CIR model: Definition, it's advantage over the Vasicek model. Mean & variance of CIR model using Ito- Doeblin formula.

Vasicek interest rate model: we start with a closed form function & show how it takes same differential form as Vasicek model using Ito-Doeblin formula, distribution of the interest rate. It's merit (mean reverting)- demerits (P[R(t) < 0] > 0)

Generalized Geometric Brownian Motion: Ito-Doeblin formula, Distribution for the simplistic parameter case (α(t) = α; σ(t) = σ), finding the Martingale. The distribution of Ito integral for deterministic integrand using Moment Generating function.

Ito-Doeblin formula for Ito Process: Integral form and Differential form.

Ito process: Definition. Quadratic Variation of Ito Process and it's proof.

Ito- Doeblin formula; Ito- Doeblin formula for Brownian motion and it's proof. A simple example of Ito- Doeblin formula.

Using various properties calculate Ito integral of W(t) with respect to W(t). How Stochastic calculus is different from Ordinary calculus. Stratonovich integral: how mid-point sum is not equal to the Left-hand sum, unlike Riemann sum in ordinary calculus. Why left-hand sum definition is more preferable in finance rather than the Mid-point sum definition?

Ito integral for general integrands and it's properties.

Properties of Ito integral: Ito Isometry, Quadratic variation of Ito integral up to time t.

Introduction to Ito integral, Ito integral for simple integrands, construction of Ito integral. Properties of Ito integral: Ito integral is a martingale

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.

First passage time distribution of symmetric random walk. Proof that the symmetric random walk reaches the level m with probability 1.

Volatility of Geometric Brownian motion and Markov Property of Brownian motion.

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.

Quadratic variation of Brownian motion converges to T; Proved Probability convergence, Mean square convergence and with an additional condition almost sure convergence.

Quadratic Variation: definition, Proof that QV of a function which has continuous derivative is zero. What if the continuous derivative is non-zero?

The First Order Variation- two definitions: integration and summation. Equivalence of two definitions using MVT and Riemann sum.

The Filteration for Brownian motion and the Martingale property of Brownian motion

Mathematical definition of Brownian motion, Distribution of Brownian motion, mean, covariance matrix and Moment Generating Function, alternative definitions of Brownian motion.

Limiting distribution of Binomial asset pricing model

Binomial asset pricing model, brief definition of arbitrage pricing and risk neutral probability, Movement of wealth, Delta hedging.

Initial knowledge for the construction of Binomial Asset Pricing Model, Risky and Risk free financial institutions, The relationship between- risk free interest rate (r), upward factor (u) and downward factor (d).

Introduction, Intuition of Brownian Motion using scaled Random Walks: Symmetric random walk, Increments of symmetric random walks, Martingale property of symmetric random walks, Quadratic variation, Scaled Random Walk- Expectation and Variance, Limiting distribution of Scaled Random walk.

What is unexpected loss? The concept of Economic Capital. Loss distribution of a portfolio. Estimation of Economic Capital using parametric approximation and using Monte Carlo simulation.

What is Credit Risk Management and how important is it? Expected Loss, Default Probability, Exposure at Default, Loss given Default.