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This depends on how solid a trader you are and your margin requirements. But, 7k-8k is minimum I typically recommend. It doesn't really change when it becomes more volatile because you can just change your risk/reward. It is actually easier to trade with a tight risk then normal: so if you know what you are doing you can often risk very little. In normal/lower volatility markets, playing those games is more difficult because the market won't move enough when you are right. Even so, the cost of making mistakes is higher and so it is easier to lose an entire account during volatile times. The market is less forgiving of mistakes. As an aside, this goes for discretionary strategies only: quantitative strategies typically take more risk per trade and so do become riskier to run in abnormal markets but depends on the strategy.
There is another ballpark figure I use, and that is to assume you need to risk $x per day and don't want to risk more then 2% to say 7% per day. Assuming your $x risk is say $700 (I think regardless of stop loss size this is near the minimum) you would have $700/.03 = ~23k. I like also looking at 12 losing days in a row and what the capital requirement is for that, it is 8k in this case. If you think you can trade consistently and you have more then 12 losing days in a row then that would be very abnormal. But, I might add most traders cannot do this-- the large traders hold for longer periods of time but if you think you can do it then 8k is enough to give you a decent chance. There is one final way to do this which is to assume a virtual account and assume 25k is reasonable and assume say a 40% drawdown would be the max you anticipate to take: that yields 10k in this case.
So, to recap, if you want the best chance of success 25k. If you are experienced and you go with a broker offering day trading margins 7k-10k is enough to take a shot. This doesn't mean you wont lose the account: it just means, that provided no mistakes are made it might be enough.
If you are strong intraday guy, you need
($500 for the intraday margin) + ($250 * 20 = $5000) = $5500
If you are a strong enough guy, you need
Just 2x more, i.e. $10000
If you are a middle intraday guy, you need
2x more again, i.e. $20000
