Essays on Lookback and Barrier Options - A Malliavin Calculus Approach

Abstract: This thesis consists of four theoretical essays on contingent claim analysis and its connection to Malliavin calculus. The first three papers are analyzed in the famous Black and Scholes model, while the setup of the fourth paper involves an international environment and the presence of exchange rates. In the first essay the tractability of the Malliavin calculus approach to derive the replicating portfolio of a contingent claim is examined. Here we specifically study lookback and partial lookback options. We show that the Malliavin calculus approach is indeed very convenient to work with, even though the underlying theory is rather abstract. We also relate the Malliavin calculus approach to the famous delta-hedging formula and give necessary conditions for the delta-hedging approach to be valid. It turns out that for path-dependent contingent claims in general, it is usually quite hard to verify these necessary conditions as the Markovian nature of the corresponding price processes is lost. The second essay studies the generality of the Malliavin calculus approach. It is shown that the original approach can only be used when the contingent claim to be replicated is sufficiently smooth, which is the case with the lookback options for instance. However, barrier options in general do not share this property. In this paper, we show that the original Malliavin calculus approach can be extended to hold for any square integrable contingent claim. The implication of the result is that the Malliavin calculus approach to derive the replicating portfolio of a contingent claim is indeed a general approach, which is not the case with the delta-hedging approach. However, depending on the situation the delta-hedging approach may in some cases have computational advantages. As an illustrative example when this is the case, we consider barrier and partial barrier options. In the third essay new contingent claims are introduced to show how to reduce the price of the rather expensive lookback options. This is in many cases desirable from an investor's point of view in order to increase the leverage effect of his portfolio. In the paper, closed form solutions to the prices of the extreme spread options and of the look-barrier options are presented. Furthermore, we explicitly derive the replicating portfolios of the extreme spread options by using the Malliavin calculus approach, and we show that the look-barrier options can be replicated by the well-known delta-hedging approach. The fourth essay concerns the pricing of barrier options in an international environment. Here we basically consider ordinary stock options that might become worthless if the exchange rate hits some predefined barrier level during a first part of the time to maturity. Such options can be important parts in contracts between companies located in different countries. In order to derive a closed form solution to the price of these contingent claims, we use the method of changing numéraire together with Bayes' formula. Moreover, we show how Malliavin calculus can be used to directly identify the dynamics of the underlying securities under the new equivalent probability measure. As a result, we observe that Malliavin calculus is just as convenient to work with in a two-dimensional setup as in a one-dimensional setup. Finally, some aspects on the timing of rebates are analyzed.