03 - Stochastic Calculus - Part III

Recall

Function	Itô Lemma
$F(X_t)$	$dF=\frac{1}{2}\frac{d^{2}F}{d{X}^2}dt+\frac{dF}{dX}dx$
$F(t,X_t)$	$dF=(\frac{\partial F}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}F}{\partial X^2})dt+\frac{\partial F}{\partial X}dX$
$V(S)$ when $dS=\mu Sdt+\sigma SdX$	$dV=(\mu S\frac{dV}{dS}+\frac{1}{2}{\sigma }^2{S}^2\frac{d^{2}V}{d{S}^2})dt+\left(\sigma S\frac{dV}{dS}\right)dX$

Itô Integrals as non-anticipatory

Let’s consider the stochastic integral of the form $${\int }_0^{T}f(t,X(t))dX(t)$$ where $X_t$ is a Brownian motion. We’ll shorten this form to ${\int }_0^{T}f(t,X)dX$

We define this integral as $${\int }_0^{T}f(t,X)dX=\underset{N\to \mathrm{\infty }}{lim}\sum _{i=0}^{N-1}f(t_i,X_i)\cdot \underset{dX}{\underbrace{(X_{i+1}-X_i)}}$$

It’s important to define it this way in order for the itô integral to stay non-anticipatory. We know everything up to time $t_i$ and so the only uncertainties left is $X_{i+1}-X_i$ which is $dX$

Product rule within Stochastic Calculus

When dealing with Stochastic Differential Equations, we can adapt some of the rules of classical calculus such as the product rule: $d(xy)=xdy+ydx$

Let’s say we have 2 stochastic processes: $$d(X(t))={\mu }_{1}X(t)dt+{\sigma }_{1}X(t)d{W}_t$$ $$d(Y(t))={\mu }_{2}Y(t)dt+{\sigma }_{2}Y(t)d{W}_t$$

And we define a function $F$ which is a product of these 2 stochastic processes such that $F=F(X,Y)=XY$.

Using a Taylor Series Expansion, we can write: $$F(X+dX,Y+dY)\approx F(X,Y)+\frac{\partial F}{\partial X}dX+\frac{\partial F}{\partial Y}dY+\frac{1}{2}\frac{{\partial }^{2}F}{\partial X^2}d{X}^2+\frac{1}{2}\frac{{\partial }^{2}F}{\partial Y^2}d{Y}^2+\frac{{\partial }^{2}F}{\partial X\partial Y}dXdY+\dots$$

Hence, $$\begin{array}{}\text{(1)}& dF=\frac{\partial F}{\partial X}dX+\frac{\partial F}{\partial Y}dY+\frac{1}{2}\frac{{\partial }^{2}F}{\partial X^2}d{X}^2+\frac{1}{2}\frac{{\partial }^{2}F}{\partial Y^2}d{Y}^2+\frac{{\partial }^{2}F}{\partial X\partial Y}dXdY+\dots \end{array}$$

Now, we can calculate all these partial derivatives and plugged them back in the above equation. $\frac{\partial F}{\partial X}=Y$ and $\frac{{\partial }^{2}F}{\partial X^2}=0$.

Similarly $\frac{\partial F}{\partial Y}=X$ and $\frac{{\partial }^{2}F}{\partial Y^2}=0$.

Finally: $\frac{{\partial }^{2}F}{\partial X\partial Y}=1$

Plugging it all back in Equation 1: $$\begin{array}{}\text{(2)}& dF=YdX+XdY+dXdY\end{array}$$

Integral by parts

In classical calculus, we re-use the product rule to come up with the integration by part: $d(xy)=xdy+ydx$. That is $xdy=d(xy)-ydx$ which we can integrate for and get: $\int xdy=\int d(xy)-\int ydx$ which is the same as $\int xdy=xy-\int ydx$.

Let’s bring this to stochastic calculus. Again $F$ is a function of the product of 2 stochastic processes: $F=F(X,Y)=XY$ Using the same logic and our previous result Equation 2, we write $$d(XY)=YdX+XdY+dXdY$$ $$XdY=d(XY)-YdX-dXdY$$ $${\int }_0^t{X}_{s}d{Y}_s={\int }_0^{t}d(X_s{Y}_s)-{\int }_0^t{Y}_{s}d{X}_s-{\int }_0^{t}d{X}_{s}d{Y}_s$$ $${\int }_0^t{X}_{s}d{Y}_s=X_t{Y}_t-X_0{Y}_0-{\int }_0^t{Y}_{s}d{X}_s-{\int }_o^{t}d{X}_{s}d{Y}_s$$

Quotient Rule within Stochastic Calculus

We will re-use the Taylor Series Expansion (Equation 1) except this time the function $F$ is a function of the quotient of 2 stochastic processes: $F=F(X,Y)=\frac{X}{Y}$. Calculating all the partial derivatives: $\frac{\partial F}{\partial X}=\frac{1}{Y}$ and $\frac{{\partial }^{2}F}{\partial X^2}=0$.

Similarly $\frac{\partial F}{\partial Y}=\frac{-X}{Y^2}$ and $\frac{{\partial }^{2}F}{\partial Y^2}=\frac{2X}{Y^3}$.

Finally: $\frac{{\partial }^{2}F}{\partial X\partial Y}=\frac{-1}{Y^2}$

Putting it all back together: $$dF=\frac{1}{Y}dX+\frac{-X}{Y^2}dY+\frac{1}{2}\frac{2X}{Y^3}d{Y}^2+\frac{-1}{Y^2}dXdY$$ Which we can re-write as: $$\begin{array}{}\text{(3)}& dF=d\left(\frac{X}{Y}\right)=\frac{X}{Y}\cdot (\frac{1}{X}dX-\frac{1}{Y}dY-\frac{1}{XY}dXdY+\frac{1}{Y^2}d{Y}^2)\end{array}$$

In the quant world.

we can word these results in the following way - taken from here:

• Itô product rule: we buy correlation when we have a product
• Itô quotient rule: we sell correlation when we have a ratio, and we are long vol of the denominator.