Affine-structure models and the pricing of energy commodity derivatives

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1 Affine-structure models and the pricing of energy commodity derivatives Nikos K Nomikos Cass Business School, City University London Joint work with: Ioannis Kyriakou, Panos Pouliasis and Nikos Papapostolou

2 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 2

3 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. 3

4 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. 4

5 Spot Price Model Formulation (MRJSV) Decoupled spot price model with mean reverting diffusion and spike components S t = f t + exp(x t ) f t is a predictable seasonal component f t = δ 0 + δ 1 sin(2π(t + τ 1 )) +δ 2 sin(2π(t + τ 2 ))+δ 3 t X t is a Gaussian Ornstein-Uhlenbeck process: dx t = k (ε X t )dt + V t db t +dl t The evolution of the spot price variance V t is modelled by a Heston (1993) square-root diffusion : dv t = a (β V t )dt +γ V t dw t We also consider restricted versions of MRJSV as in MRSV, MRJ and MR 5

6 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 μ J 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

7 Estimation Methodology (Step 2) Second, we estimate volatility parameters and market price of risk from end-of-day futures prices 7

8 8

9 δ 0 δ 1 τ 1 δ 2 τ 2 δ 3 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 Model Heating Oil (HO) Calibration markets. Panel B reports the estimated annualized parameters of the MRSV and MRJSV models. CB GO CL HU HO Panel A: predictable component parameters Panel B: spot price & variance model parameters ϵ k σ λ µ J σ J h α β γ ρ V

10 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 Statistical model, by employing the Hansen Fit (2005) of test and the stationary Models... bootstrap of Politis and Romano (1994) using 5,000 bootstrap simulations. Aggregate Pricing Errors Pricing Errors T<0.5 Pricing Errors T>0.5 RRMSE RMSE ($) RRMSE RMSE ($) RRMSE RMSE ($) Panel A: Brent Crude Oil (CB) MR MRJ 0.131* 7.508* 0.096* 6.264* 0.151* 8.144* MRSV * MRJSV 0.129* 7.499* 0.095* 6.258* 0.149* 8.141* Panel B: Gasoil (GO) MR MRJ 0.142* 61.43* 0.100* 50.53* 0.168* 68.03* MRSV * MRJSV 0.140* 61.33* 0.099* 50.44* 0.166* 67.98* Panel C: WTI Crude Oil (CL) MR MRJ 0.103* 6.960* 0.083* 5.980* 0.112* 7.413* MRSV MRJSV 0.102* 6.957* 0.080* 5.977* 0.110* 7.412* Panel D: Gasoline (HU) MR MRJ 0.162* 0.225* 0.116* 0.193* 0.188* 0.237* MRSV * MRJSV 0.160* 0.223* 0.113* 0.192* 0.185* 0.234* Panel E: Heating Oil (HO) MR MRJ 0.134* 0.193* 0.095* 0.159* 0.157* 0.211* MRSV * MRJSV 0.133* 0.192* 0.092* 0.157* 0.158* 0.210* 10

11 empirical statistic. Table entries correspond to values in the range 0 to 1: e.g., a value of indicates that in 75% of 100,000 simulation runs, the simulated statistic has been within the bootstrap confidence interval. Abbreviations: standard Simulation Study: True Data-Generating Process 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. std std skew kurt perc1 perc99 ES1 ES99 K-S test Brent Crude Oil (CB) MR MRJ MRSV MRJSV Gasoil (GO) MR MRJ MRSV MRJSV WTI Crude Oil (CL) MR MRJ MRSV MRJSV Gasoline (HU) MR MRJ MRSV MRJSV Heating Oil (HO) MR MRJ MRSV MRJSV

12 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 12

13 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 13

14 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 14

15 CVMC Simulation Scheme 15

16 Arithmetic Asian Option Prices 16

17 Model-implied distributions of log-returns and hedging errors 17

18 18 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. CB GO CL HU HO Panel A: hedges without model misspecification (1) MR hedging error mean std (2) MRJ hedging error mean std (3) MRSV hedging error mean std (4) MRJSV hedging error mean std Panel B: % changes in std of hedging error under model misspecification (less risk factors) (1) true MRJ model MR (2) true MRSV model MR (3) true MRJSV model MR MRJ MRSV Panel C: % changes in std of hedging error under model misspecification (additional risk factors) (1) true MR model MRJ MRSV MRJSV (2) true MRJ model MRJSV (3) true MRSV model MRJSV Hedge Performance Comparisons

19 Changes in Hedging Error 19

20 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 20

21 Thank You! Affine-structure models and the pricing of energy commodity derivatives 21

22 Derivation and proofs 1/3 We consider a change of measure as in Benth (2011) 22

23 23Derivation and proofs 2/3

24 24Derivation and proofs 3/3

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