Hongjun Ha

Abstract A piecewise linear double barrier option generalizes classical double barrier options because of its versatility in designing various double boundaries. This paper discusses how to price piecewise linear double barrier options. To this purpose, we derive the probability that an underlying process does not cross a given piecewise linear double barrier, where the underlying process follows the Brownian motion of piecewise constant drift. Using the established non‐crossing probability, we

Abstract Barrier options have been instrumental in satisfying various market demands. This paper introduces piecewise linear barrier options and provides their pricing formulas. To this end, we establish the analytical piecewise linear boundary crossing probability and explain how to approximate arbitrary boundary crossing probabilities. In addition, we show that a financial instrument with early exercise is decomposable into a knock‐out barrier option and immediate rebate, which casts a new ill

This dissertation consists of two chapters. The first chapter establishes an algorithm for calculating capital requirements. The calculation of capital requirements for financial institutions usually entails a reevaluation of the company's assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. The current chap

Calculating risk measures is challenging due to the complexity of the loss random variable, which depends on multiple state variables over a risk horizon. A common simplification uses a quadratic approximation of the loss random variable to construct an empirical loss distribution. However, this approach may fail to capture extreme events over longer horizons. A more robust method involves representing the loss as a finite linear combination of higher-degree polynomial basis functions. This rais