Generalizing VWAP and TWAP with Time Changes and Weighting Measures
Summary
The document asks whether VWAP and TWAP can be combined into a more general benchmark, suggesting an exponentially decayed volume-weighted price as one possible construction. The answer reframes the problem: TWAP can be viewed as VWAP after changing the time scale according to cumulative traded volume. This connects the two familiar benchmarks through their weighting of observations.
It then describes a broader family of benchmarks based on transforming time or transforming the price-volume measure. The appropriate transformation depends on the feature a benchmark is intended to capture, and the resulting measure can be interpreted as an expectation of price under a chosen weighting distribution. This is a conceptual framework rather than a validated execution strategy: the document supplies no empirical comparison, calibration guidance, or evidence that a particular weighting improves trading outcomes. It also does not give a complete list of mathematical requirements for benchmark measures.
Key ideas
- A cumulative-volume time change makes a time-weighted average interpretable as a volume-weighted average.
- A generalized benchmark can be built by choosing a transformation that changes how observations are weighted.
- VWAP can be understood as the expected price under a volume-at-price distribution.
- The useful weighting depends on the market feature the benchmark is intended to capture.
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Full text
# Is there any measure that is a non-trivial combination of VWAP and TWAP?
# Is there any measure that is a non-trivial combination of VWAP and TWAP?
Is there any measure that is a non-trivial combination of VWAP and TWAP? For example:
\begin{equation} \textrm{VTWAP} = \frac{\textrm{VWAP}+\textrm{TWAP}}{2} \end{equation}
I'm thinking about something like this:
\begin{equation} \textrm{VTWAP}_{\textrm{exp}}(\alpha,T) = \frac{\sum{P_i * V_i * e^{-i*\alpha}}}{\sum{V_i * e^{-i*\alpha}}} \end{equation}
where $P_i$ is the price at time $T-i+1$ and $V_i$ is the volume at time $T-i+1$.
Influence of past volumes is exponentially decayed with factor $\alpha$.
We can see that $\textrm{VTWAP}_{\textrm{exp}}(0,T)=\textrm{VWAP}(T)$.
I think that good point to start to analyse this problem is to find out types of existing TWAPs.
Second part of the question:
Are there any mathematical requirements or equations that measures like TWAP and VWAP should meet?
Something like that, but more advanced: $\textrm{VWAP}(T+1)=\textrm{VWAP}(T)$ for $V_T=0$ which state that there was no trade at time $T$.
## Answer by lehalle (score 7)
https://quant.stackexchange.com/a/3261
In fact if you make the time change $$t\rightarrow \int_{\tau\leq t} V_\tau d\tau$$ a TWAP is a VWAP.
So just define the FWAP associate to a transform F: (you should ask to F to be an adapted stochastic process if you want to use models) $$t\rightarrow \int_{\tau\leq t} F(\tau) d\tau$$
You will have a new benchmark.
The real question is "what do you want to capture?"
You can also see a VWAP as the expectation of a volume at price density $d\mu(P)$: $${\rm VWAP} = \mathbb{E}_\mu (P)$$
In such a case just define a GWAP (associating a measure to a measure) as: $${\rm GWAP} = \mathbb{E}_G(\mu) (P)$$
For $G$ transforming a measure into the uniform one over its support: a GWAP is a TWAP (and for G being identity, it is a VWAP).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.