René Carmona, Princeton University
American Options with Multiple Exercises: Theory and Numerics.
Many financial transactions include multiple American exercise features: we use examples from the energy markets as motivation. We review the theory of optimal stopping in the classical case of a single stopping opportunity, and we present some recent results when stopping opportunities are multiple. We treat the geometric Brownian motion case in detail.We then recast the problem in the framework of potential theory for Markov processes, we use Rost's filling scheme to reformulate it as a linear program, and we discuss numerical implementation issues. This approach seems to be new, even in the case of standard American options.

Heath Windcliff, TD Securities
Pricing and Hedging in Incomplete Markets with Basis Risk
In this talk we extend the Black-Scholes framework to allow for the pricing and hedging of option contracts when it is not possible to trade directly in the underlying asset. For example, some popular insurance products include guarantee features on mutual funds. If the insurer is not able to short the underlying mutual fund, then hedging must be performed by trading in an alternative, but similar, asset. Using simulation we show that substantial errors can accumulate if one uses the Black-Scholes model as a proxy for the appropriate pricing problem. As this is an incomplete markets problem, it results in a non-linear PDE model that includes a "market price of basis risk" and a bid-ask spread. In simple (but common) cases, the non-linearity disappears and the pricing problem can be solved analytically. In the general setting, there exist robust numerical techniques that can be applied to the valuation of complex derivative products in this setting.

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