Generalizing the reflection principle of Brownian motion, and closed-form pricing of barrier options and autocallable investments

In this paper, we intend to generalize the well-known reflection principle, one of the most interesting properties of the Brownian motion. The essence of our generalization lies in its ability to stochastically eliminate arbitrary number of partial maximums (or minimums) in the joint events associated with the Brownian motion, thereby allowing us to express the joint probabilities in terms of the multivariate normal distribution functions. Due to the simplicity and versatility, our generalized reflection principle can be used to solve many probabilistic problems pertaining to the Brownian motion. To illustrate, we consider evaluating barrier options and autocallable structured product. Using the basic inclusion-exclusion principle, we obtain integrated pricing formulas for various barrier options under the Black-Scholes model, and derive an explicit pricing formula for the autocallable product, which is not known yet despite its popularity. These formulas are explored through numerical examples. The method of Esscher transform demonstrates its time-honored value during the derivation process.