A semi-analytic valuation of two-asset barrier options and autocallable products using Brownian bridge

Barrier options based upon the extremum of more than one underlying prices do not allow for closed-form pricing formulas, and thus require numerical methods to evaluate. One example is the autocallable structured product with knock-in feature, which has gained a great deal of popularity in the recent decades. In order to increase numerical efficiency for pricing such products, this paper develops a semi-analytic valuation algorithm which is free from the computational burden and the monitoring bias of the crude Monte Carlo simulation. The basic idea is to combine the simulation of the underlying prices at certain time points and the exit (or non-exit) probability of the Brownian bridge. In the literature, the algorithm was developed to deal with a single-asset barrier option under the Black–Scholes model. Now we extend the framework to cover two-asset barrier options and autocallable product. For the purpose, we explore the non-exit probability of the two-dimensional Brownian bridge, which has not been researched before. Meanwhile, we employ the actuarial method of Esscher transform to simplify our calculation and improve our algorithm via importance sampling. We illustrate our algorithm with numerical examples.     

Multi-step barrier products and static hedging

This paper examines multi-step barrier options with an arbitrary payoff function using extended static hedging methods. Although there have been studies using extended reflection principles to obtain joint distribution functions for barrier options with complex barrier conditions, and static hedging methods to evaluate limited barrier options with well-known payoff functions, we obtain an explicit expression of barrier option price which has a general payoff function under the Black–Scholes framework assumption. The explicit multi-step barrier options prices we discuss in this paper are not only useful in that they can handle different levels and time steps barrier and all types of payoff functions, but can also extend to pricing of barrier options under finite discrete jump–diffusion models with a simple barrier. In the last part, we supplement the theory with numerical examples of various multi-step barrier options under the Black–Scholes or discrete jump–diffusion model for comparison purposes.     

Empirically-transformed linear opinion pools

The linear opinion pool (LOP) produces potentially non-Gaussian combination forecast densities. In this paper, we propose a computationally convenient transformation for the LOP to mirror the non-Gaussianity exhibited by the target variable. Our methodology involves a Smirnov transform to reshape the LOP combination forecasts using the empirical cumulative distribution function. We illustrate our empirically transformed opinion pool (EtLOP) approach with an application examining quarterly real-time forecasts for U.S. inflation evaluated on a sample from 1990:1 to 2020:2. EtLOP improves performance by approximately 10% to 30% in terms of the continuous ranked probability score across forecasting horizons.     

联合谐波小波与递归神经网络的股市时间序列预测

为了提高神经网络对股市时间序列的预测精度，首先利用谐波小波对股票市场数据进行多尺度分解，将其分解为不同尺度且具有平移不变特征的谐波小波分量；然后根据股市时间序列的特点，构建递归神经网络模型进行短期预测，以不同尺度的谐波小波分量为输入数据，对股市数据进行多尺度预测；最后对不同尺度的预测结果进行谐波小波重构，得到最终的股市预测数据。对我国股票市场进行了实验分析，结果表明：股市时间序列经谐波小波分解后，股市数据中不同投资时间水平的价格波动可以被较好的分离，有效地提高了股票市场数据的预测精度。     

Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps

We investigate analytical solvability of models with affine stochastic volatility (SV) and Lévy jumps by deriving a unified formula for the conditional moment generating function of the log-asset price and providing the condition under which this new formula is explicit. The results lay a foundation for a range of valuation, calibration, and econometric problems. We then combine our theoretical results, the Hilbert transform method, various interpolation techniques, with the dimension reduction technique to propose unified simulation schemes for solvable models with affine SV and Lévy jumps. In contrast to traditional exact simulation methods, our approach is applicable to a broad class of models, maintains good accuracy, and enables efficient pricing of discretely monitored path-dependent derivatives. We analyze various sources of errors arising from the simulation approach and present error bounds. Finally, extensive numerical results demonstrate that our method is highly accurate, efficient, simple to implement, and widely applicable.