Intuition of scaled Random Walk, Systematic Random walk and it's increments, Martingale and Quadratic variation of random walk, scaled symmetric random walk and it's limiting distribution.
Preliminary knowledge about One period Binomial model
Binomial Asset Pricing Model, brief definition of arbitrage pricing, Risk neutral probability, Option derivatives, movement of wealth, Delta hedging...
Limiting distribution of Binomial asset pricing model
Mathematical definition of Brownian motion, distribution and MGF, alternative characterization of Brownian motion...
Filteration of Brownian motion and the Martingale property of Brownian motion...
The First Order Variation- two definitions: integration and summation. Equivalence of two definitions using MVT and Riemann sum...
Quadratic Variation: definition, Proof that QV of a function which has continuous derivative is zero. What if the continuous derivative is non-zero?
Quadratic variation of Brownian motion converges to T; Proved Probability convergence, Mean square convergence and with an additional condition almost sure convergence...
Brownian motion accumulates quadratic variation at rate one per unit time. i.e. dW(t).dW(t) = dt. Cross variation of W(t) and t is zero i.e. dW(t).dt = 0. Quadratic variation of t is zero i.e. dt.dt = 0.
Volatility of Geometric Brownian motion and Markov Property of Brownian motion...
First passage time distribution of symmetric random walk. Proof that the symmetric random walk reaches the level m with probability 1...
Distribution of first passage time distribution of Brownian motion using exponential martingale an Optional Sampling Theorem. Moment generating function of first passage time distribution of Brownian motion. C.D.F. and P.D.F. of first passage time distribution of Brownian motion using the reflection principle...
