approaching the course stochastic calculus for...

A.Y. 2015/2016

Approaching the course

Stochastic Calculus forFinance

Silvia Faggian

Department of Economics

Evolution in (continuous) time of the price of a bond

B′(t) = rB(t), ∀t ≥ 0 (1)

written also

dB(t)dt

= rB(t) or dB(t) = rB(t)dt

I B(t) price of the bond (risk-free asset)I r rate of interest (deterministic, constant)

Mathematically speaking:

I (1) is an ordinary differential equationI the unknown of the equation is a function B(t) of one

variable (time).

Evolution in (continuous) time of the price of a bond

B′(t) = rB(t), ∀t ≥ 0 (1)

written also

dB(t)dt

= rB(t) or dB(t) = rB(t)dt

I B(t) price of the bond (risk-free asset)I r rate of interest (deterministic, constant)

I (1) is an ordinary differential equationI the unknown of the equation is a function B(t) of one

variable (time).

Evolution in (continuous) time of a stock price

dS(t) = µS(t)dt + σS(t)dW (t), t ≥ 0 (2)

I S(t) price of the stock (risky - non deterministic)I µ expected return on stockI σ volatilityI dW (t) is "white noise" (a stochastic disturbance)

I (2) is an stochastic differential equation (a geometricBrownian Motion)

I the unknown of the equation is a stochastic process S(t)(a family of random variables S(t)(ω) of parameter t)

Evolution in (continuous) time of a stock price

dS(t) = µS(t)dt + σS(t)dW (t), t ≥ 0 (2)

I S(t) price of the stock (risky - non deterministic)I µ expected return on stockI σ volatilityI dW (t) is "white noise" (a stochastic disturbance)

I (2) is an stochastic differential equation (a geometricBrownian Motion)

I the unknown of the equation is a stochastic process S(t)(a family of random variables S(t)(ω) of parameter t)

The Black-Scholes-Merten equation

(Used to compute the theoretical price of a derivative)

∂f (t ,S)

∂t+ rS

∂f (t ,S)

12σ2S2∂

2f (t ,S)

∂S2 = rf (t ,S) (3)

I t time, S value of the underlying financial assetI f (t ,S) is the price of a call option or other derivative

contingent on S.I σ volatilityI r risk-free rate of interest

I (3) is an partial differential equationI the unknown of the equation is a function of two variables

The Black-Scholes-Merten equation

(Used to compute the theoretical price of a derivative)

∂f (t ,S)

∂t+ rS

∂f (t ,S)

12σ2S2∂

2f (t ,S)

∂S2 = rf (t ,S) (3)

I t time, S value of the underlying financial assetI f (t ,S) is the price of a call option or other derivative

contingent on S.I σ volatilityI r risk-free rate of interest

I (3) is an partial differential equationI the unknown of the equation is a function of two variables

Scope of the Course

I This is an advanced course in quantitative Economics,aiming to study mathematical tools that are needed inFinance.

I Covered topics include:I ordinary differential equations;I stochastic differential equations;I stochastic integral;I partial differential equations;I applications to Finance (pricing of options).

I The goal is accomplished if a student is able, by the end ofthe course, to read and understand enough a maths text inFinance to interpret premises and results (without gettingnecessarily into maths details).

Scope of the Course

Prerequisites

I A first year Mathematics course (of course). In particularcalculus and integration (on finite and infinite intervals).

I (If student were able to compute double integrals, thatwould be a blessing.)

I Probability in finite and infinite spaces.I Preferably: students are acquainted with contents of the

course "Derivatives and insurance" and "Stochasticmodels for finance".

Prerequisites

I Probability in finite and infinite spaces.

I Preferably: students are acquainted with contents of thecourse "Derivatives and insurance" and "Stochasticmodels for finance".

Prerequisites

Textbooks

I Steven E. Shreve (2000), "Stochastic Calculus for FinanceII. Continuous Time Models", Springer, (Chapters 1 - 4)(maths is carefully explained)

I Tomas Bjork, "Arbitrage Theory in Continuous Time" (thirdedition), Oxford University Press (application oriented, lessmaths)

I John C. Hull, "Options, futures, and other derivatives",Pearson-Prentice Hall, New Jersey (much applicationoriented, little math explanation)

I Lawrence C. Evans, "Introduction to Stochastic DifferentialEquations", AMS (mathematics oriented, little application)

I Other books?

Textbooks

I Other books?

Textbooks

I Other books?

Textbooks

I Other books?

Textbooks

I Other books?

Homework

I During the course, five Homework will be proposed.Students are supposed to seriously tackle all in order toreach proficiency in study.

I Homework, as well as other teaching material, will bemade available on the course web page

I ...or on a shared Dropbox folder, accessible and modifiableby any member (then be careful not to delete files...).

Examination policy

I Written exam and optional oral exam.I A score 18/30 to pass the exam.I With score 16/30 or more at the written exam, one accedes

to the oral exam.1

I Written exam contains 3 problems or theoretical questions;oral exam is, mainly but not only, a discussion about thewritten exam.

I 4 calls in a year: December, January, June, September.

Approximative schedule

I 1st week: Review of Integration of real functions of oneand more variables; Ordinary differential equations.

I 2nd week: Review on Infinite Probability spaces.I 3rd Week: Brownian MotionI 4th Week: Stochastic CalculusI 5th Week: Applications to Option pricing

How to reach meI Office hours

Monday 14:30 - 15:30,Room 005 - Ground floor - Building C2See my web page for changes of schedule, or thewebpage of the course.

I email: [email protected](Please, before writing check if the answer is already onthe course webpage)

I My webpage:http://venus.unive.it/faggian/teaching.html

I Course Webpage:http://venus.unive.it/faggian/2015-16.SCFF.html

Happy attendance!

approaching the course stochastic calculus for...

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