Hedging options with an illiquid underlying and non-probabilistic option pricing in practice

We compare three different dynamic hedging strategies for the purchase or sale of a bundle of European options to profit from volatility arbitrage. The investor will "cross hedge" with a stock highly correlated with the untraded underlying. The first strategy maximizes terminal utility, the second minimizes the variance of the incremental profit, and the third is the adjusted Black-Scholes strategy. We note that the nature of cross hedging results in significant potential for losses. We study the robustness of the strategies to misspecification of parameters by the investor and find that the first two strategies are more robust to parameter misspecification. On a different subject, we then attempt to find profit opportunities by pricing options using a simple non-probabilistic model. We find a situation where an investor willing to take risks can profit, but a more cautious investor cannot. We also derive basic non-probabilistic volatility arbitrage results.