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
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