Mathematical Finance

This is a zero credit course to be held after the office hour. This is open for all students. However, those who wish to receive a letter of recognition of attending the same, should have 100% attendance at the end of this course.

Schedule: Weekly one lecture on Thursday at 6:30 PM in LHC 203.

Lectures

1. 09/01 Introduction to the financial market. Various financial products, Time-varying nature of their prices, The basic theory of interest.

2. 31/01 Return of an asset and return of a portfolio. Random return and its expectation and dispersion. The need for diversification.

3. Mean-SD diagram of the random return of a portfolio of two assets for any given correlation structure.

4. The shape of the feasible curve (with no short selling). The shape of the feasible set for portfolios consisting of more than two assets.

   1. Minimum Variance Point, Efficient frontier.

5. 07/02 Mathematical formulation of portfolio optimization problem using mean-variance criterion where short selling is allowed.

6. The relevant constrained optimization problem using Lagrange multiplier. Derivation of a system of linear equations. Representation of

7. the coefficient matrix as a block matrix. Proof of existence (provided all assets do not have an identical mean return) and uniqueness (provided the assets do not have a deterministic dependence) of the solution to the linear system. Use of Schur complement for writing inverse of the coefficient block matrix. Derivation of a formula for the optimal portfolio weight given a target expected return. An example.

8. 09/02 Discussions on portfolio optimization problem using mean-variance criterion where short selling is not allowed. Illustration using some examples.

9. 14/02 Two Fund Theorem and One Fund Theorem. Proof of one fund theorem. Closing the discussion on portfolio theory and starting ``Derivative Pricing''. Meaning of derivatives. Definition of Call Option. Meaning of arbitrage and Fair Price. Fair price of a call option at the maturity. Fair price of any deterministic future payoff at present.
   21/02 No class due to midsem exams.

10. 28/02 Fair price of any random future payoff at present, if the payoff depends on a deterministic way on a random asset price dynamics. Introduction to the Black-Scholes-Merton (BSM) model, the SDE. A historical note. Definition of Brownian motion and geometric Brownian motion.

11. 07/03 Visualization of Girsanov transformation, and equivalent martingale measure. Fairness of the market. Completeness of the market.

12. 15/03 The class of 14th March is rescheduled on 15th. The option price PDE under BSM assumption. Proof that the solution to the PDE gives fair price of call option using Ito's lemma. The put-call parity.
    21/03 Holiday

13. 28/03 Heston model and option pricing: an extension of Black-Scholes theory. Venue 41. This is the last lecture.