# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## [PDS] Ito Calculus: Main Results

Ito Calculus: Main Results

1. Definition of the Ito Integral: Given the Brownian Motion $X(t)$, the Ito Integral of a function $F(X(t))$ is defined by the limit:

\begin{aligned} \displaystyle \int_{0}^{t}F(X(\tau))dX(\tau) &= \lim_{n \to \infty} \sum_{i = 1}^{n}F(X(t_{i - 1}))(X(t_i) - X(t_{i - 1})) \end{aligned}

It is important that the limit be over the ‘lower sum’, i.e. the increment of the Brownian Motion $X(t)$ is considered after the function evaluation at $F(X(t_{i - 1}))$. This is a crucial point. In finance, the hedgers and speculators are worried about the change in their portfolio \textit{after} the trade has been made. One can think of $F(X)$ as the trading strategy, and $(X(t_i) - X(t_{i - 1}))$ as a ‘shock’ to the portfolio (the change in the prices of underlying securities).

2. Ito’s Lemma: Given a function $F(X(t))$ of the Brownian Motion $X(t)$, Ito’s Lemma is a relationship between the ‘value’ of the function of the Brownian Motion $F(X(t))$ in terms of its derivatives:

\begin{aligned}\displaystyle F(X(t)) &=F(X(0)) +\int_{0}^{t}\frac{\partial F}{\partial X}dX(t) +\frac{1}{2}\int_{0}^{t}\frac{\partial^2 F}{\partial X^2}dt\end{aligned}

Note that integral $\displaystyle \int_{0}^{t}\frac{\partial F}{\partial X}dX(t)$ is an Ito integral and has to be interpreted in the Ito sense (as above).

3. The SDE (‘short-hand’) Version of Ito’s Lemma: Since the Brownian Motion $X(t)$ is not differentiable anywhere, technically one can’t differentiate the integral version of Ito’s Lemma on both sides to write the corresponding SDE. That said, since the SDE version of Ito’s Lemma ‘looks like’ the ever-so-familiar Taylor series, it is sometimes easier to remember Ito’s Lemma as an SDE, as long as one keeps in mind that the correct way is the integral way.

$\displaystyle dF(X(t)) =\frac{\partial F}{\partial X}dX(t)+\frac{1}{2}\frac{\partial^2 F}{\partial X^2}dt$

4. Ito’s Lemma for $F(t, X(t))$: If there is an explicit dependence of $F$ on time $t$, Ito’s Lemma changes predictably as:

$\displaystyle dF(X(t)) = \frac{\partial F}{\partial t}dt + \frac{\partial F}{\partial X}dX(t) + \frac{1}{2}\frac{\partial^2 F}{\partial X^2}dt$

5. Ito’s Lemma for $F(t, S(t, X(t)))$: Given the stock price process $S(t, X(t))$:

\begin{aligned}\displaystyle dS &=\mu S dt +\sigma S dX(t)\end{aligned}

Ito’s Lemma for a function of the stock price and time, $F(t, S(t, X(t)))$ is written as:

\begin{aligned} \displaystyle dF &=\frac{\partial F}{\partial t}dt +\frac{\partial F}{\partial S}dS +\frac{1}{2}\frac{\partial^2 F}{\partial S^2} (dS)^2\\&= \frac{\partial F}{\partial t}dt + \frac{\partial F}{\partial S}(\mu S dt + \sigma S dX(t)) + \frac{1}{2}\frac{\partial^2 F}{\partial S^2} (\mu S dt + \sigma S dX)^2 \\&= \Big( \frac{\partial F}{\partial t} + \mu S \frac{\partial F}{\partial S} \Big) dt + \sigma S \frac{\partial F}{\partial S} dX(t) + \frac{1}{2}\frac{\partial^2 F}{\partial S^2} (\mu^2 S^2 dt^2 + \sigma^2 S^2 (dX(t))^2 + 2 \mu \sigma S^2 dt dX(t)\end{aligned}

Ignoring all terms $> O(dt)$, and setting $(dX)^2 = dt$ in the mean-square sense:

\begin{aligned}\Rightarrow dF = \Big( \frac{\partial F}{\partial t} + \mu S \frac{\partial F}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 F}{\partial S^2} \Big) dt + \sigma S \frac{\partial F}{\partial S} dX(t)\end{aligned}

In general, Ito’s Lemma for a function $F(t, S(t))$ of any linear SDE of the form:

$dS = A(t, S(t))dt +B(t, S(t))dX(t)$

takes the form:

\begin{aligned} dF &= \Big( \frac{\partial F}{\partial t} + A \frac{\partial F}{\partial X} + \frac{1}{2}B^2 \frac{\partial^2 F}{\partial S^2} \Big) dt + B \frac{\partial F}{\partial S} dX(t)\end{aligned}

6. Ito’s Lemma for $F(t, S_1(t), S_2(t))$: Given two processes $S_1(t)$ and $S_2(t)$:

\begin{aligned} dS_1 &= A_1(t, S(t))dt + B_1(t, S(t))dX_1(t)\\ dS_2 &=A_2(t, S(t))dt +B_2(t, S(t))dX_2(t)\end{aligned}

Ito’s Lemma for a function $F(t, S_1(t), S_2(t))$ is given as:

\begin{aligned} dF = \displaystyle \Big(\frac{\partial F}{\partial t} + A_1\frac{\partial F}{\partial S_1} + A_2\frac{\partial F}{\partial S_1} + \frac{1}{2}B^2_1\frac{\partial^2 F}{\partial S^2_1} + \frac{1}{2}B^2_2\frac{\partial^2 F}{\partial S^2_2} + \rho B_1 B_2 \frac{\partial^2 F}{\partial S_1 \partial S_2} \Big) dt \\ + B_1\frac{\partial F}{\partial S_1} dX_1(t) + B_2\frac{\partial F}{\partial S_2} dX_2(t)\end{aligned}

Later we use this result to write the Chain Rule in Ito Calculus.

7. Ito integrals are Martingales: Given any process $g(t)$, under mild integrability conditions (see Bjork or Shreve for details), the following holds:

\begin{aligned} E\Big[\int_{t}^{T}g(\tau)dX(\tau) \Big\rvert F_t \Big] &= 0\end{aligned}

We can generalize to write that, modulo an integrability condition, $\displaystyle X(t) = E[\int_{0}^{t}g(\tau)dX(\tau)]$ is a martingale. That is, every Ito integral is a martingale.

Written by Vineet

February 11, 2013 at 11:46 pm

Posted in Teaching: PDS

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