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Liquidity Providers on Automated Market Makers generate millions of USD in transaction fees daily. However, the net value of a Liquidity Position is vulnerable to price changes in the underlying assets in the pool. The dominant measure of loss in a Liquidity Position is Impermanent Loss. Impermanent Loss for Constant Function Market Makers has been widely studied. We propose a new metric to measure Liquidity Position…
This study derives the expected liquidity cost when performing the delta hedging process of a European option. This cost is represented by an integration formula that includes European option prices and a certain function depending on the delta process. We first define a unit liquidity cost and then show that the liquidity cost is a multiplication of the unit liquidity cost, stock price, supply curve parameter, and t…
In volatile financial markets, balancing risk and return remains a significant challenge. Traditional approaches often focus solely on equity allocation, overlooking the strategic advantages of options trading for dynamic risk hedging. This work presents DeltaHedge, a multi-agent framework that integrates options trading with AI-driven portfolio management. By combining advanced reinforcement learning techniques with…
We investigate whether the volatility risk premium is negative by examining the statistical properties of delta-hedged option portfolios (buy the option and hedge with stock). Within a stochastic volatility framework, we demonstrate a correspondence between the sign and magnitude of the volatility risk premium and the mean delta-hedged portfolio returns. Using a sample of S&P 500 index options, we provide empirical t…
This paper studies empirical deep hedging for S&P 500 index options under a local downside-shortfall reward. It moves beyond performance comparison by asking what the learned hedge does, when it fails, and whether it can be made auditable. TD3 agents are compared with a daily-updated Black-Scholes delta hedge on the same option episodes. In walk-forward tests from 2015 to 2023, the agents usually learn a systematic d…
The recent work of Horikawa and Nakagawa (2024) claims that under a complete market admitting statistical arbitrage, the difference between the hedging position provided by deep hedging and that of the replicating portfolio is a statistical arbitrage. This raises concerns as it entails that deep hedging can include a speculative component aimed simply at exploiting the structure of the risk measure guiding the hedgin…
Fukasawa introduced in [Fukasawa, Math Financ, 2012] two necessary conditions for no butterfly arbitrage which require that the $d_1$ and $d_2$ functions of the Black-Scholes formula have to be decreasing. In this article we characterize the set of smiles satisfying these conditions, using the parametrization of the smile in delta. We obtain a parametrization of the set via one real number and three positive function…
This study deals with the problem of pricing European currency options in discrete time setting, whose prices follow the fractional Black Scholes model with transaction costs. Both the pricing formula and the fractional partial differential equation for European call currency options are obtained by applying the delta-hedging strategy. Some Greeks and the estimator of volatility are also provided. The empirical studi…
Modelling stock prices via jump processes is common in financial markets. In practice, to hedge a contingent claim one typically uses the so-called delta-hedging strategy. This strategy stems from the Black--Merton--Scholes model where it perfectly replicates contingent claims. From the theoretical viewpoint, there is no reason for this to hold in models with jumps. However in practice the delta-hedging strategy is w…
This paper develops a robust mathematical framework for Constant Function Market Makers (CFMMs) by transitioning from traditional token reserve analyses to a coordinate system defined by price and intrinsic liquidity. We establish a canonical parametrization of the bonding curve that ensures dimensional consistency across diverse trading functions, such as those employed by Uniswap and Balancer, and demonstrate that …
Reinforcement learning trading systems published in the academic literature overwhelmingly rely on price-aggregate state representations (OHLCV bars) or limit-order-book depth features, leaving microstructure pattern theories from the practitioner literature, namely Auction Market Theory and Market Profile, without a peer-reviewed computational instantiation. We present ViperQ, a reinforcement learning system whose s…
This paper makes the Millennium Prize problem P vs NP operational in quantitative finance by studying cardinality-constrained portfolio selection. Starting from the convex Markowitz mean-variance program with CAPM-based expected returns (Rf plus beta times ERP), we impose a hard sparsity rule that limits the portfolio to K assets out of approximately 94 industry portfolios (Damodaran). The constraint couples discrete…
In this paper, we present an artificial neural network framework for portfolio compression of a large portfolio of European options with varying maturities (target portfolio) by a significantly smaller portfolio of European options with shorter or same maturity (compressed portfolio), which also represents a self-replicating static hedge portfolio of the target portfolio. For the proposed machine learning architectur…
This paper acts as a collection of various trading strategies and useful pieces of market information that might help to implement such strategies. This list is meant to be comprehensive (though by no means exhaustive) and hence we only provide pointers and give further sources to explore each strategy further. To set the stage for this exploration, we consider the factors that determine good and bad trades, the noti…
We derive the stochastic price process for tokens whose sole price discovery mechanism is a constant-product automated market maker (AMM). When the net flow into the pool follows a diffusion, the token price follows a constant elasticity of variance (CEV) process, nesting Black-Scholes as the limiting case of infinite liquidity. We obtain closed-form European option prices and introduce liquidity-adjusted Greeks. The…
We present a reinforcement-learning (RL) framework for dynamic hedging of equity index option exposures under realistic transaction costs and position limits. We hedge a normalized option-implied equity exposure (one unit of underlying delta, offset via SPY) by trading the underlying index ETF, using the option surface and macro variables only as state information and not as a direct pricing engine. Building on the "…
This paper mathematically models a constant-function automated market maker (CFAMM) position as a portfolio of exotic options, known as perpetual American continuous-installment (CI) options. This model replicates an AMM position's delta at each point in time over an infinite time horizon, thus taking into account the perpetual nature and optionality to withdraw of liquidity provision. This framework yields two key t…
Retail proprietary-trading firms sell a two-stage product: a paid evaluation that must reach a profit target before breaching a trailing drawdown, then a funded account that must survive a minimum window and a consistency rule before a payout. We show the geometry of this contract creates incentives that differ by stage and make passing a poor standalone signal of skill. Under end-of-day trailing the evaluation rewar…
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.
Delta is the first derivative of option value to the underlying — the hedge ratio and a moneyness label.
Delta Hedging — Continuous rebalancing of directional exposure that links options markets to underlying liquidity.
Gamma is the sensitivity of delta to the underlying — how fast the hedge ratio moves, and who is chasing whom.
Gamma Hedging — Delta adjustments by dealers that can accelerate trends or pin prices near strikes.
Greeks are the sensitivities of option value to spot, vol, time, and rates — the risk report of a non-linear book.
Realized Volatility — Historical return variation that determines PnL for delta-hedged option positions.
Long the convertible and short the delta in the stock — harvest cheap implied vol / credit, with funding and squeeze risk.
Scanner code fragment for put/call ratio and open-interest activity on the ATM chain window.
Strategy bridge from Blackboard scanner cards into Options Lab templates without implying live broker execution.
Educational IV × Greeks × surface sandbox already mounted on the Blackboard — synthetic regimes for pedagogy beside the live scanner.
Long the convertible and short the delta in the stock — harvest cheap implied vol / credit, with funding and squeeze risk.
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.
Delta is the first derivative of option value to the underlying — the hedge ratio and a moneyness label.
Delta Hedging — Continuous rebalancing of directional exposure that links options markets to underlying liquidity.
Gamma is the sensitivity of delta to the underlying — how fast the hedge ratio moves, and who is chasing whom.
Gamma Hedging — Delta adjustments by dealers that can accelerate trends or pin prices near strikes.
Realized Volatility — Historical return variation that determines PnL for delta-hedged option positions.