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The author expresses deep gratitude to Abbas Mirakhor, Abdoulaye Bio-Tchané, Genevieve Labeyrie, Robert Flood, Saeed Mahyoub, U. Jacoby, A. Kovanen, C. Steinberg (all IMF), and Ibrahim Gharbi (the Islamic Development Bank) for their help with this paper.
When BP shut down the Prudhoe Bay field in Alaska for pipeline maintenance.
Futures’ contracts on Brent, three-month delivery; the sample contains 1130 observations. The source is Reuters.
The recessionary effect of high oil prices has been studied by Hamilton (1983). A considerable literature thereafter has dealt with the relationship between oil shocks and real GDP. By causing a general increase in the price level, an oil shock, ceterus paribus, reduces real cash balances and therefore aggregate demand.
Investors and speculators, through opening and closing positions on the futures markets, affect price dynamics and increase price volatility. However, their role is limited to the short run. Given the sample period under study, underlying fundamentals were key determinants of the oil price process. Incidentally, the IMF World Economic Outlook, September 2006, could not establish evidence for a long-term effect of speculation on oil prices.
See, for instance, The International Energy Agency, Oil Market Report, September, 2006. This figure includes natural gas liquids.
World economy was reported to have grown at about 4-5 percent in real terms during 2002-2006. See International Monetary Fund, World Economic Outlook, September, 2006.
The relationship between oil prices and economic fundamentals was studied in an IMF working paper (WP/06/62). Besides estimating demand and supply functions for crude oil, the paper analyzed the influence of monetary policy on crude oil prices.
The frequency of jumps exceeding ±3 percent was estimated from the sample at 23 percent.
Implied volatility is the volatility which equates the Black-Scholes (1973) call option pricing formula with the call option’s market value.
GARCH stands for Generalized Autoregressive Conditional Heteroskedasticity. A GARCH (1,1) model is defined as follows: The mean equation:
A solution to this SDE can be written as
Ball and Torous (1985) modeled the jump component in Merton’s model as a Bernoulli process. In this respect, either one or no abnormal event occurs during the time interval (t, t + Δ), with Prob[one abnormal event]= λΔ, Prob[no abnormal event]= 1 − λΔ, and Prob[more than one abnormal event]=0. The density function for the log-return becomes: f (x) = N (μΔ + nβ, σ2 Δ + nδ2)(λΔ) + N (μΔ, σ2Δ) (1 − λΔ)
The characteristic function φX (u) is related to the moment generating function GX (u) GX(u) = E[exp(uXt)] = ∫exp(uXt)dF (Xt)by change of the transform variable u → −iu, namely GX (iu) = φX (u), and GX (u) = φX (−iu).
Parzen (1962), Feuerverger and Mureika (1977), Feuerverger and McDunnough (1981a and 1981b)) suggested the use of the CF to deal with the estimation of density functions. Madan and Seneta (1987) proposed a CF-based approach to estimate the J-D model. In the same vein, Bates (1996), Duffie et al. (2000), Chacko and Viceira (2003), and many other authors have proposed the use of CF for estimating affine J-D models.
Note that Xn = elog(Xn) =enlog(X). Therefore,
A Levy process (LP) (Xt)t≥0 has a value X0 = 0 at t = 0 and is characterized by independent and stationary increments, and stochastic continuity, i.e., discontinuity occurs at random times. The CF of a LP is given by the Levy-Khintchine formula:
The probability density of the Gamma process with mean rate t and variance υt is well known:
Cont and Tankov (2004) argued that the inverse problem could be an ill-posed problem and proposed the use of relative entropy, which is the Kullback-Leibler distance for measuring the proximity of two equivalent probability measures, as a regularization method with the prior distribution estimated from the statistical data via the maximum likelihood method. This regularization will enable to find a unique martingale measure.
The cumulants are also equivalently defined in terms of the characteristic function, which is the Fourier transform of the probability density function: