Daniel Eliasson

The other day, Stephanie and I went to see District 9, and damn, this movie is excellent. The story in short: one day a large spaceship comes to earth, and stops over Johannesburg. The ship contains almost 2 million starving aliens that are brought to a refugee camp, which quickly turns into a township that is given the name District 9. A company is given the job of transferring the aliens from District 9 to a sort of concentration camp conveniently located far outside of Johannesburg. This operation is headed up by Wikus van de Merwe, who quickly becomes infected with an alien virus and finds himself on the run from his former comrades.

The charm of this movie is that it is so unmistakably South African. Partially filmed in a documentary style, we get to see anti-hero Wikus explain the procedures of the eviction to the cameras, in his thick Afrikaans accent, liberally sprinkled with the word “fok”.

There is a political tone to the movie, the most obvious being, of course, the references to apartheid, with the aliens being kept separate from the humans, living in shacks. The title, Disctrict 9, is of course a reference to District Six of Cape Town, a coloured area that was declared white-only in 1966 and from which the residents were evicted during the late 60s to the 80s.

But there is also the xenophobia of South African blacks; many township residents haven’t taken too kindly to immigrants from other African countries, and violence has been ripe.

Other aspects of South African culture that feature in the movie is muti, and of course Nigerian gangs.

But despite the political undercurrent, the movie does not preach, not even a little bit, but instead just gets on its merry sci-fi way, and there is a lot of lekker fighting and special effects to be seen, as well as many laughs. “Don’t point your fokken tentacles at me!”

Highly recommended.

For some reason, when I have too much to do, I find ways to do semi-productive things that aren’t the things I should be doing. Some people will know this phenomenon as tentastädning or similar.

In my case, I took some back-of-a-napkin calculations that I made the other day and turned them into two little calculators for personal finance stuff.

The first one lets you calculate the annuity on a loan, given the principal, interest and maturity, or to calculate the maximum principal you can get for a certain annuity.

The second calculator lets you see what your monthly savings would add up to by saving for a specified period, and then holding the money in the bank during another period. You can also compare two different strategies for saving to see the difference. As an example, it might be more lucrative to save for a shorter period but start early, than to save during a longer period later in life.

Anyway, if anyone wants to try them out, they’re available at http://financecalcs.danieleliasson.com/

As we saw in the last post, a stock option can be characterised by the following:

• The underlying stock.
• The maturity, being the day on which the option expires.
• The strike price, being the price at which the underlying can be bought (call option) or sold (put option).

Recall that a European option is one which can be exercised only at the maturity, while an American option can be exercised at any time up to and including the maturity.

A game option generalises the American option. It has all the characteristics of the American option, but it is also possible for the writer (seller) of the option to terminate it at any time up to maturity. If the writer terminates the option, he must pay to the holder of the option the same amount as if the holder had exercised the option, plus a penalty.

As an example, consider a game call option on SASOL, with maturity 1 October 2009, strike price R300, and the fixed penalty R10. Let’s say I’ve sold you this option. On 20 September, SASOL trades for R315. If you exercise your option, you will make a profit of R15, since you’re buying a stock worth R315 for only R300. If, on the other hand, I decide to terminate the option, you will get R15 + R10 = R25, since I must pay you an extra penalty.

When would the holder of an option want to exercise? When he believes that the payoff from exercising right now is bigger than the expected winnings of holding on to the option.

Similarly, the writer of a game option will terminate the option if he believes that the cost of terminating the option is less than the cost of letting it continue, and be exercised later.

Imagine us now, me the writer and you the holder of abovementioned game option. Every morning we pick up our newspapers, and see what price SASOL is trading for. This tells you how much money you could make by exercising the option, and it tells me how much it would cost me to terminate it. Then we do this:

• I will ask myself if the cost of terminating the option today is less than what I expect I’ll have to pay later. If I believe so, I will terminate the option, and pay you.
• You ask yourself if the payoff from exercising the option today is higher than what you expect you’ll get later. If so, you’ll exercise the option, and receive payment.

Now, you would prefer that I terminate the option, rather than you exercise it, since you receive more money that way. Conversely, I’d rather have you exercise than have to terminate myself. Still, I’d rather terminate today than have you exercise later, if the stock price goes up further. Conversely, you’ll rather exercise if you think the stock will drop in value.

This can be seen as a zero-sum game, where you and I both try to find the best time to “stop” the game. These stochastic games were originally described by E. B. Dynkin in 1969, and are subsequently called Dynkin games. This is also where the name game option comes from.

So what is it that I’ve been working on? The main result of my work is a Monte-Carlo algorithm to determine the fair price of a game option. Essentially, I simulate a large number of ways the stock may move, and determine what the optimal stopping times are for the writer and holder of a game option. This gives a price for the game option in that specific scenario. These prices are then averaged over the large number of scenarios, and out comes an approximation of the true fair price.

To visualise, imagine that a stock begins at R100 on 1 Januari, and we follow it for a year. The following picture shows 20 simulated paths that the stock might take. The simulations were done using the famous Black-Scholes model.

Click for larger image.

Now, imagine a game call option on a stock, with strike price slightly above R100, and penalty R10. The next picture shows the termination and exercise payoffs of that option for one simulated stock price path. The optimal exercise moment for the holder of the option is marked out. Note that whenever the stock trades for less than the strike price, the option payoff is zero, since it makes no sense to exercise at a loss. This shows how an option might be worthless, but once the premium is paid, you cannot actually lose any more money by buying an option.

Click for larger image.