Option Valuation Using Intraday Data

Option Valuation Using Intraday Data

Peter Christoffersen

Rotman School of Management, University of Toronto, Copenhagen Business School, and

CREATES, University of Aarhus

Model Free Realized Volatility versus Model Free VIX We should Use RV to Estimate / Model Option Prices…

Overview: Potential Approaches

• RV Only: QMLE using RV and Kalman filter. Works much better than QMLE using Returns. Barndorff-Nielsen 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).

Estimating SV Models on Options Only

• Consider the following simple iterative two-step 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.

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

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)

Assumption 4: Measurement Errors

• Assume IV measurement errors • With the following properties

• Conditional independence versus unconditional independence.

Optimization Problem

• Choose state vector and structural parameters to minimize

• Penalty on (RV less V(t))2 (in pure SV). How to choose the λ_n 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.

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 (λ_n) vanishes and has no first-order asymptotic effect in estimation. • Testing framework is the core contribution:

– Corollary 1: Option fit test

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.

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?

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. Spread-weighted errors?

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?

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

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

The Economic Value of Realized

Volatility: Using High Frequency

Returns for Option Valuation

Peter Christoffersen Bruno Feunou

Kris Jacobs Nour Meddahi

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.

Heston-Nandi (2000) Affine GARCH

(Note: timing convention)

• Returns: • Moments: • Dynamics: • Rewritten:

New GARV Model

GARV Component Structure

• Expected variance: • Where:

Quasi MLE On Returns and RV

• Returns Likelihood:

• RV Likelihood:

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GARV Risk Neutralization

• MGF of Physical is of the form

• Assume CEFJ style linear 2-shock pricing kernel

Risk Neutral Process

• Using the physical process and the pricing kernel above, we get the RN return process

Risk Neutral MGF and Option Prices

• Using the above parameter mapping we get

MLE on Options

• Assume the following option valuation error structure:

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Realizing Smiles: Pricing Options

with Realized Volatility

Fulvio Corsi Nicola Fusari Davide La Vecchia

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

HARGL Variance Distribution Dynamics

• Assume an autoregressive Gamma (ARG) process for RV_t+1 with shape parameter δ,

scale c, and HAR location dynamic of the form • We can write

• This implies that variance moments are linear

Leverage Effect

Conditional Laplace Transform (MGF)

• Conditional ONE STEP LT for RV (under P)

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:

Estimation on Returns and RV

• MLE for RV

• Truncated infinite sum at 90. • Regression for returns

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:

Overall Option Valuation Results

-S&P500 options, 1996-2004,

-RMSE on IV and on dollar prices. -RMSE and ratio RMSE to HARGL model.

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.