Fundamentals Of Derivatives Pricing
In some of the last posts, we talked about how to price different derivatives. But we had no unifying theme. In this post I am going to describe the three general categories of how to price derivatives. Remember, derivatives are assets whose value is dependent on the value of another asset, such as an option or a future.
When hedging, we build a risk neutral portfolio. That
means we buy other assets, such that the risks cancel.
Suppose we play the following game: I have a coin and if it shows heads, you win 1$. If it shows tails, you get nothing in return. How much would you pay to play this game? In this case, the coin itself is the asset, and the return of our game is the derivative. When hedging, assume there is another possible game. Assume there is the game where you win 1$ on tails and win nothing on heads. Obviously this should have the same price as the previous game.
But now, if you play both games, you get 1$ and this return does not depend on which side the coin landed. So it is a riskless asset. So both games together are worth 1$. But since each game is worth the same, one game is worth half a dollar.
That is how hedging works.
In risk-neutral valuation you disregard the risks and only look at the
For example in our coin-flip above, one game has a return expectation of half a dollar. So it is worth half a dollar. This seems like an obvious way of pricing instruments, but let us take another example. Suppose I take a deck of cards, give it a good shuffle, take one card out and if it is the ace of spades, you win 26 $. On all other cards, you gain nothing. For simplicity, let us assume we have no jokers and it is a regular deck of cards, so it has 52 different cards in it. Then the expected value of your return is the same. But the risk is higher.
We can also throw a dice with 100 sides and if it lands on 5, you get 50$. On all other sides, you get nothing. The expected return is the same, but again, risk has increased. So normally you expect to be compensated for this additional risk. When you try to price assets in a risk-neutral way, you are not compensated for risk. So, now I may have convinced you that just looking at the expected value is not a trivial and obvious wya to price assets. But then, why does it work? There is a thing called the fundamental theorem of asset pricing, which roughly says that in an arbitrage-free market this works.
Replication works by composing the derivative by “smaller” assets which are easier to price. Let us take as example a forward contract that I will next year sell 1000$ for euros at the exchange rate next year. How much should this contract be worth? Well, I lose 1000$, so I can instead sell a zero-coupon bond over 1000$ which expires next year. I can proceed to exchange the amount I have gained in euros and buy another zero-coupon bond which grants me euros with all the money I have just made by selling the 1000$-bond. Then my account is at zero again and next year, I lose 1000$ dollars and get some euros. So the effect is the same and therefore the forward-agreement should be worth the price of the euro-bond minus the price of the 1000$-bond. You can do this more rigorously by computing the forward exchange rate, if you want to. The idea of replication is, that if there is a mispricing between the current price of the derivative and the price of some replication thereof which behaves in exactly the same way, then some clever arbitrageurs will take care of it and the prices will again converge.