英文标题：
《Asymptotic indifference pricing in exponential L\\\'evy models》
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作者：
Cl\\\'ement M\\\'enass\\\'e and Peter Tankov
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最新提交年份：
2015
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英文摘要：
  Financial markets based on L\\\'evy processes are typically incomplete and option prices depend on risk attitudes of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the nonlinear partial integro-differential equation associated to the indifference price. In this work, we develop closed form approximations to exponential utility indifference prices in exponential L\\\'evy models. To this end, we first establish a new non-asymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity. Next, we use this formula to derive a closed-form approximation of the indifference price by treating the L\\\'evy model as a perturbation of the Black-Scholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of L\\\'evy processes (A. \\v{C}ern\\\'y, S. Denkl and J. Kallsen, arXiv:1309.7833) to nonlinear and non-smooth functionals. Our closed formula represents the indifference price as the linear combination of the Black-Scholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying L\\\'evy process, and the derivatives of the Black-Scholes price. As a by-product, we obtain a simple explicit formula for the spread between the buyer\'s and the seller\'s indifference price. This formula allows to quantify, in a model-independent fashion, how sensitive a given product is to jump risk in the limit of small jump size.
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中文摘要：
基于列维过程的金融市场通常是不完整的，期权价格取决于个体代理人的风险态度。在这种背景下，效用无差异价格的概念在学术界得到了广泛的应用。尽管理论上非常有吸引力，但由于求解与无差异价格相关的非线性偏积分微分方程的计算成本较高，这种定价方法仍然难以在实践中应用。在这项工作中，我们发展了指数L趵evy模型中指数效用无差异价格的封闭形式近似。为此，我们首先建立了无差异价格的一个新的非渐近近似，它推广了以前关于这个数量的小风险规避渐近的结果。接下来，我们使用这个公式推导出无差异价格的闭合形式近似，方法是将列维模型视为Black-Scholes模型的扰动。这将最近一篇论文（a.\\v{C}erny，S.Denkl和J.Kallsen，arXiv:1309.7833）中介绍的L挈evy过程的光滑线性泛函的方法扩展到非线性和非光滑泛函。我们的闭合公式将无差异价格表示为Black-Scholes价格和修正项的线性组合，这些修正项取决于基础Lāevy过程的方差、偏度和峰度，以及Black-Scholes价格的导数。作为副产品，我们得到了一个简单明确的公式，用于计算买方和卖方的无差异价格之间的价差。该公式允许以独立于模型的方式量化给定产品在小跳跃范围内跳跃风险的敏感性。
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分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
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一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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