We present a tree algorithm, called the willow tree, for financial derivative pricing. The setup of the tree uses a fixed number of spatial nodes at each time step. The transition probabilities are determine by solving linear programming problems. The willow tree method is radically superior in numerical performance when compared to the binomial tree method.

The objective of this research proposal is to explore the extent to which credit derivatives can be used effectively by domestic financial institutions, in particular, Commercial Banks to hedge the credit risk associated with lending to the Small, Medium and Micro enterprise (SMME) market segment, thereby making lending to this market segment an attractive and viable banking proposition. The financial services sector in South Africa has come under severe criticism from Government, trade unions and the unbanked, low income earners for not fulfilling their social responsibility, in terms of, not banking the Small, Medium and Micro enterprise (SMME) customer base. In particular, financial institutions have been accused of ignoring or not giving sufficient attention to the financial/credit needs of this market segment. These parties have argued that many of the domestic financial institutions are applying standard credit criteria to this market segment, which they feel is incorrect. This has often resulted in SMME's having their requests for credit facilities declined by domestic financial institutions and then having to resort to approaching unscrupulous "loan sharks" for credit facilities, which facilities are often made available to them at exorbitant interest rates. The alleged reluctance on the part of domestic financial services institutions to make available credit facilities, in the form of start-up business loans and asset-based finance to the SMME segment has possibly hindered economic growth, productivity, employment and resulted indirectly in a host of other social anomalies. Alister Ruiters of the Department of Trade and Industry has been publicly vociferous in his attack on domestic financial institutions (Business Day, August18, 1999). It would appear these financial institutions are only prepared to do business with this market segment in partnership with Government, where Government bears a large proportion of the risk by providing guarantees or indemnities on behalf of the client. Examples of such guarantees include Khula and Sizabantu guarantees issued by agencies controlled within the ambit of the Department of Trade and Industry. Financial service institutions have defended their actions by countering that the credit risk attached to making loans available to the SMME market segment is often unacceptable to them. Many of these potential clients are characterised by adverse credit records, show little stability, in terms of, employment and domicilium and often do not have any tangible collateral available to support their loan requests. That is, the risk from lending to this market segment far outweighs the potential returns. Further, these financial institutions have argued that with South Africa having been accepted into the international fold and following the accelerated pace of globalisation, new markets have opened up for their shareholders. Hence, shareholders are requiring improved returns (capital gains and/or dividends); else they are at liberty to move their funds to other investment destinations. The pressure on domestic financial institutions to deliver consistently better returns on equity has been and continues to be a difficult one. This is exacerbated by the increasing competitive pressure from both retail competitors who are now offering financial services, such as Pick 'n Pay Financial Services, Woolworth's, and foreign financial institutions, who have entered the domestic scene. For many of the retail competitors the offering of financial services is seen merely as an extension of their product line. Existing infrastructure, in the form of, branches /outlets and technology are largely already in place. Further, they are not bound by the same liquidity reserve requirements imposed by the South African Reserve Bank (SARB), as are the domestic financial institutions they now compete against. Hence, the retail competitors' profit margins are likely to be higher. Further, as many of the foreign financial institutions are not constrained by the same social responsibility obligations local financial institutions face and as they have not invested substantially in branch networks and other infrastructure in South Africa, their profit margins are higher and hence their returns on equity is likely to be significantly higher than the domestic financial institutions. Following the increasing popularity of Credit Derivatives in countries, such as, the United States of America, the United Kingdom and India, it is my intention to explore whether this instrument can be used effectively by domestic financial institutions as an hedging tool to insure against what they might otherwise consider unacceptable risk in the SMME market segment. That is, will the use of credit derivatives make the lending of funds to this client base an acceptable or attractive proposition to domestic financial institutions. However, we first need to define credit risk and credit derivatives before we proceed further. Creditex (Commentary, May 2001) defines credit risk as: "the risk of loss following default. " PriceWaterhouseCoopers defines a credit derivative as : "a credit risk management instrument that allows a financial institution to transfer credit risk to another party". Having, in simple terms, defined what we mean by credit risk and credit derivatives, we proceed by suggesting how credit derivatives can be used as an effective hedging tool and also some of the possible shortcomings that may be associated with the use of credit derivatives in South Africa. Cheow and Chiu (Managing Credit Risks, May 23,2001) suggest credit derivatives have the potential to transform the way in which Commercial Banks do business. The impact of credit derivatives is likely to result in changes in Bank's operating and credit models of assessment, pricing policies and offer insight into how products and services may be developed and implemented. Traditionally Banks appear to have been involved in all aspects of lending from origination to administration, monitoring and collection. These authors suggest the resulting credit model emanating from the use of credit derivatives is likely to only concentrate on origination of the loan with the view of later selling-off the book itself or insuring the credit risk. This latter alternative involves credit derivatives. We turn our attention to highlighting some possible constraints to the effective use of credit derivatives in South Africa. These are as follows :  Lack of effective infrastructure  Lack of liquidity  Lack Of Transparency  Restrictive Central Bank regulations and exchange controls  limited number of large financial institutions. / Thesis (MBA)-University of Natal, Durban, 2002.

Thesis (M.Sc.)--Victoria University of Wellington, 2007. / Includes bibliographical references.

Thesis (Ph. D.)--University of North Texas, Dec., 2007. / Title from title page display. Includes bibliographical references.

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 120-131). Also available in electronic version. Access restricted to campus users.

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2005. / Includes bibliographical references (leaves 107-111). Also available in electronic version.

Thesis (M.S.) -- Worcester Polytechnic Institute. / Keywords: Finite Difference; Pricing; C++. Includes bibliographical references (leaf 21).

M.Comm.

The research defines derivatives as private contracts, with future rights and obligations imposed on all parties, used to hedge or transfer risk, which derives value from an underlying asset price or index, which asset price or index may take on various forms. The nature of derivatives is that the instruments are intended to be risk management tools. The objectives of derivatives is either to hedge a risk, or to speculate. Derivatives may be classified by the manner in which they are traded, either over the counter (OTC) or on exchange. Alternatively, derivatives may be classified on the basis of structure and mechanisms, i.e. forwards, futures, options or swaps. Risk and risk management are defined in the third chapter with the focus on merchant banking. The nature of risk is that it is inherent in all activities. The nature of risk management is that it aims to ensure that the risks faced by the merchant bank are managed on a daily basis. The objective of risk management is to ensure that losses are minimised and the appropriate level of risk is taken in order to maximise profits. Risk may be classified as operational, operations, market, systemic, credit and legal risk. A comprehensive discussion of credit risk is presented, as it pertains to the legal risk in derivatives in a merchant bank. This includes insolvency, set-off, netting, credit derivatives and collateral. Legal risk is defined as the risk of loss primarily caused by legal unenforceability (i.e. a defective transaction, for instance a contract), legal liability (i.e. a claim) or failure to take legal steps to protect assets (e.g. intellectual property). The nature of legal risk is that it is caused by jurisdictional and other cross-border factors, inadequate documentation, the behaviour of financial institutions, a lack of internal controls, financial innovation or the inherent uncertainty of the law. The objectives of legal risk management in derivatives is to avoid the direct and indirect costs associated with legal risk materialising. This includes reputational damage. Derivatives attract specific legal risks due to the complexity of the instruments as well as the constant innovation in the market. There remains some legal uncertainty regarding derivatives in terms of gaming, wagering and gambling, as well as insurance. The relationship between risk and derivatives is that due to the complexity and constant innovation associated with derivatives, there are some inherent risks to trading in derivatives. It is therefore important to ensure that there is a vested risk management culture in the derivatives trading environment. Chapter four gives an overview of derivatives legislation in foreign jurisdictions and in South Africa. The contractual and documentation issues are discussed with reference to ad hoc agreements, master agreements and ISDA agreements. The practical implementation issues of master agreements and ad hoc agreements are also discussed. The recommendations are that legal risk management be approached in a similar manner to credit, market and other risk disciplines. A legal risk management policy needs to be developed and implemented. The second recommendation is that a derivative to manage the legal risk in derivatives be developed. / Prof. P. Sutherland Dr. C. van der Bijl

Magister Scientiae - MSc / Numerical methods form an important part of the pricing of financial derivatives and especially in cases where there is no closed form analytical formula. We begin our work with an introduction of the mathematical tools needed in the pricing of financial derivatives. Then, we discuss the assumption of the log-normal returns on stock prices and the stochastic differential equations. These lay the foundation for the derivation of the Black Scholes differential equation, and various Black Scholes formulas are thus obtained. Then, the model is modified to cater for dividend paying stock and for the pricing of options on futures. Multi-period binomial model is very flexible even for the valuation of options that do not have a closed form analytical formula. We consider the pricing of vanilla options both on non dividend and dividend paying stocks. Then show that the model converges to the Black-Scholes value as we increase the number of steps. We discuss the Finite difference methods quite extensively with a focus on the Implicit and Crank-Nicolson methods, and apply these numerical techniques to the pricing of vanilla options. Finally, we compare the convergence of the multi-period binomial model, the Implicit and Crank Nicolson methods to the analytical Black Scholes price of the option. We conclude with the pricing of exotic options with special emphasis on path dependent options. Monte Carlo simulation technique is applied as this method is very versatile in cases where there is no closed form analytical formula. The method is slow and time consuming but very flexible even for multi dimensional problems. / South Africa