How to Hedge an Option Against an Adversary: Black-Scholes Pricing is Minimax Optimal
We consider a popular problem in finance, option pricing, through the lens of an online learning game between Nature and an Investor. In the Black-Scholes option pricing model from 1973, the Investor can continuously hedge the risk of an option by trading the underlying asset, assuming that the asset's price fluctuates according to Geometric Brownian Motion (GBM). We consider a worst-case model, in which Nature chooses a sequence of price fluctuations under a cumulative quadratic volatility constraint, and the Investor can make a sequence of hedging decisions. Our main result is to show that the value of our proposed game, which is the "regret" of hedging strategy, converges to the Black-Scholes option price. We use significantly weaker assumptions than previous work-for instance, we allow large jumps in the asset price-and show that the Black-Scholes hedging strategy is near-optimal for the Investor even in this non-stochastic framework.
This paper considers the problem of option pricing in 
      finance. Using a minimax approach, the authors construct a game between 
      Nature and an Investor, prove that the game value of this game converges 
      to the classic Black-Scholes option price, and give an explicit hedging 
      strategy that achieves this value. Clarity: This is a very 
      math-heavy paper. Unfortunately, I am not very knowledgeable in the area 
      of stochastic calculus, so I am unable to verify the correctness of the 
      proofs in the paper and in the 14-page supplementary material. The authors 
      do provide a reference to a standard book in stochastic calculus, but 
      unfortunately I do not have the time to familiarize myself with the 
      material. This is not a critisism of the paper. In fact, the authors 
      explain the results in words very well and outline proof steps clearly for 
      anyone who may look to build off these results further in the future. 
      Quality: As mentioned above, I could not verify the correctness of 
      the results, but rigorous proofs are provided for all of the theorems. The 
      authors clearly state the strengths of their results (explicit strategy, 
      weaker assumptions required) over previous work. Originality: The 
      Nature vs. Investor game approach is not new, which the authors make clear 
      in the introduction. This paper is basically an extension of Abernethy et 
      al. [1] in two ways. Firstly, the assumptions required for the main result 
      are strictly weaker than the assumptions used by Abernethy et al. 
      Secondly, the authors provide an explicit hedging strategy for achieving 
      the game value, whereas such a strategy was absent from [1]. While the 
      mechanism may not be new, the analysis appears to be sufficiently 
      original. Significance: Given the statement above, I would 
      describe the contributions here as incremental as the results are not 
      extremely game-changing compared to previous work. However, the analysis 
      appears quite difficult and novel, and the problem does appear to be 
      addressed in a better way than previous research. On the other 
      hand, I am somewhat concerned about this paper's relevance to the NIPS 
      community. After a quick scan through the technical areas listed in the 
      call for papers, I had a hard time placing this paper in any of the given 
      categories (although there is a "but not limited to" clause). Given the 
      paper is purely analytical with no empirical results, perhaps this paper 
      is more appropriate for a mathematics / ecommerce journal or conference. 
      On that note, I was wondering if there are any other lessons to be learned 
      here for a practitioner (Investor) beyond the fact that we now have more 
      evidence that Black-Scholes hedging is robust. Typo: - line 
      227: "to to" RESPONSE TO AUTHOR FEEDBACK: While option 
      pricing does not seem to fit the typical NIPS audience, I did fail to make 
      the connection with stochastic optimization and sequential decision making 
      algorithms. These are certainly appropriate areas for 
      NIPS.