Endpoint Choices in Discounted Futures Cash-Flow Integrals
Summary
The document addresses whether a discrete discounted cash-flow sum for a futures contract should use the discount factor at the beginning or end of each interval when passing to a continuous-time expression. One response relates the choice of sampling point to stochastic integration: Itô sums use values known at the left endpoint, while a Stratonovich integral uses midpoint values. Another response argues that the apparently later discount factor can still be known at the left endpoint under the book’s indexing and filtration assumptions.
The central lesson is to check measurability and time indexing before identifying a sum as an Itô integral. If the discount factor labeled at the later time is already determined by information available at the earlier time, the notation does not imply right-endpoint sampling. The discussion is brief and leaves the precise discount-factor convention dependent on the model setup; it provides no numerical example or independent derivation of the futures cash-flow valuation formula.
Key ideas
- Itô integrals are defined through left-endpoint adapted sums, while Stratonovich sums use midpoints.
- A time label alone does not establish when a discount factor becomes known.
- Under the cited indexing, the later-labeled discount factor may be measurable at the earlier endpoint.
- Check the filtration and rate convention before interpreting the sum as a right-endpoint integral.
Tags
Full text
# Value of cash flow for a future in Shreve's book
# Value of cash flow for a future in Shreve's book
In Shreve's book, the value of cash flow for a future of discrete case is
$$\dfrac{1}{D(t)}E\Big[\sum\limits_{j=k}^{n-1}D(t_{j+1})(\textrm{Fut}_S(t_{j+1},T)-\textrm{Fut}_S(t_j,T))\Big|\mathcal{F}(t)\Big]$$ The continuous version is
$$\dfrac{1}{D(t)}E\Big[\int_t^T D(u)\textrm{d} \textrm{Fut}_S(u,T) \Big|\mathcal{F}(t)\Big]$$
But you know that Ito integral chooses the left-hand endpoint, we should replace $D(t_{j+1})$ by $D(t_j)$ in the first equation, the version in Shreve's book is actually the right-hand endpoint, but the author always regards as a Ito integral. So where is my misunderstanding? Here $D(t)$ is discounted factor.
## Answer by user16651 (score 1)
https://quant.stackexchange.com/a/31768
Let $\mathcal {V} =\mathcal {V}(t,T)$ be the class of functions $$f(t,\omega):[0,\infty)\times\Omega\to\mathbb{R}$$ such that
- $(t,\omega)\to f(t,\omega)$ is $\mathcal{B}\times\mathcal{F}$ where $\mathcal{B}$ denotes the Borel algebra on $[0,\infty)$.
- $f(t,\omega)$ is $\mathcal{F}_t$ adapted.
- $\mathbb{E}\left[\int_{t}^{T}f^2(s,\omega)ds\right]<\infty$
Suppose $f\in\mathcal V(t,T)$ and that $t\to f(t,\omega)$ is continuous . Let $I=\{u_i\}_{i=0}^{n}$ is a sequence of partitions of $[t,T]$, Indeed $t=u_0<u_1<\cdots<u_n=T$ .By definition of Ito integral, we have
$$\int\limits_t^T f(u,\omega)dW(u,\omega)=\lim_{\Delta u_j\to0}\sum_{j=0}^{n-1}f(u_j,\omega)(\,W(u_{j+1},\omega)-W(u_{j},\omega)\,)\qquad,\quad\text{ in }\, L^2(P).$$
Similarly we define the Stratonovich integral of $f$ by
$$\int\limits_t^T f(s,\omega)\circ dW(u,\omega)=\lim_{\Delta u_j\to0}\sum_{j=0}^{n-1}f(u_j^*,\omega)(\,W(u_{j+1},\omega)-W(u_{j},\omega)\,)$$
where $u_j^*=\frac12(u_j+u_{j+1}),$ whenever the limit exists in $L^2(P)$. In general these integrals are different.
I think the value of cash flow for a future of discrete case is
$$\dfrac{1}{D(t)}\mathbb{E}\left[\sum\limits_{j=k}^{n-1}D(t_{j})(\,\textrm{Fut}_S(t_{j+1},T)-\textrm{Fut}_S(t_j,T)\,)\Big|\mathcal{F}_t\right]$$
thus the continuous version is
$$\dfrac{1}{D(t)}\mathbb{E}\Big[\int_t^T D(u)\textrm{d} \textrm{Fut}_S(u,T) \Big|\mathcal{F}(t)\Big]$$
## Answer by zer0hedge (score 1)
https://quant.stackexchange.com/a/32821
I believe the confusion is due to indexes. $D(t_{j+1})$ is $\mathcal{F}(t_j)$ measurable, so it is known on the left-hand endpoint. If you substitute the formula (6.2.2) for $D(t_{j+1})$:
$$D(t_{j+1}) = \frac{1}{(1+R(t_0))(1+R(t_1)+ \cdots + (1+R(t_j))}$$
you will recognize an Ito integral:
$$\dfrac{1}{D(t)}E\Big[\sum\limits_{j=k}^{n-1}\frac{1}{(1+R(t_0))+ \cdots + (1+R(t_j))}(\textrm{Fut}_S(t_{j+1},T)-\textrm{Fut}_S(t_j,T))\Big|\mathcal{F}(t)\Big]$$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.