https://studiegids.vu.nl/en/courses/2025-2026/X_400352After this course, the student will:be acquainted with the notion of pricing derivatives and arbitrage, and with the basics of discrete stochastic processes, including the concepts of martingales; be able to price derivatives in the binomial model. be familiar with the basics of continuous time stochastic processes, concepts of measure theory, Brownian motion, martingales; be able to verify whether the process is a Brownian motion and/or is a martingale. be familiar with the notions of stochastic integral Ito and SDE, the Ito's formula; be able to compute stochastic integrals, to use the Ito formula, and to derive properties of stochastic integrals and the solutions of SDEs. be familiar with the Black-Sholes model; be able to price options by the no-arbitrage and Girsanov's theorem, to compute the hedging portfolio and the Greeks, and to make connections to the corresponding Black-Scholes PDEs. be familiar with the extended Black-Sholes models; be able to price derivatives in those models, to make connection with the corresponding PDEs, and to apply the Black-Scholes machinery to the interest rates models.Financial institutions trade in risk, and it is therefore essential to measure and control such risks. Financial instruments such as options, swaps, forwards, etc. play an important role in risk management, and to handle them one needs to be able to price them. This course gives an introduction to the mathematical tools and theory behind risk management. A "stochastic process" is a collection of random variables, indexed by a set T. In financial applications the elements of T model time, and T is the set of natural numbers (discrete time), or an interval in the positive real line (continuous time). "Martingales" are processes whose increments over an interval in the future have zero expectation given knowledge of the past history of the process. They play an important role in financial calculus, because the price of an option (on a stock or an interest rate) can be expressed as an expectation under a so called martingale measure. In this course we develop this theory in discrete and continuous time. Most models for financial processes in continuous time are based on a special Gaussian process, called Brownian motion. We discuss some properties of this process and introduce "stochastic integrals" with Brownian motion as the integrator. Financial processes can next be modelled as solutions to "stochastic differential equations". After developing these mathematical tools we turn to finance by applying the concepts and results to the pricing of derivative instruments. Foremost, we develop the theory of no-arbitrage pricing of derivatives, which are basic tools for risk management.Lectures (2 x 45min) and Tutorials (2x45). About 13 meetings of each are planned.Two assignments covering computation and theory (up to 30% of final grade) and written examination/resit examination (at least 70% of final grade). There is no resit possibility for the assignments.Lecture notes contain all of the material. Additional, optional literature:Shreve, "Stochastic Calculus for Finance I: The Binomial Asset Pricing Model", Springer; Shreve, "Stochastic Calculus for Finance II: Continuous-time models", Springer.mBA, mBA-D, mMath, master Econometrics.A significant part of the course is used to introduce mathematical subjects and techniques like Brownian motion, stochastic integration and Ito calculus. In view of this, the course is NOT meant for students who already followed the master course "Stochastic Integration" or "Stochastic differential equations". On the other hand, after completing this course, students may be motivated to follow other courses (like the two mentioned above) where stochastic calculus is treated in a deeper and more rigorous way.Probability Theory (e.g., X_400622), Mathematical Analysis (e.g., XB_0009), Measure Theory (e.g., X_401028 or Mastermath course Measure Theoretic Probability)