Weighted Scoring Rules and Convex Risk Measures

In Weighted Scoring Rules and Convex Risk Measures, Dr. Zachary J. Smith and Prof. J. Eric Bickel (both at the University of Texas at Austin) present a general connection between weighted proper scoring rules and investment decisions involving the minimization of a convex risk measure. Weighted scoring rules are quantitative tools for evaluating the accuracy of probabilistic forecasts relative to a baseline distribution. In their paper, the authors demonstrate that the relationship between convex risk measures and weighted scoring rules relates closely with previous economic characterizations of weighted scores based on expected utility maximization. As illustrative examples, the authors study two families of weighted scoring rules based on phi-divergences (generalizations of the Weighted Power and Weighted Pseudospherical Scoring rules) along with their corresponding risk measures. The paper will be of particular interest to the decision analysis and mathematical finance communities as well as those interested in the elicitation and evaluation of subjective probabilistic forecasts.

From Bachelier to Dupire via optimal transport

Famously, mathematical finance was started by Bachelier in his 1900 PhD thesis where – among many other achievements – he also provided a formal derivation of the Kolmogorov forward equation. This also forms the basis for Dupire’s (again formal) solution to the problem of finding an arbitrage-free model calibrated to a given volatility surface. The latter result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article, we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.

UNIVERSAL RISK BUDGETING

I juxtapose Cover’s vaunted universal portfolio selection algorithm ([Cover, TM (1991). Universal portfolios. Mathematical Finance, 1, 1–29]) with the modern representation of a portfolio as a certain allocation of risk among the available assets, rather than a mere allocation of capital. Thus, I define a Universal Risk Budgeting scheme that weights each risk budget, instead of each capital budget, by its historical performance record, á la Cover. I prove that my scheme is mathematically equivalent to a novel type of [Cover, TM and E Ordentlich (1996). Universal portfolios with side information. IEEE Transactions on Information Theory, 42, 348–363] universal portfolio that uses a new family of prior densities that have hitherto not appeared in the literature on universal portfolio theory. I argue that my universal risk budget, so-defined, is a potentially more perspicuous and flexible type of universal portfolio; it allows the algorithmic trader to incorporate, with advantage, his prior knowledge or beliefs about the particular covariance structure of instantaneous asset returns. Say, if there is some dispersion in the volatilities of the available assets, then the uniform or Dirichlet priors that are standard in the literature will generate a dangerously lopsided prior distribution over the possible risk budgets. In the author’s opinion, the proposed “Garivaltis prior” makes for a nice improvement on Cover’s timeless expert system, that is properly agnostic and open to different risk budgets from the very get-go. Inspired by [Jamshidian, F (1992). Asymptotically optimal portfolios. Mathematical Finance, 2, 131–150], the universal risk budget is formulated as a new kind of exotic option in the continuous time Black–Scholes market, with all the pleasure, elegance, and convenience that entails.

Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

The pricing and hedging of financial derivatives have become one of the hot research issues in mathematical finance today. In the case of non-risk neutrality, this article uses the martingale method and probability measurement method to study the pricing method and hedging strategy of financial derivatives. This paper also further studies the hedging strategy of financial derivatives in the incomplete market based on the BSM model and converts the solution of this problem into solving a vector on the Hilbert space to its closure. The problem of space projection is to use projection theory to decompose financial derivatives under a given martingale measure. In the imperfect market, the vertical projection theory is used to obtain the approximate pricing method and hedging strategy of financial derivatives in which the underlying asset follows the martingale process; the projection theory is further expanded, and the pricing problem of financial derivatives under the mixed-asset portfolio is obtained. Approximate pricing of financial derivatives; in the discrete state, the hedging investment strategy of financial derivatives H in the imperfect market is found through the method of variance approximation.

Sensitivity analysis of Wasserstein distributionally robust optimization problems

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first-order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide an explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least-squares regression. We consider robustness of call option pricing and deduce a new Black–Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples.

General drawdown of general tax model in a time-homogeneous Markov framework

Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.

Jonathan Michael Borwein 1951 − 2016: Life and Legacy

Jonathan M. Borwein (1951−2016) was a prolific mathematician whose career spanned several countries(UK, Canada, USA, Australia) and whose many interests includedanalysis, optimization, number theory, special functions, experimental mathematics, mathematical finance, mathematical education,and visualization. We describe his life and legacy, and give anannotated bibliography of some of his most significant books and papers.

The Theory of Uncertaintism

The stock jumps of the underlying assets underpinning the Margrabe options have been studied by Cheang and Chiarella [Cheang, GH and Chiarella C (2011). Exchange options under jump-diffusion dynamics. Applied Mathematical Finance, 18(3), 245–276], Cheang and Garces [Cheang, GHL and Garces LPDM (2020). Representation of exchange option prices under stochastic volatility jump-diffusion dynamics. Quantitative Finance, 20(2), 291–310], Cufaro Petroni and Sabino [Cufaro Petroni, N and Sabino P (2020). Pricing exchange options with correlated jump diffusion processes. Quantitate Finance, 20(11), 1811–1823], and Ma et al. [Ma, Y, Pan D and Wang T (2020). Exchange options under clustered jump dynamics. Quantitative Finance, 20(6), 949–967]. Although the authors argue that they explored stock jumps under Hawkes processes, those processes are the Poisson process in their applications. Thus, they studied Hawkes processes in-between two assets while this study explores Hawkes process within any asset. Furthermore, the Poisson process can be flipped into Hawkes process and vice versa. In terms of hedging, this study uses specific Greeks (rho and phi) while some of the mentioned studies used other Greeks (Delta, Theta, Vega, and Gamma). Moreover, hedging is carried out under static and dynamic environments. The results illustrate that the jumpy Margrabe option can be extended to complex barrier option and waiting to invest option. In addition, hedging strategies are robust both under static and dynamic environments.