SUMMARY:
• fourth act of Lecture 4 title slide (slide 138)
• a call option priced via a discrete model with over 2.5 million subperiods, setup and payoff function (slide 139)
• the risk-neutral world for this option (slide 140)
• the model of the underlying, the ending underlying price and the contingent claim (slide 141)
• the price as an expected value (slide 142)
• the corresponding coin-flipping game (slide 143)
• an easier coin-flipping problem; some notation (slide 144)
• restatement of the easier problem; more notation; a strategy for tackling the easier problem via a sequence of even easier problems, but of increasing difficulty (slide 145)
• the distribution of the first difference; intuitive definition of a random variable; another random variable with the same distribution (slide 146)
• the generating function and Fourier transform of the distribution of the first difference; the inverse Fourier transform; what happens on dividing the first difference by 7, start (NOTE: Contrary to what is said in the slide, this generating function is not typically called the "moment generating function", but the moment generating function is closely related) (slide 147)
• what happens on dividing the first difference by 7, end (slide 148)
• the distribution of the second difference (slide 149)
• the generating function and Fourier transform of the distribution of the second difference (slide 150)
• the generating function and Fourier transform of the distribution of the Nth difference (slide 151)
• what happens on dividing the Nth difference by the square root of N; the renormalized Nth power of cosine (slide 152)
• estimation via the limit of this renormalized Nth power (slides 153-156)
• the key idea of the Central Limit Theorem: after this estimation, take the inverse Fourier transform (slide 157)
• the distribution of the variable Z (slide 158)
• some probabilities for Z (slides 159-160)
• the generating function and Fourier transform of the distribution of Z (slides 161-162)
• Z approximates X, which is the Nth difference over the square root of N (slide 163)
• our restated easier problem, and an approximate solution (slide 164)
• a new easier problem, involving expected values (slide 165)
• another easier problem, involving expeceted values (slide 166)
• another easier problem, involving expected values of a function of Z (slide 167)
• another easier problem, involving expected values of a function of X, and its approximate solution using the fact that Z is close to X (slides 168-169)
• another easier problem, involving any "reasonable" function g of X, with the hope that a good choice of function will solve achieve our goal (slide 170)
• finding the right choice for g (slides 171-175)
• reduction of our goal to an integral (slide 176)
• remainder of lecture: pricing problem - calculation of the integral; the Central Limit Theorem - statement (slide 177)
• fifth act of Lecture 4 title slide (slide 178)
• resttatement of the integral we need to calculate (slide 179)
• getting rid of the positive part (slide 180-181)
• splitting the problem into two integrals (slide 182)
• computation of the two integrals (slides 183-186)
• plugging in the numbers (slide 187)
• approximate solution of the original coin-flipping game and option pricing problem (slide 188)
• a more realistic restatement of the bank assumption (slide 189)
• a more realistic restatement of the stock assumptions (slides 190-193)
• the 50-50 drift and volatility equations (slide 194)
• a discrete model with only 10 subperiods, and with the same stock assumptions (slides 195-197)
• more and more subperiods, limiting on the Black-Scholes model (slide 198)
• the Black-Scholes Option Pricing Formula (slide 199)
• moving from 50-50 probabilities to 65-35 gives the same answer, in the limit (slides 200-204)
• remainder of lecture: the Central Limit Theorem - statement (slide 205)
• sixth act of Lecture 4 title slide (slide 206)
• reminder of our approximate solution, as an integral (slide 207)
• the 50-50 Central Limit Theorem, motivation and intuitive statement (slide 208)
• toward a more concise intuitive statement (slides 209-212)
• exponentially bounded (exp-bdd) functions (slides 213-214)
• a more rigorous statement of the Central Limit Theorem (slide 215)
• exact formulas for the expected values appearing in the theorem, for small values of N (slides 216-217)
• first fully rigorous satement of the Central Limit Theorem, based on expected value formulas; future goal of formulating a good mathematical model for coin-flipping, via "independent" random variables (slide 218)
• a plan for seven future lectures:
  • Random variables
  • The Central Limit Theorem Redux
  • Girsanov's Theorem
  • First Derivation of Black-Scholes
  • Stochastic Processes
  • Ito's Lemma
  • Black-Scholes Redux
  MATH RULES!! (slide 219)