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Register a new userSub-sampling and other considerations for efficient risk estimation in large portfolios
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Computing risk measures of a financial portfolio comprising thousands of options is a challenging problem because (a) it involves a nested expectation requiring multiple evaluations of the loss of the financial portfolio for different risk scenarios and (b) evaluating the loss of the portfolio is expensive and the cost increases with its size. In this work, we look at applying Multilevel Monte Carlo (MLMC) with adaptive inner sampling to this problem and discuss several practical considerations. In particular, we discuss a sub-sampling strategy that results in a method whose computational complexity does not increase with the size of the portfolio. We also discuss several control variates that significantly improve the efficiency of MLMC in our setting.
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We investigate the problem of computing a nested expectation of the form $mathbb{P}[mathbb{E}[X|Y] !geq!0]!=!mathbb{E}[textrm{H}(mathbb{E}[X|Y])]$ where $textrm{H}$ is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested expectations with the idea of adaptively selecting the number of samples in the approximation of the inner expectation, as proposed by (Broadie et al., 2011). We propose and analyse an algorithm that adaptively selects the number of inner samples on each MLMC level and prove that the resulting MLMC method with adaptive sampling has an $mathcal{O}left( varepsilon^{-2}|logvarepsilon|^2 right)$ complexity to achieve a root mean-squared error $varepsilon$. The theoretical analysis is verified by numerical experiments on a simple model problem. We also present a stochastic root-finding algorithm that, combined with our adaptive methods, can be used to compute other risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), with the latter being achieved with $mathcal{O}left(varepsilon^{-2}right)$ complexity.
We propose a novel approach to infer investors risk preferences from their portfolio choices, and then use the implied risk preferences to measure the efficiency of investment portfolios. We analyze a dataset spanning a period of six years, consisting of end of month stock trading records, along with investors demographic information and self-assessed financial knowledge. Unlike estimates of risk aversion based on the share of risky assets, our statistical analysis suggests that the implied risk aversion coefficient of an investor increases with her wealth and financial literacy. Portfolio diversification, Sharpe ratio, and expected portfolio returns correlate positively with the efficiency of the portfolio, whereas a higher standard deviation reduces the efficiency of the portfolio. We find that affluent and financially educated investors as well as those holding retirement related accounts hold more efficient portfolios.
Consider a financial market with nonnegative semimartingales which does not need to have a num{e}raire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deflator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite variation part of the multiplicative decomposition of the deflator.
We derive representations of local risk-minimization of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian rnstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two parameters: volatility risk premium $beta$ and leverage effect $rho$. Arai and Suzuki (2015, arxiv:1503.08589) dealt with the same problem under constraint $beta=-frac{1}{2}$. In this paper, we relax the restriction on $beta$; and restrict $rho$ to $0$ instead. We introduce a Malliavin calculus under the minimal martingale measure to solve the problem.
Markowitz (1952, 1959) laid down the ground-breaking work on the mean-variance analysis. Under his framework, the theoretical optimal allocation vector can be very different from the estimated one for large portfolios due to the intrinsic difficulty of estimating a vast covariance matrix and return vector. This can result in adverse performance in portfolio selected based on empirical data due to the accumulation of estimation errors. We address this problem by introducing the gross-exposure constrained mean-variance portfolio selection. We show that with gross-exposure constraint the theoretical optimal portfolios have similar performance to the empirically selected ones based on estimated covariance matrices and there is no error accumulation effect from estimation of vast covariance matrices. This gives theoretical justification to the empirical results in Jagannathan and Ma (2003). We also show that the no-short-sale portfolio is not diversified enough and can be improved by allowing some short positions. As the constraint on short sales relaxes, the number of selected assets varies from a small number to the total number of stocks, when tracking portfolios or selecting assets. This achieves the optimal sparse portfolio selection, which has close performance to the theoretical optimal one. Among 1000 stocks, for example, we are able to identify all optimal subsets of portfolios of different sizes, their associated allocation vectors, and their estimated risks. The utility of our new approach is illustrated by simulation and empirical studies on the 100 Fama-French industrial portfolios and the 400 stocks randomly selected from Russell 3000.
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