The computational hardness of pricing compound options

Abstract

It is generally assumed that you can make a financial asset out of any underlying event or combination thereof, and then sell a security. We show that while this is theoretically true from the financial engineering perspective, compound securities might be intractable to price. Even given no information asymmetries, or adversarial sellers, it might be computationally intractable to put a value on these, and the associated computational complexity might afford an advantage to the party with more compute power. We prove that the problem of pricing an option on a single security with unbounded compounding is PSPACE hard, even when the behavior of the underlying security is computationally tractable. We also show that in the oracle model, even when compounding is limited to at most k layers, the complexity of pricing securities grows exponentially in k.