Quantitative Finance and Insurance

 The purpose of the course is to discuss:

1) how to construct and manage a  fixed-income portfolio

2) how to model the term structure of interest rates

3) how to price and hedge interest rate derivatives

Fundamental mathematical tecniques will be presented. Practical applications will be greatly discussed.

At the end of the course, the student is expected to be capable of:

• understanding fixed-income markets and instruments
• evaluating the impact of the determinants of interest rate movements
• constructing and maintaining a fixed-income portfolio, and assessing its credit risk
• constructing interest rate curves from marked data
• pricing and hedging interest rate derivatives
• applying the basic course knowledge to theoretical issues and concrete market situations
• approaching the subject in a critical manner through the examination of different approaches in the literature and practice of the fixed-income market
• having gained communication skills, through the debate during the lectures
• having gained learning abilities, through a variety of learning tools (teaching material, class discussion, lab sessions, homework and tests)

Lectures, class discussion and lab sessions. 

Slides and other course material will be made available on Moodle in due time.

Written exam 80% (24 points), homeworks and group project, 20% (6 points). The written exam is a closed-book exam lasting 2 hours.  Homeworks and projects must be uploaded on Moodle. Deadlines will be posted on Moodle. 

 

Part 1 [Martina, 16 hours] 

BOND ANALYSIS:
- Bond definition and characteristics
- Bond types
- Bond structure and priority
- Bond valuation
- Bond Return Measures (CY, YTM, YTC)
- Yield curve
- The effect of interest rate changes on bond prices
- Duration
- Determinants of Interest Rates
- Bond ratings and CRAs
- Bond spread
- Bond Yields
- Government Bonds

PORTFOLIO MANAGEMENT:
- Portfolio competition and market/strategy comments

Part 2 [Marena, 16 hours]

INTEREST RATES DERIVATIVES: PRODUCTS AND MARKET MODELS

1. Spot and forward contracts
   - Taxonomy of rates
2. Interest rate linear derivatives
   - FRA and swaps
3. Bootstrapping/interpolating the interest rate curve
4. Interest rate options
   - Options on bonds
   - Caps and floors
   - Swaptions
5. The change of measure technique
6. Market models:
   - The Black model
   - Market models
   - Beyond Black's model
7. Hedging interest rate risk

REFERENCES
Björk, T.. Arbitrage theory in continuous time, OUP, 2009
Brigo, D., Mercurio, F., Interest Rate Models: Theory and Practice With Smile, Inflation and Credit, Springer, 2006.
Kienitz, J., 2013&2017. Interest rates explained 1&2, Palgrave

 

Part 3 [Roncoroni, 16 hours]

INTEREST RATES DERIVATIVES AND SHORT RATE MODELS

Part I: Linear Interest Rate Derivatives.

1. Time value
- Time measurement
- Accrual and discount factors
- Properties

2. Interest Rates
- Interest as a frequency
- Simple, discrete, and continuously compounding
- Forward time value and rates

3. Linear Derivatives
- Definition and classes
- Single payment: Zero, floating rate bond, forward rate agreement
- Multiple payments: coupon bonds, floating rate note, swap

4. Training on Interest Rate Calculus
- Selected exercises (with solution)

Part II: Nonlinear Interest Rate Derivatives.

1. Nonlinear Derivatives
- Definition
- Bond options,
- Cap(let), floor(let), swaption

2. Reduction to Bond Options
- Jamshidian trick
- Application to swaption
- Caplet as a put option on a zero

3. Change of Numéraire
- Problem statement and solution heuristics
- Mathematical prerequisites: Change of measure, abstract Bayes' theorem, Girsanov theorem
- Change of numéraire: technique and effect on Itô dynamics; Example: the forward measure

4. Option Pricing under Stochastic Time Value
- The pricing problem
- General option pricing formula
- Geman-El Karoui-Rochet formula

Part III: Short Rate Modeling

1. Arbitrage-Free Bond Markets Generated by a Short Rate Model.
- Portfolio dynamics and risk premium condition
- Pricing function indeterminacy
- Risk neutral expectation formula

2. Affine Bond Market and Short Rate Models
- Definition and characterization of affine term structures
- Application I: Model calibration to a quoted discount function
- Application II: Zero-coupon bond option pricing

3. The Hull-White Model: Calibration and ZBO Pricing
- Model definition
- Calibration to the quoted term structure
- Bond option formula

4. Introduction to Forward Rate Curve Modeling (time permitting)
- Interest rate relations
- The Heath-Jarrow-Morton framework
- Pricing with a HJM model

REFERENCES
Björk, T., Arbitrage Theory in Continuous Time (3rd ed.), Oxford University Press, 2009.
Brigo, D., Mercurio, F., Interest Rate Models: Theory and Practice, Springer Finance, 2006.
Roncoroni, A., Fixed Income Lecture Notes 1-3, 2021.

Material and lecture notes will be made available via Moodle.

References for FIXED INCOME DERIVATIVES:
- Björk, T.. Arbitrage theory in continuous time, OUP, 2009
- Brigo, D., Mercurio, F., Interest Rate Models: Theory and Practice With Smile, Inflation and        Credit, Springer, 2006.
- Kienitz, J., 2013&2017. Interest rates explained 1&2, Palgrave

 

 The course should be taken together with "Derivatives".  Students who cannot attend classes are kindly requested to contact instructors at the beginning of the course.