As I explained in another answer, we can not meaningfully use measures of dormancy from P2P currencies in Quantity Theory of Money comparisons relative to GDP (or in Visa's case total
Q over the period).
I also explained in that linked answer why it is unlikely that Bitcoin would obtain the same velocity multiple of 8 per quarter as with fiat.
Expounding, in order to get that many round-trips per quarter, requires the network effects of a wide diversity of available goods and services which only comes with economies-of-scale. So it is a chicken-and-egg dilemma.
Based on that, the true quarterly velocity of Bitcoin may be on the order of a intuitive guess of 1, so with 10% of the a $200bln online economy (as proposed by the prior answer), with 21 millions coins, I calculate:
$200bln × 0.1 ÷ (4 × 1 x 21 million) = $238
A lower velocity means a higher FX price for Bitcoins, but a low velocity also probably means low adoption for transacting, thus I am skeptical of the 10% assumption.
One could make a wild guess and claim either 1% share with 1 velocity or 10% share with 10 velocity thus:
$200bln × 0.01 ÷ (4 × 1 x 21 million) = $23.8
$200bln × 0.1 ÷ (4 × 10 x 21 million) = $23.8
Note we are dividing by M × V (inverted the Quantity Theory of Money equation), because we want the exchange price in dollars, not the price of goods and services in bitcoin.
It should be noted the Quantity Theory of Money is probably not the main factor that will drive the Bitcoin's exchange price, because Bitcoin is predominantly a speculation system and subject to volatility manipulation. So (irrational) confidence will likely drive the value of Bitcoin (as is often the case with illiquid small cap “investments” or pseudo-pyramid schemes). In a speculative frenzy, the marketcap of Bitcoin could rise well above any realistic estimate of its Quantity Theory of Money value.