# Minisymposium “On Optimal Strategies of Multivariate Passport Options”

• Jörg Kampen
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 12)

American, and Passport Options. These options are modelled by free boundary equations and optimal stopping problems, and by HJB-equations and stochastic optimal control problems. A Bermudean Option contract allows early exercise only at discrete values of time prescribed in the contract. Bermudean Options are popular in high-dimensional fixed income markets and treated typically by Monte—Carlo simulations. At each possible date of expiration theholder of a Bermudean Option has to decide between the value of the product upon exercise and the value of the product upon non-exercise. The latter value is given in terms of conditional expectations. The approximation by conditional estimators involves Monte—Carlo errors. Christian Fries investigates the foresight bias of the Bermudean Option which he interpretes as an Option on the Monte—Carlo error of the conditional estimator. He shows how to apply an analytical correction on the foresight bias which allows for simplifications in coding and more efficient pricing. As the number of exercise dates increase and the maximal distance of two consecutive exercise times decreases, Bermudean Options approach American Options which can be exercised at any time up to expiration. The contribution of Etienne Chevalier provides a lower bound for the difference between the value function of a multivariate American Option and the payoff function. From this he obtains a convergence rate of the Bermudean exercies region to the American one. This result is important because up to now we have to rely on Monte—Carlo methods in order to price and exercise higher-dimensional American Options. A uniform approach for American options and some related early exercise problems is presented in the contribution of John Chadam. He summarizes a bunch of recent works concerning analytical and numerical treatment of American options, prepayment of mortgages, and shows that that the underlying approach can also be applied to the inverse first crossing problem of a default barrier. The approach is based on the representations of prices of early exercise options by fundamental solutions. The fundamental solution plays also a fundamental role in the contribution by Jörg Kampen. He determines the optimal strategy of a multivariate call option on a traded account where the option holder pays a premium upfront and is allowed to choose short and long positions of a portfolio within certain position limits. The so-called passport options is modelled by HJB-equations and optimal stochastic control problems.

## Keywords

Optimal Control Problem Fundamental Solution American Option Analytical Correction Early Exercise
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.