The stochastic differential
\begin{align}
dS_t &= S_{t+dt} \;-\; S_t\\
dX_t &= X_{t+dt} \;-\; X_t\\
dW_t &= W_{t+dt} \;-\; W_t\\
df(W_t) &= f(W_{t+dt}) \;-\; f(W_t)\\
df(W_t,t) &= f(W_{t+dt}, t+dt) \;-\; f(W_t, t)
\end{align}
Now if $X_t = f(W_t, t)$, its stochastic differential $dX_t = f(X_t, t)$ and it follows that
$$df(X_t,t) = f(X_{t+dt}, t+dt) \;-\; f(X_t, t)$$
We can use Itô’s second lemma to deduce an expression for this stochastic differential.
Easy Itô’s Lemma (Itô’s second lemma for two variables)
A common form or model for the infinitesimal increment in the value of an asset is $$dX_t= a_tdt + b_tdW_t \tag{1}$$ This is a generalised SDE of the increment in asset $S(X_t,t)$ price over an infinitesimally small duration $dt$. It is somewhat logical by observation of historical stock prices in general, that a primitive asset’s price will be comprised of a time-trending deterministic mean $\mu S_t = a_t$ and a random volatile component $\sigma S_t = b_t$. Both $a$ and $b$ being functions of $W_t, t$. It is reasonable and necessary to assume that the SDE above is at least twice differentiable for a primitive asset such as a stock. The stock is then said to follow an Ito process driven by a single Brownian motion $W_t$ such that
\begin{align}df(X_t,t) = \partial_t f dt + \partial_x f dX_t + \frac{1}{2} \partial_{xx}^2 f (dX_t)^2 \tag{2} \end{align}
which is simply a Taylor expansion of $df$ up to second order of $dX_t$ but truncated at $dt$. By the axioms of Brownian motion, in the Taylor series expansion of $df$, terms containing $dt^\alpha$ in which $\alpha > 1$ are zero. This is the essence of Ito’s rules. Keep in mind that $dt$ is equivalent to the Brownian motion increment generally denoted as $h$, or as $h = \dfrac{t}{n}$ in a quadratic varation of Brownian motion comprised of $n$ Brownian motions. When applied to the model $(1)$ above, it might be easier to think of the lemma as the sum of three main terms of the second order Taylor expansion:
- partial of $f$ wrt $t$ times $dt$ (higher order partials are zero)
- partial of $f$ wrt $X_t$ times $dX_t$
- Half the second partial of $f$ wrt $X_t$ (the $W_t$ random variable) times $(dX_t)^2$
There are now two important substitutions to make, one for each infinitesimal Brownian increment. First for the $(dX_t)^2$ we must invoke Itô’s multiplication table
as $dt^2$ and $dtdW_t$ are both zero. Therefore, $$df(X_t,t) = \partial_t f dt + \partial_x f dX_t + \frac{1}{2} b_t^2 \partial_{xx}^2 f dt $$ where $X_t$ is only allowed to depend on $W_s$ for $s \leq t$.
For the second infinitesimal Brownian increment $dX$ we simply substitute the process, the model assumed for the incremantal price of the primitive asset $$df(X_t,t) = \mu X_t dt + \sigma X_t dW_t$$.
The model in $(1)$ above can also be obtained directly by assuming a stock price model $X_t(W_t, t)$ based on geometric Brownian motion where stochastic process $X_t$ is a martingale. Applying Ito’s second lemma to the increment $dX_t$ over infinitessimal time increment $dt$ gives the SDE for the increment as in $(1)$ above.
Non-primitive, Non-linear and non-differentiable assets. Derivatives and Replicating Portfolios
Consider the pay-off of a European call. The very definition of this derivative contract suggests that if we had a suitable mechanism or machine (if such could be invented), we could replace or replicate the derivative instrument with the primitive stock $S$ and cash of some amount greater or less than $K$. Such a machine would of course need to be fast and flexible such that as the stock price changes, the amount of stock $S$ held and the amount of cash, multiples or fractions of $K^+$ or $K^-$ can change instantaneously with the stock price fluctuation. It has been shown that the stock $S$ can be transformed as $S=Ke^x$. This would be necessary for such an imaginary machine to be even worth having and to avoid complete financial ruin as soon as it is switched on. Why? Because if this were not the case, there would be an unfair opportunity for any one in possession of the real derivative contract to profit from the machine (and invariably it’s onwer’s) sluggishness by making profitable trades in the time lag during which the machine owner is perhaps unaware that the market price of the contract has changed. On the contrary, if the machine owner was a genius from another world, there is every possibility they could do better than the owner of the real asset if the owner of the real derivative asset trades using a slow paper system. In the days when computers were very slow and before the arrival of the personal computer, such a genius might have existed. But in today’s world where millions have been spent on infrastructure just to achieve a few milliseconds of advantage by traders with their computers located on the west coast of England just to reach the New York exchanges ahead of someone in London, Paris or Frankfurt, such genius opportunities are practically non-existent.
The unfair scenarios described above are what is often known as arbitrage. Any stock and cash substitute for a derivative must be completely devoid of arbitrage opportunities. The very nature of derivative contracts makes them in their direct mathematical form non-differentiable. However, if we can substitute a cash and stock combination then we may be able to create a model for the derivative such that we can create a useful mathematical model of the derivative. This is the essence of replicating portfolios.
So let us just assume that we have a replicating portfolio that mimicks a derivative contract but which is made up entirely of underlying stock and cash. It turns out that by applying Ito’s lemma to the replicating portfolio, making identical $dX_t$ and $df(X_t, t)$ substitutions and grouping the (riskless $dt$ and risky $dW_t$ terms) the model for replicating portfolio yields a PDE. The risk-neutral assumption of the portfolio (assummed by constantly relancing the amount of $S$ and cash held means that the risky $dW$ terms all vanish in this replicating model. The stochastic differential equation of the underlying primitive asset has given way to a partial differential equation of its derivative asset. This PDE was first discovered by Black and Scholes.