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 deltahedging formula and give necessary conditions for the deltahedging approach to be valid. It turns out that for pathdependent 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 deltahedging approach. However, depending on the situation the deltahedging 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 lookbarrier 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 lookbarrier options can be replicated by the wellknown deltahedging 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 twodimensional setup as in a onedimensional setup. Finally, some aspects on the timing of rebates are analyzed.
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 deltahedging formula and give necessary conditions for the deltahedging approach to be valid. It turns out that for pathdependent 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 deltahedging approach. However, depending on the situation the deltahedging 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 lookbarrier 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 lookbarrier options can be replicated by the wellknown deltahedging 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 twodimensional setup as in a onedimensional setup. Finally, some aspects on the timing of rebates are analyzed.
Original language  English 

Qualification  Doctor 
Awarding Institution 

Supervisors/Advisors 

Award date  1998 Sept 18 
Publisher  
Publication status  Published  1998 
Bibliographical note
Defence detailsDate: 19980918
Time: 13:00
Place: HCE III, Room 210
External reviewer(s)
Name: Øksendal, Bernt
Title: Professor
Affiliation: Oslo University, Norway

Subject classification (UKÄ)
 Economics
Free keywords
 Girsanov transformations.
 lookback options
 Contingent claims
 barrier options
 hedging
 pricing
 arbitrage
 complete markets
 selffinancing portfolios
 BlackScholes formula
 ClarkOcone formula
 Malliavin calculus
 Financial science
 Finansiering