Local Covariance in the Forward–Futures Price Relationship
Summary
The document asks how Cox, Ingersoll, and Ross move from a discrete sum involving changes in forward prices and bond discount factors to a continuous-time integral involving local covariance. The author rewrites the summand in terms of relative changes and tries to connect it to a covariance calculated across a time series. This raises a question about what covariance means when the expression uses values that vary with time inside an integral.
The author is specifically concerned that a covariance treated as one constant produces an extra time-span factor and does not match the cited integral. The source is described as an empirical relationship between forward and futures prices, not the better-known interest-rate model. The document contains no resolution or derivation, and its attempted transformation is presented as a question rather than a proven result. It therefore identifies a conceptual distinction to investigate: an overall sample covariance across observations is not automatically the same object as local covariance of percentage changes at each time.
Key ideas
- The question concerns a discrete-to-continuous step in a forward–futures pricing derivation.
- The discrete expression combines forward-price changes with changes in bond discount factors.
- The author questions whether a sample covariance can represent time-varying local covariance inside an integral.
- The proposed algebra is not confirmed, and the document provides no answer to the derivation question.
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# 33139
# Deriving Cox, Ingersoll and Ross expression for the relationship between forwards and futures, how do they conclude a specific step?
I'm trying to derive a specific relationship about the relationship between forwards and futures from "The relationship between forward and futures prices", written 1981 by Cox, Ingersoll and Ross (not the interest rate model!). The relationship has basically been empirically debunked, but nonetheless provides relevant info to me. I have issues understanding some steps in the proof. Basically, Cox, Ingersoll and Ross go from:
$$ -\frac{1}{B(t)}\sum_{i=t}^{T-1} \left[f(i+1) -f(i)\right]\left[\frac{B(i)}{B(i+1)} -1 \right] \tag 1$$
directly to
$$\frac{1}{B(t)}\int_{t}^{T} f(w)\text{Cov}[\tilde{f}(w),\tilde{B}(w)]dw \tag 2$$
by assuming a continuous-state where the covariance of $\tilde{f}$ and $\tilde{B}$ refers to the local covariance of the percentage change of $B$ and $f$.
I can easily write $(1)$ as:
$$ \frac{1}{B(t)}\sum_{i=t}^{T-1} f(i+1) \left[\frac{f(i+1) -f(i)}{f(i+1)}\right]\left[\frac{B(i)-B(i+1)}{B(i+1)} \right] \tag 3$$
which, with
$$\text{Cov}(X,Y) = \frac{1}{n^2}\sum_{i=1}^{n-1}\sum_{j>i}^{n} (x_i - x_j)(y_i - y_j)$$
will get me to: (right??)
$$-\frac{T^2\text{Cov}_{t,T}[\tilde{f},\tilde{B}]}{B(t)}\int_{t}^{T} f(w)dw \tag 4$$
But, this is not very close to $(2)$. For example, my covariance is a constant, and I do not know how to get $T^2$ to disappear. What I do 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 $(2)$?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.