MODELLING VOLATILITY SURFACES WITH GARCH Robert G. Trevor Centre for Applied Finance Macquarie University robt@mafc.mq.edu.au October 2000
MODELLING VOLATILITY SURFACES WITH GARCH WHY GARCH? stylised facts about asset price returns spot markets changes not normally distributed relatively more very small changes and more very large changes leptokurtic changes are hard to predict very noisy movements around changing mean but not independent or identically distributed large changes followed by large changes of either sign volatility clustering risk is partly predictable large amount of mean reversion, with small amount of persistence options markets historical vol underestimates option prices at-the-money vol underestimates away-from-the-money vol smiles and now smirks in equities suggest that some other source of risk is being priced by market (implicit, not explicit) Applied Finance Centre 2 A/Prof Rob Trevor
explaining stylised facts GARCH and Stochastic Vol? spot markets enormous and growing literature on GARCH-type models suggests that they can explain much of the stylised facts more recent interest in stochastic vol models options markets but estimation much more demanding => much less empirical evidence some emerging evidence for GARCH-type models less research for stochastic vol have been lots of implementation problems Applied Finance Centre 3 A/Prof Rob Trevor
implementation problems GARCH what is risk-neutralised or no-arbitrage probability measure how estimate parameters how calculate option prices Monte Carlo not American these problems have now been solved stochastic volatility what is risk-neutralised or no-arbitrage probability measure how estimate parameters discrete data, continuous specification how calculate option prices Monte Carlo not American finite difference Applied Finance Centre 4 A/Prof Rob Trevor
GARCH OPTION PRICING MODEL assumed process for spot prices ln S t+ = r + λ h t+t 2 h t+t + h t+t υ t+ S t 2 GARCH h t+2 t+ = β 0 + β h t+t + β 2 h t+t υ t+ NGARCH h t+2 t+ = β 0 + β h t+t + β 2 h t+t ( υ t+ c) 2 2 GJR h t+2 t+ = β 0 + β h t+t + β 2 h t+t υ t+ + β 3 h t+t max( 0, υ t+ ) 2 where the innovation υ t+ is iid standard normal and Var[ ln(s t+ S t )I t ] h t+t risk-neutralised/arbitrage-free process ln S t+ = r 2 h t+t + h t+t ε t+ S t GARCH h t+2 t+ = β 0 + β h t+t + β 2 h t+t ( ε t+ λ) 2 NGARCH h t+2 t+ = β 0 + β h t+t + β 2 h t+t ( ε t+ (c + λ) ) 2 GJR h t+2 t+ = β 0 + β h t+t + β 2 h t+t ( ε t+ λ) 2 + β 3 h t+ t max( 0, ε t+ + λ) 2 where the innovation υ t+ has been replaced by ε t+ λ can show that this process is arbitrage free what about λ? (See JC Duan (995), "The GARCH Option Pricing Model", Mathematical Finance, 5, pp3-32 and J Kallsen and M Taqqu (998), "Option Pricing in ARCH-type Models", Mathematical Finance, Vol 8, pp3-26) Applied Finance Centre 5 A/Prof Rob Trevor
A HELPFUL RE-PARAMETERISATION some restrictions on the parameters of the GARCH process are required ensure that conditional variance is non-negative GARCH h 0 > 0,β 0 > 0,β > 0, β 2 > 0 NGARCH h 0 > 0,β 0 > 0,β > 0, β 2 > 0 GJR h 0 > 0,β 0 > 0,β > 0, β 2 > 0,β 3 > 0 ensure that unconditional variance is bounded GARCH β + β 2 < NGARCH β + β 2 (+ c 2 ) < GJR β + β 2 + β 3 2 < unfortunately parameters NOT independent of observation frequency β and β 2 depend on frequency, but don't vary much across assets β 0 can be difficult to understand, best to recast as an annualised steady state or unconditional vol GARCH σ s = 365 β 0 ( β β 2 ) NGARCH σ s = 365 β 0 ( β β 2 ( + c 2 )) GJR σ s = 365 β 0 ( β β 2 β 3 2) for daily data annualise initial vol σ 0 = 365 h 0 for daily data Applied Finance Centre 6 A/Prof Rob Trevor
ESTIMATING THE PARAMETERS can estimate from spot data using econometric techniques (maximum likelihood) need quite a few observations at least 0 for daily frequency log-likelihood surface can be problematic often need simplex to refine initial values and BHHH (or BFGS) to estimate non-normal conditional densities can be easily handled, or can use QML to get robust standard errors many econometric packages available. My preference is RATS (http://www.estima.com). Comes with GARCH estimation routines which I wrote. recursive model dirty data such as missing observations, weekends, price limits, spikes? See Ieuan Morgan and Robert Trevor (999), "Limit Moves as Censored Observations of Equilibrium Futures Price in GARCH Processes", Journal of Business & Economic Statistics, Vol 7, pp 397-408 can also imply out from observed option prices, once you have a procedure to price options when underlying is a GARCH process need range of strikes (and/or maturities) at least as many as number of parameters pick parameters that minimise distance from observed prices/vols optimisation can be tricky/slow ideally both approaches should give same results spot data yields underlying process option data yields risk-neutralised/arbitrage-free simply related via change of measure Applied Finance Centre 7 A/Prof Rob Trevor
Implied Vol 25% GARCH AND VOLATILITY SMIRK 24 Day SPX PUT Options CBOE 3pm on 27 August 996 20% 5% 0% 5% 0. 5.00.05 Actual Implied Lattice Implied Implied NGARCH model σ 0 = 5.8%,σ s =3.87%, β = 0.7983, β 2 = 0.03, c =.6425 Stock Strike Actual ImpliedLattice Implie Difference 0.6 662.2 600.00 9.33% 2.03% -.7% 36 662.2 605.00 8.09% 20.28% -2.9% 22 662.2 60.00 9.9% 9.56% 0.36% 363 662.2 620.00 8.89% 8.5% 0.74% 54 662.2 6.00 7.27% 6.83% 0.44% 589 662.2 635.00 6.74% 6.20% 0.54% 665 662.2 640.00 5.93% 5.57% 0.36% 740 662.2 645.00 5.0% 4.96% 0.05% 86 662.2 6.00 4.49% 4.37% 0.2% 89 662.2 655.00 3.5% 3.78% -0.27% 967 662.2 660.00 3.40% 3.23% 0.8%.0042 662.2 665.00 2.85% 2.69% 0.6%.08 662.2 6.00 2.39% 2.9% 0.2%.093 662.2 675.00.86%.72% 0.4%.0269 662.2 680.00 0.73%.32% -0.59%.0344 662.2 685.00 0.07%.00% -3%.0420 662.2 6.00 7.4% 0.84% -3.69% Applied Finance Centre 8 A/Prof Rob Trevor
PICKING THE RIGHT PRICING ALGORITHM no closed form solutions available, even for Europeans key is volatility clustering => path dependent vol, so closed forms unlikely claimed closed form (SL Heston and S Nandi (2000), "A Closed-Form GARCH Option Valuation Model", Review of Financial Studies, Vol 3, pp585-625) model is not a standard GARCH formulation volatility innovation is not scaled by conditional variance need to numerically solve a bivariate system of difference equations to get coefficients of the characteristic function for price of underlying then need to numerically integrate real part of a function of this complex valued characteristic function to invert it to find European option values analytical approximations (JC Duan, G Gauthier and JG Simonato (999), "An Analytical Approximation for the GARCH Option Pricing Model", Journal of Computational Finance, Vol 2, pp75-66) can deduce approximations to moments of a GARCH process for a given maturity very messy formula (corrections from authors) moments can be used to approximate the distribution of terminal value of an asset can use this to value European claims on terminal values Monte Carlo European only, but may be useful for path dependent payouts control variates depend on option being valued BS evaluated at 365 h 0, 365 σ s 2 or at BS implied vol from GARCH option value moment corrections (JC Duan and JG Simonato (998), Empirical Martingale Simulation for Asset Prices, Management Science, Vol 44(9), pp28-233) Applied Finance Centre 9 A/Prof Rob Trevor
Markov chain approximation with sparse matrix tricks (JC Duan and JG Simonato (forthcoming), American Option Pricing under GARCH by a Markov Chain Approximation, Journal of Economic Dynamics and Control) American and European, but not as efficient as lattice lattice (Peter Ritchken and Robert Trevor (999), "Pricing Options under Generalized GARCH and Stochastic Volatility Processes", Journal of Finance, Vol 54, pp 377-402) American and European efficient procedure also handles stochastic volatility models can be modified for use with path dependent payouts once have prices for range of parameter values, strikes, maturities, etc, can use a neural network for interpolation (See M. Hanke (997), "Neural Network Approximation of Option Pricing Formulas for Analytically Intractable Option Pricing Models", Journal of Computational Intelligence in Finance, Vol 5, pp20-27) recommended methods? European, terminal distribution only Duan et al analytical approximation European, heavy path dependencies or multiple assets Monte Carlo Americans or other types of Europeans Ritchken and Trevor lattice GARCH provides accurate, efficient approximation to stochastic volatility models (bivariate diffusions) see Ritchken and Trevor paper for quality of approximation Applied Finance Centre 0 A/Prof Rob Trevor
GARCH IMPLIED VOLATILITY SURFACES smiles, smirks and grimaces controlled by parameters of GARCH equation following charts show actual fitted volatility surfaces (large) with initial volatility 20% below and above steady state volatility (pair of small charts) Applied Finance Centre A/Prof Rob Trevor
SPX Options on CBOE 0.28 0.23 0.8 0.3. 0 0.08 SPX Options on CBOE 0.23 0.8 0.3. 0 0.08 SPX Options on CBOE 0.28 0.23 0.8 0.3. 0 0.08 Applied Finance Centre 2 A/Prof Rob Trevor
AUD Options on OTC 0.4. 0 0.09 AUD Options on OTC 0.2. 0 0.07 AUD Options on OTC 0.5. 0 0.0 Applied Finance Centre 3 A/Prof Rob Trevor
HSI Options on HKSE 0. 0.25 0.20. 0 0.5 HSI Options on HKSE 0. 0.25 0.20. 0 0.5 HSI Options on HKSE 0.35 0. 0.25 0.20. 0 0.5 Applied Finance Centre 4 A/Prof Rob Trevor
WHAT REMAINS TO BE DONE? does GARCH explain the smile/smirk preliminary evidence suggests that it does strike price bias in BS maturity bias in BS key is dynamics of volatility clustering providing parameters prove to be stable, likely to provide superior hedging results more detailed, rigorous testing to be done over range of instruments and markets, especially on hedging can use GARCH not just for pricing and hedging normal options exotics that trade off the same underlying calculating the risk neutral probability distribution Applied Finance Centre 5 A/Prof Rob Trevor