Essays on Basket Options Hedging and Irreversible Investment Valuation

Abstract

<p>Basket options are one of well-known newly-generated exotic options. As its name implies, it is an option on a portfolio of several assets. As the underlying basket offers more diversification, basket options gain increasing popularity in world financial markets as a fundamental instrument to manage portfolio risks. Examples thereof are equity index options which are traded on the exchange and usually contingent on at least 15 stocks, as well as currency basket options traded over the counter and written on over two currencies. Obviously, the unique feature of basket options is the basket underlying and a complex correlation structure therefore involved. It provides investors a couple of benefits like high diversification, a lower price against a portfolio of single options and so on, and meanwhile complicates the evaluation of basket options. The inherent challenge in pricing and hedging basket options stems primarily from the analytical intractability of the distribution of the basket. Moreover, the correlation between underlying assets is observed to be volatile over time. Due to the lack of standardized basket options traded in the market, the correlation structure can be only estimated from historical time series or from scarce option data. This further prevents us from exactly pricing basket options, and more importantly, perfectly hedging basket options. As a result, a partial- or super-hedge is often pursued in the literature when hedging basket options. Apart from these difficulties, we address another difficulty resulted from a great number of underlying assets in the basket while hedging basket options. If following the standard hedging method, a hedging portfolio for basket options should be related to all underlying assets in the basket. Clearly, if the number of the underlying assets is over 15, such a dynamic hedging strategy would be not only hardly implementable in many practical situations but also create a large transaction cost. In this sense, a static or buy-and-hold hedge strategy has its advantage in cost saving and hence hedge performance. As a result, the first part of this dissertation aims to design a static hedging strategy for European-style basket options and to analyze its hedging result. <br /> The newly developed static hedging strategies consist of traded plain-vanilla options on only subset of underlying assets. The optimal hedge is either super-- or partial-replicating, depending on the objective function taken in the numerical optimization. Considering the numerical challenge in the optimization with constraints on the initial capital (or some other hedging requirements) and the maximal number of hedging assets, hedging portfolios are suggested in this thesis to be obtained in two steps, namely pre-selection of the sub-hedge-basket and determination of optimal hedging instruments, more precisely, the optimal strikes of available plain-vanilla options on the chosen subset of the basket. Especially, a multivariate statistical technique, Principal Components Analysis, is introduced to identify dominant assets in the basket by taking into account all the coefficients that greatly influence the basket value, such as weight, volatility and correlation. As demonstrated by numerical examples, such hedging portfolios work satisfactorily, generating a reasonably small hedging error though by using only several assets.<br /> Real options are defined in the literature as to describe opportunities of investment in non-financial assets with some degree of freedom in decision making against the underlying uncertainty. As many other researchers, we are also interested in this topic and are going to study irreversible investment valuation in the second part of this dissertation. An extensive literature investigates the irreversible investment problem under uncertainty Despite a high reputation in academics, the real options theory is not widely adopted by corporate managers and practitioners due to the lack of transparency and simplicity of the standard real options approaches, i.e., the contingent claim analysis and the dynamic programming method. The second part of this dissertation first develops a Shadow Net Present Value rule by using a new approach in the real options theory. The method starts with identifying the expected present value from the investment and comes to the final conclusion via representing the expected present revenue in terms of the expected present value of the running supremum of the shadow revenue of the investment. By aiming at the net profit of the investment which is the mere concern of investors, this approach thus facilitates an intuitive understanding of the real options theory and also a wider application into the practice. Meanwhile, it generalizes the elegant explicit characterization of the investment decision rule to all exponential Levy processes: The optimal investment policy is a trigger strategy such that the investment is initiated at the first time when the value of the investment project comes to a critical threshold. As two extensions, this technique is then applied to two more complicated and practical models taking into consideration gradual capacity generation and risk neutrality, respectively. In each model, both qualitative and quantitative analysis is given on the investment feature and its relationship with related parameters.</p>