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Portfolio management is an essential component of investment strategy that aims to maximize returns while minimizing risk. This paper explores several portfolio management strategies, including asset allocation, diversification, active management, and risk management, and their importance in optimizing portfolio performance. These strategies are examined individually and in combination to demonstrate how they can hel…
In the field of quantitative finance, volatility models, such as ARCH, GARCH, FIGARCH, SV, EWMA, play the key role in risk and portfolio management. Meanwhile, factor investing is more and more famous since mid of 20 century. CAPM, Fama French three factor model, Fama French five-factor model, MSCI Barra factor model are mentioned and developed during this period. In this paper, we will show why we need adjust group …
We consider a financial market where stocks are available for dynamic trading, and European and American options are available for static trading (semi-static trading strategies). We assume that the American options are infinitely divisible, and can only be bought but not sold. In the first part of the paper, we work within the framework without model ambiguity. We first get the fundamental theorem of asset pricing (…
We study exponential-utility maximization for high-frequency trading in a discretized fractional Brownian motion model. Using spectral methods for stationary Gaussian sequences, we derive the asymptotic growth rate of the optimal certainty equivalent. We also show that the suitably rescaled optimal positions converge in finite-dimensional distributions to a Gaussian white-noise-type field.
This paper studies a modular cash-overlay rule for allocating between a fixed growth-defensive risky sleeve R and interest-bearing cash C. The risky sleeve is a static 50/50 combination of equal-weight growth/technology and defensive income/value ETF baskets; the target is future R-C return, with the cash leg earning the contemporaneous cash rate. Two independent filters are tested. The slow-tail filter maps continuo…
The distribution of income and wealth in developed economies exhibits a robust two-class structure: an exponential (Boltzmann--Gibbs) bulk covering $\sim\!97\%$ of the population, and a power-law (Pareto) tail in the upper $\sim\!3\%$. We derive this structure from first principles via an explicit mechanistic chain: Gibrat's law for firm growth implies a Zipf firm-size distribution; maximum entropy applied to within-…
A solution to a portfolio optimization problem is always conditioned by constraints on the initial capital and the price of the available market assets. If a risk neutral measure is known, then the price of each asset is the discounted expected value of the asset's price under this measure. But if the market is incomplete, the risk neutral measure is not unique, and there is a range of possible prices for each asset,…
Quantum machine learning has demonstrated significant potential in solving practical problems, particularly in statistics-focused areas such as data science and finance. However, challenges remain in preparing and learning statistical models on a quantum processor due to issues with trainability and interpretability. In this letter, we utilize the maximum entropy principle to design a statistics-informed parameterize…
This paper investigates the potential of Bayesian optimization (BO) to optimize the atr multiplier and atr period -the parameters of the Supertrend indicator for maximizing trading profits across diverse stock datasets. By employing BO, the thesis aims to automate the identification of optimal parameter settings, leading to a more data-driven and potentially more profitable trading strategy compared to relying on man…
We propose an alternative linearization to the classical Markowitz quadratic portfolio optimization model, based on maximum drawdown. This model, which minimizes maximum portfolio drawdown, is particularly appealing during times of financial distress, like during the COVID-19 pandemic. In addition, we will present a Mixed-Integer Linear Programming variation of our new model that, based on our out-of-sample results a…
This paper studies an optimal consumption problem for a loss-averse agent with reference to past consumption maximum. To account for loss aversion on relative consumption, an S-shaped utility is adopted that measures the difference between the non-negative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the utility with respect to consumption, allowing us to focus …
Although maximizing median and quantiles is intuitively appealing and has an axiomatic foundation, it is difficult to study the optimal portfolio strategy due to the discontinuity and time inconsistency in the objective function. We use the intra-personal equilibrium approach to study the problem. Interestingly, we find that the only viable outcome is from the median maximization, because for other quantiles either t…
The present paper originated from a problem in Financial Mathematics concerned with calculating the value of a European call option based on multiple assets each following the binomial model. The model led to an interesting family of polytopes $P(b)$ associated with the power-set $\mathcal{L} = \wp\{1,\dots,m\}$ and parameterized by $b \in \mathbb{R}^m$, each of which is a collection of probability density function o…
Currently, pension providers are running into trouble mainly due to the ultra-low interest rates and the guarantees associated to some pension benefits. With the aim of reducing the pension volatility and providing adequate pension levels with no guarantees, we carry out mathematical analysis of a new pension design in the accumulation phase. The individual's premium is split into the individual and collective part a…
It has often been stated that, within the class of continuous stochastic volatility models calibrated to vanillas, the price of a VIX future is maximized by the Dupire local volatility model. In this article we prove that this statement is incorrect: we build a continuous stochastic volatility model in which a VIX future is strictly more expensive than in its associated local volatility model. More generally, in this…
We use supervised learning to identify factors that predict the cross-section of returns and maximum drawdown for stocks in the US equity market. Our data run from January 1970 to December 2019 and our analysis includes ordinary least squares, penalized linear regressions, tree-based models, and neural networks. We find that the most important predictors tended to be consistent across models, and that non-linear mode…
We propose a strategy for automated trading, outline theoretical justification of the profitability of this strategy and overview the hypothetical results in application to currency pairs trading. The proposed methodology relies on the assumption that processes reflecting the dynamics of currency exchange rates are in a certain sense similar to the class of Ornstein-Uhlenbeck processes and exhibits the mean reverting…
We examine the issue of sensitivity with respect to model parameters for the problem of utility maximization from final wealth in an incomplete Samuelson model and mainly, but not exclusively, for utility functions of positive power-type. The method consists in moving the parameters through change of measure, which we call a weak perturbation, decoupling the usual wealth equation from the varying parameters. By rewri…
FlashAttention computes exact attention with tiling that keeps softmax stats in SRAM, cutting HBM traffic and unlocking longer contexts at the same FLOP count.
Q-learning is an off-policy TD method that learns action values Q(s, a) toward the greedy target r + γ max_a′ Q(s′, a′), without needing the behavior policy to be optimal.
Reinforcement learning trains a policy to maximize expected return by interacting with an environment: states, actions, rewards, and (usually) a discount factor.
Softmax maps a real vector to a probability simplex: softmax(z)_i = exp(z_i) / Σ exp(z_j). It is the standard last layer for classification and attention weights.
A tokenizer splits raw text into the discrete tokens a model actually sees — bytes, characters, or learned subwords — and defines the vocabulary the softmax is over.
A VAE is a probabilistic autoencoder: the encoder outputs a distribution q(z|x), the decoder p(x|z), and training maximizes an ELBO with a KL term that keeps the latent well-behaved.
The Fed’s dual mandate is maximum employment and stable prices — two goals that agree in a demand shock and fight in a supply shock.
Shannon entropy H(P) = −Σ p log p is the expected surprise of a distribution — a measure of uncertainty used in information theory, portfolio tilts, and some max-ent priors.
PCA finds orthogonal directions of maximum variance in a covariance (or correlation) matrix — the yield-curve level/slope/butterfly and many equity ‘statistical factors’ are PCA.
The efficient frontier is the set of mean-variance-optimal portfolios — maximum expected return for each volatility, given the inputs.
Kelly is the stake that maximizes the expected log of wealth — an aggressive sizing rule that needs a true edge and a stomach.
Maximum Drawdown Control — Rules that de-risk after losses to preserve capital and investor mandates.
Short last-month’s extreme daily winners (lottery names) and long the boring residual — Bali–Cakici–Whitelaw MAX.
The Fed’s dual mandate is maximum employment and stable prices — two goals that agree in a demand shock and fight in a supply shock.
The efficient frontier is the set of mean-variance-optimal portfolios — maximum expected return for each volatility, given the inputs.
Shannon entropy H(P) = −Σ p log p is the expected surprise of a distribution — a measure of uncertainty used in information theory, portfolio tilts, and some max-ent priors.
FlashAttention computes exact attention with tiling that keeps softmax stats in SRAM, cutting HBM traffic and unlocking longer contexts at the same FLOP count.
Kelly is the stake that maximizes the expected log of wealth — an aggressive sizing rule that needs a true edge and a stomach.
Short last-month’s extreme daily winners (lottery names) and long the boring residual — Bali–Cakici–Whitelaw MAX.
Maximum Drawdown Control — Rules that de-risk after losses to preserve capital and investor mandates.
PCA finds orthogonal directions of maximum variance in a covariance (or correlation) matrix — the yield-curve level/slope/butterfly and many equity ‘statistical factors’ are PCA.
Q-learning is an off-policy TD method that learns action values Q(s, a) toward the greedy target r + γ max_a′ Q(s′, a′), without needing the behavior policy to be optimal.
Reinforcement learning trains a policy to maximize expected return by interacting with an environment: states, actions, rewards, and (usually) a discount factor.
Softmax maps a real vector to a probability simplex: softmax(z)_i = exp(z_i) / Σ exp(z_j). It is the standard last layer for classification and attention weights.
A tokenizer splits raw text into the discrete tokens a model actually sees — bytes, characters, or learned subwords — and defines the vocabulary the softmax is over.
A VAE is a probabilistic autoencoder: the encoder outputs a distribution q(z|x), the decoder p(x|z), and training maximizes an ELBO with a KL term that keeps the latent well-behaved.