Abstracts

In this paper, we consider problems of hedging barrier-style basket options using individual options based on our previously proposed optimal hedging strategy in Yamada (2010—2013) for European options. The optimal hedging problem is formulated as follows: Find optimal smooth functions of individual options to minimize the mean square error from the payoff of down-and-out basket option. For solving the problem, we shall take advantage of a necessary and sufficient condition expressed in terms of linear equations of conditional expectations involving multivariate and path-dependent underlying asset in barrier options. We provide an efficient computational procedure by performing the Brownian bridge decomposition in the expression of underlying assets, and show that the computations involving conditional expectations of basket type barrier options may be reduced to those of unconditional expectations.
Then, we apply our methodology to the Black and Cox (1976) type first passage time structural model, in which a dafaultable company possesses/runs multiple assets/projects and the default may occur the first time the asset value hits a certain lower threshold before the maturity. We discuss how to evaluate individual equity values using smooth functions of optimal hedging problem, and examine the behavior of terminal values of individual equity values approximated by the optimal smooth functions based on the conditional expectations given default or survival. We provide the notion of unbiased payoff, in which the payoff defined by the sum of individual options is said to be unbiased with respect to the barrier option payoff if their conditional expectations are equal given default or survival. We show that the unbiased payoff is constructed by adding a systematic term related to the default event for adjusting the conditional expectations. Moreover, such an unbiased payoff is shown to achieve a smaller mean square error in the objective function of the optimal hedging problem. Finally, we demonstrate how to evaluate contributions of individual equity values to default or survival using the unbiased payoff via certain normalization for conditional expectations. Numerical experiments illustrate that, in the case where the firm is still survival under default risk, the source of excess profit is mainly brought from a high volatility project (or asset), whereas a negatively larger equity value is obtained with a higher volatility project when the firm is default.