FINM345A09Lecture1corr - FINM 345/STAT 390 Stochastic...

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FINM 345/STAT 390 Stochastic Calculus, Autumn 2009 Floyd B. Hanson , Visiting Professor Email: fhanson@uchicago.edu Master of Science in Financial Mathematics Program University of Chicago Lecture 1, Corrected Post-Lecture 7:30-9:30 pm, 28 * September 2009, Kent 120 in Chicago 8:30-10:30 pm, 28 September 2009 at UBS Stamford 9:30-11:30 am, 29 September 2009 at Spring in Singapore * { Monday 28 September Yom Kippur is an official U. Chicago holiday, but since we would miss a whole week of classes, we will have the usual evening class in Chicago starting at 7:30pm. The religious holiday ends 42 minutes after sunset, so there is an overlap only in Chicago. Individuals, of course, are free to follow their conscience. Sorry, for any inconvenience. } FINM 345/STAT 390 Stochastic Calculus Lecture1–page1 Floyd B. Hanson
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0.01 Histogram of S&P500 Log-Returns 1988-2008: 0.05 0 0.05 0.1 0 10 20 30 40 50 60 70 Fig04: LR Histogram Vs. NormPDF+1, ’88 ’08 LR, Log Returns Freq. & Scaled NormPDF LR Hist. NormPDF+1 Figure 0.01: S&P500 Daily Log-Return Adjusted Closings from 1988 to 2008 (post-1987) showing long-tails of rare events. Normal kernel- smoothed graph, in red , plus one which accounts for non-central and nor- mally invisible, but financially important, rare jumps . FINM 345/STAT 390 Stochastic Calculus Lecture1–page2 Floyd B. Hanson
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0.02 Extreme Negative Tail Events for Log-Returns (’88-’08): 0.05 0.06 0.07 0.08 0.09 0 0.5 1 1.5 2 LRneg Histogram, ’88 ’08 S&P500 LRneg (pot= 0.04), Log Returns Frequency (a) Extreme Negative Tails. 0.05 0.06 0.07 0.08 0.09 0.1 2.5 3 Fig09: LRpos Histogram, ’88 ’08 S&P500 LRpos (pot=0.048), Log Returns (b) Extreme Positive Tails. Figure 0.02: Extreme Negative and Positive Log-Return Tail Events, with Thresholds POT = - 0 . 04 and +0 . 048 , respectively. POT means Peaks Over (or Under) Threshold. These represent the significant crashes or bonanzas during the time period. { Note: vertical scale differences. } FINM 345/STAT 390 Stochastic Calculus Lecture1–page3 Floyd B. Hanson
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Course Outline (tentative) 1. Introduction to Stochastic Diffusion and Jump Processes : Basic properties of Poisson and Wiener stochastic processes. Based on the calculus model, differential and incremental models are discussed. The continuous Wiener processes model the background or central part of of financial distributions, while the Poisson jump process models the extreme, long tail behavior of crashes and bubbles of financial distributions. 2. Stochastic Integration for Stochastic Differential Equations : While the stochastic differentials and increments are useful in developing stochastic models and numerically simulating solutions, stochastic integration is important for getting explicit solutions or more manageable forms. 3. Elementary Stochastic Differential Equations (SDEs) : The stochastic chain rules for jump-diffusions with simple Poisson jump processes, starting from diffusion chain rules to jump chain rules to jump-diffusion chain rules.
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FINM345A09Lecture1corr - FINM 345/STAT 390 Stochastic...

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