Robust option pricing

• We propose a non-probabilistic approach to model the stock price dynamics.
• We model heavy tails, and autocorrelation, etc. using these uncertainty sets.
• We use robust optimization to solve high-dimensional option pricing problems.
• Unlike DP, our approach scales polynomially with the dimension of the option.
• We model the “implied volatility smile” observed in market.

In this paper, we combine robust optimization and the idea of ∊∊-arbitrage to propose a tractable approach to price a wide variety of options. Rather than assuming a probabilistic model for the stock price dynamics, we assume that the conclusions of probability theory, such as the central limit theorem, hold deterministically on the underlying returns. This gives rise to an uncertainty set that the underlying asset returns satisfy. We then formulate the option pricing problem as a robust optimization problem that identifies the portfolio which minimizes the worst case replication error for a given uncertainty set defined on the underlying asset returns. The most significant benefits of our approach are (a) computational tractability illustrated by our ability to price multi-asset, American and Asian options using linear optimization; and thus the computational complexity of our approach scales polynomially with the number of assets and with time to expiry and (b) modeling flexibility illustrated by our ability to model different kinds of options, various levels of risk aversion among investors, transaction costs, shorting constraints and replication via option portfolios.