英文标题：
《Small-time asymptotics for a general local-stochastic volatility model
with a jump-to-default: curvature and the heat kernel expansion》
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作者：
John Armstrong, Martin Forde, Matthew Lorig, Hongzhong Zhang
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最新提交年份：
2016
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英文摘要：
We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model. For this we use the Bellaiche \\cite{Bel81} heat kernel expansion combined with Laplace\'s method to integrate over the volatility variable on a compact set, and (after a gauge transformation) we use the Davies \\cite{Dav88} upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. If the correlation $\\rho < 0$, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for $\\beta=1, \\rho \\le 0$. \\bl{For uncorrelated stochastic volatility models, our results also include a SABR-type model with $\\beta=1$ and an affine mean-reverting drift, and the exponential Ornstein-Uhlenbeck model.} We later augment the model with a single jump-to-default with intensity $\\lm$, which produces qualitatively different behaviour for the short-maturity smile; in particular, for $\\rho=0$, log-moneyness $x > 0$, the implied volatility increases by $\\lm f(x) t +o(t) $ for some function $f(x)$ which blows up as $x \\searrow 0$. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \\& Pascucci \\cite{LPP15}, and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov \\& Spector \\cite{AS12} for the case $\\rho=0$.
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中文摘要：
在一般的不相关局部随机波动率模型下，我们计算了隐含波动率的一个精确的小时间估计。为此，我们使用Bellaiche{Bel81}热核展开结合拉普拉斯方法对紧致集上的波动变量进行积分，并且（经过规范变换后）我们使用具有有界Ricci曲率的流形上热核的Davies{Dav88}上界来处理尾积分。如果相关性$\\rho<0$，如果波动率的漂移采用特定的函数形式，并且没有局部波动性成分，我们的方法仍然有效，我们的结果包括$\\beta=1、\\rho\\le 0$的SABR模型。\\bl{对于不相关的随机波动率模型，我们的结果还包括$\\beta=1$和仿射均值回复漂移的SABR型模型，以及指数Ornstein-Uhlenbeck模型。}后来，我们用强度$\\lm$对模型进行了一次跳转，以违约，这会对短期成熟微笑产生质的不同行为；特别是，当$\\rho=0$、对数货币性$x>0$时，某些函数$f（x）$的隐含波动率增加了$\\lm f（x）t+o（t）$，最后变成了$x\\searrow 0$。最后，我们将我们的结果与Lorig、Pagliarani和Pascucci{LPP15}中的一般渐近展开进行了比较，并用蒙特卡罗模拟和Antonov\\&Spector{AS12}中给出的$\\rho=0$情况下的精确闭式解对SABR模型的结果进行了数值验证。
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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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