Itô calculus in a nutshell

Ito^ calculus in a nutshell

Vlad Gheorghiu

Department of Physics Carnegie Mellon University Pittsburgh, PA 15213, U.S.A.

April 7, 2011

Vlad Gheorghiu (CMU)

It^o calculus in a nutshell

April 7, 2011 1 / 23

Outline

1 Elementary random processes 2 Stochastic calculus 3 Functions of stochastic variables and It^o's Lemma 4 Example: The stock market 5 Derivatives. The Black-Scholes equation and its validity. 6 References A summary of this talk is available online at

Vlad Gheorghiu (CMU)

It^o calculus in a nutshell

April 7, 2011 2 / 23

Elementary random processes

Elementary random processes

Consider a coin-tossing experiment. Head: you win $1, tail: you give me $1.

Let Ri be the outcome of the i-th toss, Ri = +1 or Ri = -1 both with probability 1/2.

Ri is a random variable.

E [Ri ] = 0, E [Ri2] = 1, E [Ri Rj ] = 0.

No memory! Same as a fair die, a balanced roulette wheel, but not

blackjack!

Now let Si =

i j

=1

Rj

be

the

total

amount

of

money

you

have

up

to

and including the i-th toss.

Vlad Gheorghiu (CMU)

It^o calculus in a nutshell

April 7, 2011 3 / 23

Elementary random processes

Random walks

This is an example of a random walk.

Figure: The outcome of a coin-tossing experiment. From PWQF.

Vlad Gheorghiu (CMU)

It^o calculus in a nutshell

April 7, 2011 4 / 23

Elementary random processes

If we now calculate expectations of Si it does matter what information we have.

E [Si ] = 0 and E [Si2] = E [R12 + 2R1R2 + . . .] = i . The random walk has no memory beyond where it is now. This is the Markov property.

The random walk has also the martingale property:

E [Si |Sj , j < i] = Sj . That is, the conditional expectation of your

winnings at any time in the future is just the amount you already hold.

Quadratic variation:

i j

=1(Sj

- Sj-1)2.

You

either

win

or

lose

$1

after each toss, so |Sj - Sj-1| = 1. Hence the quadratic variation is

always i.

Vlad Gheorghiu (CMU)

It^o calculus in a nutshell

April 7, 2011 5 / 23

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download