Multi‐step reflection principle and barrier options

Abstract

This paper examines a class of barrier options, multi‐step barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression for option prices of this type under the Black–Scholes model by deriving the multi‐step reflection principle, that is, the multi‐step boundary‐crossing probability of Brownian motion. Multi‐step barrier options are not only useful in that they can handle barriers of different levels and time steps but can also approximate options with arbitrary barriers. Moreover, they can be applied to pricing barrier options under jump‐diffusion models.

Piecewise linear boundary crossing probabilities, barrier options, and variable annuities

“…Note that the probability in (8) conforms to the multi‐step reflection principle of Lee et al (2022), which shows the scalability of the piecewise constant drift setting.…”

“…To this end, we take advantage of the Esscher transform (Esscher, 1932). The Esscher transform enjoys popularity in valuing contingent claims because it simplifies a calculation of the expectation of payoff (see e.g., Gerber & Shiu, 1994;Lee et al, 2019;Lee et al, 2021;Ng & Li, 2011). Here, we point out the focal aspect of the Esscher transform, which is related to a factorization property.…”

“…Barrier options are widely used to meet the various risk management needs and speculation demands for their cheap premium compared to the corresponding vanilla options. As the financial markets highly develop, and as the market participants' purposes of using the barrier options diversify, it is natural to observe the attempt to flourish the kinds of barrier options by contriving attractive barriers rather than flat or linear form (Hui, 1997; Lee et al, 2021, 2022). It is quick to imagine nonclassical barrier shapes, of course, but derivations of related pricing formulas can be challenging.…”

“…To our best knowledge, the analytical solution to the crossing probabilities of BM for a piecewise linear boundary is limited. In Lee et al (2021), for instance, the authors derive the non-crossing probability for a piecewise linear boundary, but their probabilities are applicable when the piecewise linear boundary is defined over three disjoint subintervals at most. We close the gap in the literature by deriving the analytical crossing probability of BM for a piecewise linear boundary given over subintervals…”

“…So, it is necessary to develop a generalized factorization formula to proceed for later discussion. In Lee et al (2021), the authors develop the socalled partial factorization formula for establishing the factorization properties of expectation when the drift of the BM is a piecewise constant only over three subintervals. We generalize the partial factorization formula for BM of a piecewise constant drift over n subintervals.…”