First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SC03111823, A.A. 2017/18

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course Second cycle degree in
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2017/18
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 7.0
Type of assessment Mark
Course unit English denomination STOCASTIC METHODS FOR FINANCE
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English

Teacher in charge MARTINO GRASSELLI SECS-S/06

Course unit code Course unit name Teacher in charge Degree course code

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Educational activities in elective or integrative disciplines MAT/06 Probability and Mathematical Statistics 4.0
Educational activities in elective or integrative disciplines SECS-S/06 Mathematics for Economics, Actuarial Studies and Finance 3.0

Mode of delivery (when and how)
Period Second semester
Year 1st Year
Teaching method frontal

Organisation of didactics
Type of hours Credits Hours of
Hours of
Individual study
Practice 3.0 24 51.0 No turn
Lecture 4.0 32 68.0 No turn

Start of activities 26/02/2018
End of activities 01/06/2018

Prerequisites: Stochastic analysis
Target skills and knowledge: The course presents some important models that are typically used in the banking industry.
The students at the end should be familiar with pricing and hedging in both discrete and continuous time and they should be able to apply stochastic methods to the pricing of equity/forex/fixed income products
Examination methods: Final examination based on: Written and oral examination.
Assessment criteria: Critical knowledge of the course topics. Ability to present the studied material.
Course unit contents: The pricing problem in the binomial models
Risk neutral pricing in the discrete time world
European and American options in the binomial model.

Arbitrage and risk neutral pricing in continuous time.
Pricing of contingent claims in continuous time: the Black&Scholes formula.
Black&Sholes via PDE and via Girsanov.
Hedging and completeness in the Black&Scholes framework.
Feynman-Kac formula and risk neutral pricing in continuous time.
Pur Call parity, dividends and static vs dynamic hedging.
The Greeks and the Delta-Gamma hedging. Delta-Gamma-Vega neutral portfolios.

Barrier options pricing in the Black&Scholes model.
Quanto option pricing in the Black&Scholes model.

Multi asset markets, pricing and hedging.
Exchange options pricing in the multi-asset Black&Scholes model.
Incomplete markets: quadratic hedging.

Smile and skew stylized facts.
Beyond the Black&Scholes model: stochastic volatility.
The Heston model.

Bonds and interest rates. Pre-crisis and multiple-curve frameworks.
Short rate models, Vasicek, CIR, Hull-White models, affine models.
Cap&Floor pricing in the short rate approaches. The pricing of swaptions.

Forward rate models: HJM approach, the drift condition and BGM models.
Change of numeraire and Forward Risk Neutral measure.
LIBOR and Swap models.
Planned learning activities and teaching methods: Lecture supported by tutorial, exercises and laboratory activities.
Additional notes about suggested reading: Lecture notes and reference books will be given by the lecturer.
Textbooks (and optional supplementary readings)
  • T. Bjork, Arbitrage theory in continuous time. --: Oxford Univ. Press, Second Edition, 2004. Suggested for: Pricing products in the Black&Scholes framework, arbitrage, barrier options, forex, interest rates Cerca nel catalogo
  • D. Lamberton and B. Lapeyre, Introduction to stochastic calculus applied to finance.. --: Cambridge University Press., 2000. Suggested for: Discrete time binomial models, Black&Scholes formula, Girsanov methodology Cerca nel catalogo
  • J. Hull, Options, Futures and Other Derivatives. --: Pearson, 8th edition, 2012. Suggested for: General introduction of option markets, Greeks, financial institutions Cerca nel catalogo