Recall
| Function | Itô Lemma |
|---|---|
| \(F(X_t)\) | \(dF = \frac{1}{2} \frac{d^2F}{dX^2} \space dt + \frac{dF}{dX} \space dx\) |
| \(F(t, X_t)\) | \(dF = \left( \frac{\partial F}{\partial t} + \frac{1}{2} \frac{\partial^2 F}{\partial X^2} \right) dt + \frac{\partial F}{\partial X} dX\) |
| \(V(S)\) when \(dS = \mu S dt + \sigma S dX\) | \(dV = \left( \mu S \frac{dV}{dS} + \frac{1}{2} \sigma^2 S^2 \frac{d^2V}{dS^2} \right)dt + \left( \sigma S \frac{dV}{dS}\right) dX\) |
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 \[\int_0^T f(t, X) dX = \lim_{N \to \infty} \sum_{i=0}^{N-1} f(t_i, X_i) \cdot \underbrace{ (X_{i+1} - X_i) }_{dX}\]
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: \[d(X(t)) = \mu_1 X(t) dt + \sigma_1 X(t) dW_t\] \[d(Y(t)) = \mu_2 Y(t) dt + \sigma_2 Y(t) dW_t\]
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: \[F(X + dX, Y + dY) \approx F(X,Y) + \frac{\partial F}{\partial X} dX + \frac{\partial F}{\partial Y} dY + \frac{1}{2} \frac{\partial^2F}{\partial X^2} dX^2 + \frac{1}{2} \frac{\partial^2F}{\partial Y^2} dY^2 + \frac{\partial^2F}{\partial X \partial Y} dXdY + \dots\]
Hence, \[dF = \frac{\partial F}{\partial X} dX + \frac{\partial F}{\partial Y} dY + \frac{1}{2} \frac{\partial^2F}{\partial X^2} dX^2 + \frac{1}{2} \frac{\partial^2F}{\partial Y^2} dY^2 + \frac{\partial^2F}{\partial X \partial Y} dXdY + \dots \tag{1}\]
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: \(\frac{\partial^2F}{\partial X \partial Y} = 1\)
Plugging it all back in Equation 1: \[dF = Y dX + X dY + dXdY \tag{2}\]
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: \(\int xdy = \int d(xy) - \int y dx\) which is the same as \(\int x dy = xy - \int y dx\).
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 \[d(XY) = Y dX + X dY + dXdY\] \[X dY = d(XY) - Y dX - dXdY \] \[\int_0^t X_s dY_s = \int_0^t d(X_sY_s) - \int_0^t Y_s dX_s - \int_0^t dX_sdY_s\] \[\int_0^t X_s dY_s = X_tY_t - X_0Y_0 - \int_0^t Y_s dX_s - \int_o^t dX_sdY_s\]
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: \(\frac{\partial^2F}{\partial X \partial Y} = \frac{-1}{Y^2}\)
Putting it all back together: \[dF = \frac{1}{Y} dX + \frac{-X}{Y^2} dY + \frac{1}{2} \frac{2X}{Y^3} dY^2+ \frac{-1}{Y^2} dXdY\] Which we can re-write as: \[dF = d \left( \frac{X}{Y} \right) = \frac{X}{Y} \cdot \left( \frac{1}{X} dX - \frac{1}{Y} dY - \frac{1}{XY} dXdY + \frac{1}{Y^2} dY^2\right) \tag{3}\]
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.