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.     

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.