It is a common misconception that "shorting stock" means a short sale of shares. "Shorting stock" simply means to take a negative position on the stock price, which can be by selling shares "short" (intending to repay the shares later after the price falls), or as Le Chiffre did, by purchasing put options. Either can be called "going short" on a stock. A profit is realized if the price falls later on. If the price of the stock actually goes up, though, the investor loses money.
A put option gives the buyer the right, but not the obligation, to sell a stock at a defined "strike price" to the party who sold the put options. By definition, the strike price is lower than the current price; the financial instrument would otherwise never be sold in the first place. The buyer initially pays the difference between the current market price and the strike price, plus a premium, hence Le Chiffre's need to get a new client and more funds: "I have the money, so short another million shares." Le Chiffre expected to exercise his put options and regain what he paid for them, plus much more, once the hired henchman destroyed Skyfleet's prototype airliner. When he was told the "ellipsis" password expires in 36 hours, he remarked, "That's all the time I have anyway." With Skyfleet's coincidental unveiling of a new airliner on the same day the options expired, Le Chiffre was cutting things very close.
Since we only know Le Chiffre's final losses, there are infinite possible combinations of shares, strike prices and market prices. As an example, let's say that Obanno's money was $10 million out of the $101,206,000 lost. The next Q&A says that Obanno gave $100 million to Le Chiffre, but the actual amount is never mentioned, and clearly the $100 million loss is not just Obanno's money. We'll also say that Skyfleet stock's market price was precisely $30, and that Le Chiffre bought put options for a total of 10 million shares, at a strike price of $20 with a premium of $0.206 per share. (This is not an impossible share volume. DAL, for example, has over 845 million shares outstanding.) The less likely the seller thinks the strike price will be reached, the less of a premium will be asked. Indeed, Le Chiffre's own broker warned, "Nobody expects this stock to go anywhere but up," so the premium on these puts would be very small. Le Chiffre would initially have to pay the difference between the market price and strike price, plus the premium: $102,206,000. As Skyfleet faced bankruptcy, Le Chiffre would buy 10 million shares on the open market, say $1 per share, then exercise his option to sell them at $20: a total of $190 million, less his $101,206,000 to buy the puts, for a net profit of $88,796,000. Quite a "reasonable rate of return" for a few days! If the market price had been $25, and the strike price $15, Le Chiffre would have netted $38,794,000. (To keep the scenario simple, we're using a price difference of $10 per share and 10 million shares.)
But Le Chiffre's plan failed, and because Skyfleet stock's price continued to stay above the strike price, there was no point to Le Chiffre exercising the puts. In those final hours, no one would buy them for any amount, so they "expired worthless," and Le Chiffre lost all the money he put into the puts (pun intended). It is very common for investors to purchase options that lose all value, and thus lose the money invested into them.
Le Chiffre probably preferred put options to selling shares short because of the odds: options allowed him to set a maximum loss, in the unlikely yet not impossible event that his scheme failed. The irony is that despite his mathematical genius, he was a poor investor. A short sale would have been unlikely to lose so much money in the course of a few days. Once the markets opened, Le Chiffre could have started buying to cover, with every dollar increase in price being a loss of "only" $10 million. On the other hand, the put options were a necessary plot device so Le Chiffre could lose a great deal of money with no recourse. Now, had Le Chiffre tried selling shares short, he'd have still needed a massive amount of money to start. No brokerage house would let a client incur such a huge potential liability without collateral, in case the market moved against the client.
It should be noted that Le Chiffre's broker was extremely incompetent. "I'm sorry, I'm not sure yet how much you've lost." A professional wouldn't have been caught off guard by an important and prominent client. He would have carefully monitored such a large position and be able to report on a moment's notice, especially with the ease that computerized records, with the cost of all his transactions, can be summoned. Perhaps it was a plot device for Le Chiffre to demonstrate his mathematical genius, though it could have been written with the broker about to disclose the final losses, and Le Chiffre interrupting with the amount.