Exotic options pricing in continuous time with Monte Carlo simulations

Exotic options are options with special properties. The holder of an exotic option may have some unusual rights, for example decide the time to start the contract or the decision to become a call or a put. The payoff of an exotic option may have unique characteristic, for example it may depend on the max or min of the underlying price in the history. Based on Black-Scholes Model, these options can be viewed from two different perspectives. One is to specify the distribution and compute the expectation. The other is using replicating portfolio. The former is related to martingale theory, while the latter involves solving a PDE. In this essay, well look at how to value various kinds of options and contrast analytical and Monte Carlo simulation pricing approaches. Chooser, Barrier, Look-back, and Asian choices are the exotic alternatives discussed in our article.