Thoughts on Hakansson's Paradox

Hakansson’s so-called paradox (JoFQA; Hakansson, 1979) poses a somewhat skeptical question regarding the value of derivatives: if options can only be priced because they can be replicated, then, since they can be replicated, why are they needed at all?

Interesting question, but the more I think about it, the less of a paradox it becomes to an intelligent financial engineer.

It is true that in some sense, the true 'value' of a derivative is in part its ability to create novel payoff patterns relative to that which could have created with the underlying. Stated another way, their value is their ability to be "non-redundant", because if you can replicate the payoff perfectly then to you it isn't really of any economic value except for the fact perhaps that you can hedge efficiently.

But it seems legitimate to say that a derivative sells, all else equal, at a price proportional to the basis risk one expects to take on when hedging that security. Thus for example, if one expects to take on a ton of basis risk should there be a market dislocation (ie. GM correlation trade or CDS selling), one will sell that at a premium to compensate for the risk. All forms of risk should be compensated for properly so this makes sense.

Even though the value of derivatives is very heavily tied to the concept of dynamic replication, this by no means that "options can only be priced because they can be replicated," as was said in Taleb (note: Taleb doesn't make the claim himself and he by all means has 100x as strong a grasp of derivatives than me). As one moves away from that which can be replicated, the basis risk goes up, as does the premium one expects to pay due to basis risk, plus the hedging costs themselves for the optimal hedge done by the optimal hedger, adjusted for liquidity concerns (if one differentiates between pure basis risk and liquidity risk, which I admittedly have blurred a bit).

Variance swaps are a good example of this. One cannot purely trade the underlying. However it can be said (if I'm wrong let me know) that their heightened popularity relative to volatility swaps is due to the fact that it's easier to hedge variance swaps with a strip of options up and down the strike scale. Theoretically if one had a continuous distribution of strikes all up and down the strike scale and all were highly liquid you would arrive at the theoretical "value" of the variance swap. The real value accounts for the fact that some strikes are less liquid, making it more difficult to hedge. The real value also accounts for the fact that one can slide right off the strike scale should the underlying stock tank like a stone because strikes simply don't exist in certain areas. Whatever residual basis risk one expects to incur relative to the optimal hedge's actual hedging costs are accounted for.

A few conclusions I've reached:
1) Theoretical value assuming continuous rebalancing is only the first step, and sometimes it's no step at all;
2) There are a ton of other things to keep track of when determining what a derivative is worth like basis risk, liquidity risk, and supply/demand factors;
3) We pay in a very real sense for the 'value' of a derivative, where 'value' is defined to by uniqueness and non-redundancy of the payoff pattern, and this is a very logical cost. To the extent that a derivative is not redundant, one must compensate the originator of the derivative with a VaR basis risk argument; to the extent that a derivative *is* redundant, however, one must compensate the originator for his hedging costs.
4) When the market pays too much attention to theoretical replication and forgets about the basis and liquidity risk, supply and demand for certain derivatives over others, and other real world stuff along those lines, watch out.

Watch out, CDS. You are a one sided market. One sided markets are one sided until they are not.

Categories: theoretical_quant