Probability.Stochastic processes in discrete and continuous time. Bownian motion, Ito formula, martingale.
The Black-Scholes model.The Cox-Ross-Rubinstein model.Pricing:The no-arbitrage price and its implications. The Black- Scholes PDE pricing formula for European options. Risk neutral valuation. Hedging: the Greeks. Pricing of American Options. Path dependent options.Some extensions of Black-Scholes model. Stochastic volatility models. Jump-diffusion models.
The term structure.
J. Hull, Options, Futures and other Derivative Securities. Prentice Hall.
T.Bjork, Arbitrage theory in continuous time. Oxford University Press.
Knolewdge of and ability to apply the fundamental quantitative tools for the analysis of financial markets.
Ability to compute securities' prices, both with discrete time models, and with continuous time models; the hedging strategies; the term structure of
Calculus and Integration
Classes and exercises
Type of Assessment
It is given as a written exam. This is intended to verify: 1) the acquired knowledges as for concepts, models and tools which have been the object of the course; 2) the following skills have been developed by the student: ability to apply the knowledges obtained, ability to derive conclusions, communication skills using an adequate language, understanding and learning ability.
30% Test about Probability
70% Final written exam
Probability: Probability spaces. The conditional probability. Stochastic Independence. Bernoulli scheme.
Discrete random variables. Examples of discrete random variables. Computation of the density of a function
of a discrete random variable. Moments. Continuous random variables. Limiting Theorems.
Sto chastic Mo dels in Finance: Introduction to stochastic processes in continuous time. The Filtration.
The Brownian Motion: Definition and Properties. The Bachelier model for asset prices. Arithmetic Brownian
Motion. The SDE model, the solution process and its distribution. Extension of the model with time varying
coefficients. The Black-Scholes model for asset prices. Itˆo formula. Itˆo table. Case 1: a function of the asset
price. Case 2: a function of time and asset price. Case 3: a function of two asset prices. Examples: discounted
asset price, the product of two asset price B&S models.
Option Pricing: The no-arbitrage price and its implications. The Black-Scholes PDE pricing formula for
European options. Risk neutral valuation. Explicit computation of the European Call and Put option price
using the risk neutrality approach. The Girsanov Theorem. Completeness and Replication. Parity Relations.
Martingale process in finance: the Brownian motion, the discounted asset price. Equivalence of the two pricing
approaches: from the pricing PDE to the risk-neutral valuation and viceversa. Pricing American Options.
Path-dependent options: pricing Lookback options, Barrier options and Asiatic options when the underlying
follows the Binomial model.
Dynamic Hedging: Dynamic Hedging, the Greeks (Delta, Gamma). Computation of Delta and Gamma
for European Call and Put options. How to construct a Delta neutral portfolio. How to construct a DeltaGamma neutral portfolio. Computation of the Vega, Rho, Theta for European Call and Put options. Relation
between the Black&Scholes PDE and the Greeks. How to construct a Delta-Gamma-Vega neutral portfolio.
The Greeks for non-European options: the case of the Digital option.
Some extensions of Black-Scholes mo del: Historical Volatility. Implied Volatility surface. The smile.
Time dependent volatility. Local Volatility models (CEV model). Stochastic volatility models (Heston model).
Merton jump-diffusion model.
Bonds and interest rates: Short rate models (Vasicek, Ho-Lee, CIR, Hull-White models). Bond Market:
forward rates, instantaneous forward rates, short rates, Zero Coupon Bond and Coupon Bonds. Complete and
Incomplete Markets. The Vasicek model (explicit solution, mean, variance, long term mean and variance). CoxIngersoll-Ross (CIR) model. Ho-Lee model. Hull-White models (generalized Vasicek and CIR). The inversion
of the curve and the problem of parameters fitting, study of the case of Ho-Lee model. The market price of
risk. The Term Structure PDE. Affine Term structure. Affine models. Riccati equations for pricing bond under
the Ho-Lee model and the Vasicek model. Explicit solution of the ZCB price for the Ho-Lee and the Vasicek
model. Change of Numeraire and the pricing of European options on a ZCB.
Textb o ok: Probability
 R. Ash, Basic Probability Theory, free downloadable at:
Textb o oks: Finance
 T. Bjork, Arbitrage Theory in Continuous Time, Oxford University Press, Third Edition. (You can find
the material covered in the chapters 2 (The Binomial model), 4 (Stochastic integrals), 7 (Arbitrage pricing), 9
(Parity relations and Delta Hedging), 23 (Short rate models), 24 (Martingale models for the short rate)).
 J. Hull, Options, Futures and Other Derivatives, Prentice Hall, N.J. (For the 8th edition: you can
find the material covered in the chapters 12 (Introduction to Binomial Trees), 13 (Wiener processes and Itˆos
lemma), 14 (Black-Scholes-Merton model), 18 (The Greeks Letters) , 25 (Exotic Options), 27 (Martingale and
Probability Measures), 30 (Interest rates derivatives, short rate models)).
 E. Rosazza Gianin and C. Sgarra, Mathematical Finance: Theory Review and Exercises, Springer