title slide (slide 1)
correction: "obligation derivative" more commonly called an "off-market forward" (slides 2-3)
notations for functions and expressions -- evaluation (slide 4)
notations for functions and expressions -- differentation (slides 5-8)
examples of differentiation of functions and expressions (slides 9-10)
deriv. of expression agrees with deriv. of function (slide 11)
continue a gauche, limite a droite for expressions and funcions (slide 12)
the product rule, or differentiation by parts (slides 13-16)
the quotient rule (slides 17-18)
the chain rule (slides 19-21)
a practice differentiation problem (slide 22)
logarithmic differentiation (slides 23-25)
basics of integration (slide 26)
review of the Fundamental Theorem of Calculus, in various forms (slides 27-36)
review of Integration by Substitution in one variable (slides 37-45)
review of general Integration by Substitution in multi-variables (slide 47)
the special case of polar coordinates (slides 48-58)
review of Integration by Parts (slides 59-60)
remainder of lecture: limits of powers of functions after renormalization, and other limits; financial mathematics integrals; pricing problem - reduction to an integral; pricing problem - calculation of the integral; the Central Limit Theorem - statement (slide 61)
second act of Lecture 4 title slide (slide 62)
some simple limits of quotients of polynomials (slide 63)
compounding with n conversions; continuous compounding givs a limit that needs to be calculated (slide 64)
one approach to computing that limit (slides 65-67)
the limit of renormalized powers of 1-x^2, the algebra (slide 68)
the limit of powers of 1-x^2, the graphs (slides 69-72)
renormalization, one of the graphs (slide 73)
the limit, the graph (slide 74)
more limits of renormalized powers (slide 75)
Maclaurin approximations (slide 76)
examples of Maclaurin approximations (slide 77)
renormalized powers of cosine, the graphs (slides 78-81)
the limit, the graph (slide 82)
the general fact: the limit of renormalized powers when the second order Maclaurin approximation is 1-x^2/2 (slide 83)
proof for x=3 (slides 84-89)
the hypothesis that f'' is continuous at 0 (slide 90)
statement of the fact, as a theorem (slide 91)
similar results, for other second-order Maclaurin approximations (slide 92)
the general theorem: the second-order Maclaurin approximation determines the limit of the renormalized nth power (slide 93)
thin tails for the bell curve, the picture (slide 94)
fat tails for another function, the picture (slide 95)
the bell curve hugs the x-axis to the left and right, the picture (slide 96)
the bell curve hugs the x-axis to the left and right, symbolic verification through limits (slide 97)
exponentials of quadratics with negative quadratic coefficient always tend to 0 at plus or minus infinity (slide 98)
extending this result to the complex plane, the limits along horizontal lines are zero (slides 99-103)
the integral along a vertical segment tends to zero as the segment moves infinitely to the right or left (slide 104)
remainder of lecture: financial mathematics integrals; pricing problem - reduction to an integral; pricing problem - calculation of the integral; the Central Limit Theorem - statement (slide 105)
third act of Lecture 4 title slide (slide 106)
the integral of e^{-x^2/2} from minus infinity to infinity (slides 107-110)
integration of an exponential of a quadratics from minus infinity to infinity; use of Cauchy's Theorem (slides 111-119)
the CDF of the standard normal distribution, called \Phi (NOTE: Contrary to what the slide says, this is *not* typically called the "error function", though the error function is related to this) (slide 120)
definite integrals of an exponential of a quadratics; completing the square in the exponent (slide 121)
exercise: a related integral (slide 122)
integration (from minus infinity to infinity) against e^{-x^2/2} of the positive part of: the difference between an exponentiated linear and a constant; preparation for development of the Black-Scholes Option Pricing Formula (slides 123-127)
integration against e^{-x^2/2} of polynomials; a recrusion formula (slides 128-136)
remainder of lecture: pricing problem - reduction to an integral; pricing problem - calculation of the integral; the Central Limit Theorem - statement (slide 137)
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)