SP 500 SP 500 logarithmic scale Asset price

S&P 500

S&P 500, logarithmic scale

Asset price model assumption S(t) = Asset price at time t In the long run, log S(t) grows linearly

Apple Inc. (AAPL)

Daily rate of return (on Closing Price) Date 12/24/18 12/26/18 12/27/18 12/28/18 12/31/18 1/2/19 1/3/19 1/4/19 1/7/19 1/8/19 1/9/19 Open 148. 15 148. 30 155. 84 157. 50 158. 53 154. 89 143. 98 144. 53 148. 70 149. 56 151. 29 High 151. 55 157. 23 156. 77 158. 52 159. 36 158. 85 145. 72 148. 55 148. 83 151. 82 154. 53 Low 146. 59 146. 72 150. 07 154. 55 156. 48 154. 23 142. 00 143. 80 145. 90 148. 52 149. 63 Close 146. 83 157. 17 156. 15 156. 23 157. 74 157. 92 142. 19 148. 26 147. 93 150. 75 153. 31 Rate 7. 04% -0. 65% 0. 05% 0. 97% 0. 11% -9. 96% 4. 27% -0. 22% 1. 91% 1. 70%

AAPL daily rate of return (2019)

Asset Price Process S(t) •

IBM

IBM daily rate of return (2019)

Asset Price: Model assumptions

LAIX Inc. (LAIX)

LAIX daily rate of return (2019)

Summary: Asset Price Model assumptions

Central Limit Theorem Binomial distribution: Number of Heads on N = 100 coin tosses, p = 0. 5

Call Options AAPL, 2019 -12 -20 Expiration 2019 -12 -27 Strike 200 280 295 Bid 77. 65 1. 75 0. 07 Ask 80. 95 1. 84 0. 11 Expiration 2020 -04 -17 Strike 200 280 295 Bid 78. 35 13. 85 7. 45 Ask 82. 70 14. 45 8. 15 • K = 200 : In-The-Money • K = 280 : At-The-Money • K = 290 : Out-Of-The-Money

Call Options AAPL “At-The-Money” Option 1 Share = 279. 44 Strike Price = 280