## _wp08133 - 1. Descriptive Statistics for Oil Price Returns

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---

### I. Introduction
- Crude oil prices maintained upward persistence since early 2003 and broke US$120/barrel in April 2008.
- Rapid oil price increases may:
  - Turn inflationary by causing other prices to rise.
  - Decelerate world economic growth.
- Recent upsurges occurred amid rising commodity prices, instability in housing, equity, and credit markets, and depreciating exchange rates.
- Fast rise in commodities including oil could reflect delayed effects of excessively expansionary monetary policies during 2001–04 when key interest rates were forced down to postwar record levels.
- Relaxation of monetary policy in August 2007─March 2008 set off renewed commodity price inflation and currency depreciation; multi-billion bail out facilities and cuts in federal funds rate were undertaken.
- Paper scope and approach:
  - Analyzed oil prices during 2000M1–2007M10.
  - Distinguished two samples: 2000M1–2003M4 and 2003M5–2007M10 to exhibit monetary policy effects.
  - Assumed oil prices driven by Levy processes (LP) of generalized hyperbolic (GH) type; used daily data to estimate parameters.
  - Emphasized GH/NIG advantages: combine features of normal and stable distributions, model asymmetry, frequent small and large jumps, and reduce option smile.
  - Applied Esscher transform to obtain martingale (risk-neutral) processes for derivative pricing.
  - Also estimated implied risk-neutral distribution from option prices on November 2, 2007 for end-December 2007, finding traders expected further oil price rise and significant right-tail probability.

### II. Recent evolution of oil prices (2000M1–2007M10)
- Data: Light crude futures prices from Reuters, 1986 observations, January 4, 2000 to October 29, 2007.
- Two distinct patterns observed:
  - 2000M1–2003M4: relative stability around a mean of US$27 per barrel.
  - 2003M5–2007M10: strong upward deterministic trend and persistence, prices rose to cross US$96/barrel in October 2007.
- ARMA representations:
  - Sample 2000M1–2003M4:
    - S = 1.325*S(-1) - 1.195*S(-2) + 0.842*S(-3) – 0.191*MA(1) + 0.778*MA(2) + 0.214*MA(3) + 0.769
    - R^2 = 0.97; DW = 2.01
  - Sample 2003M5–2007M10:
    - S = 0.990*S(-1) + 0.000537*TREND + 0.098*MA(1) + 0.307
    - R^2 = 0.99; DW = 2.00
- Interpretation: In the second sub-period the lag structure shortened and a deterministic trend became significant.

### III. Descriptive statistics for oil price returns
- Returns definition: xt = log(St) - log(St-1) = -log(St-1/St).
- Empirical moments (exact values from Table 1):
  - 2000M1–2003M4:
    - Mean = -0.005312
    - Median = 0.000000
    - Maximum = 11.23630
    - Minimum = -9.436276
    - Std. Dev. = 2.225607
    - Skewness = 0.056786
    - Kurtosis = 4.954168
    - Jarque-Bera = 133.9488
    - Probability = 0.000000
  - 2003M5–2007M10:
    - Mean = 0.118992
    - Median = 0.128184
    - Maximum = 7.356257
    - Minimum = -6.080348
    - Std. Dev. = 1.686031
    - Skewness = -0.021088
    - Kurtosis = 3.873152
    - Jarque-Bera = 36.55292
    - Probability = 0.000000
- Annualized interpretations preserved exactly:
  - Mean return -0.005 in first sub-period → decline in oil price at 1.28 (=-0.005*255) percent per year.
  - Mean return 0.12 in second sub-period → increase in oil price at 30.6(=0.12*255) percent per year.
  - Volatility (daily Std. Dev.):
    - Fell from 2.226 (annualized to 35.6(=2.226*255) percent) to 1.686 (annualized to 26.9(=1.686*255) percent).
- Distributional conclusions:
  - Skewness small in both periods → distributions approximately symmetric.
  - Kurtosis declined from 4.95 to 3.87.
  - Normality rejected in both periods → prominence of large jumps in daily oil prices.
  - Overall: markets out-of-equilibrium in second sub-sample, driven by strong upward trend reflecting rising demand, supply rigidities, and tensions.

### IV. Volatility dynamics (GARCH)
- Volatility estimated using GARCH(1,1).
- Characteristics:
  - High for daily data, exhibiting volatility clustering followed by mean reversion.
  - Annualized volatility:
    - First sub-period around 35 percent.
    - Second sub-period fell sharply to 22 percent, consistent with a solid deterministic trend.

### V. Modeling oil prices as a Lévy process and NIG specification
- SDE for asset price with LP: dSt = (ρ)St dt + σ St dZt− (notation as in source).
- LP properties:
  - Independent increments, stationary increments, stochastic continuity.
  - LPs are limits of random walks and infinitely divisible.
- Rationale: LPs account for skewness, excess kurtosis, and frequent small and large jumps.
- NIG specifics:
  - NIG is a special case of GH distribution with λ = 1/2.
  - GH constructed as normal variance-mean mixture with GIG mixing distribution.
  - NIG density, moments, MGF, and skewness/kurtosis formulas given explicitly in source.
  - NIG process: pure jump and infinite activity; representable as time-changed Brownian motion with inverse Gaussian (IG) time-change.
  - Characteristic function tractability enables option pricing via FFT and recovery of distributions via numerical inversion.

### VI. Estimation of NIG parameters for oil price returns (exact estimates)
- Software: R with HyperbolicDist, ghyp, and fBasics packages.
- Parameter estimates (Table 2, parameterization (α, β, δ, μ) and associated moments):
  - 2000M1–2003M4:
    - Alpha = 0.54
    - Beta = -0.02
    - Delta = 2.69
    - Mu = 0.08
    - Mean = -0.005
    - Variance = 4.92
    - Skewness = -0.08
    - Kurtosis = 1.72
  - 2003M5–2007M10:
    - Alpha = 0.97
    - Beta = -0.06
    - Delta = 2.76
    - Mu = 0.29
    - Mean = 0.12
    - Variance = 2.86
    - Skewness = -0.11
    - Kurtosis = 1.13
- Alternative parameterization (Table 3) results:
  - 2000M1–2003M4:
    - Lambda = -0.5
    - Alpha.bar = 1.46
    - Shape parameter = 0.08
    - Mu (location) = 2.22
    - Sigma (dispersion) = -0.08
    - Mean = -0.005
    - Variance = 4.95
    - Skewness = -0.08
    - Kurtosis = 2.06
  - 2003M5–2007M10:
    - Lambda = -0.5
    - Alpha.bar = 2.68
    - Shape parameter = 0.29
    - Mu (location) = 1.69
    - Sigma (dispersion) = -0.17
    - Mean = 0.12
    - Variance = 2.87
    - Skewness = -0.04
    - Kurtosis = 1.22
- Key estimation conclusions (preserved from source):
  - NIG fits oil price returns closely in both sub-periods.
  - Parameters changed significantly in 2003M5–2007M10:
    - Location parameter μ increased from 0.08 to 0.29.
    - Mean return increased from -0.005 (annualized to -0.005*255 = -1.28 percent) in 2000M1–2003M4 to 0.12 (annualized to 0.12*255 = 30.6 percent) in 2003M5–2007M10.
  - Decline in kurtosis and volatility consistent with greater predictability/deterministic trend in the second period.
- Additional parameter observations (exact figures from source):
  - Scale parameter δ increased from 2.69 to 2.76.
  - Shape parameter β ranged -0.02–-0.06.
  - Shape parameter α increased from 0.54 to 0.97.
  - Volatility fell from 2.2 (annualized to 35 percent) to 1.7 (annualized to 27 percent).

### VII. Market incompleteness, Esscher transform, and risk-neutral measures
- LP markets are incomplete; multiple equivalent martingale measures exist.
- Esscher transform maps P to Q via parameter h chosen so discounted asset price is a martingale.
- Definitions (preserved from source):
  - St = S(0) e^{Xt}, Xt an LP with X(0)=0.
  - MGF M(u,t) = E[e^{uXt}] = ∫ e^{ux} f(x,t) dx; infinite divisibility implies M(u,t) = [M(u,1)]^t.
  - Esscher density: f(x,t;h) = e^{hx} f(x,t) / M(h,t).
  - dQ/dP |_t = e^{hXt} / M(h,t).
- Martingale condition solved for h* (source provides algebraic condition specific to NIG).
- Computed Esscher parameters (as reported in source):
  - h = -0.4858 for 2000M1–2003M4.
  - h = -0.5152 for 2003M5–2007M10.
- Corresponding Esscher transforms (signs as in source):
  - dQ/dP = exp(0.4858 t + 0.2439 Xt) for first sub-sample.
  - dQ/dP = exp(0.5152 t + 0.2691 Xt) for second sub-sample.
- Alternative risk-neutral construction (Carr et al., 2003) and expressions for log-price and CF under risk-neutral measure given in source.

### VIII. Pricing, characteristic functions, and illustrative computation
- Example market inputs (November 2, 2007):
  - Futures oil price for end-December 2007: US$95.93/barrel.
  - US three-month treasury bill: r = 4.595 percent.
  - Time to maturity t = 57/365 = 0.16.
  - Risk-neutral parameters example: α = 0.97, β = -0.06, δ = 2.76, μ = 0.29.
- Risk-neutral CF for NIG and pricing via FFT described; numerical CF expression provided in source.

### IX. Inverse problem: inferring risk-neutral density from option prices (empirical implementation)
- Estimation objective: estimate θ = (α, β, δ, μ) by minimizing quadratic pricing error over M options subject to put-call parity and possible penalty λ ≥ 0 (formulas (24)–(26) in source).
- Option-pricing methods:
  - When density not closed-form, use characteristic function methods (Carr and Madan (1999)) and FFT.
  - ECF and Tikhonov regularization approaches detailed in source.
- Empirical implementation (November 2, 2007 market data) — two methods produced implied NIG parameters:
  - Method 1 (FFT + constrained minimization):
    - α̂ = 3.30, β̂ = 0.35, δ̂ = 5.09, μ̂ = 1.75.
  - Method 2 (ECF + GMM):
    - α̂ = 3.1, β̂ = 0.30, δ̂ = 5.41, μ̂ = 1.74.
- Implied option statistics (exact values as reported):
  - First method: mean 2.29 percent, variance 1.57 (sigma=1.25), skewness 0.08, kurtosis 0.19.
  - Second method: expected mean 2.27 percent, variance 1.77 (sigma=1.33), skewness 0.07, kurtosis 0.19.
- Market-implied average oil price at end-December 2007: US$98/barrel = (95.93*exp(0.022)).
- Interpretation from source:
  - Dispersion measured by sigma was lower than statistical values in Tables 2 and 3.
  - Positive skewness implied higher probability for oil prices to rise above expected mean.
  - Kurtosis below 3 implied flatter NIG distribution compared with normal distribution and higher than normal probability for tail events.
  - Derived risk-neutral density had low kurtosis, implying flatter distribution and significant probability for tail events.

### X. Short-term market expectations, driving fundamentals, and commodity dynamics
- Market participants’ short-term expectations influenced by:
  - Expansionary monetary policy since 2001.
  - Sharply depreciating U.S. dollar: fell by over 65 percent vis-à-vis Euro since 2001.
  - Higher world economic growth and higher demand for oil.
  - Crude oil supply rigidities leading traders to expect excess demand and further upward pressure on oil prices.
- Commodity-wide dynamics:
  - World aggregate demand for commodities expanded, resulting in double digit inflation for commodities prices, estimated at about 23 percent per year in 2003M5–2007M10.
  - High commodity inflation would erode real interest rate and stimulate further real aggregate demand.
- Financial conditions and monetary stance noted in source:
  - Sub prime market defaults, large write-offs by leading banks, and piling up of credits signaled restraints in monetary policy were not in the offing soon.
  - Further relaxation of monetary policy in August-December 2007 coincided with immediate weakening in U.S. dollar and surge in oil prices.
  - These conditions likely led traders to assume persistence in oil prices and to allow more likelihood for right tail events.

### XI. Risks, scenarios, and potential macroeconomic consequences
- Key open questions (as posed in source):
  - How far oil prices could rise without reaching critical zone that triggers a world recession or a drop in oil supply?
  - How far monetary stance can remain accommodative without exacerbating inflation and causing recession?
- Concerns and scenarios:
  - Persistence of trends (rising oil and commodity prices, depreciating currencies, weakening financial conditions) could culminate in explosive commodities prices and turn out to be un-sustainable.
  - Rapidly falling value of international reserves for oil producers could discourage oil supply.
  - Recessionary and inflationary implications of oil price shocks may have longer lags before being fully transmitted to output and prices.
- Historical literature (summarized in source) documents recessionary effects of oil price shocks and longer transmission lags.

### XII. Policy implications and recommendations (exact formulations from source)
- Monetary policy role:
  - Restoring stable oil markets is essential for durable economic growth and price stability; prudent monetary policy may be necessary for achieving longer-term oil price stability.
  - Safe conduct of monetary policy is a prerequisite for economic stability and growth.
  - Monetary policy cannot be a panacea; different economic issues require targeted instruments and solutions.
- Alternative and targeted measures highlighted:
  - Balance of payments deficits could be best and quickly addressed via monetary approach to balance of payments: reducing public and private deficit financing through credit ceilings.
  - External competitiveness should be durably achieved via productivity gains, cost reduction, and technical innovations rather than attempting to depress nominal exchange rates alone.
  - Exchange rate depreciation may not restore external competitiveness under expansionary monetary policy; to be effective, exchange rate depreciation must be supported by restrictive monetary policy.
- Policy dilemma underscored:
  - Policymakers face a trade-off between restraining monetary policy with attendant temporary recession or risking high inflation with attendant recession, financial disorder, and social unrest.
  - High inflation may discourage supply of goods and, if oil supply turns regressive, economic growth would be impeded and pressure on oil price will accelerate.

### XIII. Key statistics and exact figures cited
- Mean and variance (first method): 2.29 percent; variance 1.57 (sigma=1.25).
- Mean and variance (second method): 2.27 percent; variance 1.77 (sigma=1.33).
- Market-implied oil price end-December 2007: US$98/barrel = (95.93*exp(0.022)).
- U.S. dollar depreciation: fell by over 65 percent vis-à-vis Euro since 2001.
- Commodity inflation rate: estimated at about 23 percent per year in 2003M5–2007M10.
- NIG mean return change: rose to 0.11 from –0.005 between sub-periods.
- NIG dispersion change: fell to 1.70 from 2.22 between sub-periods.
- Expected oil price rise based on NIG parameters: about 30 percent per year.
- Variance of oil returns dropped sharply during July-October, 2007 (computation not reported in text).

*Source: _wp08133 - 1. Descriptive Statistics for Oil Price Returns (PDF chapter content).*

### 1. Descriptive Statistics for Oil Price Returns ........................................................................

### _wp08133 - 1. Descriptive Statistics for Oil Price Returns

### I. Introduction
- Crude oil prices maintained upward persistence since early 2003 and broke US$120/barrel in April 2008.
- Rapid oil price increases may:
  - Turn inflationary by causing other prices to rise.
  - Decelerate world economic growth.
- Recent upsurges occurred amid rising commodity prices, instability in housing, equity, and credit markets, and depreciating exchange rates.
- Fast rise in commodities including oil could reflect delayed effects of excessively expansionary monetary policies during 2001–04 when key interest rates were forced down to postwar record levels.
- Relaxation of monetary policy in August 2007─March 2008 set off renewed commodity price inflation and currency depreciation; multi-billion bail out facilities and cuts in federal funds rate were undertaken.
- Paper scope and approach:
  - Analyzed oil prices during 2000M1–2007M10.
  - Distinguished two samples: 2000M1–2003M4 and 2003M5–2007M10 to exhibit monetary policy effects.
  - Assumed oil prices driven by Levy processes (LP) of generalized hyperbolic (GH) type; used daily data to estimate parameters.
  - Emphasized GH/NIG advantages: combine features of normal and stable distributions, model asymmetry, frequent small and large jumps, and reduce option smile.
  - Applied Esscher transform to obtain martingale (risk-neutral) processes for derivative pricing.
  - Also estimated implied risk-neutral distribution from option prices on November 2, 2007 for end-December 2007, finding traders expected further oil price rise and significant right-tail probability.

### II. Recent evolution of oil prices (2000M1–2007M10)
- Data: Light crude futures prices from Reuters, 1986 observations, January 4, 2000 to October 29, 2007.
- Two distinct patterns observed:
  - 2000M1–2003M4: relative stability around a mean of US$27 per barrel.
  - 2003M5–2007M10: strong upward deterministic trend and persistence, prices rose to cross US$96/barrel in October 2007.
- ARMA representations (sample regressions):
  - Sample 2000M1–2003M4:
    - S = 1.325*S(-1) - 1.195*S(-2) + 0.842*S(-3) – 0.191*MA(1) + 0.778*MA(2) + 0.214*MA(3) + 0.769
    - (t-statistics shown in source)
    - R^2 = 0.97; DW = 2.01
  - Sample 2003M5–2007M10:
    - S = 0.990*S(-1) + 0.000537*TREND + 0.098*MA(1) + 0.307
    - (t-statistics shown in source)
    - R^2 = 0.99; DW = 2.00
- Interpretation: In the second sub-period the lag structure shortened and a deterministic trend became significant.

### III. Descriptive statistics for oil price returns
- Returns definition: xt = log(St) - log(St-1) = -log(St-1/St).
- Empirical change in moments between sub-periods (Table 1 highlights):
  - Sample 2000M1–2003M4:
    - Mean = -0.005312
    - Median = 0.000000
    - Maximum = 11.23630
    - Minimum = -9.436276
    - Std. Dev. = 2.225607
    - Skewness = 0.056786
    - Kurtosis = 4.954168
    - Jarque-Bera = 133.9488
    - Probability = 0.000000
  - Sample 2003M5–2007M10:
    - Mean = 0.118992
    - Median = 0.128184
    - Maximum = 7.356257
    - Minimum = -6.080348
    - Std. Dev. = 1.686031
    - Skewness = -0.021088
    - Kurtosis = 3.873152
    - Jarque-Bera = 36.55292
    - Probability = 0.000000
- Annualized interpretations preserved exactly from source:
  - Mean return -0.005 in first sub-period → decline in oil price at 1.28 (=-0.005*255) percent per year.
  - Mean return 0.12 in second sub-period → increase in oil price at 30.6(=0.12*255) percent per year.
  - Volatility (daily Std. Dev.):
    - Fell from 2.226 (annualized to 35.6(=2.226*255) percent) to 1.686 (annualized to 26.9(=1.686*255) percent).
- Distributional conclusions:
  - Skewness small in both periods → distributions approximately symmetric.
  - Kurtosis declined from 4.95 to 3.87.
  - Normality rejected in both periods → prominence of large jumps in daily oil prices.
  - Overall: markets out-of-equilibrium in second sub-sample, driven by strong upward trend reflecting rising demand, supply rigidities, and tensions.

### IV. Volatility dynamics (GARCH)
- Volatility estimated using GARCH(1,1).
- Characteristics:
  - High for daily data, exhibiting volatility clustering followed by mean reversion.
  - Annualized volatility:
    - First sub-period around 35 percent.
    - Second sub-period fell sharply to 22 percent, consistent with a solid deterministic trend.

### V. Modeling oil prices as a Lévy process
- SDE for asset price with LP: dSt = (ρ)St dt + σ St dZt− (source notation: t S, t S−, Zt is a LP, ρ and σ are drift and volatility parameters).
- Solution (Doléans-Dade exponential) and reformulations provided in source:
  - St = exp( ... ) product form including jumps (source equations (4), (5), (6)).
- Lévy process properties (from source):
  - Independent increments, stationary increments, stochastic continuity.
  - LPs are limits of random walks and infinitely divisible.
- Rationale: LPs account for skewness, excess kurtosis, and frequent small and large jumps.

### VI. Oil price process as Normal Inverse Gaussian (NIG)
- NIG is a special case of generalized hyperbolic (GH) distribution with λ = 1/2.
- GH construction: normal variance-mean mixture where mixing distribution is generalized inverse Gaussian (GIG).
- NIG density function provided in source (equation (8)) with parameters μ, δ, α, β and constraint 22 γ = α^2 − β^2.
- Moments of NIG (from source):
  - E[X] = μ + δ β / γ
  - Var[X] = δ α^2 / γ^3   (source expression preserved)
  - Skewness and Kurtosis formulas provided (source equations (11), (12)).
- MGF of NIG given in source (equation (13)), satisfying u < α − β.
- NIG process properties:
  - Pure jump and infinite activity model.
  - Can be represented as time-changed Brownian motion with inverse Gaussian (IG) time-change.
  - Characteristic function tractability enables option pricing via FFT and recovery of distributions via numerical inversion (Davies, 1973).

### VII. Estimation of NIG parameters for oil price returns
- Software used: R program with HyperbolicDist, ghyp, and fBasics packages.
- Estimation results (Table 2, parameterization (α, β, δ, μ) and associated moments):
  - 2000M1–2003M4:
    - Alpha = 0.54
    - Beta = -0.02
    - Delta = 2.69
    - Mu = 0.08
    - Mean = -0.005
    - Variance = 4.92
    - Skewness = -0.08
    - Kurtosis = 1.72
  - 2003M5–2007M10:
    - Alpha = 0.97
    - Beta = -0.06
    - Delta = 2.76
    - Mu = 0.29
    - Mean = 0.12
    - Variance = 2.86
    - Skewness = -0.11
    - Kurtosis = 1.13
- Alternative parameterization results (Table 3, (λ, Alpha.bar, Shape parameter, Mu, Sigma, Beta) and moments):
  - 2000M1–2003M4:
    - Lambda = -0.5
    - Alpha.bar = 1.46
    - Shape parameter = 0.08
    - Mu (location) = 2.22
    - Sigma (dispersion) = -0.08
    - Mean = -0.005
    - Variance = 4.95
    - Skewness = -0.08
    - Kurtosis = 2.06
  - 2003M5–2007M10:
    - Lambda = -0.5
    - Alpha.bar = 2.68
    - Shape parameter = 0.29
    - Mu (location) = 1.69
    - Sigma (dispersion) = -0.17
    - Mean = 0.12
    - Variance = 2.87
    - Skewness = -0.04
    - Kurtosis = 1.22
- Key estimation conclusions (preserved from source):
  - NIG fits oil price returns closely in both sub-periods.
  - Parameters changed significantly in 2003M5–2007M10:
    - Location parameter μ increased from 0.08 to 0.29.
    - Mean return increased from -0.005 (annualized to -0.005*255 = -1.28 percent) in 2000M1–2003M4 to 0.12 (annualized to 0.12*255 = 30.6 percent) in 2003M5–2007M10.
  - Decline in kurtosis and volatility consistent with greater predictability/deterministic trend in the second period.
- Notes on estimation methodology:
  - Characteristic function known in closed form allows empirical characteristic function estimation as alternative (Parzen 1962; Feuerverger and McDunnough, 1981).
  - Simulation algorithms for NIG reference Rydberg (1997) and Dagpunar (1989).

*Source: _wp08133 - 1. Descriptive Statistics for Oil Price Returns (PDF chapter content).*

### 30.6 percent) in 2003M5–2007M10. Scale parameter

### _wp08133 - 30.6 percent) in 2003M5–2007M10. Scale parameter

### Parameter estimates and interpretation for two sub-samples
- Scale parameter δ increased from 2.69 to 2.76; by exceeding unity, it remained high, indicating a stretched out distribution.
- Shape parameter measuring skewness β remained in the range -0.02–-0.06, indicating symmetric oil price returns distribution.
- Shape parameter measuring tail steepness α increased from 0.54 in 2000M1–2003M4 to 0.97 in 2003M5–2007M10, indicating steeper tails and therefore higher frequency of smaller jumps.
- Volatility fell from 2.2 in 2000M1–2003M4 (annualized to 35 percent) to 1.7 in 2003M5–2007M10 (annualized to 27 percent).
- In 2000M1–2003M4:
  - Oil prices exhibited high volatility.
  - Distribution mean was small and negative, indicating fluctuations around a slightly declining trend.
- In 2003M5–2007M10:
  - Oil price process showed declining volatility but was driven by a sharply upward trend, annualized to 30.6 percent per year.
  - Density was symmetric (probability of upward jumps matched downward jumps) but a powerful drift component produced a rising trend.
- Economic explanation:
  - World real GDP expanded at 4–5.5 percent per year during 2003–07.
  - U.S. dollar kept depreciating.
  - Faster growth of world oil demand plus rigidities in world oil supply created excess demand; short-term inelasticities implied small excess demand produced large price variation; positive income effect dominated negative price effect, keeping oil prices under rising pressure.

### Market incompleteness and Esscher transform (risk-neutral measures for NIG)
- Levy processes (LP) are generally incomplete markets: multiple equivalent martingale measures exist; uniqueness relates to market completeness.
- Esscher transform (Gerber and Shiu (1994)) maps statistical distribution P to an equivalent martingale measure Q via parameter h chosen so discounted asset price is a martingale.
- Definitions and relationships:
  - St = S(0) e^{Xt}, with Xt an LP with stationary and independent increments and X(0)=0.
  - MGF M(u,t) = E[e^{uXt}] = ∫ e^{ux} f(x,t) dx; infinite divisibility implies M(u,t) = [M(u,1)]^t.
  - Esscher-transformed density: f(x,t;h) = e^{hx} f(x,t) / M(h,t).
  - Esscher-equivalent measure Q defined by dQ/dP |_t = e^{hXt} / M(h,t).
- Martingale condition for parameter h*: choose h* such that discounted stock price process {e^{-rt} St} is a Q-martingale:
  - Equivalent formulation: (1,1;h*) e^{r} = M(1,1;h*) = e^{r}.
  - Logarithmic form: log[(1,1;Mh*)] = log[(1,1;Mh)] + log[( ,1;Mh)] etc. (equation (19) in source).
- For NIG, h satisfies a specific algebraic equation involving parameters μ, δ, α, β (source displays the full expressions).

- Computed Esscher parameters for Table 2 parameters:
  - h = -0.4858 for 2000M1–2003M4.
  - h = -0.5152 for 2003M5–2007M10.
- Corresponding Esscher transforms:
  - dQ/dP = exp(0.4858 t + 0.2439 Xt) for first sub-sample (signs as in source).
  - dQ/dP = exp(0.5152 t + 0.2691 Xt) for second sub-sample (signs as in source).

- Alternative risk-neutral construction (Carr et al., 2003):
  - For St = S(0) exp(Xt), enforce E[e^{-rt} St] = S(0) to determine adjustment; formulas (20)–(23) in source give log-price and characteristic function (CF) under the risk-neutral measure for NIG.
  - Risk-neutral log-price for NIG: log(St) = log(S(0)) + (1-β^2)μ t + ... (see equation (21) in source) with condition 1+βα > 0.

### Pricing and characteristic functions: illustrative computation
- Example using parameters in Table 2 and market data on November 2, 2007:
  - Futures oil price for end-December 2007: US$95.93/barrel.
  - US three-month treasury bill: r = 4.595 percent.
  - Time to maturity t = 57/365 = 0.16.
  - Risk-neutral parameters example: α = 0.97, β = -0.06, δ = 2.76, μ = 0.29.
  - Risk-neutral CF for NIG given in source yields a numeric example:
    - NIG CF = 7.47 * exp(0.0460.442   0.941   (  0.05)  ) (expression as provided in source).

### Inverse problem: inferring risk-neutral density from option prices
- Objective: estimate NIG parameters θ = (α, β, δ, μ) with constraints 0 ≤ δ, μ ∈ R, 0 ≤ β ≤ 1, by minimizing quadratic pricing error over M traded options:
  - Minimize sum_{j=1}^M (C_j(K_j, T; θ) - C_j^{market})^2 subject to put-call parity and possibly a penalty parameter λ ≥ 0 (formulas (24)–(26) in source).
- Put-call parity constraint used: S_0 + e^{-rT} P_j(K_j,T) = C_j(K_j,T) + K_j e^{-rT}.
- Option pricing methods:
  - If transition density known in closed form, C^*_{j}(K,T) = e^{-rT} E_Q[max(S_T - K,0)].
  - When density not closed-form, use characteristic function methods (Carr and Madan (1999)) and FFT pricing:
    - Modified call transform ψ_T(u) expressible in terms of φ_T(u) (equation (30)).
    - Use parameter a>0 to remove singularity at u=0; require (1+a)T φ(·) finite.
- Estimation approaches in Fourier space and empirical characteristic function (ECF):
  - Least-squares restated in Fourier space (equation (31)).
  - ECF method: estimate empirical characteristic function from implied discrete risk-neutral density q̂ obtained by deconvolution/Tikhonov regularization, then match theoretical NIG CF to ECF (equations (34)–(36)).
  - Tikhonov regularization formula for q̂: q̂ = e^{rT} (D' D + κ I)^{-1} D' V (equation (33)), with κ>0 penalty.

### Empirical implementation and results (November 2, 2007 market data)
- Market inputs:
  - Date: November 2, 2007 (market data used).
  - Futures price for end-December 2007: US$95.93/barrel.
  - Risk-free rate: r = 4.595 percent.
- Two methods used to imply risk-neutral distribution:
  - Method 1: FFT of call and put prices and constrained minimization (equation (31)).
    - Implied parameters: α̂ = 3.30, β̂ = 0.35, δ̂ = 5.09, μ̂ = 1.75.
  - Method 2: ECF (equation (36)) and General Method of Moments.
    - Implied parameters: α̂ = 3.1, β̂ = 0.30, δ̂ = 5.41, μ̂ = 1.74.
- Applying NIG moment formulas (9–12) to the first method yields implied expected mean (details in source).
- Note: Parameters for 2000M1–2003M4 did not satisfy condition 1 + β α > 0 and therefore would not yield real parameters for NIG distribution (footnote in source).

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08133.pdf*

### 2.29 percent, variance 1.57 (sigma=1.25),

### _wp08133 - 2.29 percent, variance 1.57 (sigma=1.25),

### Implied crude oil density and option-implied statistics
- For the first method: mean 2.29 percent, variance 1.57 (sigma=1.25), skewness 0.08, and kurtosis 0.19.
- For the second method: expected mean 2.27 percent, variance 1.77 (sigma=1.33), skewness 0.07, and kurtosis 0.19.
- Market-implied average oil price at end-December 2007: US$98/barrel = (95.93*exp(0.022)).
- Interpretation of distributional moments:
  - Dispersion measured by sigma was lower than statistical values in Tables 2 and 3, implying narrower interval of variation around expected mean.
  - Positive skewness implied higher probability for oil prices to rise above expected mean than to fall below this mean.
  - Kurtosis below 3 implied flatter NIG distribution compared with normal distribution and higher than normal probability for tail events.
- Derived risk-neutral density had low kurtosis, implying flatter distribution and significant probability for tail events.

### Short-term market expectations and driving fundamentals
- Market participants’ short-term expectations appeared strongly influenced by underlying fundamentals:
  - Expansionary monetary policy since 2001.
  - Sharply depreciating U.S. dollar, which fell by over 65 percent vis-à-vis Euro since 2001.
  - Higher world economic growth and consequently higher demand for oil.
  - Crude oil supply rigidities leading traders to expect excess demand and further upward pressure on oil prices.
- Commodity-wide dynamics noted:
  - World aggregate demand for commodities expanded, resulting in double digit inflation for commodities prices, estimated at about 23 percent per year in 2003M5–2007M10.
  - High commodity inflation would erode real interest rate and stimulate further real aggregate demand.
- Financial conditions and monetary stance:
  - Sub prime market defaults, large write-offs by leading banks, and piling up of credits signaled that restraints in monetary policy were not in the offing soon.
  - Further relaxation of monetary policy in August-December 2007 coincided with immediate weakening in U.S. dollar and surge in oil prices.
  - These conditions likely led traders to assume persistence in oil prices and to allow more likelihood for right tail events.

### Time-series behavior and NIG estimation results
- Comparison of two sub-periods:
  - In 2003M5-2007M10 compared to 2000M1–2003M4, oil prices parameters changed drastically under NIG distribution estimation.
  - Mean return rose to 0.11 from –0.005.
  - Dispersion fell to 1.70 from 2.22.
  - NIG for both sub-periods had low kurtosis, implying flatter than normal distribution.
- Based on NIG parameters, oil prices would be expected to rise at about 30 percent per year.
- Computation (not reported here) showed that variance of oil returns dropped sharply during July-October, 2007.
- Crude oil density forecast for end-December 2007 was extracted from option prices data on November 2, 2007; traders’ expectations aligned with prevailing fundamentals.

### Risks, scenarios, and potential macroeconomic consequences
- Key open questions posed:
  - How far oil prices could rise without reaching critical zone that triggers a world recession or a drop in oil supply?
  - How far monetary stance can remain accommodative without exacerbating inflation and causing recession?
- Concerned scenarios and risks:
  - Persistence of trends (rising oil and commodity prices, depreciating currencies, weakening financial conditions) could culminate in explosive commodities prices and turn out to be un-sustainable.
  - Rapidly falling value of international reserves for oil producers could discourage oil supply.
  - Recessionary and inflationary implications of oil price shocks may have longer lags before being fully transmitted to output and prices, as documented by prior literature cited in the text.
- Historical literature referenced (findings summarized in text):
  - Hamilton (1983) and Hamilton and Herrera (2004) found recessionary effects of oil prices and allowance for longer transmission lags.
  - Bernanke et al. (1997), Jones et al. (2004), and Lee et al. (1995) found significant recessionary and inflationary impact of oil prices on real GDP and consumer prices.

### Policy implications and recommendations
- Monetary policy role:
  - Restoring stable oil markets is essential for durable economic growth and price stability; prudent monetary policy may be necessary for achieving longer-term oil price stability.
  - Safe conduct of monetary policy is a prerequisite for economic stability and growth.
  - Monetary policy cannot be a panacea; different economic issues require targeted instruments and solutions.
- Alternative and targeted measures highlighted:
  - Balance of payments deficits could be best and quickly addressed via monetary approach to balance of payments: reducing public and private deficit financing through credit ceilings.
  - External competitiveness should be durably achieved via productivity gains, cost reduction, and technical innovations rather than attempting to depress nominal exchange rates alone.
  - Exchange rate depreciation may not restore external competitiveness under expansionary monetary policy; to be effective, exchange rate depreciation must be supported by restrictive monetary policy.
- Policy dilemma underscored:
  - Policymakers face a trade-off between restraining monetary policy with attendant temporary recession or risking high inflation with attendant recession, financial disorder, and social unrest.
  - High inflation may discourage supply of goods and, if oil supply turns regressive, economic growth would be impeded and pressure on oil price will accelerate.

### Key statistics and exact figures cited
- Mean and variance (first method): 2.29 percent; variance 1.57 (sigma=1.25).
- Mean and variance (second method): 2.27 percent; variance 1.77 (sigma=1.33).
- Market-implied oil price end-December 2007: US$98/barrel = (95.93*exp(0.022)).
- U.S. dollar depreciation: fell by over 65 percent vis-à-vis Euro since 2001.
- Commodity inflation rate: estimated at about 23 percent per year in 2003M5–2007M10.
- NIG mean return change: rose to 0.11 from –0.005 between sub-periods.
- NIG dispersion change: fell to 1.70 from 2.22 between sub-periods.
- Expected oil price rise based on NIG parameters: about 30 percent per year.
- Variance of oil returns dropped sharply during July-October, 2007 (computation not reported in text).

*Source: Excerpt from _wp08133 (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08133.pdf_