Recently, there is a growing trend to offer guarantee products where the investor is allowed to shift her account/investment value between multiple funds. The switching right is granted a finite number per year, i.e. it is American style with multiple exercise possibilities. In consequence, the pricing and the risk management is based on the switching strategy which maximizes the value of the guarantee put option. We analyze the optimal stopping problem in the case of one switching right within different model classes and compare the exact price with the lower price bound implied by the optimal deterministic switching time. We show that, within the class of log-price processes with independent increments, the stopping problem is solved by a deterministic stopping time if (and only if) the price process is in addition continuous. Thus, in a sense, the Black & Scholes model is the only (meaningful) pricing model where the lower price bound gives the exact price. It turns out that even moderate deviations from the Black & Scholes model assumptions give a lower price bound which is really below the exact price. This is illustrated by means of a stylized stochastic volatility model setup.
Primal-dual linear Monte Carlo algorithm for multiple stopping - An application to flexible caps
(2012)
In this paper we consider the valuation of Bermudan callable derivatives with
multiple exercise rights. We present in this context a new primal-dual linear
Monte Carlo algorithm that allows for ecient simulation of lower and upper price
bounds without using nested simulations (hence the terminology). The algorithm
is essentially an extension of a primal{dual Monte Carlo algorithm for standard
Bermudan options proposed in Schoenmakers et al. (2011), to the case of multiple
exercise rights. In particular, the algorithm constructs upwardly a system of dual
martingales to be plugged into the dual representation of Schoenmakers (2010).
At each level the respective martingale is constructed via a backward regression
procedure starting at the last exercise date. The thus constructed martingales are
nally used to compute an upper price bound. At the same time, the algorithm
also provides approximate continuation functions which may be used to construct
a price lower bound. The algorithm is applied to the pricing of
exible caps
in a Hull and White (1990) model setup. The simple model choice allows for
comparison of the computed price bounds with the exact price which is obtained
by means of a trinomial tree implementation. As a result, we obtain tight price
bounds for the considered application. Moreover, the algorithm is generically
designed for multi-dimensional problems and is tractable to implement.