Course Code: MT/DS3284
Course Title: Mathematical Finance
Course Coordinator: Anindya Goswami
BSMS, Int-PhD, and PhD Credits: 4 credits
Pre-requisites: None
Objectives: In this course, students will be trained in the mathematical and computational techniques required to solve problems in quantitative finance. It is suitable for third- or fourth-year BS-MS students, first- or second-year MSc students, and first-year int-PhD students.
Course content (Number of lectures):
Financial markets and instruments, risk-free and risky assets (1); Return and risk of a one-period portfolio, mean-variance portfolio theory, efficient frontier, Sharpe ratio (4); Demonstration of mean-variance portfolio computation with real data (1).
Interest rates, present and future values of cash flows, term structure of interest rates, spot rate, forward rate (2); Conditional expectation and martingales, Brownian motion and Ito calculus, Ito’s lemma and its application (2), Models for short rates and instantaneous forward rates (2).
Random payoff of financial derivatives with an example of a vanilla option (1), Discrete- and Continuous-time asset pricing models, Binomial, Trinomial, GBM, Stochastic volatility, Regime switching, Merton’s jump-diffusion (3); Discrete-time trading, notion of self-financing and market completeness, No-arbitrage principle and risk-neutral probability measure (3), Financial derivatives: Forward, futures, swaps, and options contracts - Vanilla and Exotic, Value of forwards and swaps (3). Self-financing, admissible strategy, and EMM in continuous time case, local martingale, EMM implies NA under admissible continuous-time trading, EMM in BSM model (4).
Discrete-time and Continuous-time pricing of European options under Binomial lattice, BSM, and Heston models (3); existence and computation of implied volatility, volatility smile, a stylized fact and some important properties of BSM formula (3), American option pricing (2) Monte-Carlo simulation for pricing financial derivatives (1).
Evaluation/Assessment: Quiz 30%, Mid Sem 35%, End Sem 35%
Suggested readings:
Luenberger, David. Investment science, Oxford Univ Pr; 2e, 2013
Kallianpur, Gopinath and Karandikar Rajeeva L. Introduction to Option Pricing Theory, Birkhauser, 2000
Wilmott, Paul. Paul Wilmott on Quantitative Finance, 3 Volume Set, Wiley; 2e, 2006
Shreve, Steven. Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer, 2005
Shreve, Steven. Stochastic Calculus for Finance II: Continuous-Time Models, Springer 1e, 2004
Sondermann, Dieter. Introduction to Stochastic Calculus for Finance: A New Didactic Approach, Springer-Verlag Berlin, 2006
Delbaen, Freddy and Schachermayer Walter. The Mathematics of Arbitrage, Springer, 2006
Does this course expect strong Math knowledge among students?
Ans: No
Do students not majoring in Maths also get a fair chance to top the class?
Ans: Yes
Will the treatment be mathematical?
Ans: Yes
How do the students having fewer Math electives comprehend the mathematical content?
Ans: All students are expected to have full or nearly full attendance. Everyone will have the opportunity to clarify doubts in class.
Are the applications in the field of quantitative finance industry discussed?
Ans: Yes, every topic will be justified from that angle.
How much emphasis is on the proofs of statements compared to their applications?
Ans: The emphasis will be balanced on both.
How greatly the mathematical and computational techniques, taught in the course are useful in academia as well as in industry?
Ans: This will introduce many notions with full rigor, that are considered essential in the finance industry. For this reason, this course is also very useful for academic research in Math Finance.
Is there any lecture note on stochastic calculus that introduces the topic with full mathematical details, but omits long proofs?
Ans: Find a lecture note in this hyperlink. This note includes links to video recordings of lectures too.
Financial markets and financial instruments, risk-free and risky assets.
Ideal Bank, Interest rates and frequency of compounding, Discounting, present and future values of cash flows, Fair price of fixed income instruments, Non-constant ideal bank, Term structure of interest rates, spot rate, and forward rate.
Relation between the spot rate, and forward rate. Short rate. Preliminaries for short rate models.
Brownian motion and Ito Calculus
Various continuous time models of short rates.
HJM model for the instantaneous forward rates. Notes
Return and risk of a portfolio, mean and variance of portfolio return, mathematical formulation of one-period portfolio optimization problem
Markowitz's modern portfolio theory without short selling, Efficient frontier,
Algebraic formula for minimum variance portfolio in the presence of short selling.
Numerical Example of a portfolio of five risky assets. Spread Sheet
Two fund and one fund theorems, Sharpe ratio. Notes
The usefulness of financial derivatives with an example of a vanilla option, the random payoff of a call option, Terminologies related to option contracts
3rd Feb: Quiz 1 [Max=15,Min=1,Mean=9.5,Median=9.8,STD=3.5]
Various types or classifications of derivative contracts, Forward and its current value, Futures, and marking to market.
Equality of theoretical futures and forward prices, Interest rate swap and its value.
Exotic option contracts. Notes
No-arbitrage principle. Linearity of pricing under the absence of type A arbitrage.
13th Feb: Quiz 2 [Max=15, Min=3, Mean=10.5, Median=11, STD=3.4]
Equivalent martingale measure(EMM) or risk-neutral probability.
Revisit conditional expectation and martingales. Notes
19th Feb: Mid sem [Max=33, Min=11, Mean=24.6, Median=25, STD=5.1] Solution key
Self-financing strategy, and complete and incomplete market.
Definition of volatility and historical volatility, Binomial Asset Price Model.
Arithmatic and geometric Brownian motion model for risky assets and their limitations.
Some extensions of GBM model. Notes
The presence of EMM implies the absence of arbitrage in discrete-time trading.
The right meaning of EMM in continuous time case and its justification.
Definition and an example of local martingale;
Proof of EMM implies NA under admissible continuous time trading. Derivation of EMM in BSM model. Notes
Fair price of European option and its hedging under a discrete time Binomial Lattice model. Spread Sheet
Fair price of European option and its hedging under continuous time Black-Scholes-Merton model
25th Mar: Quiz 3 [Max=13.5, Min=2, Mean=8, Median=7.5, STD=2.6]
European option price under Heston's stochastic volatility model.
A quick recapitulation of the discussed option pricing topics. Put-call parity and solution to some exercises. Some useful Greeks and implied volatility. Approximate volatility implied by ATM call option.
Existence and uniqueness of implied volatility.
Volatility smile, homogeneity property, and bounds of BSM formula for vanilla options. Notes
Price of American perpetual put option under BSM model.
Price of American style options under binomial lattice model of spot prices.
Monte-Carlo simulation for pricing financial derivatives Notes
22nd Apr: End sem [Max=35, Min=5, Mean=25.1, Median=25.8, STD=6.6]
Total score
≥89 ⇒ A+
≥83 ⇒ A
≥76 ⇒ B+
≥67 ⇒ B
≥55 ⇒ C+
≥45 ⇒ C
≥36⇒ D
Max=94, Min=28, Mean=70.3, Median=72.5, STD=14.9