{t+T} \left\| {I_t } \right. } \right] } \right\}$, where $I_{t}$ is the information available at time “$t$”. It can be expressed as:

$$\begin{aligned} F_{t,T} =\exp \left[ {ce^{AT}Z_t +g \left( T \right) } \right] \end{aligned}$$

(12)

where $g(T)=ce^{A T}\int _t^{t+T} {e^{-A s}} b^{\diamondsuit }ds+\frac{1}{2}ce^{A T} \left[ {\int _t^{t+T} {e^{-A s}R(e^{-A T}} )^{T}ds } \right] (e^{A T})^{T}c^{T}$, which is a deterministic function.

The Kalman filter technique is a recursive methodology that estimates the unobservable time series and the state variables or factors $(Z_{t})$ based on an observable time series $(Y_{t})$, which depends on these state variables.

If the difference between the current period and the initial period is one period time, $Z_{t}$ follows the discrete process:

$$\begin{aligned} Z_t =c_t +TZ_{t-1} +\psi _t\qquad t=1,\ldots ,N_t \end{aligned}$$

(13)

where $c_t =e^{At}\int _{t-1}^t {e^{-As}bds} \in \mathfrak {R}^{h}, T=e^{A}\in \mathfrak {R}^{h x h}and \psi _t \in \mathfrak {R}^{h}$ is a vector of serially uncorrelated Gaussian disturbances with zero mean and covariance matrix $Q=(e^{A})\left[ {\int _{t-1}^t {e^{-As}R(e^{-As})^{T}ds} } \right] (e^{A})^{T}$. This equation will be called, following standard conventions in the literature, the transition equation. It is worth noting that expression (13) can be derived from expression (9) and thus $c_{t}$, $T$ and $Q$ can be computed from expression (9).

The measurement equation is just the expression of the log-futures prices $(Y_{t})$ in terms of the factors $(Z_{t})$ by adding serially uncorrelated disturbances with zero mean $(\eta _{t})$ to take into account measurement errors derived from bid-ask spreads, price limits, non-simultaneity of observations, errors in data, etc. To avoid dealing with a great amount of parameters, the covariance matrix $H_{t}$ will be assumed diagonal with main diagonal entries equal to $\sigma _{\eta }$. This simple structure for the measurement errors is imposed so that the serial correlation and cross correlation in the log-prices is attributed to the variation of the unobservable state variables. The measurement equation (which can be derived from expression (12))will be expressed as:

$$\begin{aligned} Y_t =d_t +M_t Z_t +\eta _t\qquad t=1,\ldots ,N_t \end{aligned}$$

(14)

where $Y_t ,d_t \in \mathfrak {R}^{n}, M_t \in \mathfrak {R}^{n x h}, Z_t \in \mathfrak {R}^{h}$, $h$ is the number of state variables, or factors, in the model, and $\eta _t \in \mathfrak {R}^{n}$ is a vector of serially uncorrelated Gaussian disturbances with zero mean and covariance matrix $H_{t}$.

Let $Y_{t|t-1}$ be the conditional expectation of $Y_{t}$ and let $\Xi _t$ be the covariance matrix of $Y_{t}$ conditional on all information available at time $t-1$. Then, after omitting unessential constants, the log-likelihood function can be expressed as:

$$\begin{aligned} l=-\sum _t {\ln |\Xi _t |-\sum _t {(Y_t -Y_{t|t-1} )^{\prime }\Xi _t^{-1} (Y_t -Y_{t|t-1} )} } \end{aligned}$$

(15)