Option Valuation Models with HF Data a Comparative Study*

Transcription

1 Option Valuation Models with HF Data a Comparative Study* The Properties of Black Model with Different Volatility Measures. Ryszard Kokoszczyński, Natalia Nehrebecka, Paweł Sakowski, Paweł Strawiński, Robert Ślepaczuk University of Warsaw * We gratefully acknowledge government financial support via grant no. N N324336

2 Outline Motivations Literature review Methodology Data Technical issues when dealing with HF data Results Summary Further research 2

3 Motivations General objective search for the best (possibly universal) option pricing model giving outcomes close to those quoted in the market, High frequency data (0-second data interval, based on tick data), Use of WIG20 index option quotes (bid and ask) to remove nonsynchronous bias, Research hypothesis: BIV model gives the lowest pricing error as it includes the most recent observation when estimating volatility, Detailed questions: What kind of volatility should be used in Black model? What time period (n) should be used for averaging volatility in the estimation process? What is optimal interval (delta) for estimating volatility? Are errors dependent on TTM and moneyness ratio classification we use? 3

4 Literature Review Pricing models began in the 960s with seminal papers of Black and Scholes (973), Merton (973), then Heston (993), Hull and White (987) introduced SV model, and later Duan (995) implemented GARCH models. There are two types of empirical studies: First type includes direct attempts to verify option pricing models with data from stock exchanges and other markets: Duan and Zhang (200), Lehar, Scheischer and Schittenkopf (2002), Christoffersen and Jacobs (2004), Stentoft (2008), Mitra (2009), Moon (2009), Studies of the second type focus on volatility models that are later applied for pricing Hansen and Lunde (200), Martens and Zein (2002), Shu and Zhang (2006), Martens and Dijk (2007), Ślepaczuk and Zakrzewski (2008), There is a small number of empirical studies of option pricing models for the Polish capital market: Osiewalski and Pipień (2003), Kuziak (2005), Pipień and Pajor (2005), Piontek (2007), Fiszeder (2008), Bartkowiak (2009). However, we do not know of comprehensive studies covering a wide range of models for data of different intervals and comparing their properties in an attempt to find the best theoretical model for different markets. Direct motivation for our study: Fiszeder (2008), 4

5 Methodology. Black Model Index futures pricing Black model: c = p = d d e rt e rt gdzie : [ FN( d [ KN ( d ) KN ( d 2 ) FN( d 2 ln( F / K) + σ T / 2 = σ T 2 ln( F / K) σ T / 2 = = d σ σ T 2 )] )] T () (2) Black-Scholes model vs. Black model: WIG20 index futures mature exactly in the same day as WIG20 index options, and prices are set exactly in the same way, No need for calculating the dividend ratio for the index, Data from 8.30 and 9.00 am may be used, though index data start later. 5

6 Methodology 2. Pricing Models and Error Metrics Option pricing models applied in this study: Black model with historical volatility (sigma as standard deviation, n=2 (month)), Black model with realized volatility (sigma as an estimate of realized volatility RV, point estimate calculated from observations with delta interval), Black model with implied volatility (sigma as implied volatility for the previous observation, calculated separately for each TTM and moneyness class, an average for 50 groups, Error Metrics: Root Mean Squared Error (RMSE): RMSE = n i= ( Black i MID i ) 2 () Heteroscedastic Mean Absolute Error (HMAE): Heteroscedastic RMSE (HRMSE): Percentage of OverPrediction (%OP): HMAE = n HRMSE = n n % OP = OPi, where : OPi n n i= i= = 0 Blacki MIDi MID Blacki MIDi MIDi i= i i n if if i Black i 2 > MID Black < MID i (2) (3) (4) 6

7 Methodology 3. Volatility Models Historical volatility estimator (BHV model) (standard deviation for log returns): n n VAR = where: ( N * n) t ri, t = log Ci, t logci, t r = N * n Realized volatility estimator (BRV model): = i= Implied volatility estimator (BIV model) based on the most recent observation N n ( r N i, t r i, t= i= r) t 2 RV N 2, t = r i, t i= (5) (6) (7) (8) Annualization and averaging: Multi- period estimator (range, n>): annual _ std SD n = 252* N Single-period estimator (point, average)*: * VAR n (9) annual _ std [ RV ] n = 252 n n t= [ RV ], t (0) * In this study realized volatility averaging has been done for 5-minute interval and n days, where n=,2,3,5,0,2. 7

8 Data 0-second data made available by the Warsaw Stock Exchange: WIG20 index options quotes (bid-ask quote), WIG20 futures quotes, WIBOR interest rate (daily data), Time period: , observations, Quotes for 65 CALL options (C8, F8 and I8) and 63 PUT options (O8, R8 and U8). All calculations done with SAS 9.2 and Mathematica 7.0, 8

9 Technical Problems with HF Data Nature of data file: coded SIR text file with all transaction data from WSE, Size of data file (up to 200 GB, after compression), Computing power constraints -> time needed for calculations, How to compare results: absolute vs. relative measures, How to present research results?: 200 statistics for comparing models quality, Outlier identification: can we remove observations for mid <, 5, 0 or not? 9

10 Results_ We got more than 2 mn theoretical premiums predicted by the Black model. Below we show how they relate to : 6 pricing models (BHV, BRV0s, BRVm, BRV5m, BRV5m i BIV), 2 types of option (call and put), 5 classes of moneyness ratio: deep OTM (0-0,85), OTM (0,85-0,95), ATM (0,95 -,05), ITM (,05 -,5) and deep ITM (>,5) for call options and in the opposite order for put options, 5 classes for time to maturity ( [0-5 days], [6-30 days], [3-60 days], [6-90 days], [9+ days], That allows for multidimensional comparison of pricing models we use in this study. Table. BRV model* : no. of predicted premiums for different classes of moneyness TTM Option moneyness status 0-5 days 6-30 days 3-60 days 6-90 days 9+ days Total CALL deep OTM CALL OTM CALL ATM CALL ITM CALL deep ITM Total CALL PUT deep OTM PUT OTM PUT ATM PUT ITM PUT deep ITM Total PUT Total CALL and PUT * BHV model 7 mn, BIV model 5 mn observations. 0

11 Results_2. Number of predicted premiums BRV model Figure. Number of call options to TTM and moneyness ratio for active midquotes Figure 2. Number of put options to TTM and moneyness ratio for active midquotes

12 Results_3. Strike prices traded in the period of our study Figure 3. Moneyness ratio histogram for call options and available strike prices. Figure 4. Moneyness ratio histogram for call options and active mid-quotes. deep OTM OTM ATM ITM deep ITM deep OTM OTM ATM ITM deep ITM S F moneyness ratio = rt K / e = K () Figure 5. Moneyness ratio histogram for put options and available strike prices. Figure 6. Moneyness ratio histogram for put options and active mid-quotes. deep ITM ITM ATM OTM deep OTM deep OTM deep ITM ITM ATM OTM deep OTM 2

13 Results_4. Comparing Volatility Models Figure 7. Historical and realized volatility (5m, 5m_5, 5m_2) 3

14 Results_5. Comparing Volatility Models cont d. Figure 8. Implied volatility for ATM call option 4

15 Results_6. Comparing Volatility Models cont d. Figure 9. Implied volatility for ATM put option 5

16 Results_7 Figure 0. OP statistics for call options, all pricing models, TTM and moneyness classes. Model BRV0s Model 2 BRV5m Model 3 BRV5m_5 Model 4 BRV5m_2 Model 5 BHV Model 6 BIV - no significant differences for models: BRV0s, BRV5m, BRV5m_5, - BIV model has the best OP value (approx. 0.5), next one is BHV model, and then BRV5m_2 model, - all models show most often underpredicted premia for TTM=.2 (with the exception of deep_itm and BIV model), 6

17 Results_8 Figure. RMSE statistics for call options, all pricing models, TTM and moneyness classes. Model BRV0s Model 2 BRV5m Model 3 BRV5m_5 Model 4 BRV5m_2 Model 5 BHV Model 6 BIV - no significant differences for models: BRV0s, BRV5m, BRV5m_5, - error increases with TTM, but it may be due to the higher option value confirmation needed from HMAE or HRMSE => comparing these error metrics show that RMSE is a misleading one, - lowest values for BIV model, then for BHV and BRV5m_2 models, 7

18 Results 9 Figure3. HMAE statistics for call options, all pricing models, TTM and moneyness classes. Model BRV0s Model 2 BRV5m Model 3 BRV5m_5 Model 4 BRV5m_2 Model 5 BHV Model 6 BIV - no significant differences for models: BRV0s, BRV5m, BRV5m_5, - best results for BIV model, and then forbhv and BRV5m_2 models, - very high HMAE values for BRV_deep OTM_TTM-3 model seem to suggest existence of outliers in initial data we conciuosly left at this stage -> unfortunately they distort other results; those outliers are the outcome of point estimates for volatility, i.e. we think they are spurious outliers as their cause is the specific nature of the BRV model, 8

19 Results_0 Figure 4. HRMSE statistics for call options, all pricing models, TTM and moneyness classes. Model BRV0s Model 2 BRV5m Model 3 BRV5m_5 Model 4 BRV5m_2 Model 5 BHV Model 6 BIV - HRMSE results confirm conclusions derived from HMAE statistics, 9

20 Results_ Figure5. OP statistics for put options, all pricing models, TTM and moneyness classes.. Model BRV0s Model 2 BRV5m Model 3 BRV5m_5 Model 4 BRV5m_2 Model 5 BHV Model 6 BIV - no significant differences for BRV model (theoretical premium underestimates actual prices), - best results for BIV model, then for BIV, BHV, and BRV5m_2 models, - all models underestimate market prices (with the exception of deep_itm options), 20

21 Results_2 Figure 6. RMSE statistics for put options, all pricing models, TTM and moneyness classes. Model BRV0s Model 2 BRV5m Model 3 BRV5m_5 Model 4 BRV5m_2 Model 5 BHV Model 6 BIV - no significant differences for BRV0s, BRV5m, BRV5m_5 models, - lowest values for BIV model, next is BHV, and then BRV5m_2 model, - error increases with TTM (with the exception of deepitm options), but it may be due to the higher option value -> this outcome is not confirmed by HMAE and HRMSE statistics, - very high RMSE values for BRV and BHV models for deep ITM option with TTM =; again possibility of outliers, Option Valuation Models with HF Data a Comparative Study. a Comparative Study. The Properties of Black Model with Different Volatility Measures. 2

22 Results_3 Figure 7. HMAE statistics for put options, all pricing models, TTM and moneyness classes. Model BRV0s Model 2 BRV5m Model 3 BRV5m_5 Model 4 BRV5m_2 Model 5 BHV Model 6 BIV - no significant differences for BRV0s, BRV5m, BRV5m_5 models, - lowest values for BIV model, next for BHV, and BRV5m_2 models, - very high values for deep_otm and OTM options for TTM=,2,3 (with the exception of BIV model)-> Black < MID - close to zero values for ITM and deep_itm options, 22

23 Results_4 Figure 8. HRMSE statistics for put options, all pricing models, TTM and moneyness classes. Model BRV0s Model 2 BRV5m Model 3 BRV5m_5 Model 4 BRV5m_2 Model 5 BHV Model 6 BIV - similar conclusions as for the HMAE statistics, 23

24 Results_5 another dimension Figure 9. HMAE statistics for put options, all pricing models, TTM and moneyness classes - best results for BIV model, and then for BHV model, - no significant differences among BRV models, - best model fit for high TTM and moneyness ratio, - high error values for low TTM and moneyness ratio, 24

25 Results_6 another dimension, different averaging parameters and moneyness ratio Figure 20. RMSE for call options, BRV model, different averaging parameters and moneyness ratio. Model BRV5m Model 2 BRV5m_ Model 3 BRV5m_2 Model 4 BRV5m_3 Model 5 BRV5m_5 Model 6 BRV5m_0 Model 7 BRV5m_2 - model quality increases with the averaging parameter, - moneyness ratio does not influence model quality, 25

26 Results_8 another dimension, different averaging parameters and moneyness Figure 2. RMSE statistics for call options, BRV model, different averaging parameters and TTM. Model BRV5m Model 2 BRV5m_ Model 3 BRV5m_2 Model 4 BRV5m_3 Model 5 BRV5m_5 Model 6 BRV5m_0 Model 7 BRV5m_2 - continuous quality gains for pricing models with increasing averaging parameter for TTM=3,4,5, - no quality differences among models for dla TTM =,2. 26

27 Conclusions BIV model gives best results (in term of volatility estimation), BHV model is slightly worse (what is the best value of n?), BRV models give clearly the worst results, We obtained best results for BRV models for averaging with n=2, though deciding what is the best value for n requires further detailed studies, In the case of put options there is a clear relation between model error and TTM, and model error and moneyness ratio: High error values for low TTM and moneyness ratio, Best fit for high TTM and moneyness ratio, We believe that the reason for poor outcomes for BRV model is high volatility of (point estimate of) RV, range estimate for HV (in the case of BHV model), with additional averaging over n days explains better results - > cf. Figs. 20 and 2. Multidimentional presentation of raw data allowed us to notice spurious outliers, that were actually no true outliers at all, 27

28 Further Research Including other pricing models with different volatility models and estimates: GARCH model, SV model, Volatility index based on VIX design, Implied volatility calculated separately for all optionsi, Averaging volatility for different values of n and delta, Defining ways for getting rid off spurious outliers : Negative implied volatility for put options with TTM =,2 and with moneyness ratio = 5, Rapid jumps of (point estimate of) RV resulting in BRV models with TTM=3 and moneyness ratio= overestimating market prices by 40000%, Comparing results of this study with results for transaction prices, Conducting similar study for other markets in different countries and different depth. 28

29 Thank you for your attention Paweł Sakowski, Robert Ślepaczuk, 29