On the whole, I thought your project was very well done, however there are a few pints within that I found to be somewhat unclear. First, your explanation of the differences between European and American options is delayed until your final results, and is somewhat sparse. Second, the equations used for the finance Monte Carlo method are neither shown nor explained, as are the rest of the parameters in the Ising model aside from the tendency to act within the market (Ci and Si). It would have been helpful to have provided a slightly more rigorous outline of each of the equations you provided above the very good general explanation. Finally, I think a section mentioning the further possible uses of each model in finance above these relatively simple scenarios would have been interesting, not something necessarily in depth, just a section that prompts the reader to consider further, more useful applications of your option models.

To elaborate on the first critique, In you preliminary data you mentioned how the European option is easier to computationally model using the Black-Scholes model, and later, in your results, you state how the primary difference between the American model and European models are the ability to prematurely act upon options prior to reaching maturity. While you provided a convincing argument as to why the binomial trees provide a better prediction, you did not provide an argument as to why the Black-Scholes model is useless. Having read your explanations and code, it is clear that binomial trees are more complicated and volatile, and your project would be strengthened greatly by explaining why this extra complexity needs to implemented.

The second critique has to do with understanding each of the models that you use. While your project delves into depth on the background theory of Monte Carlo integration and approximation, as well as consider the general sweep of each each of the methods in your write up, some of the innards of each equation are left unclear for the reader. In the Monte Carlo method, you discussed how you were able to price options using a simpler equation that required less variables. It would have been helpful to show this equation and discuss why it is applicable to finance options. As for the Ising model, you discussed the magnetization and market tendencies of the buyers and sellers, yet you did not discuss how your approximation of the strength of the relations with nearby nodes (the “first term” of Equation 3 in your conclusion) affects the market strategies of each of the buyers.

My final critique is a direct result of the obvious simplicity of this method. Even to someone poorly versed in Economics and market theory, it is clear that this method is a loose estimate, at best. The volatility of the market is arbitrarily set and constant, the market must be modeled in isolation, and all the actors in the market have to be actively participating. In reality, markets can become more or less volatile as you have discussed and would have been nice to have seen that model implemented or elaborated upon in your conclusion even though you ran out of time. Furthermore, it would have been inspiring to talk about the possible ways to take the market out of isolation in a model, ie to model related markets through independent, co-inhabiting Ising models. This could then inspire the reader to think about the possible use of the ising model in practice, for example modeling the fracking and deepwater crude oil platform markets. However, I was taught a significant amount by your project, and thought on the whole it was extremely successful.