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1 Chapter 1Figure 1.1 A histogram for 100,000 simulated rolls for a pair of fair dice. ...Figure 1.2 A histogram for 100,000 simulated dice rolls with fair dice. Incl...Figure 1.3 (a) A histogram for 100,000 simulated dice rolls with unfair dice...Figure 1.4 A histogram for 100,000 simulated rolls with a group of fair, six...Figure 1.5 A detailed plot of the normal distribution and the corresponding ...Figure 1.6 (a) The 2D position of a particle in a fluid, moving with Brownia...Figure 1.7 The distribution of the horizontal displacements of the particle ...Figure 1.8 The (a) daily returns, (b) price, and (c) daily returns histogram...Figure 1.9 (a) QQQ returns versus SPY returns. The covariance between these ...Figure 1.10 SPY daily returns distribution from 2010–2021. Included is the V...Figure 1.11 SPY daily returns distribution from 2010–2021. Included are VaR ...

2 Chapter 2Figure 2.1 (a) The 45‐day expected move cone for SPY in early 2019. The pric...Figure 2.2 The three phases of the VIX, using data from early 2017 to late 2...Figure 2.3 IV indexes for different assets with their respective averages (d...Figure 2.4 Implied volatility indexes for different equities from 2017–2020 ...

3 Chapter 3Figure 3.1 The price of SPY in the last five months of 2019. Included is the...Figure 3.2 (a) Historical P/L distribution (% of initial credit) for short 4...Figure 3.3 Historical P/L distribution ($) for 45 DTE 16...Figure 3.4 The VIX (solid) and VXAZN (dashed) from 2015–2016. Labeled are th...Figure 3.5 SPY IV from 2010–2020. The average prices for 45 DTE 16 SPY stra...Figure 3.6 (a) SPY IV from 2005–2021. Labeled are the extreme losses for 45 ...Figure 3.7 Historical P/L distributions for 45 DTE 16 SPY strangles, held t...Figure 3.8 Historical P/L distribution for 45 DTE 16 SPY strangles, held to...Figure 3.9 P/L averages for portfolios with N number of trades, randomly sam...Figure 3.10 (a) Performances from 2017 to 2021, through the 2020 sell‐off. E...

4 Chapter 4Figure 4.1 Loss as a % of BPR for 45 DTE 16 SPY strangles held to expiratio...Figure 4.2 Data from 45 DTE 16 SPY strangles from 2005–2021. (a) BPR as a f...

5 Chapter 5Figure 5.1 Standard deviation of daily P/Ls (in dollars) for 16 SPY strangl...Figure 5.2 Graphical representation of the iron condor described in Table 5....Figure 5.3 The price of SPY in the last 5 months of 2019. Included is the 45...

6 Chapter 6Figure 6.1 (a) Two portfolios, each with $100,000 in capital invested, tradi...Figure 6.2 (a) Two portfolios, each with $100,000 in capital invested, tradi...

7 Chapter 7Figure 7.1 Performance comparison for Portfolios A and B from 2020 to 2021. ...

8 Chapter 8Figure 8.1 Standard deviation of daily P/Ls (in dollars) for 16 SPY strangl...Figure 8.2 Cumulative P/L of three different portfolios containing some comb...Figure 8.3 Portfolio performance of three different portfolios from early 20...

9 Chapter 9Figure 9.1 IV indexes for different stocks from 2017–2020. Assets include (a...

10 AppendixFigure A.1 Comparison of the log‐normal distribution (a) and the normal dist...Figure A.2 Volatility curve for OTM 30 DTE SPY calls and puts, collected on ...