Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market

Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.Performance of stochastic option pricing models and Construction of volatility smiles for option pricing in an emerging derivatives market.

MINISTRY OF EDUCATION AND TRAINING UNIVERSITY OF ECONOMICS HO CHI MINH CITY NGUYEN TRI MINH PERFORMANCE OF STOCHASTIC OPTION PRICING MODELS AND CONSTRUCTION OF VOLATILITY SMILES FOR OPTION PRICING IN AN EMERGING DERIVATIVES MARKET PhD THESIS Ho Chi Minh City – 2023 MINISTRY OF EDUCATION AND TRAINING UNIVERSITY OF ECONOMICS HO CHI MINH CITY PERFORMANCE OF STOCHASTIC OPTION PRICING MODELS AND CONSTRUCTION OF VOLATILITY SMILES FOR OPTION PRICING IN AN EMERGING DERIVATIVES MARKET Major: Finance and banking Code: 9340201 PhD THESIS Supervisor: Prof Tran Ngoc Tho Ho Chi Minh City – 2023 OPENING DECLARATION I hereby declare that the PhD thesis “Performance of Stochastic Option Pricing Models and Construction of Volatility Smiles for Option Pricing in an Emerging Derivatives Market” is my independent research, conducted under supervision of Prof Tran Ngoc Tho The research was carried out with integrity and based on reliable data sources The referenced works are properly cited within the thesis PhD student Nguyen Tri Minh Table of Contents ABSTRACT Chapter INTRODUCTION 1.1 Research background 1.2 Research objectives and contributions 1.3 Summary of methodology and results 1.4 Thesis structure 11 Chapter THEORETICAL FRAMEWORK AND LITERATURE REVIEW 12 2.1 Core concepts 12 2.1.1 What is an option 12 2.1.2 Option price/premium 15 2.2 The classic Black-Scholes option pricing model 16 2.2.1 Market model 16 2.2.2 Replicating Strategies 23 2.2.3 Risk-neutral pricing 31 2.3 Stochastic volatility and stochastic option pricing models 38 2.3.1 Stochastic volatility .38 2.3.2 Heston model’s process 43 2.4 The implied volatility surface 44 2.4.1 Implied volatility and volatility smile 44 2.4.2 Deriving an expression of implied volatility 45 2.5 Basic facts about machine learning 47 2.6 Literature review and research contributions 50 2.6.1 Stochastic option pricing models 50 2.6.2 Methods of determining implied volatility surface 52 2.6.3 Option pricing in illiquid markets 53 2.6.4 Application of machine learning in option pricing .55 Chapter METHODOLOGY AND DATA 57 3.1 Performance of stochastic option pricing models 57 3.1.1 Stochastic volatility models 57 3.1.2 The Heston model for European options 58 3.1.3 Heston and Heston++ models’ characteristic functions .61 3.1.4 Bates model’s characteristic function 62 3.1.5 Heston-Hull-White’s characteristic function 63 3.1.6 Risk-neutral moments 64 3.1.7 Model calibration 66 3.1.8 Data collection 69 3.2 Construction of implied volatility smiles for illiquid options 81 3.2.1 Data collection 81 3.2.2 Proposed methods 93 Chapter RESULTS AND DISCUSSIONS .98 4.1 Performance of stochastic option pricing models 98 4.1.1 In-sample pricing performance .98 4.1.2 Out-of-sample pricing performance 104 4.1.3 Hedging performance 106 4.1.4 Implied volatility surface characteristics 108 4.1.5 Risk-neutral return distribution characteristics 111 4.1.6 Industry characteristics .114 4.1.7 General findings 117 4.1.8 Robustness test 118 4.2 Construction of implied volatility smiles 120 4.2.1 Findings 120 4.2.2 Result figures for Correlation method 122 4.2.3 Result figures for K-nearest neighbor method (KNN) .124 4.2.4 Result figures for weighted K-nearest neighbor method (WKNN) 126 4.2.5 Backtesting 128 Chapter CONCLUSION 132 LIST OF THESIS-RELATED PUBLICATIONS 134 REFERENCES 135 LIST OF ABBREVIATIONS AR: autoregressive GDP: gross domestic product IV: implied volatility KNN: K-nearest neighbor LIBOR: London interbank offered rate RMSE: root-mean-square error RNK: risk-neutral kurtosis RNS: risk-neutral skewness RNV: risk-neutral variance TED: Treasury-Eurodollar WKNN: weighted K-nearest neighbor LIST OF TABLES Table 3.1: Starting values and parameter bounds .67 Table 3.2: Descriptive statistics per calibration 70 Table 3.3: Descriptive statistics 83 Table 3.4: Correlation values 93 Table 4.1: Median calibrated model parameters .102 Table 4.2: AR(1) coefficients of the model parameters 103 Table 4.3: Model performance and implied volatility surface characteristics ($RMSE) 109 Table 4.4: Model performance and implied volatility surface characteristics (%RMSE) 110 Table 4.5: Model performance and risk-neutral return distribution characteristics ($RMSE) 113 Table 4.6: Model performance and risk-neutral return distribution characteristics (%RMSE) 114 Table 4.7: Model performance and industry characteristics ($RMSE) 115 Table 4.8: Model performance and industry characteristics (%RMSE) 116 Table 4.9: Early exercise premium and bid-ask spread 119 Table 4.10: Early exercise premium and bid-ask spread relative to the market option prices .119 Table 4.11: Backtesting results for the correlation method 129 Table 4.12: Backtesting results for the KNN method .130 Table 4.13: Backtesting results for the Weighted KNN method 131 LIST OF FIGURES Figure 3.1: Number of call options 71 Figure 3.2: Number of put options 72 Figure 3.3: Average call moneyness 73 Figure 3.4: Average put moneyness 74 Figure 3.5: Average price 75 Figure 3.6: Average implied volatility 76 Figure 3.7: Average maturity 77 Figure 3.8: Average trading volume 78 Figure 3.9: Average open interest .79 Figure 3.10: Volatility .84 Figure 3.11: Skewness 85 Figure 3.12: Kurtosis 86 Figure 3.13: Daily return 87 Figure 3.14: Minimum return 88 Figure 3.15: Maximum return 89 Figure 3.16: Momentum 90 Figure 3.17: Reversal 91 Figure 4.1: Median root mean square error .100 Figure 4.2: Median 1-day ahead change in the root mean square error 105 Figure 4.3: Median hedging error 107 Figure 4.4: IV smile for low realized skewness and low realized kurtosis 122 Figure 4.5: IV smile for low realized skewness and high realized kurtosis 122 Figure 4.6: IV smile for high realized skewness and low realized kurtosis 123 Figure 4.7: IV smile for high realized skewness and high realized kurtosis 123 Figure 4.8: IV smile for low realized skewness and low realized kurtosis 124 Figure 4.9: IV smile for low realized skewness and high realized kurtosis 124 Figure 4.10: IV smile for high realized skewness and low realized kurtosis 125 Figure 4.11: IV smile for high realized skewness and high realized kurtosis 125 Figure 4.12: IV smile for low realized skewness and low realized kurtosis 126 Figure 4.13: IV smile for low realized skewness and high realized kurtosis 126 Figure 4.14: IV smile for high realized skewness and low realized kurtosis 127 Figure 4.15: IV smile for high realized skewness and high realized kurtosis 127 ABSTRACT This thesis has two objectives The first objective is carrying out a comparison of performance between four stochastic option pricing models (Heston, Heston++, Bates and Heston-Hull-White), based on pricing a cross-section of stock options across various industries The second is proposing a method of constructing implied volatility (IV) smiles for stock options in a new or illiquid option market (Vietnam in this case), using data from an existing market to so (the US in this case) For the first objective, the results show that the Heston model performs the best in insample pricing, as well as capturing the characteristics of the market, while Heston ++ performs the best in out-of-sample pricing and hedging For the second objective, the three proposed methods for constructing IV smiles, namely correlation, K-nearest neighbor (KNN) and weighted KNN, perform reasonably well, with weighted KNN considered the best among them 128 4.2.5 Backtesting Finally, a backtesting exercise is performed by estimating the IV smiles as of 29 August 2014 of 27 US companies belonging to the Dow Jones Industrial Average (DJIA) using the correlation, the KNN and the WKNN methods I have not been able to produce the implied volatility smiles for the remaining DJIA companies (in particular Honeywell, JPMorgan Chase and Verizon) due to missing data For the purpose of this exercise, each DJIA company for which the IV smile is estimated is removed from the universe of companies from which the most similar one according to the correlation, KNN or the WKNN method is determined I then apply all of these methods and compare the results to the actual IV smile observed in the market The results of the backtesting exercise are shown in Tables 4.11, 4.12 and 4.13 For each method, I display the true implied volatility of each US company for 75-call delta, 50-call delta and 25-call delta points as well as the respective predictions and absolute errors The correlation method achieves an average absolute error of slightly under 5.4%, whereas for the KNN method the average absolute error is under 3.5% and for the WKNN method it is under 2.4% Hence, it can be concluded that all methods perform reasonably well, with the WKNN method outperforming the other two methods In overall, Chapter provides the detailed results of the two research objectives (option model performance and construction of IV smiles), as well as the relevant robustness tests that help in increasing those results’ reliability The results show that in most cases, the Heston model performs the best among the chosen four stochastic option pricing models, and the three proposed methods for construction of IV smiles are viable, with WKNN considered the best performer among them 129 Table 4.11: Backtesting results for the correlation method Company 75delta True IV smile 5025delta delta Predicted IV smile 755025delta delta delta 75delta Absolute error 5025delta delta 3M 0.1955 0.1727 0.1594 0.1591 0.1370 0.1266 0.0364 0.0357 0.0328 American Express 0.2221 0.2052 0.1951 0.3571 0.2105 0.3690 0.1350 0.0053 0.1740 Amgen 0.2680 0.2500 0.2367 0.1846 0.1740 0.1644 0.0834 0.0760 0.0723 Apple 0.2694 0.2646 0.2635 0.2818 0.2241 0.2141 0.0124 0.0406 0.0494 Boeing 0.2324 0.2116 0.2034 0.2093 0.2093 0.2014 0.0231 0.0023 0.0020 Caterpillar 0.2368 0.2106 0.1928 0.4811 0.4811 0.2412 0.2443 0.2705 0.0484 Chevron 0.1890 0.1675 0.1565 0.2464 0.1719 0.1133 0.0574 0.0044 0.0432 Cisco Coca-Cola 0.2237 0.2121 0.2029 0.1470 0.1857 0.2298 0.0767 0.0264 0.0269 0.1714 0.1574 0.1504 0.1806 0.1215 0.2200 0.0092 0.0358 0.0695 Disney 0.2371 0.2194 0.2067 0.2515 0.2536 0.2563 0.0145 0.0343 0.0496 Dow 0.2505 0.2292 0.2212 0.1938 0.2122 0.2438 0.0567 0.0170 0.0226 Goldman Sachs 0.2146 0.1940 0.1759 0.1945 0.1867 0.1809 0.0202 0.0073 0.0050 Home Depot 0.2045 0.1836 0.1734 0.1881 0.1547 0.1602 0.0164 0.0289 0.0132 IBM Intel 0.2021 0.1857 0.1790 0.1623 0.1747 0.2082 0.0398 0.0111 0.0292 0.2564 0.2405 0.2331 0.2909 0.2017 0.2355 0.0345 0.0388 0.0024 Johnson & Johnson 0.1818 0.1634 0.1533 0.1446 0.1175 0.1354 0.0372 0.0459 0.0179 McDonald's 0.1660 0.1534 0.1546 0.1296 0.1087 0.1119 0.0364 0.0446 0.0427 Merck 0.2133 0.1941 0.1902 0.5488 0.1434 0.7016 0.3356 0.0507 0.5114 Microsoft 0.2334 0.2120 0.2021 0.2224 0.2067 0.2017 0.0110 0.0054 0.0004 Nike 0.2347 0.2225 0.2147 0.2182 0.1624 0.2373 0.0165 0.0601 0.0226 Procter & Gamble 0.1620 0.1424 0.1377 0.1180 0.1204 0.1231 0.0440 0.0220 0.0146 Salesforce 0.3231 0.3062 0.2897 0.3739 0.3854 0.4062 0.0508 0.0791 0.1165 Travelers 0.2521 0.2386 0.2270 0.1932 0.1915 0.1906 0.0590 0.0471 0.0364 UnitedHealth 0.2433 0.2223 0.2100 0.1906 0.1877 0.2021 0.0528 0.0346 0.0078 Visa Walgreens Boots Alliance 0.2235 0.2128 0.2064 0.2128 0.2227 0.2467 0.0108 0.0099 0.0403 0.2411 0.2301 0.2293 0.3295 0.2811 0.3866 0.0883 0.0510 0.1573 Walmart 0.1588 0.1429 0.1409 0.1346 0.1288 0.1306 0.0242 0.0142 0.0103 130 Table 4.12: Backtesting results for the KNN method Company 75delta True IV smile 5025delta delta Predicted IV smile 755025delta delta delta 75delta Absolute error 5025delta delta 3M 0.1955 0.1727 0.1594 0.1964 0.1659 0.1587 0.0009 0.0068 0.0008 American Express 0.2221 0.2052 0.1951 0.2045 0.1846 0.1763 0.0176 0.0206 0.0188 Amgen 0.2680 0.2500 0.2367 0.2627 0.2444 0.2405 0.0054 0.0056 0.0038 Apple 0.2694 0.2646 0.2635 0.2219 0.2240 0.2351 0.0476 0.0406 0.0284 Boeing 0.2324 0.2116 0.2034 0.2795 0.2015 0.1486 0.0471 0.0102 0.0547 Caterpillar 0.2368 0.2106 0.1928 0.1867 0.1760 0.1734 0.0501 0.0346 0.0194 Chevron 0.1890 0.1675 0.1565 0.1740 0.1408 0.1303 0.0150 0.0267 0.0262 Cisco Coca-Cola 0.2237 0.2121 0.2029 0.2068 0.1905 0.1873 0.0169 0.0216 0.0156 0.1714 0.1574 0.1504 0.1815 0.1302 0.2375 0.0101 0.0272 0.0871 Disney 0.2371 0.2194 0.2067 0.2094 0.1815 0.1901 0.0277 0.0379 0.0166 Dow 0.2505 0.2292 0.2212 0.2814 0.2568 0.2336 0.0309 0.0277 0.0123 Goldman Sachs 0.2146 0.1940 0.1759 0.2345 0.2027 0.1733 0.0198 0.0087 0.0026 Home Depot 0.2045 0.1836 0.1734 0.1794 0.1563 0.1585 0.0251 0.0273 0.0150 IBM Intel 0.2021 0.1857 0.1790 0.1904 0.1584 0.1296 0.0117 0.0274 0.0494 0.2564 0.2405 0.2331 0.5980 0.2610 0.6430 0.3416 0.0205 0.4099 Johnson & Johnson 0.1818 0.1634 0.1533 0.3011 0.1518 0.0153 0.1192 0.0116 0.1380 McDonald's 0.1660 0.1534 0.1546 0.1380 0.1253 0.1209 0.0279 0.0281 0.0337 Merck 0.2133 0.1941 0.1902 0.2080 0.1863 0.1770 0.0052 0.0078 0.0132 Microsoft 0.2334 0.2120 0.2021 0.2540 0.2357 0.2241 0.0206 0.0237 0.0220 Nike 0.2347 0.2225 0.2147 0.2163 0.2033 0.1911 0.0184 0.0192 0.0235 Procter & Gamble 0.1620 0.1424 0.1377 0.1908 0.1413 0.1119 0.0288 0.0011 0.0259 Salesforce 0.3231 0.3062 0.2897 0.4003 0.3858 0.3720 0.0772 0.0796 0.0823 Travelers 0.2521 0.2386 0.2270 0.2779 0.2530 0.2381 0.0257 0.0144 0.0111 UnitedHealth 0.2433 0.2223 0.2100 0.2202 0.1901 0.1778 0.0231 0.0322 0.0321 Visa Walgreens Boots Alliance 0.2235 0.2128 0.2064 0.2271 0.2007 0.1910 0.0036 0.0121 0.0154 0.2411 0.2301 0.2293 0.2520 0.2219 0.2054 0.0109 0.0081 0.0239 Walmart 0.1588 0.1429 0.1409 0.1423 0.1325 0.1301 0.0164 0.0104 0.0109 131 Table 4.13: Backtesting results for the Weighted KNN method True IV smile 5025delta delta Predicted IV smile 7550delta delta 25-delta 75delta Absolute error 5025delta delta Company 75delta 3M 0.1955 0.1727 0.1594 0.2528 0.19401 0.15243 0.0573 0.0213 0.0070 American Express 0.2221 0.2052 0.1951 0.21621 0.21297 0.21199 0.0059 0.0078 0.0169 Amgen 0.2680 0.2500 0.2367 0.28663 0.28493 0.28362 0.0186 0.0349 0.0469 Apple 0.2694 0.2646 0.2635 0.22669 0.21324 0.21483 0.0427 0.0514 0.0487 Boeing 0.2324 0.2116 0.2034 0.25492 0.25288 0.25329 0.0225 0.0413 0.0499 Caterpillar 0.2368 0.2106 0.1928 0.22334 0.20708 0.1937 0.0134 0.0035 0.0009 Chevron 0.1890 0.1675 0.1565 0.14816 0.14522 0.1425 0.0409 0.0223 0.0140 Cisco Coca-Cola 0.2237 0.2121 0.2029 0.23125 0.20647 0.19066 0.0075 0.0057 0.0123 0.1714 0.1574 0.1504 0.18017 0.16102 0.14583 0.0088 0.0036 0.0046 Disney 0.2371 0.2194 0.2067 0.24689 0.24791 0.25724 0.0098 0.0286 0.0506 Dow 0.2505 0.2292 0.2212 0.20892 0.21004 0.22055 0.0415 0.0191 0.0007 Goldman Sachs 0.2146 0.1940 0.1759 0.20871 0.20391 0.20097 0.0059 0.0099 0.0251 Home Depot 0.2045 0.1836 0.1734 0.16882 0.16235 0.1568 0.0357 0.0212 0.0166 IBM Intel 0.2021 0.1857 0.1790 0.19761 0.18422 0.17663 0.0045 0.0015 0.0024 0.2564 0.2405 0.2331 0.1787 0.17308 0.17024 0.0777 0.0674 0.0629 Johnson & Johnson 0.1818 0.1634 0.1533 0.17557 0.17859 0.18543 0.0062 0.0152 0.0321 McDonald's 0.1660 0.1534 0.1546 0.1186 0.11437 0.11113 0.0474 0.0390 0.0435 Merck 0.2133 0.1941 0.1902 0.21322 0.1944 0.17686 0.0000 0.0003 0.0134 Microsoft 0.2334 0.2120 0.2021 0.25853 0.27627 0.30488 0.0251 0.0643 0.1028 Nike 0.2347 0.2225 0.2147 0.2321 0.22333 0.22127 0.0026 0.0009 0.0066 Procter & Gamble 0.1620 0.1424 0.1377 0.16498 0.15757 0.1515 0.0030 0.0152 0.0138 Salesforce 0.3231 0.3062 0.2897 0.41557 0.3477 0.28556 0.0925 0.0415 0.0041 Travelers 0.2521 0.2386 0.2270 0.25821 0.25393 0.25192 0.0061 0.0153 0.0249 UnitedHealth 0.2433 0.2223 0.2100 0.30063 0.25807 0.24028 0.0573 0.0358 0.0303 Visa Walgreens Boots Alliance 0.2235 0.2128 0.2064 0.22709 0.20072 0.191 0.0036 0.0121 0.0154 0.2411 0.2301 0.2293 0.25513 0.24069 0.23814 0.0140 0.0106 0.0089 Walmart 0.1588 0.1429 0.1409 0.13861 0.13255 0.14094 0.0201 0.0104 0.0000 132 Chapter CONCLUSION The two research of objectives of this thesis are as follows Firstly, it is to evaluate the performance of four different stochastic option pricing models, namely Heston, Heston ++, Bates, and Heston-Hull-White, in pricing stock options across different industries in a chosen market (the US) Secondly, it is to come up with a viable method of constructing IV smiles for the purpose of providing IV data on new stock options that had not yet been issued, or stock options which are illiquid and/or being traded in illiquid markets For the first objective, after testing model performance on in-sample pricing, out-ofsample pricing, hedging and various other robustness tests, the results show that in general, the Heston model has the best performance out of the four chosen stochastic option pricing models However, as a model is a simplification of reality by nature, it is not a perfect choice Therefore, an alternate stochastic model that can further improve upon the Heston model while still retaining its strengths is desirable For the second objective, it can be concluded that the correlation and machine learning methods have their own pros and cons In particular, performance-wise KNN and WKNN methods are considered better than correlation, as they closely track the US company that has the behavior of return closest to the Vietnamese company in question On the other hand, the correlation method has less complex data requirements and hence can be used as an elementary method to fulfil the task of constructing IV smiles Backtesting shows that the WKNN method yields the best performance Furthermore, performance of all methods can be improved by increasing the amount of data available to be used as reference From the results of this thesis, a comprehensive option pricing scheme can be formulated in order to provide IV data with the best accuracy possible and reduce reliance on the traditional, popular but also flawed BS model so that the option prices can better reflect the real-life market conditions This will help countries who are 133 interested in building a formal option market, as well as those who have already had their own, but with limited liquidity The more efficient the 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