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This paper develops a model for option market making in which the hedging activity of the market maker generates price impact on the underlying asset. The option order flow is modeled by Cox processes, with intensities depending on the state of the underlying and on the market maker's quoted prices. The resulting dynamics combine stochastic option demand with both permanent and transient impact on the underlying, lea…
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 (…
In this paper the zero vanna implied volatility approximation for the price of freshly minted volatility swaps is generalised to seasoned volatility swaps. We also derive how volatility swaps can be hedged using a strip of vanilla options with weights that are directly related to trading intuition. Additionally, we derive first and second order hedges for volatility swaps using only variance swaps. As dynamically tra…
We consider the super-hedging price of an American option in a discrete-time market in which stocks are available for dynamic trading and European options are available for static trading. We show that the super-hedging price $π$ is given by the supremum over the prices of the American option under randomized models. That is, $π=\sup_{(c_i,Q_i)_i}\sum_ic_iφ^{Q_i}$, where $c_i \in \mathbb{R}_+$ and the martingale meas…
With model uncertainty characterized by a convex, possibly non-dominated set of probability measures, the agent minimizes the cost of hedging a path dependent contingent claim with given expected success ratio, in a discrete-time, semi-static market of stocks and options. Based on duality results which link quantile hedging to a randomized composite hypothesis test, an arbitrage-free discretization of the market is p…
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 …
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…
The principal portfolios of the standard Capital Asset Pricing Model (CAPM) are analyzed and found to have remarkable hedging and leveraging properties. Principal portfolios implement a recasting of any correlated asset set of N risky securities into an equivalent but uncorrelated set when short sales are allowed. While a determination of principal portfolios in general requires a detailed knowledge of the covariance…
In this paper we investigate novel applications of a new class of equations which we call time-delayed backward stochastic differential equations. Time-delayed BSDEs may arise in finance when we want to find an investment strategy and an investment portfolio which should replicate a liability or meet a target depending on the applied strategy or the past values of the portfolio. In this setting, a managed investment …
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…
Hedging a derivative position under transaction costs and market frictions requires a trading rule that adapts to changing conditions. Deep hedging trains a neural policy for this task but policy training does not determine whether a trading desk can afford to run the policy. We apply robust hedging valuation adjustment (HVA) as a post-training valuation-adjustment layer that evaluates tracking-loss CVaR together wit…
Shorting for hedging exposes to risk when the market dynamics is uncertain. Managing uncertainty and risk exposure is key in portfolio management practice. This paper develops a robust framework for dynamic minimum-variance hedging that explicitly accounts for forecast uncertainty in volatility and covariance estimation to achieve empirical stability and reduced turnover, further improving other standard performance …
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 "…
Finding the hedge ratios for a portfolio and risk compression is the same mathematical problem. Traditionally, regression is used for this purpose. However, regression has its own limitations. For example, in a regression model, we can't use highly correlated independent variables due to multicollinearity issue and instability in the results. A regression model cannot also consider the cost of hedging in the hedge ra…
Dealers make money by providing liquidity to clients but face flow uncertainty and thus price risk. They can efficiently skew their prices and wait for clients to mitigate risk (internalization), or trade with other dealers in the open market to hedge their position and reduce their inventory (externalization). Of course, the better control associated with externalization comes with transaction costs and market impac…
The Libor market model is a mainstay term structure model of interest rates for derivatives pricing, especially for Bermudan swaptions, and other exotic Libor callable derivatives. For numerical implementation the pricing of derivatives with Libor market models is mainly carried out with Monte Carlo simulation. The PDE grid approach is not particularly feasible due to Curse of Dimensionality. The standard Monte Carlo…
Since most of the traded options on individual stocks is of American type it is of interest to generalize the results obtained in semi-static trading to the case when one is allowed to statically trade American options. However, this problem has proved to be elusive so far because of the asymmetric nature of the positions of holding versus shorting such options. Here we provide a unified framework and generalize the …
The main result of this paper is a collateralized counterparty valuation adjusted pricing equation, which allows to price a deal while taking into account credit and debit valuation adjustments (CVA, DVA) along with margining and funding costs, all in a consistent way. Funding risk breaks the bilateral nature of the valuation formula. We find that the equation has a recursive form, making the introduction of a purely…
Black-Scholes is the European option formula under lognormal spot, constant vol, and continuous hedging — a quoting convention more than a belief about the world.
Dealer gamma positioning describes whether option dealers are structurally long or short gamma, shaping how hedging flows amplify or dampen market moves.
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.
Skew measures the relative richness of downside versus upside implied volatility, helping track hedging demand and asymmetry in market risk pricing.
Vanna Charm Flow — Second-order greek hedging flows that amplify or dampen spot moves around expiries.
A gamma squeeze is a price spiral where dealer hedging of short call (or put) gamma forces them to buy rallies and sell dips, amplifying the move that created the gamma.
Key Rate Duration — Bucketed rate sensitivity across curve points for relative-value and hedge construction.
FX Implied Volatility — Option-implied uncertainty for currency pairs, key for hedging and risk budgeting.
Yield Curve Control — Official caps on benchmark yields and the distortions they create in RV and cross-market hedging.
Tail Risk Hedging — Explicit protection against left-tail moves via options, vol, or convex instruments.
Liability Driven Investing — Pension hedging of liabilities with long duration bonds/swaps.
Black-Scholes is the European option formula under lognormal spot, constant vol, and continuous hedging — a quoting convention more than a belief about the world.
Dealer gamma positioning describes whether option dealers are structurally long or short gamma, shaping how hedging flows amplify or dampen market moves.
Delta Hedging — Continuous rebalancing of directional exposure that links options markets to underlying liquidity.
FX Implied Volatility — Option-implied uncertainty for currency pairs, key for hedging and risk budgeting.
Gamma Hedging — Delta adjustments by dealers that can accelerate trends or pin prices near strikes.
A gamma squeeze is a price spiral where dealer hedging of short call (or put) gamma forces them to buy rallies and sell dips, amplifying the move that created the gamma.
Liability Driven Investing — Pension hedging of liabilities with long duration bonds/swaps.
Skew measures the relative richness of downside versus upside implied volatility, helping track hedging demand and asymmetry in market risk pricing.
Tail Risk Hedging — Explicit protection against left-tail moves via options, vol, or convex instruments.
Vanna Charm Flow — Second-order greek hedging flows that amplify or dampen spot moves around expiries.
Yield Curve Control — Official caps on benchmark yields and the distortions they create in RV and cross-market hedging.