英文标题：
《A volatility-of-volatility expansion of the option prices in the SABR
  stochastic volatility model》
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作者：
Olesya Grishchenko, Xiao Han, Victor Nistor
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最新提交年份：
2018
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英文摘要：
  We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a \"small\" parameter. As an example, we treat in detail the important case of the SABR PDE for $\\beta = 1$, namely $\\partial_{\\tau}u = \\sigma^2 \\big [ \\frac{1}{2} (\\partial^2_xu - \\partial_xu) + \\nu \\rho \\partial_x\\partial_\\sigma u + \\frac{1}{2} \\nu^2 \\partial^2_\\sigma u \\, \\big ] + \\kappa (\\theta - \\sigma) \\partial_\\sigma$, by choosing $\\nu$ as small parameter. This yields $u = u_0 + \\nu u_1 + \\nu^2 u_2 + \\ldots$, with $u_j$ independent of $\\nu$. The terms $u_j$ are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of $u$ that are in \"closed form,\" and hence can be evaluated very quickly. Most of the other related methods use the \"time\" $\\tau$ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of $\\nu$, similar to Hagan\'s formula, but including also the {\\em mean reverting term.} We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility.
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中文摘要：
我们提出了一种用显式公式快速逼近抛物型偏微分方程（PDEs）解的通用快速方法。我们的方法还提供了解的导数的快速近似，这对许多其他方法来说是一个挑战。我们的方法基于“小”参数的可计算级数展开。作为一个例子，我们详细讨论了SABR PDE的重要情况，即$\\ beta=1$，即$\\部分{\\ tau}u=\\ sigma^2 \\大[\\ frac{1}{2}（\\部分^ 2\\u xu-\\部分{xu）+\\ nu \\ rho \\部分{x \\部分{sigma u+\\ frac{1}{2 \\ sigma u\\，\\大]+\\ kappa（\\ theta-\\ sigma）\\部分{\\ sigma，sigma通过选择$\\nu$作为小参数。这将产生$u=u\\u 0+\\nu u\\u 1+\\nu^2 u\\u 2+\\ldots$，其中$u\\u j$独立于$\\nu$。术语$u\\u j$是可显式计算的，这对许多其他相关方法也是一个挑战。截断此扩展将导致以“闭合形式”计算的美元近似值，因此可以非常快速地进行计算。大多数其他相关方法使用“time”$\\tau$作为一个小参数。我们的方法的优点是，它缩短了计算时间，因此更容易确定和推广公式。我们还获得了SABR模型中隐含波动率的显式扩展，以美元为单位，类似于Hagan公式，但也包括{em均值回复项。}我们提供了几个数值试验，证明了我们方法的性能。特别是，我们将我们的公式与哈根公式进行比较。当用于实际市场数据时，我们的结果也表现良好，并显示了波动率的均值回复特性。
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分类信息：

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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一级分类：Mathematics        数学
二级分类：Analysis of PDEs        偏微分方程分析
分类描述：Existence and uniqueness, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDE\'s, conservation laws, qualitative dynamics
存在唯一性，边界条件，线性和非线性算子，稳定性，孤子理论，可积偏微分方程，守恒律，定性动力学
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一级分类：Mathematics        数学
二级分类：Numerical Analysis        数值分析
分类描述：Numerical algorithms for problems in analysis and algebra, scientific computation
分析和代数问题的数值算法，科学计算
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