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We address the problem of asset pricing in a market where there is no risky asset. Previous work developed a theoretical model for a shadow riskless rate (SRR) for such a market in terms of the drift component of the state-price deflator for that asset universe. Assuming asset prices are modeled by correlated geometric Brownian motion, in this work we develop a computational approach to estimate the SRR from empirica…
Shadow prices simplify the derivation of optimal trading strategies in markets with transaction costs by transferring optimization into a more tractable, frictionless market. This paper establishes that a naïve shadow price Ansatz for maximizing long term returns given average volatility yields a strategy that is, for small bid-ask-spreads, asymptotically optimal at third order. Considering the second-order impact of…
We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang 2014) pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process $S=(S_t)_{0\leq t\leq T}$ satisfying the condition $(NUPBR)$ of "no unbounded profit with bounded risk". This condition requires that $S$ is a sem…
This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and on…
While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under trans…
For portfolio choice problems with proportional transaction costs, we discuss whether or not there exists a "shadow price", i.e., a least favorable frictionless market extension leading to the same optimal strategy and utility. By means of an explicit counter-example, we show that shadow prices may fail to exist even in seemingly perfectly benign situations, i.e., for a log-investor trading in an arbitrage-free marke…
We consider the problem of optimizing the expected logarithmic utility of the value of a portfolio in a binomial model with proportional transaction costs with a long time horizon. By duality methods, we can find expressions for the boundaries of the no-trade-region and the asymptotic optimal growth rate, which can be made explicit for small transaction costs. Here we find that, contrary to the classical results in c…
For utility maximization problems under proportional transaction costs, it has been observed that the original market with transaction costs can sometimes be replaced by a frictionless "shadow market" that yields the same optimal strategy and utility. However, the question of whether or not this indeed holds in generality has remained elusive so far. In this paper we present a counterexample which shows that shadow p…
In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713], stochastic control theory has also been used to solve various problems of this type in the presence of proportional transaction costs. Martingale methods, on the other hand, have so far only been u…
Myopic optimization (MO) outperforms reinforcement learning (RL) in portfolio management: RL yields lower or negative returns, higher variance, larger costs, heavier CVaR, lower profitability, and greater model risk. We model execution/liquidation frictions with mark-to-market accounting. Using Malliavin calculus (Clark-Ocone/BEL), we derive policy gradients and risk shadow price, unifying HJB and KKT. This gives dua…
In a market with one safe and one risky asset, an investor with a long horizon, constant investment opportunities, and constant relative risk aversion trades with small proportional transaction costs. We derive explicit formulas for the optimal investment policy, its implied welfare, liquidity premium, and trading volume. At the first order, the liquidity premium equals the spread, times share turnover, times a unive…
ESG ratings provide a quantitative measure for socially responsible investment. We present a unified framework for incorporating numeric ESG ratings into dynamic pricing theory. Specifically, we introduce an ESG-valued return that is a linearly constrained transformation of financial return and ESG score. This leads to a more complex portfolio optimization problem in a space governed by reward, risk and ESG score. Th…
We provide an explicit characterization of the optimal market making strategy in a discrete-time Limit Order Book (LOB). In our model, the number of filled orders during each period depends linearly on the distance between the fundamental price and the market maker's limit order quotes, with random slope and intercept coefficients. The high-frequency market maker (HFM) incurs an end-of-the-day liquidation cost result…
Hydro storage system optimization is becoming one of the most challenging tasks in Energy Finance. While currently the state-of-the-art of the commercial software in the industry implements mainly linear models, we would like to introduce risk aversion and a generic utility function. At the same time, we aim to develop and implement a computational efficient algorithm, which is not affected by the curse of dimensiona…
We formalize a single structural condition on a portfolio problem, causal separation: conditional on the realized path of a declared set of drivers through the investment horizon, asset returns are mutually independent. From this condition we derive the complete static portfolio theory it induces. Separation forces a diagonal-plus-low-rank conditional covariance through an exact tower decomposition, with the low-rank…
This paper studies convex duality in optimal investment and contingent claim valuation in markets where traded assets may be subject to nonlinear trading costs and portfolio constraints. Under fairly general conditions, the dual expressions decompose into tree terms, corresponding to the agent's risk preferences, trading costs and portfolio constraints, respectively. The dual representations are shown to be valid whe…
We consider an optimal consumption/investment problem to maximize expected utility from consumption. In this market model, the investor is allowed to choose a portfolio which consists of one bond, one liquid risky asset (no transaction costs) and one illiquid risky asset (proportional transaction costs). We fully characterize the optimal consumption and trading strategies in terms of the solution of the free boundary…
For portfolio optimisation under proportional transaction costs, we provide a duality theory for general cadlag price processes. In this setting, we prove the existence of a dual optimiser as well as a shadow price process in a generalised sense. This shadow price is defined via a "sandwiched" process consisting of a predictable and an optional strong supermartingale and pertains to all strategies which remain solven…
Shadow Banking Assets — Non-bank credit intermediation outside traditional capital rules.
Shadow Mode Deployment — Running a new model in parallel without affecting decisions.
RAG Retrieval Quality shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Chunk Overlap Strategy shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Embedding Drift shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Reranker Gain shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Tool Calling Reliability shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Agent Loop Budget shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Prompt Injection Defense shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Output Guardrail shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Eval Harness Score shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Hallucination Rate shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Citation Faithfulness shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Context Window Pressure shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Model Routing Policy shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Latency SLO shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Cost Per Query shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Cache Hit Rate shadow (AI Systems).
Shadow Deployment chat (AI Systems).
Shadow Deployment lab (AI Systems).
Shadow Deployment rag (AI Systems).
Shadow Deployment research (AI Systems).
Shadow Deployment trading desk (AI Systems).
Shadow Deployment ops (AI Systems).
Shadow Deployment batch (AI Systems).
Shadow Deployment streaming (AI Systems).
Shadow Deployment founder mode (AI Systems).
Shadow Deployment production (AI Systems).
Shadow Deployment canary (AI Systems).
Shadow Deployment shadow (AI Systems).
Canary Release shadow (AI Systems).
Human Approval Gate shadow (AI Systems).
Audit Log Completeness shadow (AI Systems).
Knowledge Freshness shadow (AI Systems).
Source Authority Score shadow (AI Systems).
Multi Agent Handoff shadow (AI Systems).
Agent Loop Budget shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Answer Consistency shadow (AI Systems).
Audit Log Completeness shadow (AI Systems).
Cache Hit Rate shadow (AI Systems).
Canary Release shadow (AI Systems).
Chunk Overlap Strategy shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Citation Faithfulness shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Context Window Pressure shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Cost Per Query shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Embedding Drift shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Eval Harness Score shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Fine Tune Regression shadow (AI Systems).
Grounding Coverage shadow (AI Systems).
Hallucination Rate shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Human Approval Gate shadow (AI Systems).
Jailbreak Resistance shadow (AI Systems).
Knowledge Freshness shadow (AI Systems).
Latency SLO shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Memory Store Hygiene shadow (AI Systems).
Model Routing Policy shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Multi Agent Handoff shadow (AI Systems).
Output Guardrail shadow — AI retrieval, agent, evaluation, or production-reliability concept.
Prompt Injection Defense shadow — AI retrieval, agent, evaluation, or production-reliability concept.
RAG Retrieval Quality shadow — AI retrieval, agent, evaluation, or production-reliability concept.