Moving from discrete sum of changes to continuous integral of local covariance - how is this done?

by MatlabNoob   Last Updated March 20, 2017 14:19 PM

So, I'm trying to derive a specific relationship about the relationship between forwards and futures. The expression comes from the paper "The relationship between forward and futures prices" written 1981 by Cox, Ingersoll and Ross if any one is interested. Basically, they go from

$ -\frac{1}{B(t)}\sum_{i=t}^{T-1} \left[f(i+1) -f(i)\right]\left[\frac{B(i)}{B(i+1)} -1 \right]$ (1)

directly to

$\frac{1}{B(t)}\int_{t}^{T} f(w)\text{Cov}[f(w),B(w)]dw $ (2),

by assuming a continuous-state where the covariance refers to the local covariance of the percentage change of B and f.

I can easily write (1) as

$ \frac{1}{B(t)}\sum_{i=t}^{T-1} f(i+1) \left[\frac{f(i+1) -f(i)}{f(i+1)}\right]\left[\frac{B(i)-B(i+1)}{B(i+1)} \right]$ (3)

which, with

$Cov(X,Y) = \frac{1}{n^2}\sum_{i=1}^{n-1}\sum_{j>i}^{n} (x_i - x_j)(y_i - y_j)$

will get me: (right??)

$-\frac{\text{Cov}_{t,T}[f,B(t)]}{B(t)}\int_{t}^{T} f(w)dw $ (4)

but from here? I know

$\text{Cov}(f,B) \int f(w) dw = \int \text{Cov}(f,B) f(w) dw $, because $\text{Cov}(f,B)$ is a constant (and f,B are vectors of timeseries). In (2) however, $\text{Cov}(f(w),B(w))$ means the covariance of $f(w)$ and $B(w)$, where $w$ will change with $dw$ from $t$ to $T$. But, at a discrete point in time (t=w), $f(w)$ and $B(w)$ are only two values, not a time-series!

Where do I go wrong when I go from (3) to (4), i.e HOW do Cox, Ingersoll and Ross go from (1) to (2)? ?

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