Continuous Time Finance
Johan Waldén, UC Berkeley, Haas School of Business, August 1-11, 2023 at Swedish House of Finance - Stockholm School of Economics
The development of derivatives markets may be the single most important innovation in financial markets in the last fifty years. The celebrated Merton, Black & Scholes option pricing formula derived in the 70's has arguably been the most successful research paper in social sciences. The formula has been widely adopted by market participants, and has initiated a new field, Financial Engineering, which occupies researchers in financial institutions and universities throughout the world.
The focus of the course is on applied stochastic calculus applied to problems within finance. We will cover the theory of noarbitrage, Brownian motion, Ito integrals and calculus (Ito's lemma), change of measure (Girsanov's theorem), and links to partial differential equations (Feynman Kac's theorem, Kolmogorov equations). We apply the theory to derive many continuous time asset pricing formulas, including the celebrated Black-Scholes formulas for pricing plain-vanilla options. We also discuss equilibrium models in continuous time (Lucas tree economy, Cox Ingersoll Ross economy)
Tomas Björk, “Arbitrage Theory in Continuous Time”, 4th edition, 2020.
Steven Shreve, “Stochastic Calculus for Finance I: The Binomial Asset Pricing Model”
Steven Shreve, “Stochastic Calculus for Finance II: Continuous-Time Models”.
Karatzas & Shreve, “Methods of Mathematical Finance”.
Darrell Duffie, “Dynamic Asset Pricing Theory”, 3rd edition.
Please attend all classes and do not arrive late.
Pass, Pass with Distinction, or Fail. Grades will be based on:
Class participation 20.0 %
Assignment: 40.0 %
Final exam: 40.0 %
Total 100.0 %
There will be an open book take-home final exam, that students should solve individually. The date for the exam is yet to be determined, but it will be sometime between August 10-20.
There will be one, quite extensive, assignment that students may solve individually or in groups of two (no more!). The assignment is due on Thursday, August 11.
Ethics and etiquette:
Students are allowed to consult all the material provided in the course (slides, lecture
notes, course books, etc.), but are not allowed to use any external material that
resembles a “solution” to an assignment. If there is any doubt, please contact the
professor. Students are also allowed to discuss course material, including assignments,
with each other. However, any help must stop far short of hinting (or providing) the
solution to an assignment, in form of explicit calculations, etc. The material in the
course (slides, lecture notes, assignments, solutions, etc.) is proprietary. Posting of
material on the internet is strictly prohibited and will lead to legal action.
All chapters refer to Björk’s textbook. This outline is tentative. Depending on the time
constraints, we may not have time to cover everything on the list.
Discrete Model Chapters 2-3
• One-period: noarbitrage, fundamental theorem of asset pricing, price systems, martingale pricing
• Binomial model & general multi-period model Continuous time modeling Chapters 4-7
• Stochastic integrals
• Relationship to differential equations
• Dynamic portfolios
• Arbitrage Pricing The Black-Scholes model Chapters 8-10
• Black Scholes PDE, market completeness Martingale pricing Chapters 11-13
• Stochastic Discount Factors Girsanov’s theorem Multidimensional Models Chapter 14
• Pricing, risk-neutral valuation, state space reduction, More derivatives Chapter 16
• Derivatives on dividend paying stocks, barrier options Term structure models Chapters 19-21
• Short rate models, Affine term structure models Change of numeraire Chapter 15
• Application to options and interest rate models Equilibrium models Notes
• Lucas one-tree model, Cox Ingersoll Ross model