Topics in Financial Mathematics

这篇摘要包含了数学金融学中两个项目的摘要。第一个主题是关于从市场价格数据中提取通胀预期的方法。第二个主题是关于SDEs的数值模拟和蒙特卡罗方法在期权定价中的应用。估计未来通货膨胀率的风险中性密度：通货膨胀预期在货币政策中起着至关重要的作用。因此，已经有几篇论文讨论了从市场信息中提取预期的方法。从写在通货膨胀率上的期权中，我们可以提取未来通货膨胀率的概率密度函数。此外，通过对不同时间间隔内通货膨胀率之间的依赖结构进行建模，我们可以提取未写期权的通货膨胀率的密度函数。例如，5y/5y通胀率可以从5年通胀率和10年通胀率的期权以及两者之间的依赖关系中推断出来。一个关键问题是如何建立不同比率之间的相关性模型。一种解决方案是使用可拟合到附加数据的copula。然而，这就产生了一些问题，如：选择合适的交配体；选择合适的数据来拟合。可能存在以“无模型”的方式强加依赖结构的可能性。这将需要额外的年度期权价格数据。由于目前的同比期权流动性不是特别强，观察到的价格很嘈杂，而且往往违反无套利条件。因此，本项目的起点将是生成合成的年度期权价格数据，并开发一种提取不同利率之间相关性的方法。扩散的方差减小方法：在逼近SDEs时，我们通常只要求逼近的期望值与真解的期望值接近。在这种情况下，我们可以依靠弱方法，它保证了这种意义上的收敛性。在弱意义下模拟SDEs时，离散化误差减小较快，误差可以通过Talay-Tubaro展开进行近似。另一个误差来源是由期望的蒙特卡罗近似引起的，它减少得很慢。由于蒙特卡罗方法的收敛速度慢，方差减少技术已经发展到减少整体误差。虽然这些技术不能提高收敛速度，但它们通过减小误差系数来减少计算时间。由于抛物型偏微分方程具有由费曼-卡茨公式给出的概率表示，因此对偏微分方程的模拟对于求解偏微分方程具有应用价值。这在高维环境中尤其重要，因为传统的求解偏微分方程的数值方法受到“维度诅咒”的困扰。减少方差的方法包括重要性抽样或控制变量，或两者的结合。理论上，方差可以减少到零，但这需要了解PDE的完整解，这是不切实际的。这激发了构建实际的方差减少方法，其中全解是近似的。这些方法的一个优点是，解的初始近似不需要保证精度，因为这些近似不会使蒙特卡罗近似产生偏差。因此，可以采用深度学习方法。首先要探索的是深度学习方法是否可以在计算成本方面与线性回归方法竞争。另一个潜在的探索领域是Dirichlet边值问题，它也有一个概率表示，可以服从方差减少技术。狄利克雷问题的求解在障碍期权定价中有一定的应用。

This abstract contains summaries of two projects in mathematical finance. The first topic is about methods of extracting inflation expectations from market price data. The second topic concerns the numerical simulation of SDEs and Monte Carlo methods which have applications in option pricing. Estimation of risk-neutral densities of future inflation rates: Inflation expectations play a crucial role in monetary policy. As such, there have been several papers on methods to extract expectations from market information. From options written on inflation rates, we can extract a probability density function of future inflation rates. Moreover, by modelling a dependence structure between inflation rates over different time intervals, we can extract density functions of rates on which options are not written. For example, the 5y/5y inflation rate can be inferred from options written on both the 5-year inflation rate and the 10-year inflation rate as well as the dependence between the two rates. A key question is how should the dependence between different rates be modelled. One solution is to use copulas which can be fitted to additional data. This, however, raises problems such as: choosing the appropriate copula; and choosing the appropriate data on which to fit the copula. There might be the possibility of imposing a dependence structure in a 'model-free' way. This would require additional year-on-year option price data. Since current year-on-year options are not particularly liquid, observed prices are noisy and often violate no-arbitrage conditions. Hence, the starting point of this project would be to generate synthetic year-on-year option price data and develop a method of extracting correlations between different rates. Variance reduction methods for diffusions: When approximating SDEs, it is often the case that we only require that the expected value of the approximation is close to the expected value of the true solution. In such cases, we can rely on weak methods, which guarantee convergence in this sense. When simulating SDEs in the weak sense, the discretisation error decreases relatively quickly and the error can be approximated via the Talay-Tubaro expansion. The other source of error, resulting from the Monte Carlo approximation of the expectation, decreases slowly. As a consequence of the slow convergence of Monte Carlo method, variance reduction techniques have been developed to decrease the overall error. While these techniques do not increase the convergence rate, they decrease computational time by reducing the coefficient of the error. Simulation of SDEs has applications for solving PDEs, since parabolic PDEs have a probabilistic representation given by the Feynman-Kac formula. This has particular importance in high-dimensional settings, where traditional numerical methods of solving PDEs suffer from the 'curse of dimensionality'. Methods of variance reduction include importance sampling or control variates, or a combination of the two. In theory, the variance can be reduced to zero but it requires knowledge of the full solution of the PDE, which is not practical. This motivates the construction of practical methods of variance reduction, where the full solution is approximated. An advantage of these approaches is that the initial approximation of the solution requires no guarantee of accuracy, since these approximations do not bias the Monte Carlo approximation. Therefore, deep learning methods could be employed. An initial area to explore is whether deep learning approaches can compete with linear regression methods, in terms of computational cost. Another potential area to explore is Dirichlet boundary value problems, which have also have a probabilistic representation and can be subject to variance reduction techniques. Solving the Dirichlet problem has applications in the pricing of barrier options.