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Intraday Volatility Surface Calibration

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Intraday Volatility Surface Calibration INTRADAY VOLATILITY SURFACE CALIBRATION Master Thesis Tobias Blomé & Adam Törnqvist Master thesis, 30 credits Department of Mathematics and Mathematical Statistics Spring Term 2020 Intraday volatility surface calibration Adam T¨ornqvist,[email protected] Tobias Blom´e,[email protected] c Copyright by Adam T¨ornqvist and Tobias Blom´e,2020 Supervisors: Jonas Nyl´en Nasdaq Oskar Janson Nasdaq Xijia Liu Department of Mathematics and Mathematical Statistics Examiner: Natalya Pya Arnqvist Department of Mathematics and Mathematical Statistics Master of Science Thesis in Industrial Engineering and Management, 30 ECTS Department of Mathematics and Mathematical Statistics Ume˚aUniversity SE-901 87 Ume˚a,Sweden i Abstract On the financial markets, investors search to achieve their economical goals while simultaneously being exposed to minimal risk. Volatility surfaces are used for estimating options' implied volatilities and corresponding option prices, which are used for various risk calculations. Currently, volatility surfaces are constructed based on yesterday's market in- formation and are used for estimating options' implied volatilities today. Such a construction gets redundant very fast during periods of high volatility, which leads to inaccurate risk calculations. With an aim to reduce volatility surfaces' estimation errors, this thesis explores the possibilities of calibrating volatility surfaces intraday using incomplete mar- ket information. Through statistical analysis of the volatility surfaces' historical movements, characteristics are identified showing sections with resembling mo- tion patterns. These insights are used to adjust the volatility surfaces intraday. The results of this thesis show that calibrating the volatility surfaces intraday can reduce the estimation errors significantly during periods of both high and low volatility. However, these results highly depend on the conducted choices when constructing and analyzing the volatility surfaces which leave room for further reasearch. ii Sammanfattning F¨or investerare p˚afinansmarknader v¨arlden ¨over ¨ar m˚alet att n˚asina ekonomiska m˚almed s˚al˚agrisk som m¨ojligt. D¨arf¨or ¨ar korrekta och precisa riskber¨akningar av h¨ogsta prioritet. Volatilitetsytor anv¨ands vid riskber¨akningar f¨or att estimera optionspriser och optioners implicita volatiliteter. Idag konstrueras volatilitet- sytor baserat p˚amarknadsinformation fr˚anen dag och anv¨ands f¨or estimation n¨asta dag. Under perioder av h¨og volatilitet blir denna sorts konstruktion l¨att inaktuell, vilket leder till felaktiga riskber¨akningar. M˚aletmed detta examensarbete var att reducera en volatilitetsytas estima- tionsfel. Detta genoma att utforska m¨ojligheten att kalibrera en volatilitetsyta baserat p˚al¨opande information under dagen. Genom att analysera hur vola- tilitetsytor r¨or sig ¨over tid identiferades karakt¨arsdrag och m¨onster som kan anv¨andas f¨or att kalibrera volatilitetsytor l¨opande under en dag. Resultatet i detta examensarbete visar att kalibrering av volatilitetsytor int- ra dag kan reducera volatilitetsytors estimationsfel oavsett perioder av h¨og eller l˚agvolatilitet. Detta resultat ¨ar dock beroende av hur volatilitetsytorna ¨ar ska- pade och analyserade, vilket ger utrymme f¨or vidare studier inom omr˚adet. iii Acknowledgements Firstly, we would like to thank Nasdaq for bringing the idea of this thesis to us and letting us do our thesis at the Ume˚aoffice. Under the circumstances of the COVID-19 pandemic, we are deeply thankful for letting us use Nasdaq's equipment to complete this thesis from home. Secondly, we thank Jonas Nyl´enand Oskar Janson at Nasdaq for providing valuable insights, discussions and stunning supervision. We would also like to direct our gratitude to Markus Nyberg for useful expertise and assistance on how to write an academical report. Thirdly, we would like to thank our supervisor Xijia Liu, at Ume˚aUniver- sity, for knowledge on data analysis and remarks on the structure and content in this thesis. Finally, we direct a thank you to our fiancees Victoria Bertilsson and Melina Ahlenius˚ for your support. Without you we would probably still be analysing the volatility surfaces. A special thanks to you Victoria for letting us redo the apartment to a home office and to you Melina, for your help in report writing and the idea proposal of dividing the volatility surfaces into sections. //Adam & Tobias iv Contents 1 Introduction 1 1.1 Background . .1 1.2 Problematization . .3 1.3 Project goal . .4 1.4 Datasets . .4 1.4.1 LME Copper . .4 1.4.2 WTI NYMEX . .4 1.5 Limitations . .5 1.6 Literature review . .5 1.7 Software . .5 2 Theory 6 2.1 Options . .6 2.2 Implied volatility . .7 2.3 The volatility surface . .8 2.4 Moneyness . .9 2.5 Total implied volatility and variance . .9 2.6 Construction of a volatility surface . 10 2.7 Arbitrage . 12 2.7.1 Arbitrage conditions for options . 13 2.7.2 Arbitrage conditions for implied volatility . 14 2.8 K-Means clustering . 15 2.9 Transition matrix . 16 3 Method 17 3.1 Calculate implied volatility . 17 3.2 Arbitrage: tests and how to eliminate it . 17 3.2.1 Calendar spread arbitrage . 18 3.2.2 Butterfly arbitrage . 18 3.3 Construction of a volatility surface . 19 3.3.1 Calculate change and error between market data and the surface . 25 3.4 Analysis of the surface . 25 3.4.1 Part 1: Implied volatility time series analysis . 26 3.4.2 Part 2: Magnitude of change analysis . 28 3.5 Intraday calibration . 30 3.6 The Intraday Calibration Model (ICM) . 33 3.6.1 Model description . 33 3.6.2 Model evaluation . 34 3.7 Parallel shift . 35 v 4 Results 36 4.1 Review of assumptions and approach . 36 4.2 LME Copper . 36 4.3 WTI NYMEX . 40 4.4 Comparison and conclusion of results . 43 4.5 Volatile periods . 45 5 Discussion 47 5.1 Results . 47 5.2 Construction of the volatility surfaces . 47 5.3 Analysis of the volatility surface . 48 5.4 Future studies . 50 vi List of Figures 1 S&P 500 from April 28th 2015 to April 28th 2020. .2 2 S&P 500's intraday price process from March 23rd to March 25th 2020. .3 3 Example of a volatility surface. .9 4 A raw SVI parameterization fitted to market data. 20 5 Visualization of Durrleman's condition. Arbitrage opportunities are introduced as g(x) falls below the dashed line. 21 6 Polynomial regression interpolation in the time to maturity di- rection for log-moneyness = 0. 22 7 Volatility smiles for different time to maturities T where no cal- endar arbitrage opportunities are introduced. 22 8 A volatility surface expressed in total implied variance, !, in (a) and expressed in implied volatility, σimp, in (b). 24 9 An example of relative day-to-day changes, δ, within a volatility surface. 26 10 Example of a regression model of 4th degree fitted to a time series (i;j) dt ................................... 27 11 A volatility surface divided into sections using K-means clustering of y(i;j), here using K =6....................... 28 (i;j) 12 Relative day-to day changes of the time series dt shown in Figure 10. The solid line represent the mean change over the time period. 28 13 A section from Figure 11 divided into subsections using K-means based on standard deviation, here using K =4 .......... 29 14 Intraday calibration adjustments of a volatility surface based upon 20% of available data. 32 15 Relative day-to-day changes within a volatility surface. This fig- ure should be examined as a reference to Figure 14. 33 16 Daily T SS in the LME Copper volatility surface for the second year. 37 17 Method performances for the first 30 days in the second year. 37 18 Error increases for the parallel shift. The occasions are indicated by red dots. 38 19 Error increases for the Intraday Calibration Model. The occa- sions are indicated by red dots. 39 20 Daily T SS in the WTI NYMEX volatility surface for the second year. 40 21 Method performances for the first 30 days in the second year. 41 22 Error increases for the parallel shift. The occasions are indicated by red dots. 42 23 Error increases for the Intraday Calibration Model. The occa- sions are indicated by red dots. 42 24 Daily T SS in the volatility surfaces for the second year. 43 vii 25 Relative day-to-day changes within a volatility surface. 44 26 Method performances on LME Copper for the six highest volatile days in the second year. 45 27 Method performances on WTI NYMEX for the six highest volatile days in the second year. 46 28 Example of a volatility surface divided into sections and subsec- tions. 50 viii List of Tables 1 Examples of estimated coefficients of the regression models fitted to different implied volatility time series. 27 2 Example of a transformation matrix. 30 3 Hyper-parameters of ICM for the LME Copper dataset. 36 4 Method performances for LME Copper for the second year. 38 5 Results on the occasions when error is increased for LME Copper. 39 6 Hyper-parameters of ICM for the WTI NYMEX dataset. 40 7 Method performances for WTI NYMEX for the second year. 41 8 Results on the occasions when error is increased for WTI NYMEX. 43 9 Method performances for the 95% quantile of the most volatile days of the second year. 46 ix 1 INTRODUCTION 1 Introduction This section provides a background story about the financial market and exam- ples of volatility in the market. Further in, problematization, project goal, the data available and limitations of this thesis are presented. 1.1 Background On the financial markets, henceforth the market, participants aim to achieve their economic goals while being exposed to minimal risk. As the rapid evo- lution of technology, there exists a wide ever-increasing set of assets, financial contracts and derivatives available on the market.
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