  We consider a collection of derivatives that depend on the price of an
underlying asset at expiration or maturity. The absence of arbitrage is
equivalent to the existence of a risk-neutral probability distribution on the
price; in particular, any risk neutral distribution can be interpreted as a
certificate establishing that no arbitrage exists. We are interested in the
case when there are multiple risk-neutral probabilities. We describe a number
of convex optimization problems over the convex set of risk neutral price
probabilities. These include computation of bounds on the cumulative
distribution, VaR, CVaR, and other quantities, over the set of risk-neutral
probabilities. After discretizing the underlying price, these problems become
finite dimensional convex or quasiconvex optimization problems, and therefore
are tractable. We illustrate our approach using real options and futures
pricing data for the S&P 500 index and Bitcoin.
