Back to the past: option pricing using realized volatility

Transcription

1 Back to the past: option pricing using realized volatility F. Corsi N. Fusari D. La Vecchia Swiss Finance Institute and University of Siena Swiss Finance Institute, University of Lugano University of Lugano 12th October, Kellogg School of Management

2 Outline Introduction Motivation Previous work Stylized facts under the historical measure Model properties Risk-neutral measure Database description Conclusions

3 Introduction Motivation Outline Introduction Motivation Previous work Stylized facts under the historical measure Model properties Risk-neutral measure Database description Conclusions

4 Introduction Motivation Motivation High frequency data are nowdays wide available and easily tractable; considerable interest has been recently devoted to the use of high frequency (HF) data for measuring and forcasting volatility; volatility is a key ingredient of every option pricing model; only few papers try to employ high frequency data to price options or other derivatives. GOAL: definition of a discrete-time model for option pricing using Realized Volatility (RV) as a proxy for the unobservable underlying volatility.

5 Introduction Previous work Outline Introduction Motivation Previous work Stylized facts under the historical measure Model properties Risk-neutral measure Database description Conclusions

6 Introduction Previous work Previous work Realized volatility measure and forcast: Andersen et al. (21) and (23), Barndorff-Nielsen and Shephard (21) (22a) (22b) and (25), Comte and Renault (1998); Option pricing using RV: Stentoft (28).

7 Stylized facts Outline Introduction Motivation Previous work Stylized facts under the historical measure Model properties Risk-neutral measure Database description Conclusions

8 Stylized facts General idea A well established result in the financial econometrics literature is that, daily log returns do not have a Gaussian distribution: leptokurtic heavy-tailed distributions. Clark (1973) and Ane and Geman (2): for an underlying semi-martingale process, rescaling the log returns by an appropriate measure of the market activity allows to recover the standard Gaussian distribution. Here we focus on Realized Volatility as a measure of market activity (which is per se not observable).

9 under the historical measure Outline Introduction Motivation Previous work Stylized facts under the historical measure Model properties Risk-neutral measure Database description Conclusions

10 under the historical measure Model set-up Methodological approach: We cast our model in the general framework proposed by Bertholon, Monfort and Pegoraro (28). Specifically, in pricing derivatives, one can follow three different approaches: direct modeling (P and SDF); Risk-neutral constrained modeling (Q and P); back modeling (Q and SDF). Here we follow the first alternative (direct modeling).

11 under the historical measure under the historical measure Log-returns We assume the following dynamics for the log returns: ( ) St+1 ln := y t+1 = µ t+1 + RV t+1 ɛ t+1, (1) S t ɛ t+1 RV t+1 N(, 1). In our notation, S t+1, y t+1, and RV t+1 are respectively the price, the log returns and the RV at time t + 1. For the drift of the log returns (i.e. µ t+1 ) we propose the following specification: µ t+1 = r + ( γ 1/2) RV t+1.

12 under the historical measure under the historical measure (cont d) Realized volatility dynamics Which process to describe the RV dynamics? - Empirically we observe that the main feature of the RV is the strong persistence. = Therefore we follow Corsi (29) and we model the conditional mean of the RV (given its past values) using the conditional expected value of an HAR (Heterogeneous AutoRegressive) process.

13 under the historical measure under the historical measure (cont d) Realized volatility dynamics We model the RV as an Auto Regressive Gamma process (ARG) (see Gourieroux and Jasiack 26). This implies that: RV t+1 F t γ(δ, β (RV t, L t ), c) δ and c describe, respectively, the shape and the scale of the distribution, whereas β (RV t, L t ) is the location parameter given by: β (RV t, L t ) = β 1 RV t + β 2 ( 4 RV t i ) + β 3 ( i=1 21 i=5 RV t i ) + ΛL t. (2) We label this model Heterogeneous AutoRegressive Gamma (HARG) process.

14 Model properties Outline Introduction Motivation Previous work Stylized facts under the historical measure Model properties Risk-neutral measure Database description Conclusions

15 Model properties Model simulation 1 S&P5 log returns Simulated log returns Figure: Log-returns. Top panel: S&P 5 log-returns from January 1st 199 to December 31th 27 (4218 observations). Bottom panel: log-returns simulated path from HARG(3).

16 Model properties Variance term structure Term structure of GARCH(1,1) model 2 High h t+1 Term structure of component model 2 2 Term structure of HARG model Low h t+1 Normalized varance Normalized varance Normalized varance Horizon in days Horizon in days Horizon in days

17 Model properties Skewness-Kurtosis term structure Skewness Skewness Skewness Days to maturity Component GARCH skewness.5 GARCH(1,1) skewness Days to maturity HARG skewness Simulated.5 Fitted Days to maturity Excess kurtosis Excess kurtosis Excess kurtosis Days to maturity Component GARCH excess kurtosis 1.5 GARCH(1,1) excess kurtosis Days to maturity HARG excess kurtosis 1 Simulated Fitted Days to maturity

18 Model properties Volatility and Correlation Daily standard deviation Daily standard deviation Daily standard deviation x 1 5 GARCH(1,1): Stdev of h x 1 5 Component GARCH: Stdev of h x HARG: Stdev of RV Daily correlation Daily correlation Daily correlation.5 GARCH(1,1): Corr(R,h) Component GARCH: Corr(R,h) HARG: Corr(R,h)

19 Risk-neutral measure Outline Introduction Motivation Previous work Stylized facts under the historical measure Model properties Risk-neutral measure Database description Conclusions

20 Risk-neutral measure The stochastic discount factor The HARG(3) is an affine process under P. In order to maintain the analytical tractability of the model under Q we set up an exponential affine stochastic discount factor (SDF): M t,t+1 = exp( ν ν 1RV t+1 ν 2RV d t ν 3RV w t 1 ν 4RV m t 5 ν 5y t+1 ν 6L t). This SDF is compatible with the no-arbitrage conditions (parameter restrictions - in the paper).

21 Risk-neutral measure The (resulting) model under the risk neutral measure Under Q, the log returns follow a discrete-time stochastic model, with risk premium γ = 1/2. The RV is an HARG(3) process, featuring a transition density given by a non central gamma, with parameters ρ, δ, µ. with λ = ν 1 + γ ρ = cβ (1 + cλ) 2, (3) δ = δ, (4) c = c 1 + cλ, (5)

22 Database description Dataset We employ a large dataset of options taken from OptionMetrics from 1st January 1996 to 27th December 24. We consider only OTM Call-Put options with volatility less then 7%, maturity between 1 and 365 days and whose price is not less than.5 dollars.

23 Outline Introduction Motivation Previous work Stylized facts under the historical measure Model properties Risk-neutral measure Database description Conclusions

24 Competing models The direct modeling approach allows to estimate the HARG(3) model under the historical measure using the time series of log-returns and realized volatility. Just one parameter (ν 1 ) has to be calibrated. From a practical point of view we use the most liquid ATM Call and Put (minimizing the RMSE IV ). Competitor models: HN GARCH(1,1), Component GARCH.

25 Option pricing performace - Global HARG HN GARCH(1,1) Component GARCH RMSE p RMSE IV Table: Option pricing performance.

26 RMSE dynamics IV RMSE (%) IV RMSE dynamics

27 Nonparametric volume surface

28 HARG vs HN GARCH(1,1) Maturity Moneyness Less then 2 2 to 6 6 to 16 More then 16 < [ 2, 1) [ 1,.5) [.5,.5) [.5, 1) >= Table: HARG(3) vs HN GARCH(1,1).

29 HARG vs Component GARCH Maturity Moneyness Less then 2 2 to 6 6 to 16 More then 16 < [ 2, 1) [ 1,.5) [.5,.5) [.5, 1) >= Table: HARG(3) vs Component GARCH.

30 Option pricing performance - ATM options bias GARCH(1,1) Component GARCH HARG Implied volatility bias Jan95 Jan Jan5 Implied volatility bias Jan95 Jan Jan5 Implied volatility bias Jan95 Jan Jan5

31 Volatility regimes 5 VIX 45 4 High volatility VIX level (%) Medium volatility Low volatility

32 Option pricing performance - ATM options term structure Low implied volatility Medium implied volatility High implied volatility Days to maturity Days to maturity Days to maturity

33 Option pricing performance - cross-section 6<DTM<2 2<DTM<6 6<DTM<16 16<DTM< Low volatility period Moneyness Moneyness Moneyness Moneyness 6<DTM<2 2<DTM<6 6<DTM<16 16<DTM< Medimu volatility period Moneyness Moneyness Moneyness Moneyness 6<DTM<2 2<DTM<6 6<DTM<16 16<DTM< High volatility period Moneyness Moneyness Moneyness Moneyness

34 Option pricing performance - cross-section Implied Volatility Data cross section 1 Month 6 Months 1 Year Moneyness Implied Volatility HARG corss section 1 Month 6 Months 1 Year Moneyness Implied Volatility CJOW cross section 1 Month 6 Months 1 Year Moneyness Implied Volatility Models comparison 1 month Data CJOW HARG Implied Volatility Models comparison 6 month Data CJOW HARG Implied Volatility Models comparison 12 month Data CJOW HARG Moneyness Moneyness Moneyness

35 Option pricing performance - IV surface dynamics Level dynamics Level Term structure Term structure dynamics Skew dynamics Skew

36 Conclusions Conclusions We develop a stochastic volatility option pricing model that exploits the historical information contained in the HF data. Straightforward estimation. No need to calibrate all the parameters. The HARG(3) outperform competing GARCH models.