Topics in Financial Mathematics

金融数学专题

• 批准号: 2530250
• 金额: --
• 依托单位: University of Nottingham
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2530250
• 关键词: Topics; Financial; Mathematics

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.

本摘要包含数学金融领域两个项目的摘要。第一个主题是从市场价格数据中提取通胀预期的方法。第二个主题涉及在期权定价中应用的 SDE 和蒙特卡罗方法的数值模拟。未来通胀率风险中性密度的估计：通胀预期在货币政策中发挥着至关重要的作用。因此，已经有几篇关于从市场信息中提取预期的方法的论文。从通货膨胀率的期权中，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的通货膨胀率之间的依赖结构进行建模，我们可以提取未写入期权的利率的密度函数。例如，5年/5年通货膨胀率可以从5年通货膨胀率和10年通货膨胀率的选项以及两个通货膨胀率之间的依赖性来推断。一个关键问题是如何对不同利率之间的依赖关系进行建模。一种解决方案是使用可以拟合附加数据的联结函数。然而，这会带来一些问题，例如：选择合适的系词；并选择适合联结的适当数据。可能存在以“无模型”方式强加依赖结构的可能性。这将需要额外的同比期权价格数据。由于当前的同比期权流动性不是特别高，观察到的价格很嘈杂，并且经常违反无套利条件。因此，该项目的出发点是生成综合的同比期权价格数据，并开发一种提取不同利率之间相关性的方法。扩散的方差减少方法：在逼近 SDE 时，通常情况下我们只要求逼近的期望值接近真实解的期望值。在这种情况下，我们可以依靠弱方法，从这个意义上保证收敛。当模拟弱意义的SDE时，离散化误差减小得相对较快，并且可以通过Talay-Tubaro展开来近似误差。另一个误差源是由期望的蒙特卡罗近似产生的，它缓慢减小。由于蒙特卡罗方法收敛速度慢，已经开发出方差减少技术来减少总体误差。虽然这些技术不会提高收敛速度，但它们通过减少误差系数来减少计算时间。 SDE 的模拟可应用于求解 PDE，因为抛物线 PDE 具有由 Feynman-Kac 公式给出的概率表示。这在高维环境中尤其重要，在高维环境中，求解偏微分方程的传统数值方法会遭受“维数灾难”的困扰。减少方差的方法包括重要性抽样或控制变量，或两者的组合。理论上，方差可以降到零，但这需要知道偏微分方程的完整解，这是不切实际的。这激发了构建方差减少的实用方法，其中近似完整的解决方案。这些方法的优点是解的初始近似不需要精度保证，因为这些近似不会使蒙特卡洛近似产生偏差。因此，可以采用深度学习方法。首先要探索的领域是深度学习方法在计算成本方面是否可以与线性回归方法竞争。另一个值得探索的潜在领域是狄利克雷边值问题，它也具有概率表示，并且可以采用方差减少技术。解决狄利克雷问题可应用于障碍期权的定价。