Risk-Hedging Strategies and Portfolio Management

Abstract

This chapter considers first the risk factors impacting jointly the Clean Development Mechanism and the European Union Emissions Trading Scheme. Second, it provides an econometric modelling exercise of risk premia in CO₂ allowance spot and futures prices under the EU ETS (based on Bessembinder and Lemmon, J. Finance 57:1347–1382, 2002), which takes into account the specificities of carbon allowances in terms of storage. Third, carbon price risks in the power sector are identified econometrically by analyzing the factors that influence fuel-switching. Fourth, a stylized portfolio management exercise is proposed by using mean-variance optimization in a broadly diversified portfolio composed of stocks, bonds, weather, energy and carbon assets. The Appendix details how to compute implied volatility series from options, which can then be used to represent the risk-neutral probability distribution of carbon prices.

Appendix: Computing Implied Volatility from Option Prices

In the financial economics literature, risk-hedging strategies may also be designed by resorting to option prices (see [20] for an application to EUAs by using a dataset of plain vanilla European option prices at-the-money). We briefly recall here how to derive implied volatility measures from option pricing.

Let C(τ,K) _obs be the observed call option price, and C(τ,K,σ) _BS the Black-Scholes (BS) price computed using the implied volatility σ. By definition, we have C(τ,K) _obs =C(τ,K,σ) _BS . The implied volatility of the strike price is obtained by numerically inverting the BS formula, which can be done by solving:

$$ \min_{\sigma} \bigl(C(\tau,K)_{obs} - C(\tau,K,\sigma)_{BS} \bigr)^2$$

(5.13)

Reference [20] shows that option prices on the carbon market lead to pricing errors that are usual for commodity or equity markets (see also [5]). Implied volatility series can then be used to represent the risk-neutral probability distribution of a given asset in financial economics.

5.1.1 Problems

Problem 5.1

(Calibrating GARCH Models for Carbon Prices)

• (a) Consider the descriptive statistics for the CO₂ price series in Table 5.9: how well do they seem to fit the GARCH(p,q) models specifications?

  Table 5.9 Descriptive statistics for CO₂ prices

• (b) Following the Box-Jenkins methodology, we configure the ARMA(p,q) processes that provide the best fit to the CO₂ time-series. Then, we estimate the corresponding GARCH(p,q) model for the CO₂ price series:

  $${{Y}_{t}}=\theta {{{X}'}_{t}}+{{\varepsilon }_{t}}$$

  (5.14)
  $$\sigma _{t}^{2}=\omega +\sum\limits_{i=1}^{p}{{{\alpha }_{i}}\varepsilon _{t-i}^{2}}+\sum\limits_{j=1}^{q}{{{\beta }_{j}}\sigma _{t-j}^{2}}$$

  (5.15)

  with $\sigma_{t}^{2}$ the conditional variance, which is function of a constant term ω, the ARCH term $\varepsilon^{2}_{t-i}$, and the GARCH term $\sigma^{2}_{t-j}$.

  In Table 5.10, comment on:

  • the statistical significance of the parameters obtained in the mean and variance equations;

  • the value of the Ljung-Box test;

  • the value of the ARCH test.

  Table 5.10 GARCH estimates for CO₂ prices

  In Fig. 5.10, comment on:

  • the plot of the standardized innovations of the GARCH(p,q) model;

  • the plot of the conditional standard deviations of the GARCH(p,q) model.

  Fig. 5.10

  

  Standardized innovations (left panel) and conditional standard deviations (right panel) of the GARCH(p,q) model

• (c) We consider other models than the standard GARCH(p,q) model with a Gaussian conditional probability distribution. First, we use the specification for the conditional variance proposed by Nelson (1991, [36]):

  $$\log(\sigma_t^2)=\omega+\sum_{i=1}^p \alpha_i \left|\frac{\varepsilon _{t-i}}{\sigma_{t-i}}\right|+\sum_{j=1}^q \beta_j \log (\sigma_{t-j}^2)+ \sum_{k=1}^r \gamma_k \frac{\varepsilon_{t-k}}{\sigma_{t-k}} $$

  (5.16)

  where γ tests for the presence of the leverage effect. Recall that the leverage effect implies a higher level of volatility associated to decreasing prices in the financial economics literature. The EGARCH model is estimated with a Student t distribution.

  Second, we use the asymmetric TGARCH model by Zakoian (1994, [45]):

  $$\sigma_t^{2}=\omega+ \sum_{i=1}^p \alpha_i \varepsilon_{t-i}^2+ \sum _{j=1}^q \beta_j \sigma_{t-j}^2 + \sum_{k=1}^r \gamma_k \varepsilon _{t-k}^2 \varGamma_{t-k} $$

  (5.17)

  where Γ _t =1 if ε _t <0, and 0 otherwise. ε _t−i >0 and ε _t−i <0 denote, respectively, good and bad news. The TGARCH model is estimated with a Generalized Error Distribution (GED).

  Consider the Tables 5.11 and 5.12: do you observe any improvement with the alternative specification structures in the variance equation?

  Table 5.11 EGARCH estimates for CO₂ prices

  Table 5.12 TGARCH estimates for CO₂ prices