Affine-structure models and the pricing

of energy commodity derivatives

Joint work with:

Ioannis Kyriakou, Panos Pouliasis and Nikos Papapostolou

Nikos K Nomikos

n.nomikos@city.ac.uk

Cass Business School, City University London

Introduction

Price of crude oil is one of the world’s most important global economic

indicators.

Evidently, petroleum prices are highly volatile and occasionally exhibit

drastic shocks.

Stylized features: seasonality, jumps, stochastic volatility etc.

This is a combination of supply construction lags and inelastic demand

Apart from being an important input into production, petroleum

commodities serve as the underlying assets in a growing financial market.

Growth of a paper market on energy commodities

Entrance of new players in the energy markets

Financialization

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Objectives

The aim of this paper is to conduct a comprehensive analysis of stochastic dynamic

modelling of European and US petroleum commodity prices and enrich existing

literature with some new insights in several applications such as futures pricing,

options pricing and hedging.

We estimate a one-factor spot model from the affine class which captures well:

jumps, mean reversion and stochastic volatility in the behaviour of the spot

petroleum prices.

We obtain expressions for the theoretical futures prices.

We obtain closed-form solutions for geometric average options.

We set up delta hedge portfolios for the Asian option and investigate their

performance under various incorrect hedge models that omit the jump and/or

stochastic volatility factor.

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Motivation

We derive the bivariate characteristic function of the suggested jump diffusion model

with stochastic volatility.

We fit the models to spot and futures prices of Brent crude oil and gasoil from the

European market and light sweet crude oil, gasoline and heating oil from the US

market and find that ignoring jumps and/or stochastic volatility leads to a less realistic

description of the true (DGP). The flexibility of the proposed general model

specification is also confirmed by its ability to accurately fit the observed futures

curves in the different markets.

We apply this model to average (Asian) options, which are very popular in the energy

commodity markets

e.g. as a means of managing price exposure and potential impact on transactions, due to the time

elapsed until a tanker vessel completes its route from the production site or refinery to its

destination.

This way we extend earlier contributions, by Kemna and Vorst (1990) and Fusai and Meucci (2008),

to the more general affine class.

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Spot Price Model Formulation (MRJSV) Decoupled spot price model with mean reverting diffusion and spike components

𝑆𝑡 = 𝑓𝑡 + exp(𝑋𝑡)

𝑓𝑡 is a predictable seasonal component

𝑓𝑡 = 𝛿0 + 𝛿1sin(2𝜋(𝑡 + 𝜏1)) +𝛿2sin(2𝜋(𝑡 + 𝜏2))+𝛿3t

𝑋𝑡 is a Gaussian Ornstein-Uhlenbeck process:

𝑑𝑋𝑡 = 𝑘 (ε− 𝑋𝑡)dt + 𝑉𝑡𝑑𝐵𝑡+𝑑𝐿𝑡

The evolution of the spot price variance 𝑉𝑡 is modelled by a Heston (1993) square-root diffusion :

𝑑𝑉𝑡 = 𝑎 (β− 𝑉𝑡)dt +γ 𝑉𝑡𝑑𝑊𝑡

We also consider restricted versions of MRJSV as in MRSV, MRJ and MR

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Estimation Methodology (Step 1) Spot and Constant Maturity Futures Prices for up to 12 months for Brent (CB), GasOil

(GO), WTI (CL), RBOB Gasoline (HU) and Heating Oil (HO) from March 12, 2009 to

March 11, 2013 (1,043 daily observations)

Estimate deterministic seasonal component from spot prices

2-Stage Estimation process for the Jump component as in Clewlow and Strickland (2000)

First, using the log deseasonalized spot prices we obtain the spot parameters for the

MRJSV, MRJ and MR models

We define a jump as an observation in the log deseasonalized returns that is greater in absolute value

than a market-specific threshold given by a multiple of the sample standard.

The prices on the identified ‘jump dates’ are substituted by the averages of the two adjacent prices, the

standard deviation of the updated series is recalculated and the same procedure is repeated until no

more jumps are identified.

We estimate the jump arrival rate by the average number of identified jumps per year; the estimates of

the mean μJ and standard deviation of the jump size distribution are given by the average and standard

deviation of the jump returns, respectively.

The remaining parameters , k and , of the spot model are estimated using OLS regression.

6

Estimation Methodology (Step 2) Second, we estimate volatility parameters and market price of risk from end-of-day

futures prices

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8

Model Calibration

9

δ0

δ 1

τ1

δ2

τ2

δ3

ϵ 4.252 4.253 6.318 6.319 4.279 4.281 0.608 0.609 0.554 0.559

k 3.563 3.344 3.281 2.999 5.002 4.278 6.385 4.703 4.088 3.221

σ 0.34 0.327 0.343 0.329 0.361 0.332 0.467 0.401 0.374 0.348

λ 3.4 3.2 5 7.6 4.6

µJ -0.026 -0.013 -0.002 -0.007 -0.002

σ J 0.064 0.067 0.077 0.092 0.079

h 0.599 0.389 0.643 0.533 0.223 0.215 1.101 0.983 0.546 0.53

α 36.68 31.52 30.71 16.76 37.33 21.92 84.23 52.32 42.12 40.33

β 0.263 0.23 0.349 0.245 0.315 0.216 0.7 0.478 0.35 0.344

γ 1.259 0.989 1.86 1.084 1.531 1.114 3.694 1.95 1.817 1.742

ρ 0.204 0.181 0.254 0.186 0.236 0.172 0.452 0.3 0.247 0.245

V0 0.208 0.186 0.262 0.196 0.24 0.178 0.498 0.359 0.266 0.262

Panel B: spot price & variance model parameters

14.86 141.2 8.566 0.411 0.446

1.836 -2.673 1.328 1.073 1.085

-2.406 -23.62 -0.877 0.06 0.062

1.204 3.711 1.273 1.059 0.774

3.479 -20.56 3.871 0.156 -0.069

Panel A: predictable component parameters

-5.592 -40.64 -4.325 -0.187 -0.136

Table 1: Model Calibration

This table presents the model calibration results. Panel A reports the estimated annualized parameters of the

predictable component for each of the Brent Crude Oil (CB), Gasoil (GO), WTI Crude Oil (CL), Gasoline (HU)

and Heating Oil (HO) markets. Panel B reports the estimated annualized parameters of the MRSV and MRJSV

models.

CB GO CL HU HO

Statistical Fit of the Models...

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RRMSE RMSE ($) RRMSE RMSE ($) RRMSE RMSE ($)

MR 0.136 7.782 0.098 6.489 0.156 8.351

MRJ 0.131* 7.508* 0.096* 6.264* 0.151* 8.144*

MRSV 0.135 7.767 0.098 6.438* 0.157 8.369

MRJSV 0.129* 7.499* 0.095* 6.258* 0.149* 8.141*

MR 0.146 63.18 0.103 52.26 0.172 69.36

MRJ 0.142* 61.43* 0.100* 50.53* 0.168* 68.03*

MRSV 0.146 62.89 0.102 51.56* 0.173 69.47

MRJSV 0.140* 61.33* 0.099* 50.44* 0.166* 67.98*

MR 0.104 7.106 0.087 6.213 0.114 7.501

MRJ 0.103* 6.960* 0.083* 5.980* 0.112* 7.413*

MRSV 0.105 7.105 0.087 6.208 0.114 7.506

MRJSV 0.102* 6.957* 0.080* 5.977* 0.110* 7.412*

MR 0.171 0.238 0.124 0.208 0.195 0.243

MRJ 0.162* 0.225* 0.116* 0.193* 0.188* 0.237*

MRSV 0.179 0.239 0.121 0.196* 0.21 0.256

MRJSV 0.160* 0.223* 0.113* 0.192* 0.185* 0.234*

MR 0.139 0.202 0.102 0.172 0.16 0.217

MRJ 0.134* 0.193* 0.095* 0.159* 0.157* 0.211*

MRSV 0.141 0.202 0.101 0.167* 0.165 0.22

MRJSV 0.133* 0.192* 0.092* 0.157* 0.158* 0.210*

Panel B: Gasoil (GO)

Panel C: WTI Crude Oil (CL)

Panel D: Gasoline (HU)

Panel E: Heating Oil (HO)

The Table reports the error statistics computed for the entire term structure of futures prices under the

optimal parameter set , (no. of maturities is 12 and no. of days in sample period March 12, 2009 to

March 11, 2013, i.e., 1,043). In addition, we test the null hypothesis that none of MRJ, MRSV and

MRJSV models leads to reduction in futures pricing errors (RRMSE and RMSE) relative to the MR

model, by employing the Hansen (2005) test and the stationary bootstrap of Politis and Romano (1994)

using 5,000 bootstrap simulations.

Table 2: Futures Contracts Pricing Errors

Aggregate Pricing Errors Pricing Errors T<0.5 Pricing Errors T>0.5

Panel A: Brent Crude Oil (CB)

Simulation Study: True Data-Generating

Process

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std std skew kurt perc1 perc99 ES1 ES99 K-S test

MR 0.731 0.031 0.295 0.28 0.89 0.144 0.803∗ 0.621

MRJ 0.866 0.556∗ 0.513 0.708 0.951∗ 0.779 0.564 0.833

MRSV 0.789 0.223 0.963∗ 0.388 0.766 0.242 0.747 0.646

MRJSV 0.871∗ 0.515 0.787 0.744∗ 0.86 0.857∗ 0.509 0.884∗

MR 0.575 0.122 0.189 0.263 0.817 0.109 0.881∗ 0.565

MRJ 0.896∗ 0.556 0.53 0.608 0.937∗ 0.777 0.772 0.811

MRSV 0.788 0.577 0.72 0.282 0.862 0.266 0.857 0.724

MRJSV 0.767 0.606∗ 0.772∗ 0.696∗ 0.821 0.800∗ 0.786 0.928∗

MR 0.679 0.479 0 0.243 0.834 0.003 0.341 0.551

MRJ 0.855 0.474 0.222 0.621 0.868∗ 0.728∗ 0.763 0.77

MRSV 0.836 0.428 0.283 0.442 0.836 0.455 0.970∗ 0.619

MRJSV 0.876∗ 0.636∗ 0.446∗ 0.737∗ 0.829 0.693 0.901 0.821∗

MR 0.733 0.515 0 0.062 0.970∗ 0.009 0.204 0.597

MRJ 0.824 0.559 0.352 0.597 0.932 0.804 0.743 0.76

MRSV 0.788 0.595 0.814∗ 0.242 0.752 0.426 0.656 0.685

MRJSV 0.855∗ 0.755∗ 0.577 0.719∗ 0.908 0.819∗ 0.918∗ 0.801∗

MR 0.514 0.487 0 0.213 0.937 0.004 0.408 0.357

MRJ 0.872 0.578 0.481∗ 0.568 0.974 0.642 0.846 0.804

MRSV 0.781 0.442 0.364 0.888∗ 0.956 0.548 0.948∗ 0.817

MRJSV 0.892∗ 0.701∗ 0.465 0.872 0.996∗ 0.672∗ 0.941 0.844∗

We test whether the proposed models (MR, MRJ, MRSV, MRJSV) can accurately represent the true price dynamics of each

commodity (CB, GO, CL, HU, HO). The table reports the relative performance across models in terms of percentage number

of simulated log-return statistics of a given type lying within the corresponding 90% bootstrap confidence interval of the

empirical statistic. Table entries correspond to values in the range 0 to 1: e.g., a value of 0.750 indicates that in 75% of

100,000 simulation runs, the simulated statistic has been within the bootstrap confidence interval. Abbreviations: standard

deviation (std), skewness (skew), kurtosis (kurt), 1st and 99th percentiles (perc1 and perc99) and expected shortfalls at the

1% and 99% levels (ES1 and ES99). In addition, for each simulation, we employ the two-sample Kolmogorov–Smirnov

(K–S) test for equality of the empirical and model-implied log-return distributions and report the percentage number of times

the null hypothesis cannot be rejected at the 10% significance level. Asterisks (∗) highlight best relative performance across

models.

Table 3: True data-generaing process testing

Brent Crude Oil (CB)

WTI Crude Oil (CL)

Heating Oil (HO)

Gasoil (GO)

Gasoline (HU)

Discretely monitored Asian options

Payoff of Asian option is based on average level

Case of commodities

Prevents wild fluctuations from impacting transactions related to

large exchanged quantities or volumes

Hard to manipulate and relatively straightforward to hedge

Prevalent case: discrete monitoring and arithmetic average

There is no exact closed-form solution for pricing Average Price Asian

options.

Lack of analytical tractability: the probability distribution of the

arithmetic average is not known

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Pricing problem : discretely monitored

arithmetic Asian option Option price of the arithmetic Asian option

Option price of the geometric Asian option

Probability distribution of the geometric average can be derived and

the expectation can be computed with high accuracy

Solution to the arithmetic Asian option pricing problem: use Monte

Carlo simulation with geometric Asian option as control variate

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Pricing problem : discretely monitored

arithmetic Asian option (cont’d)

Pricing using Monte Carlo simulation with control variates: Need to simulate

Simulated arithmetic & geometric Asian discounted payoff

E(V): true price of geometric Asian option (known)

b = Cov(C,V)/Var(V): estimated optimal control variate coefficient

Assume M simulations. Control variate estimate of arithmetic Asian option price

is given by sample mean

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CVMC Simulation Scheme

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Arithmetic Asian Option Prices

Model-implied distributions of log-returns

and hedging errors

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CB GO CL HU HO

mean –0.228 –1.812 –0.070 –0.010 –0.005

std 0.557 4.361 0.559 0.019 0.015

mean –0.165 –1.674 –0.075 –0.008 –0.005

std 0.568 4.918 0.702 0.021 0.017

mean –0.201 –1.344 –0.023 –0.007 –0.004

std 0.765 6.928 0.797 0.033 0.022

mean –0.145 –1.714 –0.064 –0.007 –0.003

std 0.742 6.021 0.813 0.028 0.023

MR 4.19 3.04 0.64 13.57 3.3

MR 8.74 12.11 1.38 35.06 12.12

MR 12.07 10.42 1.55 35.23 12.85

MRJ 2.66 4.64 0.51 14.2 6.23

MRSV 3.42 0.17 0.49 3.15 1.38

MRJ –4.21 –2.48 –0.20 –10.06 –2.73

MRSV –4.07 –2.79 –0.20 –9.81 –3.07

MRJSV –3.64 –2.73 –0.04 –8.12 –2.58

MRJSV –0.38 –1.45 –0.25 –6.04 –1.39

MRJSV –3.47 –0.23 –0.66 –3.73 –1.44

Table 4: Hedging performance comparisons

Panel A reports for each market the mean and standard deviation (std) of the simulated hedging

error distribution without model misspecification in monetary terms (CB - $/bbl, GO - $/mt, CL -

$/bbl, HU - $/gal, HO - $/gal). Panels B & C report % increases (positive signs) or decreases

(negative signs) in the standard deviation of the hedging error when the hedge portfolios are

misspecified, i.e., formed based on alternative models. In Panel B (C) the incorrect hedge model

contains fewer (more) risk factors than the true model. Hedging error is defined as the difference

between the value of the delta hedge portfolio and the value of a long 1-month to maturity ATM

arithmetic Asian option with daily monitoring for a 1-week hedge period.

Panel A: hedges without model misspecification

Panel C: % changes in std of hedging error under model misspecification (additional risk factors)

(1) true MR model

(2) true MRJ model

(3) true MRSV model

(2) true MRSV model

(1) true MRJ model

(3) true MRJSV model

(4) MRJSV hedging error

(3) MRSV hedging error

(2) MRJ hedging error

(1) MR hedging error

Panel B: % changes in std of hedging error under model misspecification (less risk factors)

Hedge Performance Comparisons

Changes in Hedging Error

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Conclusions Under the assumption of the spot price model with mean reverting diffusion and jumps

with stochastic volatility

Simulated log-return distributions and individual statistics closely match the empirical estimates.

Futures pricing errors and biases are significantly lower than the nested model specifications

We price Average Price options in the petroleum markets

Provide a flexible framework for modelling market movements

The pricing algorithm is fast and accurate

Failing to account for price jumps and stochastic volatility leads to relatively lower option premia.

For different possible representations of the true DGP, the MRJSV hedge systematically achieves reduced standard deviation of the hedging error

Stochastic volatility in the hedge model plays primary role in reducing the variance of hedging error arising from model misspecification.

Extension and work in progress

Options on Futures rather than the Spot

Extension to other Commodities (e.g. Brent Crude oil, Natural Gas, Petroleum products)

Trading strategies based on extracted implied volatilities

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Thank You!

Affine-structure models and the pricing of energy

commodity derivatives

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Derivation and proofs 1/3 We consider a change of measure as in Benth (2011)

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