Partha Vemulapalli May 18, 2019

1. Using the Black-Scholes equation to determine the difference between historical and implied volatility in various sectors Partha Vemulapalli May 18, 2019

2. Why did I choose this project?: • Interests: • Finance • Mathematics • I wanted to adventure into a new interesting topic • Eventually,I ended up being drawn to implied volatility in option contracts.

3. Buying a call option gives you the right but not the obligation to buyunderlying asset at a pre-determined price. Buyinga put gives you the right but not the obligation to sellthe underlying asset a predetermined price also known as the strike price. What are option contracts? Derivatives “derive” their value from an underlying asset. My underlying asset is an Exchange-Traded Fund (ETF). Options are a sort of derivatives that come in two types: puts and calls.

4. Historical vs. Implied Volatility • Volatilityin finance is “astatistical measure of the dispersion of returns for a given security or market index” (Investopedia.com) • Implied Volatilityisderived from current option prices • Historical Volatilityis the standarddeviation of an asset’s priceover a certain time-period • Black-Scholes is a Nobel prize winning equation for the fair valuing of call and put options. Created by Myron Scholes and Fischer Black.

5. What did I do? Weeks 1-8 • Learned about trading options through simulators. • Intuitively began understanding Black-Scholes. • Basic Black-Scholes: Option Pricing and Trading • Good foundational book on option valuation.

6. Assumptions for Black-Scholes (Basic) : • Efficient Markets • Zerodividend payments • European Style Option

7. What did I do (Part 2)? Weeks 9-12: • Tracked values, calculated implied volatility, and analyzed data. • Created: • IV Spreadsheet • Data collection Spreadsheet • This presentation • Paper detailing my project

8. My Procedure: • Pick 9 different ETFs to track. • For three weeks every Friday, note down the price of the call option for the 9 different ETFs which expire the next Friday. • Choose six different strike prices (3 below the stock price and 3 above.) • Use a weighted averageof theimplied volatilities at the different strikes based on the open interest (number of contracts filled). • Obtain Risk-free rates from Treasury.gov • Obtainthe 30-Day Historical Volatility provided by the Chicago Board Options Exchange. • Back-calculate implied volatility.

9. My Implied Volatility Calculator (Spreadsheet): Formula: Spot Price (C3), Strike Price (C4), Volatility (C5), Risk-Free Rate(C6), Term (C7) = inputs d1 = (LN(C3/C4)+(C6+C5^2/2)*(C7))/(C5*SQRT(C7)) d2 = C9-C5*SQRT(C7) Call = C3*NORM.S.DIST(C9,1)-C4*EXP(-C6*C7)*NORM.S.DIST(C10,1) Put = C4*EXP(-C6*C7)*NORM.S.DIST(-C10, 1)-C3*NORM.S.DIST(-C9, 1)

11. Findings: • InformationTechnology: XLK • ConsumerDiscretionary : XLY • Financials : XLF • Industrials : XLI • Energy: XLE • ConsumerStaples : XLP • Materials : XLB • Utilities : XLU • Healthcare : XLV

12. Analysis: • No statistically significant trend between Historical Volatility and Implied Volatility in any of the sectors. • Calculated by the t-test. • More details in my paper.

13. Errors, Issues and Future Plans: • Initially tracked wrong values for the first week. • Implied volatility calculations were not perfect. I didn’t fit the assumptions Black-Scholes made. • Reasons: • The ETFs paid dividends but change in value of the option was small enough to ignore. • The ETFs were American. • Future Plans: • If I were to do a similar project in the future, I would like to try different models such as Cox-Ross-Rubinstein model also known as the binomial tree model. • Automate data collection and so this project with more Strike Prices and for different historical volatilities.

14. What I learned in this project: • A whole lot about options. • Doing this project has really made me think about the technical side of options. • Practical use of options. • How hedge funds and institutional investors use options to hedge their risk.

15. I would like to thank: • My outside advisor: ArunAthavale. • A program manager at Cisco Systems Inc • An experienced options trader • My school advisor: Matthew Linhares • Pre-Calculus, Calculus, and astronomy • Matthew McCorkle • Game theory, AP Economics, Corporate Finance • Contact with my advisors: • At least once a week

16. Thank You

17. References • Geske, Robert, and Richard Roll. “On Valuing American Call Options with the Black-Scholes European Formula.” The Journal of Finance, vol. 39, no. 2, 1984, p. 443., doi:10.2307/2327870. • Macbeth, James D., and Larry J. Merville. “Tests of the Black-Scholes and Cox Call Option Valuation Models.” The Journal of Finance, vol. 35, no. 2, 1980, p. 285., doi:10.2307/2327386. • Folger, Jean. "Options Pricing." Investopedia. 26 Mar. 2018. Investopedia. 29 Apr. 2019 <https://www.investopedia.com/university/options-pricing/>. • Kuepper, Justin. "Volatility." Investopedia. 18 Apr. 2019. Investopedia. 29 Apr. 2019 <https://www.investopedia.com/terms/v/volatility.asp>. • Crack, Timothy Falcon. Basic Black-Scholes: Option Pricing and Trading, 2017.