Option Valuation Using Intraday Data Peter Christoffersen Rotman School of Management, University of Toronto, Copenhagen Business School, and CREATES, University of Aarhus 2nd Lecture on Thursday 1
Model Free Realized Volatility versus Model Free VIX We should Use RV to Estimate / Model Option Prices 2
Overview: Potential Approaches RV Only: QMLE using RV and Kalman filter. Works much better than QMLE using Returns. BarndorffNielsen and Shephard (JRSS, 2002). Options Only: Estimating SV models using options only. Surprisingly easy. 1) Estimating SV on options and RV. Andersen, Fusari and Todorov (WP, 2011) 2) Develop a new class of affine models for returns, RV, and options. CFJM (WP, 2011) 3) Develop another new class of non-affine models for returns, RV and options. Corsi, Fusari and La Vecchia (JFE, 2013). 3
Estimating SV Models on Options Only Consider the following simple iterative twostep estimator: Step 1: For a given state vector: Choose structural parameters to min option errors across all weeks. Step 2: For given structural parameters: Each week choose V(t) to min weekly option errors. Iterate between Step 1 and Step 2. Used for example in Bates (JEconm, 2000), Huang and Wu (JF, 2004), CHJ (MS, 2009). 4
Parametric Inference and Dynamic State Recovery from Option Panels Torben Andersen Nicola Fusari Victor Todorov
Overview of AFT Summary of Paper Assumptions Optimization Problem Key Theoretical Results Monte Carlo Discussion Points Option errors Loss function Joint P and Q estimation Model misspecification Potential application 6
Assumption 1: Option Data Panels Need to allow for the ugliness of options data Strikes vary over time Maturity telescoping issue Contracts are born and die Necessary data filters (zero-volume, arbitrage, data errors, wide bid-ask spreads, etc) 7
Assumption 4: Measurement Errors Assume IV measurement errors With the following properties Conditional independence versus unconditional independence. Note: No structure on cov matrix of errors. Compare with Bates (JEconm, 2000). 8
Optimization Problem Choose state vector and structural parameters to minimize Penalty on (RV less V(t))2 (in pure SV). How to choose the λn penalty parameter? RV noisy proxy for true V(t). Log V(t) penalty? Seemingly huge dimensionality issue, but Iterative 2-step estimation (from above) can be used. 9
Key Theoretical Results Theorem 1: Consistent estimates of state vector and structural parameters. Theorem 2: Parameter estimates have asymptotically mixed Gaussian distribution. Option errors can be conditionally heteroskedastic. Penalty term (λn) vanishes and has no first-order asymptotic effect in estimation. Testing framework is the core contribution: Corollary 1: Option fit test Corollary 2: Parameter stability test Theorem 3: State vector fit test 10
Monte Carlo Study Very impressive. Assumptions on starting values is key. Start at true values (?) How much do the parameters move around? Size (Tables 3-4) is great. What about power? SVJ versus SV? Two-component versus one-component SV? Read the online supplementary appendix if you are contemplating work in this area. 11
P and Q Estimation Is Q-only estimation really a virtue? Don t we want to know P and Q and thus pricing kernel parameters, risk premia, etc.? For hedging for example? Can theory be extended to the case of joint estimation on returns and options? Bates (2000), Pan (2002), CJO (JFE,2013) 12
Data based Measurement Errors? Option valuation models have no errors and so we have to make up our own error structure. CJ (JFE, 2004). IV versus dollar prices Relative versus absolute errors: IV-based estimates may be driven mainly by high-volatility episodes. Log IV versus level? Volume weighted errors? Mid-prices issues: Bid-ask spreads are wide. Spreadweighted errors? Potential added layer of model misspecification? 13
Error Structure Motivated by Loss Function? Economic loss function to guide us in choice of error structure? What do we care about? Hedging errors? Replication errors? Incomplete markets. Which conditions do the loss function need to meet in order for the estimation and testing theory to hold? Does it hold more generally? 14
Case of Model Misspecification What are properties of model prices under model misspecification? MSE optimal? Possible to get result a la White (1982)? Amemiya: Provide conditions for parameters to converge asymptotically to their (population) objective optimal values. Need well-behaved objective function. Particularly relevant because misspecification could arise from error structure! It is not clear to me if the penalty on RV errors is enough to keep V(t) path consistent with the assumed variance dynamics in the model? 15
An Application I Would Like to See Affine versus non-affine SV for the purpose of option valuation. The estimator objective function is well-suited to address this problem rigorously. Econometric literature: 95% non-affine models Finance literature: 95% affine models My Intuition: Jumps are likely to appear spuriously important in affine models where V(t) is too smooth. 16
The Economic Value of Realized Volatility: Using High Frequency Returns for Option Valuation Peter Christoffersen Bruno Feunou Kris Jacobs Nour Meddahi 17
Motivation Realized volatility (RV) uses more information and provides better volatility forecast than GARCH. How can we use RV in option valuation? Could estimate an SV model with RV using GMM We instead incorporate RV directly into the model. Three model estimations: 1) R+RV, 2) Options, 3) Options+R+RV 18
Heston-Nandi (2000) Affine GARCH (Note: timing convention) Returns: Moments: Dynamics: Rewritten: 19
New GARV Model Returns: Moments Variance components: R-based Var: RV-based Var: Allow for correlation ρ between the two epsilons. 20
GARV Component Structure Expected variance: Where: Variance of Variance and Leverage effect: 21
Quasi MLE On Returns and RV Returns Likelihood: RV Likelihood: Joint Quasi Likelihood: 22
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24 Table 1: MLE on Returns and RV
Figure 4: Daily Conditional Vol. 25
Figure 5: Daily Vol. of Variance 26
Figure 6: Daily Leverage Effect 27
GARV Risk Neutralization MGF of Physical is of the form Assume CEFJ style linear 2-shock pricing kernel Q drift must be r: Keep Q affine via: 28
Risk Neutral Process Using the physical process and the pricing kernel above, we get the RN return process And the RN variance dynamics 29
Risk Neutral MGF and Option Prices Using the above parameter mapping we get Which can be used to get option prices via 30
MLE on Options Assume the following option valuation error structure: And option likelihood: 31
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33 Table 3: MLE on Options Only. Note: Only Q parameters
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35 Table 5: Joint MLE on Options, Returns and RV, 1996-2009 Note: P and Q Parameters
Figure 7: Weekly IVRMSE from ATM Options. (Table 5 MLEs) 36
Realizing Smiles: Pricing Options with Realized Volatility Fulvio Corsi Nicola Fusari Davide La Vecchia 37
Asset Return Process Assume a Log return process of the form With a return drift specification This can be viewed as a normal mixture model of the form 38
HARGL Variance Distribution Dynamics Assume an autoregressive Gamma (ARG) process for RVt+1 with shape parameter δ, scale c, and HAR location dynamic of the form Leverage Effect We can write This implies that variance moments are linear 39
Conditional Laplace Transform (MGF) Conditional ONE STEP LT for RV (under P) Conditional LT for K = {y, RV} jointly (under P) 40
Risk Neutral Distribution Assume a standard log-linear SDF of the form Then the model will still be of the HARGL form under Q with parameter mapping: Free parameter 41
Estimation on Returns and RV MLE for RV Truncated infinite sum at 90. Regression for returns so 42
Estimates of Physical Measures 43
Option Valuation Four steps 1) Estimate P process (above) 2) Calibrate ν1 to one-year implied vol. 3) Use above mapping to get from P to Q parameters 4) Use Monte Carlo pricing to get call price on L=50,000 simulated Q paths for S as follows: Quasi-analytical pricing is slow for long-maturity options. Therefore MC. 44
Overall Option Valuation Results -S&P500 options, 1996-2004, -RMSE on IV and on dollar prices. -RMSE and ratio RMSE to HARGL model. 45
Error Decomposition 46
Discussion Points RV has economic value Other models? Affine versus non-affine Normal distribution? RV versus SV? Measurement error in RV Modelling of leverage effect is key. Term Structure properties. Economic drivers of volatility 47