Daniel Eliasson

Now that the report of my holiday is over (a mere month and a half after the holiday itself was…), it might be fitting to turn to more academic topics. The topic of my thesis, for instance, are Game Contingent Claims (GCCs), a type of financial derivative instrument. Let me try to explain what they are, and what it is I’m doing with them. I’ll assume that everyone is familiar with the concept of stock shares, and know that these are traded on exchanges, so we can move straight on to looking at stock options. I will start by explaining briefly what stock options are, and the basic idea of how to price them.

Introducing stock options

An option is a contract that gives the holder a right, but not an obligation, to buy or sell an underlying stock at a pre-specified price, called the strike price. An option that gives the right to sell the underlying is called a put option, whilst one that gives the right to buy the underlying is called a call option. Using your right to buy or sell is known as exercising the option.

A European option is one where the right to exercise the option is given on a single specific time, the maturity. As an example, one could have a call option in SASOL with a strike price of R290 and maturity on 1 October 2009. If you buy this option, it means that on 1 October, you have the right to buy a share in SASOL for R290. If SASOL happens to be trading at R300 that day, you could buy the share for R290, then sell it for R300 and net yourself a profit of R300-R290 = R10. If, on the other hand, SASOL is trading at a low price of R280, you have no reason to exercise your option; you could buy the share for a lower price on the market. In this case, you’d just leave the option to expire.

This latter part shows the attraction of an option: even if the price of the underlying moves against you, you don’t lose. The option gives you the ability to profit from a movement in the right direction, without you losing on a movement in the wrong direction. Of course, such a sweet deal isn’t free, you have to pay to obtain the option in the first place. The price of the option is called the premium, and that is the amount of money you stand to lose if your option expires worthless.

An option that would give profit if exercised right now is said to be in the money, whilst one that would not give a profit is out of the money. If the option would just break even (underlying price = strike price), the option is at the money.

An American option works like a European option, except it can be exercised at any time up to and including the maturity date. It’s easy to see that such an option must be worth at least as much as the corresponding European option, since it has more opportunities than the European one does. As an example, let’s assume you bought an American put option on SASOL with strike price R290 and maturity 1 October 2009. This is an option that allows you to sell one share of SASOL stock for R290 at any time of your choosing up to and including 1 October.

The following section shows the basics of so called risk-neutral valuation, and is a bit more mathematical. It’s not necessary to fully understand this section, but it might be interesting to at least read through it once. Then again, it might not be. It interests me, anyway, if that says anything.

Option pricing

What is the fair premium to pay for a European option? Let’s say you sell me a European call option on SASOL with strike price R290. You are now obligated to sell me a share of SASOL stock for R290 on 1 October if I should decide to exercise my option. This means that you have a liability towards me which is associated with some risk. The fair price to ask me for this option would be the amount of money you need to remove this risk entirely. The process of offsetting your risk in a position by means of other positions is called hedging.

How will you hedge the option? One way would be to simply buy a share of the underlying. In that case, if I exercise my option, you can simply hand over the share. On the other hand, if I do not exercise my option, you’ll be left holding the share. This is not entirely ideal, because I will want to exercise if the stock price rises, since I will then profit, but I’ll not want to exercise if the price drops. In other words, if the stock becomes more valuable, I’ll take it from you, but if it instead becomes less valuable, you get to keep it. The maximum loss you make is the price of the stock when you buy it, since you might have to give it to me. But if you ask me to pay you the price of the stock for my option, I would obviously rather just buy the stock itself.

The solution is to buy stock, but not a whole share. What you really want is to remove the risk entirely from your side. In other words, you want to know from the start precisely what your costs are. Let us consider an example.

Today the price of SASOL is R300, and on 1 October one of two different situations can occur: either the stock price goes up to R320, or it drops to R280. You set up the following portfolio:

• Short one call option, worth .
• Long shares of SASOL, worth .

(Notation: a short position is one where you’ve sold the instrument, a long is one where you’ve bought it.) Since you sold the call and bought the stock, your portfolio is worth . Let us now consider the value of the portfolio in each of the two possible 1 October.

If the share price rises to R320:

• You have the money you got from selling the option, .
• The call option is exercised, and you must pay .
• You are still long shares of SASOL, which are now worth .

In total, your portfolio is now worth .

If instead the share price drops to R280:

• You have the money you got from selling the option, .
• The option expires without exercise, you pay nothing.
• Your shares are now worth .

In total, this position is worth .

As previously stated, you’d like to remove your risk entirely, so you would like to choose and in such a way as to make the portfolios at the end worth the same amount, regardless of whether the stock goes up or comes down in price. Further, since we said that the fair price is the cost for you to set up your hedge, your original portfolio must be worth zero, or in other words, the price you charge me for the option is the price you need to pay for your shares of SASOL. This gives us two equations:

and . Solving this equation system gives

and . In other words, by buying half a share of SASOL for R150 and charging me that same price for the option, you know that no matter what, come 1 October, you can rest assured that your position closes without losing you any money.

Somewhat surprisingly, the price of the option does not in any way depend on how likely the two different scenarios are. One would maybe expect that the option should be more expensive if the stock is expected to go up with 90 % probability than if the stock is expected to go up with only 40 % probability. It turns out that these expectations are already taken into account in the current price of the stock itself, and must not be considered when pricing the option.

During our trip, we had the opportunity to cross some borders:

1. South Africa/Botswana
2. Botswana/Zambia
3. Zambia/Namibia
4. Namibia/Botswana
5. Botswana/South Africa

On top of this, we also crossed the actual border between Zambia and Zimbabwe, but we didn’t leave the border post area (the Vic falls bridge), so no stamp in the passport for that one.

On the whole, the border crossings were less eventful than I’d feared. It was mostly a matter of filling in one little form in the departure country’s border post, then driving 150 meters, and filling in another form in the country to be entered’s border post. South Africa, Namibia and Botswana are all rather well-ordered countries, and border posts were rather organised and streamlined.

The same, regretfully, cannot be said of Zambia. If SA is the last outpost of Europe in Africa, then Namibia and Botswana are Africa Light, and Zambia is the first part of Africa proper. As we entered the border post to take a ferry over the Zambesi and cross into Zambia, we were immediately swarmed by a dozen men who wanted to act as our agents at the border (it later turned out that they were insurance salesmen working for commission). After Rean had lost his patience and chased away the crowd, we boarded the Zambesi Cruiser, a lekker little barge that looked like this:Once on the other side, we were met by chaos; there were cars and lorries everywhere, and little huts, some official and some inofficial. I imagine that a European country would’ve put up a large board with a map of the area and clear instructions on which places to go to obtain the correct permits. This thought probably never crossed the mind of anyone in Zambia, and even if it did, it would probably be waved away—how are you supposed to get away with corruption in such an ordered system?

There were the regular immigrations papers to be filled out; I had to get a visa for USD50 (they don’t even take their own currency, the Kwacha, at the border), but otherwise that part was quite uneventful. It was worse for Rean to get the car and trailer cleared. He had to buy road permits, insurance, reflective stickers, get a police clearance, and who knows what else? In the end, it took several hours, and the stack of papers collected had an impressive size.

At least we didn’t have to bribe anyone; the system is simply complex enough that no one will ever know where money is being skimmed off the top.

I visited a few national parks/wildlife parks during my holiday. The first was the Kruger National Park, which is South Africa’s largest game reserve. The KNP has its origin in the Sabie Game Reserve, which was created in 1898 by Transvaal Republic president Paul Kruger. The park is quite famous, and I enjoyed our day in it, but in retrospect it was somewhat disappointing; we didn’t see all that many animals, and precious few up close. But why complain?

We saw loads and loads of impala, which is a particularly pretty little buck, crocodiles, a number of eagles and vultures, a few buffalo and the odd elephant. Giraffes are very chilled out animals, so we got to see a few up close as they grazed on the side of the road. Hippos show their African origins by just lying around all day long, while the half million or so birds that were attracted to our lunch were rather more ambitious. Particularly the spreeus (a sort of sterling/mynah) were keen.

Me and Stephanie respectfully posing with Paul Kruger et al.

Alet and Reane in the KNP.

Lauren, Reane, Stephanie and the pimpmaster.

The coolest bat ever.

Hippos and their cleaning staff. Don't be fooled by their calm appearance: hippos kill more people than any other animal in Africa.

A spreeu looking to share our meal. Not in picture: his 57 000 brothers, sisters and cousins.

Buffalos are quite buff, and very dangerous if upset.

Crocodiles look ancient and dangerous, but rarely put on much of a show, preferring instead to simulate a tree trunk.

Baboons tend to act like buffoons. One couple decided to get down to business right in the middle of the road. Luckily for you, dear reader, I didn't get a picture of that.

A chill-raffe.

The most common animal in the park is still a favourite of mine: the impala. Beautiful, slender, and with a graceful way of moving around. Probably tastes good, too.

The next park visit was Chobe National Park in northern Botswana. Chobe is Botswana’s first national park, and despite being only third largest, it is the most diverse. The park has a huge population of elephants, around 50 000 of the large animals roam the park, and their presence is evident in the destruction of the trees; elephants have a tendency to knock trees over to get access to their leaves, or sometimes just because they can. We saw so many elephants of all sizes in Chobe that we ended up almost tired of them. Luckily, the park also offered sightings of warthogs, impala, buffalo, crocodile, kudu and a number of birds. Most notably, we got to see the exceptionally beautiful Rooikeelbyvreter (white-fronted bee-eater, the internet tells me).

Spotted at the entrance of the park. Ah, the smell of freshly printed Euro notes...

Warthogs ain't pretty, but pretty cute. Especially the piglets, which I failed to get a good shot of. Here's a galt, whatever that is called in English.

Young impala practice fighting over their girlfriends.

Buffalo can apparently enjoy a good roll in the mud just as much as any pig can.

Dian gets a close encounter with the elephant. (It didn't steal his peanuts.)

Five-legged elephant...

A male kudu shows off his horns.

Civilisation!

The last wildlife park we visited was in Caprivi in Namibia, just next to the Botswana border. The park consisted mostly of very dry land, but have a portion of wetlands, the very beginning of the magnificent Okovango delta. I can’t recall the name of this park, which was very pretty, but where we saw rather little wildlife except for bucks. The only real notable was the first glimpse of zebras and a couple of impressive baobab trees.

Zebra crossing outside a zebra crossing. Rebels.

Giant baobab tree. These trees are also known as upside-down trees, for obvious reasons.

If you get tired of animal spotting, maybe you prefer animal droppings.