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Market liquidity plays a vital role in the field of market micro-structure, because it is the vigor of the financial market. This paper uses a variable called convexity to measure the potential liquidity provided by order-book. Based on the high-frequency data of each stock included in the SSE (Shanghai Stock Exchange) 50 Index for the year 2011, we report several statistical properties of convexity and analyze the a…
This paper studies a type of periodic utility maximization problem for portfolio management in incomplete stochastic factor models with convex trading constraints. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon, featuring the dynamic adjustments in portfolio decision according to past achievements. Under power utility, we transform the …
In this article, we look at the effect of volatility clustering on the risk indifference price of options described by Sircar and Sturm in their paper (Sircar, R., & Sturm, S. (2012). From smile asymptotics to market risk measures. Mathematical Finance. Advance online publication. doi:10.1111/mafi.12015). The indifference price in their article is obtained by using dynamic convex risk measures given by backward stoch…
Traditional portfolio management methods can incorporate specific investor preferences but rely on accurate forecasts of asset returns and covariances. Reinforcement learning (RL) methods do not rely on these explicit forecasts and are better suited for multi-stage decision processes. To address limitations of the evaluated research, experiments were conducted on three markets in different economies with different ov…
We study superhedging of contingent claims with physical delivery in a discrete-time market model with convex transaction costs. Our model extends Kabanov's currency market model by allowing for nonlinear illiquidity effects. We show that an appropriate generalization of Schachermayer's robust no arbitrage condition implies that the set of claims hedgeable with zero cost is closed in probability. Combined with classi…
Portfolio optimization under cardinality constraints transforms the classical Markowitz mean-variance problem from a convex quadratic problem into an NP-hard combinatorial optimization problem. This paper introduces a novel approach using THRML (Thermodynamic HypergRaphical Model Library), a JAX-based library for building and sampling probabilistic graphical models that reformulates index tracking as probabilistic in…
In economic analysis, rational decision-makers often take actions to reduce their risk exposure. These actions include purchasing market insurance and implementing prevention measures to modify the shape of the loss distribution. Under the assumption that the insureds' actions are fully observed by the insurer, this paper investigates the interaction between self-protection and insurance demand when insurance premium…
Data-driven decision-making processes increasingly utilize end-to-end learnable deep neural networks to render final decisions. Sometimes, the output of the forward functions in certain layers is determined by the solutions to mathematical optimization problems, leading to the emergence of differentiable optimization layers that permit gradient back-propagation. However, real-world scenarios often involve large-scale…
A highly relevant problem of modern finance is the design of Value-at-Risk (VaR) optimal portfolios. Due to contemporary financial regulations, banks and other financial institutions are tied to use the risk measure to control their credit, market, and operational risks. Despite its practical relevance, the non-convexity induced by VaR constraints in portfolio optimization problems remains a major challenge. To addre…
We propose a new iteration scheme, the Cauchy-Simplex, to optimize convex problems over the probability simplex $\{w\in\mathbb{R}^n\ |\ \sum_i w_i=1\ \textrm{and}\ w_i\geq0\}$. Specifically, we map the simplex to the positive quadrant of a unit sphere, envisage gradient descent in latent variables, and map the result back in a way that only depends on the simplex variable. Moreover, proving rigorous convergence resul…
We consider the problem of choosing a portfolio that maximizes the cumulative prospect theory (CPT) utility on an empirical distribution of asset returns. We show that while CPT utility is not a concave function of the portfolio weights, it can be expressed as a difference of two functions. The first term is the composition of a convex function with concave arguments and the second term a composition of a convex func…
We propose a portfolio allocation method based on risk factor budgeting using convex Nonnegative Matrix Factorization (NMF). Unlike classical factor analysis, PCA, or ICA, NMF ensures positive factor loadings to obtain interpretable long-only portfolios. As the NMF factors represent separate sources of risk, they have a quasi-diagonal correlation matrix, promoting diversified portfolio allocations. We evaluate our me…
The rise of Ethereum and other blockchains that support smart contracts has led to the creation of decentralized exchanges (DEXs), such as Uniswap, Balancer, Curve, mStable, and SushiSwap, which enable agents to trade cryptocurrencies without trusting a centralized authority. While traditional exchanges use order books to match and execute trades, DEXs are typically organized as constant function market makers (CFMMs…
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…
There is an observed basis between repo discounting, implied from market repo rates, and bond discounting, stripped from the market prices of the underlying bonds. Here, this basis is explained as a convexity effect arising from the decorrelation between the discount rates for derivatives and bonds. Using a Hull-White model for the discount basis, expressions are derived that can be used to interpolate the repo rates…
We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest…
This paper characterizes the equilibrium in a continuous time financial market populated by heterogeneous agents who differ in their rate of relative risk aversion and face convex portfolio constraints. The model is studied in an application to margin constraints and found to match real world observations about financial variables and leverage cycles. It is shown how margin constraints increase the market price of ri…
The performance of trend following strategies can be ascribed to the difference between long-term and short-term realized variance. We revisit this general result and show that it holds for various definitions of trend strategies. This explains the positive convexity of the aggregate performance of Commodity Trading Advisors (CTAs) which -- when adequately measured -- turns out to be much stronger than anticipated. W…
A convertible is a bond plus an embedded call on the issuer’s stock — credit with equity convexity, or equity with a coupon, depending on the delta.
Crisis alpha is return earned from persistent trends that form after a market crisis starts — not a prediction of the crash day, and not a put that pays on a two-day dip.
Whipsaw is the range-bound regime where trend signals flip, scratch, and bleed — the ordinary cost of owning tail convexity.
Slow trend: lookbacks of roughly 6–12 months, low turnover, fewer whipsaws, later entries, and the bulk of classic CTA crisis convexity.
A managed-futures book that is structurally long options or long VIX-curve convexity — pays carry, aims to print in jumps and persistent stress.
A butterfly is long one wing, short two bodies, long the other wing — a bet on a pin or on the curvature of the smile.
A call option is the right, not the obligation, to buy the underlying at a strike by expiry — convex upside for a premium.
Gamma is the sensitivity of delta to the underlying — how fast the hedge ratio moves, and who is chasing whom.
A put option is the right to sell the underlying at a strike — convex downside, or a hedge that costs carry.
Skew measures the relative richness of downside versus upside implied volatility, helping track hedging demand and asymmetry in market risk pricing.
Theta is the sensitivity of option value to the passing of time — the daily rent of holding convexity.
Convexity Risk — Non-linear price response to yield changes, especially relevant in MBS and long bonds.
A convex optimization problem minimizes a convex function over a convex set — local minima are global, and the dual/KKT machinery is reliable. Most honest portfolio problems try to stay here.
Tail Risk Hedging — Explicit protection against left-tail moves via options, vol, or convex instruments.
The 2s10s Treasury curve measures the spread between 10-year and 2-year Treasury yields and is a key indicator of growth expectations, policy path, and term structure dynamics.
A call option is the right, not the obligation, to buy the underlying at a strike by expiry — convex upside for a premium.
A convertible is a bond plus an embedded call on the issuer’s stock — credit with equity convexity, or equity with a coupon, depending on the delta.
A convex optimization problem minimizes a convex function over a convex set — local minima are global, and the dual/KKT machinery is reliable. Most honest portfolio problems try to stay here.
Convexity Risk — Non-linear price response to yield changes, especially relevant in MBS and long bonds.
Whipsaw is the range-bound regime where trend signals flip, scratch, and bleed — the ordinary cost of owning tail convexity.
Slow trend: lookbacks of roughly 6–12 months, low turnover, fewer whipsaws, later entries, and the bulk of classic CTA crisis convexity.
A managed-futures book that is structurally long options or long VIX-curve convexity — pays carry, aims to print in jumps and persistent stress.
A put option is the right to sell the underlying at a strike — convex downside, or a hedge that costs carry.
Tail Risk Hedging — Explicit protection against left-tail moves via options, vol, or convex instruments.
Theta is the sensitivity of option value to the passing of time — the daily rent of holding convexity.