# Ito's Formula

## Ito's formula for one Brownian Motion

We want a rule to "differentiate" expression of the form , where is a differentiable function. If were also differentiable, then the ordinary chain rule would give

which could be written in differentiable notation as

However, B(t) is not differentiable, and in particular has nonzero quadratic variation, so the correct formula has an extra term, namely,

This is Ito's formula in differential form. Integrating this, we obtain Ito's formula in integral form:

## Derivation of Ito's formula

Consider so that

Let x be numbers. Taylor's formula implies

Fix and be a partition of Using Taylor's formula, we write

We let to obtain

This is Ito's formula in integral form for the special case

Homework 6. Page 251, #1, #3. Due 4/4.

## Geometric Brownian Motion

Geometric Brownian motion is

where and are constant. Define

so

Then

According to Ito's formula,

Thus, Geometric Brownian motion in differential form is

and Geometric,Brownian motion in integral form is

## Quadratic variation of Geometric Brownian motion

In the integral form of Geometric Brownian motion,

the Riemann integral

is differentiable with This term has zero quadratic variation. The Ito integral

is not differentiable. It has quadratic variation

Thus the quadratic variation of S is given by the quadratic variation of G. In differentiable notation, we write

## First derivation of the Black-Scholes formula

### Wealth of an investor.

An investor begins with nonrandom initial wealth X and at each time t, holds shares of stock. Stock is modelled by a geometric Brownian motion:

can be random, but must be adapted. The investor finances his investing by borrowing or lending at interest rate r. Let X(t) denote the wealth of the investor at time . Then

### Value of an option

Consider an European option which pays at time T. Let denote the value of this option at time t if the stock price is . In other words, the value of the option at each time is

The differential of this value is

A hedging portfolio starts with some initial wealth X and invests so that the wealth at each time tracks We saw above that

To ensure that for all t, we equate coefficients in their differentials. Equating the dB coefficients, we obtain the -hedging rule:

Equating the dt coefficients, we obtain

But we have set and we are seeking to cause to agree with . Making these substitutions, we obtain

which simplies to

In conclusion, we should let be the solution of the Black-Scholes partial differential equation

satisfying the terminal condition

If an investor starts with and uses the hedge

then he will have for all t, and in particular,

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