46 Numerical Illustration of the Hedging Strategy
Figure 4.2: Simulation of the Basket Option and Minimum–Expected–Shortfall Hedge Portfolio with Constraint V0 =V aR0.10 (for the Case of T = 3, K = 0.9)
4.3. REMARKS 47
−0.50 0 0.5 1 1.5 2
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
(Σj=17 ωj S
j(T) − K) and Σj=14 ωj(S j(T) − k
j)
Distribution
Distribution of the Basket Option and Hedging Portfolios
Basket Option Super−hedge Variance Hedge−BC ES Hedge−HP7 ES Hedge−VaR0.10
Figure 4.3: Distribution of the Underlying Basket and the Hedging Portfolios (for the Case of T = 3, K = 0.9)
basis, dynamic hedging would also be possible by duplicating the basket option with the hedging assets. This would be an extension to be considered in future works.
48 Numerical Illustration of the Hedging Strategy
Chapter 5 Conclusion
In summary, the first part of this dissertation investigates how to hedge basket options whose final payoff is related to more than one asset. A basket of underlying assets instead of a single one on one hand benefits investors from the diversification effect, on the other hand poses hedgers a big problem especially when the number of underlying assets is large. Thus, this work is intended to design a hedging portfolio related to only subset of underlying assets. Hedging basket options based on only several assets can not only reduce transaction costs if combined with other hedging strategies, but also become prac-tical and essential when some of the underlying assets are illiquid or not even available for trading.
The newly–designed hedging strategy is a static one and the hedging instruments are plain–vanilla options of thoseN1 < N individual assets in the basket with the most sig-nificant effect on basket options price. Thus, the hedging portfolio is obtained in two steps, aiming to determine the dominant assets and optimal strikes, respectively. In the first step, Principal Components Analysis, a popular multivariate statistical method for dimension reduction, is applied to basket options hedging to select only a subset of under-lying assets. The selection procedure is completed mainly by decomposing the covariance structure of the underlying basket into eigenvalues in the order of significance and eigen-vectors. Those eigenvectors in essential specify the underlying factors with decreasing significance on the basket value. By checking the correlation of each underlying asset and the first several important factors, we finally pick out those assets that contribute mostly to the factors and hence the basket. The optimal strikes are in the second step chosen by solving a numerical optimization problem with some economic optimality objective.
The optimality criterion depends on the risk attitude of hedgers. As given in the thesis, the first objective is to eliminate all the risks that the basket option is exposed to. It is nevertheless in some cases not possible by using only several assets. Alternatively, opti-mal strikes are obtained by minimizing a particular risk measure, e.g., the variance of the hedging error or the expected shortfall, given a constraint on the hedging cost.
Considering liquidity of the hedging instruments, the numerical optimizations are mod-ified by imposing another constraint that the strikes are only from the set of traded
49
50 Conclusion assets. The optimization problems become then more complicated. A simple and com-putationally efficient calibration procedure, convexity correction, is therefore designed to achieve the super–replication hedging portfolio. Basically, the optimal options which are not available are approximated by the linear combination of the two options on the same underlying asset with the neighboring strikes.
As observed from the numerical results, the static hedging method achieves the trade–off between reduced hedging cost and overall super–replication. It also demonstrates that hedging with only a subset of assets works quite well even without considering reduced transaction costs, generating a reasonably small hedging error by investing the same capital as the super–hedging portfolio on all the underlying assets which is difficult to construct or is even not available in the market. Actually, its performance will become more satisfactory if the underlying basket is large and illiquid. Since the hedging per-formance is sensitive to the subset of the selected assets, it is recommended to examine hedging costs, involved transaction costs as well as hedging errors of several subsets. To achieve a better performance, hedging basket options with a subset of assets could be improved by reallocating weights of the sub–hedge–basket to approximately match the distribution of the original basket.
Part II
Irreversible Investment Valuation
51
Chapter 6
Introduction and Overview
As an essential activity for firms and economic growth, investment has attracted a great deal of academic attention for decades. In general, investment is defined as an action of purchasing some goods (financial or physical) in hope of favorable future returns. It occurs at every moment and everywhere around us. For instance, merchandisers raise an inventory for sales, manufactories install new equipments for producing and firms put up new buildings or new plants. Even when we visit a museum, we are making an investment in the sense that some knowledge or fun is expected as a return.
One of the fundamental issues in the investment theory lies in the decision if and when, if yes, investment should be undertaken for a project. Traditionally, the Net Present Value (NPV) method is utilized to value a potential investment. The NPV of a project is according to Ross, Westerfield and Jaffe (2008) defined as the present value of its ex-pected future incremental cash flows. The investment is then undertaken only when the NPV is nonnegative. However, the NPV rule as widely acknowledged in the literature corresponds to the assumption of zero volatility of the underlying stochastic state vari-able. Most importantly, it neglects the possibility to delay the project, as well as other alternatives to subsequently expand or contract the project. These flexibilities can be nevertheless valued in the real options theory, which is the topic of this dissertation.
We first provide an introduction of the real options analysis as a solution to the challenges inherent in investment decision problems. This chapter briefly defines the real option and its analysis, and then profiles different types of common real options. Furthermore, we give an overview of two standard real option methods and summarize the difference between these two techniques. The standard irreversible investment model used for the method illustration and the analysis of the approaches serve as the benchmark for the future discussions on irreversible investment.