Published on: **Mar 4, 2016**

- 1. FIBE, BERGEN, Thursday - Friday 8 – 9 January 2015 Page: 1 In the Discussion Paper series: Volatility Re-Projection for European Carbon Markets: Implied Volatilities, Risk Premiums and Market Pricing Errors by Per Bjarte Solibakkea & Kai Erik Dahlenb a) Department of Economics and Social Sciences, Molde University College b) Department of Economics and Social Sciences, Molde University College and Institute of Industrial Economics and Technology Management, Norwegian University of Science and Technology Introduction Research Design (how) Market Implied Volatilities Summary
- 2. Page: 2 Volatility Re-Projection and Option Pricing FIBE, BERGEN, Thursday - Friday 8 – 9 January 2015 OUTLINE 1. Introduction and Research Design 2. Methodology, Stochastic Volatility (SV) Models and Re-projection i. Extracting Conditional Volatility (Smoothing) ii. Forecasting Conditional Volatility (Filtering) 3. Option Prices i. Option Market Structures and Option Market Prices ii. SV-model Options prices (Re-Projection) iii. Model errors: MPE and MAPE iv. Modelling results 4. Modelling Summary and Conclusions Introduction Research Design (how) Market Implied Volatilities Summary
- 3. Introduction and motivation 1. An enhanced understanding of energy derivative pricing 2. The extra information from conditional volatility employing contemporaneous and lagged returns 3. An evaluation of market pricing under liquid/illiquid markets with arbitrage conditions 4. Possible pattern in the (un-)conditional volatility (i.e. clustering / non-normal densities) 5. Implied volatilities and extracted risk premiums from market prices and from normal and moment-based market models 6. Commodity Markets microstructure and trading preferences ((il-)liquidity) 7. Systematic market pricing errors Page: 3 Volatility Re-Projection and Option Pricing FIBE, BERGEN, Thursday - Friday 8 – 9 January 2015 Introduction Research Design (how) Market Implied Volatilities Summary
- 4. For any contract Procedure: (1) Projection, (2) Estimation and (3)Re-Projection 1. Projection: The Scores/Moments generator (A Statistical Model) Serial Correlation in the Mean (AR-model) Volatility Clustering, Asymmetry and Level effects in the Latent Volatility (GJR- (G)ARCH-model) Hermite Polynomials for higher order features (mostly leptokurtosis) 2. Moments Estimation: The Scientific Model – A Stochastic Volatility Model 0 1 1 0 1 0 1 1 0 2 1 1 2 2 1 2 exp( ) 1 t t t t t t t t t t t t y a a y a u b b b u u z u s r z r z where z1t , z2t and (z3t ) are iid Gaussian random variables. The parameter vector is: 0 1 0 1, , , , ,a a b b s r Page: 4 0 1 1 0 1, 2, 1 1, 0 1 1, 1 0 2 2, 0 1 2, 1 0 3 1 1 2 2 1 1 2 3 2 3 exp( ) 1 t t t t t t t t t t t t t t t t t t y a a y a u b b b u c c c u u z u s r z r z u s z 0 1 0 1 0 1 1 2, , , , , , , ,a a b b c c s r s Volatility Re-Projection and Option Pricing Design and some Theory (how): Introduction Research Design (how) Market Implied Volatilities Summary
- 5. Page: 5 Volatility Re-Projection and Option Pricing Design and some Theory (how): 3. Re-Projection from the MCMC SV Model: i. Re-projected conditional returns using the standard scientific SV model, extracting: One-step-ahead conditional Mean and Volatility (smoothing) Volatility Forecasting (one-step-ahead) evaluated at data values (filtering) Multi-step-ahead Forecasting, Mean and Volatility persistence ii. Re-projected Conditional Volatility (filtered volatility) extracting: The Conditional Volatility Densities for the pricing of any functional form of Option contracts giving us Re-projected Volatility versus Market and Black’76 Option prices / Implied volatilities Markets Risk Characteristics. Calculations and Adjustments from observed market prices MPE/MAPE (errors) calculated from Market prices, Re-projected Volatility and Black’76 Prices (underlying is a future contract) Will not be reported, assumed known from previous publications Introduction Research Design (how) Market Implied Volatilities Summary
- 6. Page: 6 First, the General Theory of Re-Projection Having the SV model coefficients estimate for at our disposal, we can elicit the dynamics of the implied conditional density of the observables: ˆ n 0 1 0 1 ˆˆ | ,..., | ,..., ,L L np y y y p y y y Analytical expressions are not available, but an unconditional expectation: 0ˆ 0 0 ˆ... ,..., ,..., , ...Ln L L n y yE g g y y p y y d d can be computed by generating an simulation ˆ N t t L y from the system with parameters set to and using .ˆ n Define now: ˆ 0 1 ˆ arg max log | ,..., , nK K K LE f y y y where is the projection 0 1| ,..., ,K Lf y y y density (the scores/moments). We now let the estimated 0 1 0 1 ˆ ˆ| ,..., | ,..., ,K L K L Kf y y y f y y y N 0t tLtˆ yˆ,...,yˆg N 1 gE n Volatility Re-Projection and Option Pricing FIBE, BERGEN, Thursday - Friday 8 – 9 January 2015 Design and some Theory (how): Introduction Research Design (how) Market Implied Volatilities Summary
- 7. Page: 7 Convergence in their norm implies that as well as its partial derivatives in converges uniformly over , to those of . ˆ Kf 1 0,..., ,Ly y y , 1M L ˆp Theorem 1 of Gallant and Long (1997) states that 0 1 0 1 ˆ ˆlim | ,..., | ,...,K L L K f y y y p y y y Convergence is with respect to a weighted Sobolev norm that they describe. Volatility Re-Projection and Option Pricing FIBE, BERGEN, Thursday - Friday 8 – 9 January 2015 The General Theory of Re-Projection Design and some Theory (how): Introduction Research Design (how) Market Implied Volatilities Summary
- 8. The calculation of Market Risk Premiums At day t-1 we calculate all call- and put options implied volatilities At day t we have an estimate of the underlying optimal SV model unconditional volatility The Risk formula becomes: For the Black’76 formula and the re-projection methodology the risk adjustment is constant (dec_Rt-1) for a steps/calculations at time t . Page: 8 𝑑𝑒𝑐_𝑅𝑡−1 = 𝐶𝑎𝑙𝑙𝑀𝑉𝑜𝑙 + 𝑃𝑢𝑡𝑀𝑉𝑜𝑙 2 − 𝑈𝑛𝑐𝑜𝑛𝑑_𝑉𝑜𝑙 𝑈𝑛𝑐𝑜𝑛𝑑_𝑉𝑜𝑙 Volatility Re-Projection and Option Pricing FIBE, BERGEN, Thursday - Friday 8 – 9 January 2015 Design (how): Introduction Research Design (how) Market Implied Volatilities Summary
- 9. Page: 9 Option Pricing (filtering and re-projected volatility): The predominant application from re-projection is option pricing and implied volatilities. We estimate an unobserved state variable conditional upon past and present observables . Using a long simulation of from the optimal SV structural model and performing a projection to get , where y* is the unobserved volatility and x* is the observed (returns) variables, the conditional re-projected volatility is estimated. Any functional option complexity can now be calculated. In the general case, we obtain asset prices St at time t from a simulation labelled by t = 1, 2, …, N. 𝑆𝑡 = 𝑆𝑡−1 ∙ 𝑦𝑡 ∗ ∙ 𝑅𝑡−1 where * * ,t ty x * * * *ˆ ( | )Ky f y x dy * ty * tx 𝑦𝑡 ∗ = 𝑡 𝑡+𝑇 𝑒𝑥𝑝 𝛽10 + 𝛽12 ∙ 𝑈2𝑡 + 𝛽13 ∙ 𝑈3𝑡 𝑑𝑡 𝑅𝑡−1 = 𝐶𝑎𝑙𝑙𝑀𝑉𝑜𝑙 + 𝑃𝑢𝑡𝑀𝑉𝑜𝑙 2 − 𝑈𝑛𝑐𝑜𝑛𝑑_𝑉𝑜𝑙 𝑈𝑛𝑐𝑜𝑛𝑑_𝑉𝑜𝑙 Volatility Re-Projection and Option Pricing Introduction Research Design (how) Market Implied Volatilities Summary
- 10. and would be estimated for every maturity time-step between t and T as: where X is the strike price, T is maturity, N is the number of time-steps between t and maturity T, St,T is the risk-adjusted day-ahead underlying contract price measure and r is the relevant risk-free interest rate at time t. Similarly, for a put at time t: Page: 10 max ,0 ( )rT Q rT T Q X c e E S X e x X f x dx The fair price for a call at time t is now generally (T is the option contract maturity) Option Prices from filtering and re-projected volatility: Volatility Re-Projection and Option Pricing 1 1 max ,0 N rT N T t c N e S X 1 1 max ,0 N rT N T t p N e X S Introduction Research Design (how) Market Implied Volatilities Summary
- 11. The underlying December Future Prices: NASDAQ OMX (NOMX: NEDECX): (TIP-format) ICE (EOD_Futures_390_2014): (CSV-format) Page: 11 Market Option Prices http://www.nasdaqomx.com/commodities/markets/marketprices/ NASDAQ OMX (NOMX) market: For quarter and year contracts liquidity is relatively high. Lower liquidity at NordPool than at the ICE (London) energy contracts. The NOMX option market is mainly an OTC market. The InterContinental Exchange (ICE) market: Strongly higher liquidity. The ICE has 92% of the EU ETS international trading volume. Electronic platform. http://www.theice.com/emissions.jhtml Volatility Re-Projection and Option Pricing Introduction Research Design (how) Market Implied Volatilities Summary
- 12. Page: 12 The Optimal SV Model Parameters for NOMX and the ICE: Model Parameters diagnostics (biased upwards (Newey, 1985 and Tauchen, 1985), biased downward relative to 2.0: Volatility Re-Projection and Option Pricing Carbon Front December General Scientific Model. Parallell. Statistical Model SNP-11114000 - fit model Parameter values Scientific Model. Standard Parameters Semiparametric-GARCH. Mode Mean error h Mode Standard error 1 , a0 0.0038743 0.0077154 0.0469340 h 1 a0[1] 0.0024100 0.0127200 2 , a1 0.0412730 0.0409200 0.0174200 h 2 a0[2] -0.1234100 0.0184000 3 , b0 0.7994600 0.7572900 0.1229600 h 3 a0[3] -0.0303600 0.0136400 4 , b1 0.9784500 0.9097200 0.0802810 h 4 a0[4] 0.1242100 0.0139300 5 , s1 0.0869930 0.1305500 0.0524810 6 , s2 0.2673000 0.2168800 0.0724770 h 6 B(1,1) 0.0323500 0.0270900 7 , r1 -0.4172900 -0.3238600 0.1553400 h 7 R0[1] 0.1092300 0.0150800 h 8 P(1,1) 0.4010900 0.0260400 log sci_mod_prior 0.4302133 h 9 Q(1,1) 0.9410500 0.0073900 log stat_mod_prior 0 c2 (2) = h 10 V(1,1) -0.0005200 1180265.8 log stat_mod_likelihood -1893.14590 -2.6397 log sci_mod_posterior -1892.71569 {0.267175} Score diagnostics: Moments normalized standard Index mean score error t-statistic descriptor 1 -0.25436 1.99497 -0.1275 a0[1] 1 2 0.31626 1.99774 0.15831 a0[2] 2 3 -0.011 1.84787 -0.00596 a0[3] 3 4 -0.66444 1.89119 -0.35134 a0[4] 4 5 0 0 0 A(1,1) 0 0 6 0.32974 0.95977 0.34356 B(1,1) 7 -0.93452 2.69404 -0.34688 R0[1] 8 -3.12558 3.49944 -0.89316 P(1,1) s 9 -10.93826 14.00918 -0.78079 Q(1,1) s 10 0 0 0.10818 V(1,1) s Score diagnostics: normalized standard Index mean score error t-statistic descriptor 1 -0.34393 1.94874 -0.17649 a0[1] 1 2 0.64331 1.88078 0.34205 a0[2] 2 3 -1.05049 1.86885 -0.56211 a0[3] 3 4 -1.16356 1.92289 -0.60511 a0[4] 4 5 -0.24448 1.80654 -0.13533 a0[5] 5 6 0.48163 1.61429 0.29835 a0[6] 6 7 0 0 0 A(1,1) 0 0 8 0.01012 0.98019 0.01033 B(1,1) 9 -0.25238 2.26205 -0.11157 R0[1] 10 -0.57609 2.02835 -0.28402 P(1,1) s 11 -2.06059 11.48005 -0.17949 Q(1,1) s 12 0.15145 0.9244 0.16384 V(1,1) s Introduction Research Design (how) Market Implied Volatilities Summary
- 13. Page: 13 One-step-ahead conditional volatility (density) (conditional on xt-1 = -10%...+10% of data (filtering)): Volatility Re-Projection and Option Pricing Introduction Research Design (how) Market Implied Volatilities Summary
- 14. Page: 14 Market, Re-projection model (with conf.int.) and Black’76 model prices: Volatility Re-Projection and Option Pricing Introduction Research Design (how) Market Implied Volatilities Summary
- 15. Page: 15 Volatility Re-Projection and Option Pricing Option Pricing using implied volatilities (March 2014 and June 2014): Introduction Research Design (how) Market Implied Volatilities Summary
- 16. Page: 16 Volatility Re-Projection and Option Pricing Option Pricing using implied volatilities (September 2014 and December 2014): Introduction Research Design (how) Market Implied Volatilities Summary
- 17. Page: 17 Volatility Re-Projection and Option Pricing Implied volatilities towards Maturity (November and December 2014): Introduction Research Design (how) Market Implied Volatilities Summary
- 18. Page: 18 Volatility Re-Projection and Option Pricing Implied volatilities towards Maturity (November and December 2014): Introduction Research Design (how) Market Implied Volatilities Summary
- 19. Page: 21 Volatility Re-Projection and Option Pricing MAPE from December 2013 to December 2014: Introduction Research Design (how) Market Implied Volatilities Summary
- 20. Page: 22 Volatility Re-Projection and Option Pricing Risk Premium September 2013 to December 2014): Introduction Research Design (how) Market Implied Volatilities Summary
- 21. Page: 23 Volatility Re-Projection and Option Pricing Risk Premium November 2014 to December 2014 (last 17 days of trading): Introduction Research Design (how) Market Implied Volatilities Summary
- 22. Page: 24 Volatility Re-Projection and Option Pricing Market correlation for risk premium December 2013 to December 2014: Introduction Research Design (how) Market Implied Volatilities Summary
- 23. Page: 25 Volatility Re-Projection and Option Pricing Model diagnostics suggest that the Gaussian asymmetric multi-factor models are appropriate in modelling short-term kurtosis of front December futures returns. Conditional (xt-1) volatility densities contain extra information The re-projected conditional volatility densities (log-normal) produce the observed market volatility smiles and show consistent errors towards maturity Market implied volatilities (market option prices) are sensitive to forward looking information and towards maturity (with higher liquidity) show some interesting features (increases with an increasing volatility smile). An anomaly? Risk premiums show characteristics from implied volatilities Introduction Research Design (how) Market Implied Volatilities Summary
- 24. Page: 26 Volatility Re-Projection and Option Pricing The MAPE/MPE errors are stable and relative similar in size over time but for Black’76 explosive errors toward maturity For the markets: Relative to the ICE, the NOMX reports clearly lower MRE/MARE for the Black’76 model. The implication is that the NOMX market seems to rely more on the use of the Black’76 model for option pricing than the ICE market. For the methodologies: For at least the 6 last months before maturity, the ICE seems to rely more on the re-projection model for option pricing. Introduction Research Design (how) Market Implied Volatilities Summary
- 25. Page: 27 Volatility Re-Projection and Option Pricing General Modelling Conclusions (European carbon markets): One mean and two Gaussian stochastic volatility factors seem to capture relevant market characteristics The re-projected volatility seem to work well for general option pricing based on underlying density characteristics The re-projected volatility is clearly superior to Black76’ towards maturity and for markets showing high liquidity. This pricing information towards maturity may also suggest that the re-projected volatility methodology also produces better fundamental pricing for the whole life of the option (long before maturity). High liquidity seems to price options closer to one-step-ahead conditional volatility projection (fundamentals). For very short time to maturity the re-projected volatility report implied volatilities very close to market implied volatilities for any strike. Risk premiums are quite stable, is reflected in market implied volatilities (not contemporaneously reflected in the underlying future) and the small risk premium changes over time may emerge from general market information flow The MAPEs’ report mispricing from Black’76. For markets the NASDAQ OMX low moneyness makes direct comparison difficult. However, high correlation suggest that arbitrage conditions holds. Introduction Research Design (how) Market Implied Volatilities Summary