Quadratic Variation
Quadratic Variation is a central concept in option pricing. Quadratic Variation is the foundation for much of my intuition. By adjusting the degree of “determinism” of Quadratic Variation we can move along a spectrum of different option pricing models.
This post is a brief introduction to the conceptual idea of Quadratic Variation with a minimal (and very handwavy) mathematical discussion of its properties.
But before we move on to Quadratic Variation, we should first understand what we mean by Variation in general.
What exactly is Variation?
Intuitively we know what it means for things to “vary”, it means they change. If we think a bit deeper, variation as a concept has a few basic properties.
- That the quantity/process/function (denoted as $f$ below) we are measuring moves by some amount
- We care about the magnitude of the move and not its direction
- It is implied that “variation” occurs over a period of time say a minute or a day
Now that we have some conceptual idea of the properties of Variation we can begin to define it more formally
Definition
Imagine some price process (though it need not be) over some period of time from $0$ to $T$ or stated more succinctly
$f(t) \in \mathbb{R} \> \forall t \in [0,T]$
Now take the interval $[0,T]$ and divide it into n different subintervals. It’s often easier to think about them as being the same size as it does lead to nicer interpretations later.
So you have
$0 = t_{0} < t_{1} < … < t_{n} = T$
Consider the evolution of our process $f(t)$ over these different values of $t_{i}$ that partition our time interval.
If we wanted a measure of how much variation occurred over the period it would naturally be some function of the differences in $f(t)$
$f(t_{i}) – f(t_{i-1})$
In finance, it’s often easier to think of changes in price or level of the process, so the amount of variation in a price series over our time interval $[0,T]$ can be expressed as
$$s_{\alpha}(f, p) = \sum_{i=1,n} \left | f(t_{i}) – f(t_{i-1}) \right| ^{\alpha} $$
Where
- $f$ is our function
- $p$ is a partition of $T$ for example it’s the etnire set of all t’s. $ p_{n} = { t_{i} } > \forall i = 0 … n $
- $\alpha$ is the power of our absolute difference
Let’s take a step back and think about exactly what we’re doing here, I discussed the $p$ partition piece a bit already, but it would make sense to make it more explicit.
$p_{n}$ is only one such partition, one that has ${n}$ subdivisions. Different values of $n$ can exist such as $p_{4}$ cuts it into fourths, $p_{5}$ into fifths, etc.
Therefore the value $s_{\alpha}(f, p)$ depends not only on $f$ the random stochastic process. But also how we choose to observe it. Therefore, for different choices of $n$ and by extension $p_{n}$ the value of $s_{\alpha}(f, p)$ will change. In general $s_{\alpha}(f, p)$ can be thought of as a random variable.
This lack of “uniqueness” presents a bit of a problem. For any given $[0,T]$ we can have infinitely many valid values of “Variation” depending on what we choose for $n$ this makes it quite difficult to compare different stochastic processes or make any meaningful progress in examining them from this point of view.
To remedy this (but mostly because using calculus is so mathematically convenient) we should try to make $s_{\alpha}(f, p)$ partition independent. If we let take the limit as $n \to \infty$ and have finer and finer partitions we end up with
$$ v_{\alpha} = \lim_{n \to \infty} {s_{\alpha}(f, p)} = \lim_{n \to \infty} \sum_{i=1,n} \left | f(t_{i}) – f(t_{i-1}) \right| ^{\alpha} $$
What is $\alpha$ ?
If the limit exist for $v_{\alpha}(f)$ then your choice of $\alpha$ gives you different levels of Variation
$\alpha$ = 1 is the first variation (linear variation)
$\alpha$ = 2 is the second (quadratic variation) and so on
Quadratic Variation is important for us because of its relation to Brownian Motion. In particular, Brownian Motion is a diffusion with deterministic volatility. Under these constraints, it can be completely characterized by its Variation
Suppose that for our given process $f$ the logrithm is given by $y = ln(f(..))$
$dy = \mu_{y} dt + \sigma_{y} dz$
The quadratic variation of $y = ln(f)$ is given by
$$ \nu_{2} = \lim_{n \to \infty} \sum_{i=1,n} | y(t_{i}) – y(t_{i-1}) | ^{2} = \lim_{n \to \infty} \sum | \Delta y_{i} | ^{2} $$
We can take advantage of the fact that we are working with a diffusion with deterministic volatility i.e. $\sigma_{y}$ is constant
$$ \nu_{2} = \lim_{n \to \infty} \sum_{i=1,n} | \Delta y_{i} | ^{2} = \lim_{n \to \infty} \sum_{i=1,n} \sigma^{2} \Delta t $$
This equality is due to the fact that we don’t care about the sign of the move, and since the brownian process has a mean of zero (and constatn variance) we have that the square of its absolute difference $| \Delta y_{i} | ^{2} $ is the variance over the period $[i-1, i]$
Now in the infinite limit
$$ \nu_{2} = \lim_{n \to \infty} \sum_{i=1,n} \sigma^{2} \Delta t = \int_{0}^{T} \sigma(\mu)^{2} d\mu = \hat{\sigma}^{2} T $$
$\sigma(\mu)^{2}$ is the instantaneous variance of the process and $\hat{\sigma}^{2} $ is the root-mean-squared variance and $\hat{\sigma}$ is the Implied or Black Volatility. The thing to keep in mind is that in classical diffusion case with a deterministic volatility it’s Quadratic Variation $ \nu_{2} $ is directly linked to its root-mean-squared volatility.
Despite it’s handwavy nature this “proof” is sufficient for our conceptual understanding. For an actual proof that is less handwavy, I suggest any number of the books on Stochastic Calculus.
Intuition
Now that we’ve talked about Variation a bit, it would be wise to take a step back and examine why we went through all that trouble. I’ve said that Quadratic Variation is the “glue” that underlies all models. Or at the very least, is a useful conceptual device for intuiting their properties. But why is that so?
One general idea is the level of “determinism” of the Quadratic Variation $\nu_{2}$ is related to the degree of replication that is assumed in any given “model”. In almost all cases a deterministic Quadratic Variation implies the ability for perfect replication.
We know that there are a number of conditions that must be met in order for perfect replication to be possible and by relaxing certain parts of the Black Scholes assumptions the replicating framework falls apart quite quickly. If we don’t hedge continuously, if there are jumps, if the volatility isn’t deterministic but rather stochastic, if we don’t know the true volatility, if the market is incomplete, etc. The point is, that all of these affect the Quadratic Variation of the process. If any of these are true, the Quadratic Variation is no longer deterministic. So, we can distill all the assumption-breaking events into one concept.
In the perfect Black-Scholes world, there are no “lucky” or “unlucky” paths. Since the Quadratic Variation is the same for all paths, there cannot be paths that are “unexpected” as a rule. But in other “non-Black” worlds where “lucky” paths are the norm Quadratic Variation itself becomes a stochastic quantity. When Quadratic Variation itself becomes stochastic we make the jump into the world of Volatility Smiles.