Mathematical methods in finance

提出项目:1。我们感兴趣的是交易成本成比例的金融模型框架。本文拟建立连续时间市场的适当对偶理论，并证明连续时间市场中适当的无套利概念与对偶变量的存在性之间的等价性。我们对延迟金融模型的校准问题很感兴趣。我们希望得到一个由正则化校准问题提供的近似解。研究了一类具有可能奇异核的抛物型积分微分方程的有限差分解法。我们将重点放在有限域的定位上，并给出了定位误差的估计。我们将提供跳跃-扩散模型的希腊值计算的期望公式。我们应用Malliavin微积分中的方法，并假设连杆算子是可逆的。我们将这些思想应用于跳跃-扩散模型。我们将分析在解除管制的电力市场中电价的演变。我们使用来自加拿大和美国市场的历史数据来估计模型的参数，并通过蒙特卡洛模拟演示如何将其用于期权定价。使用鞍点近似技术，我们计算了由Levy过程驱动的模型的各种欧式期权的价格。我们期望得到一个更接近高斯基的近似。

Proposed projects:1. We are interested in the framework of financial models with proportional transaction costs. We would like to build up an appropriate duality theory for markets in continuous time, and to prove an equivalence between an appropriate concept of no arbitrage and the existence of dual variables in continuous time.2. We are interested in the calibration problem of delay financial models. We would like to obtain an approximate solution provided by a regularized calibration problem. We consider a finite difference method for solving parabolic integro-differential equations with possibly singular kernels; we focus on localization to a finite domain and provide an estimate for the localization error.3. We shall provide expectation formulas for the calculation of Greeks for jump-diffusion models. We apply methods from Malliavin calculus, and we assume that the linkage opertaor is invertible. We apply these ideas to the jump-diffusion models.4. We will analyze the evolution of prices in deregulated electricty markets. We estimate the parameters of the model using historical data from the Canadian and US markets, and demonstrate how it can be used for option pricing via Monte-Carlo simulation.5. Using the saddlepoint approximation technique, we compute the prices of various European-style options for models driven by Levy processes. We expect to obtain a better approximation to the Gaussian base.