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Stochastic Calculus and Finance

Stochastic Calculus and Finance
type: Vorlesung (V) links:
semester: WS 19/20
lecturer: Dr. Mher Safarian
sws: 2
lv-no.: <a target="lvn" href="https://campus.studium.kit.edu/events/cwsMM3rPQQWzLqw4qaCWtQ">2521331</a>
BeschreibungThe course will provide rigorous yet focused training in stochastic calculus and finance. The program will cover modern approaches in stochastic calculus and mathematical finance. Topics to be covered:
  1. Stochastic Calculus. Stochastic Processes, Brownian Motion and Martingales, Stopping Times, Local martingales, Doob-Meyer Decomposition, Quadratic Variation, Stochastic Integration, Ito Formula, Girsanov Theorem, Jump-diffusion Processes. Stable and tempered stable processes. Levy processes.
  2. Mathematical Finance: Pricing Models. The Black-Scholes Model, State prices and Equivalent Martingale Measure, Complete Markets and Redundant Security Prices, Arbitrage Pricing with Dividends, Term-Structure Models (One Factor Models, Cox-Ingersoll-Ross Model, Affine Models), Term-Structure Derivatives and Hedging, Mortgage-Backed Securities, Derivative Assets (Forward Prices, Future Contracts, American Options, Look-back Options), Option pricing with tempered stable and Levy-Processes and volatility clustering, Optimal Portfolio and Consumption Choice (Stochastic Control and Merton continuous time optimization problem), Equilibrium models, Consumption-Based CAPM, Numerical Methods.
LiteraturhinweiseWird in der Vorlesung bekannt gegeben.

Weiterführende Literatur:

  • Dynamic Asset Pricing Theory, Third Edition. by Darrell Duffie, Princeton University Press, 1996
  • Stochastic Calculus for Finance II: Continuous-Time Models, by Steven E. Shreve , Springer, 2003
  • An Introduction to Stochastic Integration (Probability and its Applications) by Kai L. Chung , Ruth J. Williams , Birkhaueser,
  • Methods of Mathematical Finance by Ioannis Karatzas , Steven E. Shreve , Springer 1998
  • Kim Y.S. ,Rachev S.T. ,Bianchi M-L, Fabozzi F. Financial market models with Levy processes and time-varying volatility, Journal of Banking and Finance, 32/7,1363-1378, 2008.
  • Hull, J., Options, Futures, & Other Derivatives, Prentice Hall, Sixth Edition, (2005).
LehrinhaltThe course will provide rigorous yet focused training in stochastic calculus and finance. The program will cover modern approaches in stochastic calculus and mathematical finance. Topics to be covered:
  1. Stochastic Calculus. Stochastic Processes, Brownian Motion and Martingales, Stopping Times, Local martingales, Doob-Meyer Decomposition, Quadratic Variation, Stochastic Integration, Ito Formula, Girsanov Theorem, Jump-diffusion Processes. Stable and tempered stable processes. Levy processes.
  2. Mathematical Finance: Pricing Models. The Black-Scholes Model, State prices and Equivalent Martingale Measure, Complete Markets and Redundant Security Prices, Arbitrage Pricing with Dividends, Term-Structure Models (One Factor Models, Cox-Ingersoll-Ross Model, Affine Models), Term-Structure Derivatives and Hedging, Mortgage-Backed Securities, Derivative Assets (Forward Prices, Future Contracts, American Options, Look-back Options), Option pricing with tempered stable and Levy-Processes and volatility clustering, Optimal Portfolio and Consumption Choice (Stochastic Control and Merton continuous time optimization problem), Equilibrium models, Consumption-Based CAPM, Numerical Methods.

Stochastische Prozesse (Poisson-Prozess, Brownsche Bewegung, Martingale), Stochastisches Integral (Integral, quadratische und Kovariation, Ito-Formeln), stochastische Differentialgleichung für Preisprozesse, Handelsstrategien, Optionspreise (Feynman-Kac), risikoneutrale Bewertungen (äquivalentes Martingalmaß, Theoreme von Girsanov), Zinsstrukturmodelle.

ZugangsvoraussetzungenKeine.
AnmerkungFür weitere Informationen: http://statistik.econ.kit.edu/
ArbeitsbelastungDer Gesamtarbeitsaufwand für diese Lerneinheit beträgt ca. 150 Stunden (5.0 Credits).

\begintabular|l|c|r|
\hline
Aktivität & & Arbeitsaufwand \\
\hline
\itshape Präsenzzeit & & \\
Besuch der Vorlesung & 15 x 90min & 22h 30m \\
Besuch der Übung & 15 x 45min & 11h 15m \\
\hline
Vor- / Nachbereitung der Vorlesung & & 22h 30m \\
Vor- / Nachbereitung der Übung & & 11h 15m \\
Skript 2x wiederholen & 2 x 20h & 40h 00m \\
Klausurvorbereitung & & 40h 00m \\
\hline
Summe & & 147h 30m \\
\hline
\endtabular
\captionArbeitsaufwand für die Lerneinheit "Stochastic Calculus and Finance"

ZielNach erfolgreichem Besuch dieser Vorlesung werden viele gängige Verfahren zur Preisbestimmung und Portfoliomodelle im Finance verstanden werden. Der Fokus liegt aber nicht nur auf dem Finance alleine, sondern auch auf der dahinterliegenden Theorie.
PrüfungDie Erfolgskontrolle erfolgt in Form einer schriftlichen Prüfung (Klausur) nach §4, Abs. 2, 1 SPO und eventuell durch weitere Leistungen als Erfolgskontrolle anderer Art nach §4, Abs. 2, 3 SPO.

Excercises

Exercises