On the condition that both futures and options exist in the markets for hedging, this paper examines the optimal hedging strategy under price risk and background risk. Compared with the previous research, which has studied options hedging against basis risk and production risk being extended to options and futures hedging against price risk and background risk, we proposed a model and have taken the budget of buying options into consideration. The model is fairly general and some existing models are special cases of it. We firstly derive the necessary and sufficient conditions that guarantee the optimality of an under-hedge, a full-hedge and an over-hedge of futures for the risk-averse utility. Then, sufficient conditions are stipulated under which an over-hedge is optimal. Furthermore, we propose a program minimizing of tail conditional expectation (TCE), which is inherently equivalent to the risk measure of expected shortfall risk (ES) or the conditional VaR (CVaR) under the continuous-time framework. Finally, we find that ES, in our proposed model, is significantly smaller than the one in the model of options hedging only. Therefore, the results emphasize the need for combining futures hedging and options hedging, and it also shows that imposing background risk, whether it be additive or multiplicative, always has a great impact on the hedging efficiency. We also present some sensitivities of the relevant parameters to provide some suggestions for the investors.
This paper constructs the evaluation index system of sustainable development level of the city,using TOPSIS method and cluster analysis to quantitative research for sustainability development level. The calculation result shows that the sustainable development level of Xi’an is 0.55456, in 2011, which is the highest one than the former ones, such as 1999 in Kungming and 2006 in Shenyang. From the longitudinal comparison, influence on the sustainable development of the world horticultural exposition for Xi'an is the best. And from the horizontal contrast, the result is as the same.
A basket option is an option on a portfolio of multiple risky assets. There is no analytical solution for basket option pricing, because the payoff of a basket option is determined by the weighted average of the prices of the multiple underlying assets. This study presents an approximation approach for valuing basket options by the algorithms of piecewise lognormal interpolation. The idea is to partition the time axis into collection of small intervals, in which the multiple lognormal average price process is approximated by a simple lognormal with the same first and second moments at the endpoints of the time intervals. Numerical examples of basket option with two stocks illustrate the accuracy of the approach when compared with results from Monte Carlo (MC) simulations.This method can be extrapolated to other complex option pricing, such as combination of options.
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