Convex optimization over risk-neutral probabilities

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