The teaching is organized as follows:

Learning outcomes

The Mathematical Finance course for the internationalized Master's Degree (delivered completely in English) aims to introduce the main concepts of stochastic discrete and continuous time part of the modern theory of financial markets. In particular, the fundamental purpose of the course is to provide the mathematical tools characterizing the setting of Itȏ stochastic calculus for the determination, the study and the analysis of models for options and / or interest rates determined by stochastic differential equations driven by Brownian motion. Basic ingredients are the foundation of the theory of continuous-time martingale, Girsanov theorems and the Faynman-Kac theorem and their applications to the theory of option pricing with specific examples in equities, also considering path-dependent options, and within the framework of interest rates models.

Program

Discrete time models
• Contingent claims, value process, hedging strategies, completeness, arbitrage
• Fundamental theorems of Asset Pricing (in discrete time)

The Binomial model for Assset Pricing
• One period / multiperiod Binomial model
• A Random Walk (RW) interlude (scaled RW, symmetric RW, martingale property and quadratic variation of the symmetric RW, limiting distribution)
• Derivation of the Black-Scholes formula (continuous-time limit)

Brownian Motion (BM)
• review of the main properties of the BM: filtration generated by BM, martingale property, quadratic variation, volatility, reflection properties, etc.

Stochastic Calculus
• Itȏ integral
• Itȏ-Döblin formula
• Black-Scholes-Merton Equation
• Evolution of Portfolio/Option Values
• Solution to the Black-Scholes-Merton Equation
• Sensitivity analysis

Risk-Neutral Pricing
• Risk-Neutral Measure and Girsanov's Theorem
• Pricing under the Risk-Neutral Measure
• Fundamental Theorems of Asset Pricing
• Existence/uniqueness of the Risk-Neutral Measure
• Dividend/continuously-Paying
• Forwards and Futures

Binomial model: numerical approaches
•Parameters for the binomial model
•The binomial model implementation
•Hedging for the binomial model


Black and Scholes: review and numerical implementation(s)
•The Itȏ-Döblin formula: a review
•Pricing Hedging and Risk for the Plain Vanilla Options
•Delta-Vega-Gamma Approsimations: Numerical Experiments
•Morte Carlo pricing princples: some easy cases
•The leverage effect of an option


Functionals of the Brownian motion and geometric Brownian motion
•The first hittting time distribution and density
•The occupation time and the Takacs formula
•The Local time definition, related quantities and density


Applications of the BM functional to the option pricing
•Barrier Options
•Digital Options
•Accumulators Options
•Implementations in VBA and MATHCAD


The Asian style options
•Impact of the fixing frequency
•Monte Carlo Evaluations
•Monte Carlo accuracy: how to build a confidence level for the pricer
•The Matching moment method (MMM): principles
•The MMM implementation for the Asian option


Advanced probabilistic tools for exotic options
•The best-of and the worst-of options.
•The worst and the best distribution for uncorrelated basket
•Extension to correlated basket


Credit Risk
•Default definition
•The relevant parameters for the credit risk: PD, EAD, LGD
•The credit portfolio Loss
•The Basel Gordy model
•A Monte Carlo framework for the loss distribution

Paolo Guasoni mini course: the didactic material concerning this part is the subset of the following topics that have been concretely treated during the 8 hours mini course given by Prof. Paolo Guasoni (DCU-Dublin) [ ask to Prof. Di Persio for further infos] :

1. Classical Theory.
The discussion starts with a review of the Merton consumption-investment problem with constant investment
opportunities, and with its asset-pricing counterpart, the Lucas model.
2. Long-run, state variables, and stochastic investment opportunities.
The discussion starts with the general model of a market with several state variables, the objectives of
equivalent safe rate and equivalent annuity, their corresponding HJB equations, and nite-horizon bounds.
Applications to models of return predictability and stochastic volatility conclude. [4].
3. Transaction Costs.
A market with transaction costs and constant investment opportunities is equivalent to another market,
found explicitly, in which investment opportunities are stochastic, but transaction costs absent. [1].
4. Price Impact.
If trading speed aects execution prices, portfolio weights are no longer controls, but state variables. The
optimal trading speed follows an autonomous diusion process, interpreted as trading volume. Short and
levered positions are endogenously banned by this friction. [5]
5. High-water marks and hedge-fund fees.
In a model of hedge fund compensation, the state variable is the ratio between the fund's assets and its
historical maximum. The long-run solution leads to a simple optimal portfolio, which shows the interplay
between fees and risk aversion. [3, 2]
6. Path-dependent Preferences and Shortfall Aversion.
A model in which the marginal utility from increases in consumption above its historical maximum is lower
than the marginal utility of marginal decreases in consumption (shortfall aversion) can explain high asset
prices and low interest rates, as well as smooth consumption with volatile wealth. [6]
References
[1] S. Gerhold, P. Guasoni, J. Muhle-Karbe, and W. Schachermayer. Transaction costs, trading volume, and the
liquidity premium. Finance and Stochastics, 18(1):1{37, 2014.
[2] P. Guasoni and J. Muhle-Karbe. Long horizons, high risk-aversion, and endogenous spreads. Mathematical
Finance, 25(4):724753, 2011.
[3] P. Guasoni and J. Obloj. The incentives of hedge fund fees and high-water-marks. Mathematical Finance,
2015, 2009.
[4] P. Guasoni and S. Robertson. Portfolios and risk premia for the long run. The Annals of Applied Probability,
22(1):239{284, 2012.
[5] P. Guasoni and M. Weber. Dynamic trading volume. Mathematical Finance, 2015. forthcoming.
[6] Paolo Guasoni, Gur Huberman, and Dan Ren. Shortfall aversion. Available at SSRN 2564704, 2015.

Examination Methods

Final Exam : the exam will consists in an oral session plus a case study developed according with prof. Michele Bonollo with respect to the following list of case studies:

#1: Stress Test of derivatives portfolios
#2: Derivatives portfolio evaluation and management
#3: Credit Portfolio Risk
#4: Exotic Options Pricing