استراتژی های معاملاتی با امکان دسترسی به بازار مشتقات

This research analyzes trading strategies with derivatives when there are several assets and risk factors. We investigate portfolio improvement if investors have full and partial access to the derivatives markets, i.e. situations in which derivatives are written on some but not all stocks or risk factors traded on the market. The focus is on markets with jump risk. In these markets the choice of optimal exposures to jump and diffusion risk is linked. In a numerical application we study the potential benefit from adding derivatives to the market. It turns out that e.g. diffusion correlation and volatility or jump sizes may have a significant impact on the benefit of a new derivative product even if market prices of risk remain unchanged. Given the structure of risk investors may have different preferences for making risk factors tradable. Utility gains provided by new derivatives may be both increasing or decreasing depending on the type of contract added.

Optimal portfolio choice certainly belongs to one of the most extensively studied problems in finance. Merton, 1969, Merton, 1971 and Merton, 1973 considers continuous time economies in which individuals dynamically adjust portfolio positions in order to maximize expected utility. More recent contributions addressing allocation include e.g. Liu et al., 2003, Wu, 2003, Das and Uppal, 2004, Munk and Sorensen, 2004, Rudolf and Ziemba, 2004, Zha, 2007, Adam et al., 2008, Huang and Milevsky, 2008 and Pelizzon and Weber, 2009. Although portfolio selection is of major concern to portfolio managers it has taken some time until there were attempts to include derivatives into dynamic portfolio optimization. This might be partially explained by the fact that initially, derivatives were seen as redundant securities which can be replicated by implementing a dynamic trading strategy in stocks and bonds. The standard Black/Scholes option pricing model supports this view. However, more advanced option pricing models take additional risk factors such as stochastic volatility into account. In these models, markets are incomplete and derivatives are no longer replicable by stocks and bonds alone. Instead they provide opportunities to earn additional risk premiums. Liu and Pan, 2003 and Branger et al., 2008 address this idea and develop models for single stock economies and analyze implications of derivatives on portfolio management in the presence of stochastic jumps and volatility. In practice several asset classes and risk factors are relevant for portfolio choice though. For instance Fama and French (1993) show that three factors are necessary to reflect that small stocks and stocks with high book-to-market-ratio tend to earn additional returns. Another example from the context of international portfolio selection are emerging markets. Several studies including e.g. Bekaert and Urias, 1999, Bekaert et al., 1998 and De Santis and Imrohoroglu, 1997 suggest that emerging markets yield high returns and low correlations to stock markets of industrialized countries indicating potential benefits from considering them for international portfolio investment. This paper contributes to the literature in several aspects. First, we solve the portfolio-planning problem in a jump-diffusion model when there are several stocks and the market is complete (“full access to the derivatives market”). Second, we analyze the case when the investor has access to some derivatives but still faces an incomplete market (“partial access to the derivatives market”). Third, in a numerical application, we study utility gains that can be obtained by successively introducing one derivative after the other. This leads to implications concerning (a) the number of derivatives needed and (b) how these derivatives should look like. For both full and partial access to the derivatives market we determine optimal factor exposures as the solution to a system of ordinary differential equations. In the presence of jump risk the numerical analysis highlights that the utility gain due to introducing yet another derivative can be increasing or decreasing in the number of derivatives which are already traded if the derivatives to be introduced and their characteristics are pre-specified. Potential utility gains follow from a complex relationship of jump probabilities, risk aversion, and the structure of risk factors available. Intuitively, the latter can be explained by the matter of fact that derivatives make more attractive “packages” of risk factors feasible. The amount by which an investor profits from the introduction of a new standard derivative depends on how suboptimal the current relation between exposures to risk factors is. As such even if market prices of risk are constant the utility gain from a new derivative depends on diffusion correlation and volatility as well as jump sizes. In extreme cases the introduction of a new derivative does not lead to an improvement at all. Depending on the packages of risk factors available, investors have different preferences for derivatives which allow for exposure optimization of individual risk factors. This is highlighted when derivatives are considered which provide an exposure to a single risk factor only. For instance, we analyze insurance contracts which make protection with respect to individual jump-risk factors tradable. Our numerical results indicate that there might be situations in which investors benefit more from trading insurance contracts on and thus sensitivities to a particular jump factor than derivatives which cover several or even all other risk factors combined. Hence, if an investor could decide on which type of derivative to introduce next he would always chose a contract which provides exposure to a the risk factor he wants to trade most. In this case, utility gains would become a decreasing function in the number of derivatives. These considerations might also be taken into account by futures and options exchanges or other financial intermediaries when developing new financial products/derivatives in order to attract the interest of portfolio managers. The remainder of the paper is structured as follows. Section 2 establishes the general model and the approach to asset allocation. Section 3 deals with full access to the derivatives market. Section 4 addresses the question of partial access in the presence of jumps. Numerical applications are discussed in Section 5. Finally, Section 6 concludes.

In this paper we address portfolio choice with derivatives in a generalized framework. We consider a model with several stocks and state variables which might be subject to jump risk. In the context of portfolio optimization we end up with a system of ordinary differential equations which can be solved either analytically or by standard numerical procedures. Furthermore, we distinguish between full and partial access to the derivatives market in order to measure utility gains from the introduction of individual derivative securities. In a numerical application we consider standard derivatives first, i.e. derivatives with pre-specified characteristics. In the presence of jumps, investors cannot pick exposures with respect to jump and diffusion factors independently. The analysis demonstrates that the importance of additional derivatives depends on several characteristics. For instance the larger the conditional probability of a joint jump the less additional value is created by introducing a second standard derivative security. However, in order to investigate the overall impact of adding standard derivatives we need to analyze the whole structure of diffusion and jump risk. In the case of perfect correlation including a second standard derivative does not improve portfolio choice at all. Nevertheless, there are also correlations in which adding the first standard derivative does not lead to improvements while the second derivative delivers utility gains. In a similar way investors should take into account volatilities and jump amplitudes when analyzing investments in derivatives. Utility gains of standard derivatives can thus be both an increasing or decreasing function of the number of contracts. In a next step we investigate insurance contracts which provide exposure to individual jump-risk factors. The results suggest that depending on the structure of risk factors investors have different preferences for making individual factors tradable. There are situations in which an investor even benefits more from trading insurance contracts on a single jump factor than from a derivative which covers several or even all other risk factors combined. If an investor could decide on the type of derivative to be introduced next he certainly would chose a contract first which provides the highest utility gain, i.e. which covers the risk factor he wants to trade most. Our results have implications for the number and design of derivatives that futures and options exchanges or other financial intermediaries make available for trade. We analyze under which circumstances derivatives might be of interest to portfolio managers. Our focus is on portfolio optimization within a continuous time model and a frictionless market. Certainly, these requirements are not directly applicable to most market participants. We may argue that portfolio managers would not include derivatives in their portfolios if improvements are too small and thus likely to be eaten up by transaction costs. Although this gives us some intuition about how market participants should analyze product innovations a more explicit consideration of market frictions would shed additional light on this matter. Furthermore, in our model we make the simplifying assumption that there are no limits on single portfolio positions. However, especially the use of derivatives is highly regulated in reality leading to limits on the amounts to be spent on derivatives for many funds. Therefore, another extension would be to take into account these limits on the position in derivatives more explicitly. In addition to that we assume that optimal factor exposures are constant. Considering state-dependent factor exposures certainly would be an important generalization. Finally, an interesting new research would also be an empirical application of our results.