Why would someone buy a way out-of-the-money call option that's expiring soon?

I'm a little late to this but I hope you get it. I think joebloggs is right this looks like an Implied Volitility play.

Here's a screenshot for Tesla far OTM calls as an example of this:

I did this today riding IV from about 123% to 156% on 1000 Strike puts and 3500 Calls expiring in 4 days and took profits on both at a combined 52% and even then the IV percentile was still low.

It's interesting you can use a Black-Scholes calculator to play around with this volatility spike affect and see just how great it can be on both sides of the option chain. They might not have even been looking for the underlying to move up period because even if it suddenly plummets, they might be able to sell those .05 cent contracts not far from a dollar. It's also important to note the contact size indicates a retail investors which even collectively is a poor indicator for macro trends.

I agree that a buy to close is a likely explanation of what happened. The buyer was willing to forego a few cents worth of time value in order to close out a covered call and perhaps "roll" the call further out (and perhaps up as well).

I think the best answer that doesn't make the buyer look like a moron is this.

Buyer had previously sold a covered call. They wanted to act on a different opportunity so they did a closing buy/write with a spread of a couple cents below asking for the stock, but it dipped a couple cents and the purchase of those options to close resolved at 4 cents due to lack of sellers.

It could be that the contracts were bought at cheaper prices such as $.01 earlier in the day.

What you see there with the bid and ask is the CURRENT bid and CURRENT ask. The high ask price means there is no current liquidity, as someone is quoting a very high ask price just in case someone really wants to trade that price. But as you said, no one would buy this with a better price on a closer strike price. The volume likely occurred at a different price than listed on the current ask.

Perhaps it was to close a short position. Suppose the seller had written the calls at some time in the past and maybe made a buck or two off of them. By buying the calls now they can close out the position and go away on vacation, or at least have one less thing they have to pay attention to. If they were covered calls, perhaps the buyer wants to sell the underlying and in order to do so has to get out of the calls.

The most likely explanation is that the calls are being bought as a part of a spread trade.

It doesn't have to be a super complex trade with a bunch of buys or sells. In fact, I bought a far out of the money option this morning in YHOO as a part of a simple vertical spread.

Like you said, it wouldn't make sense and wouldn't be worth it to buy that option by itself.

Out of the money options often have the biggest changes in value, when the stock moves upward.

This person could also gain, by the implied (underlying) volatility of the stock rising if it moves erratically to either side.

Still seems to be a very risky game, given only 4 days to expiry.

I suggest you look at many stocks' price history, especially around earnings announcements. It's certainly a gamble. But an 8 to 10% move on a surprise earning announcement isn't unheard of. If you look at the current price, the strike price, and the return that you'd get for just exceeding the strike by one dollar, you'll find in some cases a 20 to 1 return.

A real gambler would research and find companies that have had many earnings surprises in the past and isolate the options that make the most sense that are due to expire just a few days after the earnings announcement. I don't recommend that anyone actually do this, just suggesting that I understand the strategy.

Edit - Apple announced earnings. And, today, in pre-market trading, over an 8% move. The $550 calls closed before the announcement, trading under $2.