MA5018 Stochastic Calculus for Finance


Course Details

Infinite Probability Spaces, Random Variables and Distributions, Expectations, Computation of Expectations, Change of Measure, General Conditional Expectations Martingale Property for the Symmetric Random Walk, Quadratic Variation of the Symmetric Random Walk, Scaled Symmetric Random Walk, Limiting Distribution of the Scaled Random Walk, Definition of Brownian Motion, Distribution of Brownian Motion, Filtration for Brownian Motion, Martingale Property for Brownian Motion, Distribution of Brownian Motion and Its Maximum Ito's Integral for Simple Integrands, Construction of the Integral, Properties of the Integral, Ito's Integral for General Integrands, Ito-Doeblin Formula, Formula for Brownian Motion, Formula for Ito Processes Evolution of Portfolio Value, Evolution of Option Value, Equating the Evolutions, Solution to the Black-Scholes-Merton Equation, The Greeks, Put-Call Parity. Multiple Brownian Motions, Ito-Doeblin Formula for Multiple Processes, Recognizing a Brownian Motion, Brownian Bridge, Gaussian Processes, Brownian Bridge as a Gaussian Process, Brownian Bridge as a Scaled Stochastic Integral, Multidimensional Distribution of the Brownian Bridge, Brownian Bridge as a Conditioned Brownian Motion Multidimensional Market Model, Existence of the Risk-Neutral Measure, Uniqueness of the Risk-Neutral Measure. Continuously Paying Dividend, Continuously Paying Dividend with Constant Coefficients, Lump Payments of Dividends Lump Payments of Dividends with Constant Coefficients, Forward Contracts, Futures Contracts, Forward-Futures Spread

Course References:

TextBooks:
Shreve, S. E. (2004). Stochastic calculus for finance II: Continuous-time models (Vol. 11). New York: springer.
ReferenceBooks:
Karatzas, I., & Shreve, S. (2012). Brownian motion and stochastic calculus (Vol. 113). Springer Science & Business Media. Shreve, S. (2005). Stochastic calculus for finance I: the binomial asset pricing model. Springer Science & Business Media.
Prerequisite: Probability Theory