As a last example of the application of the idea that the world is probably free of easy arbitrage opportunities I here provide a brief introduction of the idea of financial engineering. Assuming the absence of arbitrage is all one needs to price financial derivatives. A financial derivative, perhaps a bit narrowly defined, is a financial product – that is a risky investment possibility – with payoffs that depend exclusively on other “basic” financial products such as bonds and stocks. You may want to google what bonds and stocks are if you do not yet know. For our purposes all we need to know is that a stock of a company has a value or price that substantially varies over time. The future price of a stock is uncertain today and this uncertainty can be quite large.
A call option on a stock of some company is an example of a financial derivative on this stock. It is the right (but not an obligation) to buy the stock at a pre-specified future date and at a pre-specified price, the so-called strike price.
Before I discuss pricing such a call option in a hugely simplified model of stock prices, let me provide an example of a financial derivative on a sport bet. The nice thing about this example is that I do not have to hugely simplify the world because a sport bet is already much simpler than a stock. While the price of a stock at some future date can probably take one of many possible prices, a sport event can usually result only in a few possible outcomes.
Let me return to the football (soccer) games from the previous part of the second lecture and let me introduce a financial derivative. In fact let us take the game between Benfica Lisbon and Manchester United. Suppose that, for some reason, I would like to buy an asset that pays me 3 Euros if Benfica wins, 2 Euros if there is a draw, and 1 Euro if Man U wins. What should be the price of this asset? You might think you have to think deeply about the likelihood of Benfica winning, of a draw, and of Man U winning. This is not necessary. Through the betting odds we already know how the market assesses these probabilities. But we do not even need to think about it like this. We can simply construct the same payoff schedule as this new asset has by placing appropriate bets on the three possible outcomes of this game. Recall that the odds were 4,75 on Benfica winning, 3,6 on a draw, and 1,78 on Man U winning. So if we place 3/4,75 = 0,63 Euros on Benfica winning, 2/3,6 = 0,56 Euros on a draw, and 1/1,78 = 0,56 on Man U winning, we get the exact same payoff schedule as the asset I am interested in. To do so I have to place a total of 0,63 + 0,56 + 0,56 = 1,75 Euros. This is, therefore, the price of this new asset. If the price were any different people could do arbitrage. If the price of the new asset was to be below 1,75 Euros then one could do arbitrage by buying the new asset and selling (“shorting”) the three bets in the proportion I gave. If the price of the new asset were above 1,75 Euros one could do arbitrage by selling (“shorting”) the asset and buying the three bets (by placing the above stated Euro amounts on these three bets). This is perhaps a little be more easily said than done, but if there was an arbitrage opportunity I am sure people would find a way to exploit it.
Let me now turn to a simplified world in which we derive the price of a call option on a stock. The simplification is this. I assume that the stock can only have one of two prices at the time at which the call option can be exercised. These values are 98 or 100 Euros. Let’s call a price of 98 the bad state and a price of 100 Euros the good state. In addition let us also assume that one can borrow and save money with an interest rate of zero. In other words the savings technology is to keep money under your mattress. Finally, let us suppose we are interested in a call option with a strike price of 99 Euros. This means that the holder of the call option has the right to buy the stock at 99 Euros at the exercise date if she so wishes. Finally let us assume that the stock price today is 98,50 Euros. We can summarize all this in a table.
Why does the call option have monetary payoffs of 1 and 0 in the respective two states? Suppose that the stock turns out to have a price of 100 when you can exercise your option. We are in the good state. Then you will exercise your option, because you can buy something for 99 Euros that you can immediately sell again for 100 Euros. So you gain 1 Euro in this case. Now suppose that the stock turns out to have a price of 98 Euros. We are in the bad state. Then you will not exercise your option as paying 99 for something that is only worth 98 is foolish. So you get nothing from your option in the bad state.
So how can we determine the price of the call option, denoted by C in the table? Again, you do not need to think about the likelihood of the good and the bad state. All you need is to see that there are no arbitrage opportunities. How could you make arbitrage? Well, you realize that if you buy half a stock and borrow 49 Euros (sell or “short” money), then you have the following payoffs at the exercise date: If the state is good you can sell your one half of a stock, now worth 50 Euros, and return the 49 Euros to whoever you borrowed it from. So you receive a total payment of 1 Euro. If the state is bad you do the same thing but now, as half of one stock is worth only 49 Euros you come out with zero. If you follow this plan you get the exact same payoff schedule as the call option provides: 1 Euro in the good state, zero in the bad state. How much does it cost you to follow this plan? You borrow 49 Euros and pay 49,25 Euros (half of 98,50 Euros). So you have to pay 25 Euro-cents today to get the exact same payoff schedule as the call option delivers at the exercise date. And so only a price of 25 Euro-cents for the call option would make this world arbitrage-free.
“Financial engineering”, the determination of arbitrage-free prices, which gave rise to the “Black-Scholes” formula for option prices and its generalizations, is based on this simple idea. Financial engineering is mathematically more difficult because stocks can have more than two values at the exercise date, many options can be exercised at any time up to a final date, and the creation of a payoff equivalent portfolio of other assets requires many minute portfolio adjustments over time. But conceptually this is what is going on.
The video (in German) is here: