This post explores the financial dynamics faced by an options market maker (MM), particularly within the volatile context of Zero Days to Expiration (0DTE) options. We will delve into the mechanics of delta hedging, the financial repercussions of imperfect hedging and gamma exposure, particularly as expiration approaches, the MM’s Profit and Loss (P&L) characteristics given their current hedge, and the potential incentives for price manipulation as expiration approaches. While we employ the Black-Scholes framework for its analytical tractability, we acknowledge its inherent limitations in accurately modeling 0DTE scenarios. For parts of this analysis, we assume a single (monopolist) MM to simplify the connection between their options book and market observables. The insights from this analysis can be valuable for non-MM participants seeking to understand and potentially anticipate price dynamics influenced by MM activities, especially as 0DTE options approach expiry.

1. Options Pricing and Greeks in the Black-Scholes Framework

The Black-Scholes model provides benchmark prices for European call and put options.

A call option gives the holder the right, but not the obligation, to buy an underlying asset $S$ at a specified strike price $K$ on or before a specified expiration date. Its Black-Scholes price $C(S, \tau)$ is:

$$C(S, \tau) = S N(d_1) - K e^{-r\tau} N(d_2)$$

A put option gives the holder the right, but not the obligation, to sell an underlying asset $S$ at a specified strike price $K$ on or before a specified expiration date. Its Black-Scholes price $P(S, \tau)$ is:

$$P(S, \tau) = K e^{-r\tau} N(-d_2) - S N(-d_1)$$

Where:

• $S$ is the current price of the underlying asset.
• $K$ is the strike price of the option.
• $\tau$ is the time to expiration (in years).
• $r$ is the continuously compounded risk-free interest rate.
• $\sigma$ is the annualized volatility of the underlying asset’s returns.
• $N(\cdot)$ is the cumulative distribution function (CDF) of the standard normal distribution.

The terms $d_1$ and $d_2$ are given by:

$$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)\tau}{\sigma\sqrt{\tau}}$$ $$d_2 = d_1 - \sigma\sqrt{\tau}$$

Key Greeks

The “Greeks” are sensitivities of the option price to changes in underlying parameters.

• Delta ($\Delta$): Measures the rate of change of the option price with respect to a dollar change in the underlying asset’s price.

  $$\Delta = \frac{\partial V}{\partial S}$$

  where $V$ is the option price. For calls, $\Delta_C = N(d_1)$, and for puts, $\Delta_P = N(d_1) - 1 = -N(-d_1)$.

• Gamma ($\Gamma$): Measures the rate of change of Delta with respect to a dollar change in the underlying asset’s price.

  $$\Gamma = \frac{\partial^2 V}{\partial S^2}$$

  For both calls and puts, $\Gamma = \frac{N'(d_1)}{S\sigma\sqrt{\tau}}$, where $N'(x)$ is the probability density function (PDF) of the standard normal distribution.

For 0DTE options, as $\tau \to 0$, Gamma for at-the-money (ATM) options (where $S \approx K$) becomes extremely large. This implies that Delta can change dramatically with even minor movements in the underlying price $S$.

2. Delta Hedging: The Ideal and The Intense Reality of 0DTE

Market makers typically aim to profit from the bid-ask spread on options they trade, rather than from directional bets on the underlying asset. They manage the price risk of their options positions primarily through delta hedging.

2.1. The Continuous Delta Hedging Ideal

The goal of delta hedging is to maintain a portfolio whose value is insensitive to small changes in the underlying asset’s price. Let $V(S,t)$ be the aggregate market value of a portfolio of options identical to those the MM is net short (i.e., the ‘long equivalent’ value of their short book). The MM’s actual options holding thus has value $-V(S,t)$. To hedge this position, the MM aims to hold $H = \Delta_V = \partial V / \partial S$ shares of the underlying asset $S$. The MM’s portfolio value is then $\Pi = -V(S,t) + H S = -V + \Delta_V S$. The change in value of this portfolio, $d\Pi$, over an infinitesimal time interval $dt$, assuming continuous re-hedging, is:

$$d\Pi = -dV + H dS$$

By Ito’s lemma, the change in the ‘long equivalent’ option book value $dV$ is:

$$dV = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS + \frac{1}{2} \frac{\partial^2 V}{\partial S^2} (dS)^2$$

In Black-Scholes, $(dS)^2 = S^2 \sigma^2 dt$. Let $\Theta_V = \frac{\partial V}{\partial t}$, $\Delta_V = \frac{\partial V}{\partial S}$, and $\Gamma_V = \frac{\partial^2 V}{\partial S^2}$ be the theta, delta, and gamma of this ‘long equivalent’ options book $V$. Then, $dV$ can be written as:

$$dV = \Theta_V dt + \Delta_V dS + \frac{1}{2} \Gamma_V S^2 \sigma^2 dt$$

If the MM maintains a delta-neutral hedge, $H = \Delta_V$. So, the change in the MM’s portfolio value is:

$$d\Pi = -(\Theta_V dt + \Delta_V dS + \frac{1}{2} \Gamma_V S^2 \sigma^2 dt) + \Delta_V dS$$ $$d\Pi = -\left(\Theta_V + \frac{1}{2} \Gamma_V S^2 \sigma^2\right) dt$$

Since an option holder (long $V$) experiences time decay (typically $\Theta_V < 0$) and benefits from convexity ($\Gamma_V > 0$), the MM, by being short $V$, collects theta ($-\Theta_V dt > 0$) but pays for being short gamma ($-(\frac{1}{2} \Gamma_V S^2 \sigma^2) dt < 0$). This $d\Pi$ represents the MM’s P&L from decay and gamma exposure under perfect hedging.

2.2. 0DTE Hedging: Challenges and Inevitable Discrepancies

Perfect, continuous delta hedging is a theoretical ideal. For 0DTE options, practical challenges are significantly amplified:

• Extreme Gamma: As noted, ATM Gamma skyrockets as $\tau \to 0$. Delta changes explosively with minor price moves in $S$, demanding extremely frequent, precise, and potentially large re-hedges.
• Transaction Costs & Market Impact: Frequent re-hedging incurs substantial transaction costs. Furthermore, the large volumes required, especially when re-hedging rapidly due to high gamma, can exert significant market impact, pushing execution prices away from the MM and increasing hedging costs.
• System Latency & Execution Risk: A finite time lag invariably exists between detecting a delta change requirement and executing the hedge. The underlying price $S$ can move significantly within this lag.
• Execution Challenges with Large Volumes: Even if the underlying stock is generally liquid, the sheer size and rapid succession of hedging trades required to manage extreme gamma can strain market absorptive capacity. This can lead to increased slippage, difficulty in sourcing liquidity for very large orders without further adverse price movements, and challenges in executing hedges at desired prices or in the required timeframe.

These factors mean an MM’s hedge can easily become mismatched from the true current delta of their options book. As expiration draws nearer, the rapidly increasing gamma and escalating transaction costs can make continuous, perfect re-hedging practically unattainable, leading to a situation where the MM’s established hedge becomes effectively “stuck” or, at best, imperfectly managed relative to the fast-changing ideal.

3. P&L Consequences of a “Last” Hedge Near Expiration

As established, market makers (MMs) face escalating difficulties in maintaining a perfect delta hedge for 0DTE options as expiration nears. Extreme gamma, transaction costs, and liquidity constraints can lead to a situation where the MM’s hedge becomes misaligned. Let $V(S,t)$ continue to represent the ‘long equivalent’ market value of the MM’s net short options book, so the MM’s options holding has value $-V(S,t)$. They also hold an effectively “stuck” share hedge, $H_{\text{last}}$. Suppose the underlying price moves from $S_{\text{last}}$ (where the hedge $H_{\text{last}} = \Delta_V(S_{\text{last}}) = (\partial V/\partial S)|_{S_{\text{last}}}$, the delta of the ‘long equivalent’ book $V$ at $S_{\text{last}}$, was appropriate) to a new price $S_c$ over a short period where the hedge could not be perfectly adjusted. The change in the MM’s portfolio value due to this unhedged component of price movement is:

$${\text{P\&L}}_{\text{gamma\_exposure}} = \left[-V(S_c, t_1) + H_{\text{last}} S_c\right] - \left[-V(S_{\text{last}}, t_0) + H_{\text{last}} S_{\text{last}}\right]$$

Rearranging terms:

$${\text{P\&L}}_{\text{gamma\_exposure}} = -\left(V(S_c, t_1) - V(S_{\text{last}}, t_0)\right) + H_{\text{last}} (S_c - S_{\text{last}})$$

Assuming the time interval $t_1 - t_0$ is very small, so $t_1 \approx t_0$, and the primary impact on the ‘long equivalent’ option value $V$ comes from the change in $S$. We approximate $V(S_c, t_1)$ using a Taylor series expansion of $V(S, t_0)$ around $S_{\text{last}}$. Recalling that $H_{\text{last}} = \Delta_V(S_{\text{last}})$ (the delta of the ‘long equivalent’ options book $V$ at $S_{\text{last}}$), and letting $\Gamma_V(S_{\text{last}})$ be the gamma of this ‘long equivalent’ book $V$ at $S_{\text{last}}$, the expansion gives:

$$V(S_c, t_1) - V(S_{\text{last}}, t_0) \approx \Delta_V(S_{\text{last}}) (S_c - S_{\text{last}}) + \frac{1}{2} \Gamma_V(S_{\text{last}}) (S_c - S_{\text{last}})^2$$

Since $H_{\text{last}} = \Delta_V(S_{\text{last}})$, we can write:

$$V(S_c, t_1) - V(S_{\text{last}}, t_0) \approx H_{\text{last}} (S_c - S_{\text{last}}) + \frac{1}{2} \Gamma_V(S_{\text{last}}) (S_c - S_{\text{last}})^2$$

Substituting this back into the ${\text{P\&L}}_{\text{gamma\_exposure}}$ equation:

$${\text{P\&L}}_{\text{gamma\_exposure}} \approx -\left( H_{\text{last}} (S_c - S_{\text{last}}) + \frac{1}{2} \Gamma_V(S_{\text{last}}) (S_c - S_{\text{last}})^2 \right) + H_{\text{last}} (S_c - S_{\text{last}})$$ $${\text{P\&L}}_{\text{gamma\_exposure}} \approx -\frac{1}{2} \Gamma_V(S_{\text{last}}) (S_c - S_{\text{last}})^2$$

If the MM is net short options, their ‘long equivalent’ book $V$ typically has positive gamma ($\Gamma_V(S_{\text{last}}) > 0$). The MM’s overall portfolio is therefore effectively short gamma with respect to unhedged movements. Consequently, any significant unhedged price movement $(S_c - S_{\text{last}})^2 > 0$ results in a negative ${\text{P\&L}}_{\text{gamma\_exposure}}$. This represents the realized cost due to the combination of being effectively short gamma and the inability to perfectly adjust the hedge $H_{\text{last}}$ during the price movement.

This P&L is largely a sunk cost by the time the MM evaluates their position at $S_c$. The key takeaway is that due to hedging frictions near expiration, such gamma-driven P&L events are likely. Following such an event, or simply acknowledging the current “stuck” nature of their hedge, the MM must manage their position into expiration.

4. MM P&L Profile with a “Last” Hedge into Expiration

Having arrived at a price $S_c$ with a hedge $H_{\text{last}}$ that is no longer perfectly optimal (or perhaps was the result of a recent, best-effort re-hedge), the MM faces the final moments to expiration. The P&L incurred due to prior gamma exposure (as discussed in Section 3) is now a sunk cost. The critical question is how their P&L will behave from this point ($S_c$ with hedge $H_{\text{last}}$) until the final expiration price $S_T$.

For the remainder of the 0DTE period (from $S_c$ with hedge $H_{\text{last}}$ until expiration at price $S_T$), the MM is effectively “stuck” with their existing options book (quantities $Q_k$) and its inherent gamma profile, alongside their current hedge $H_{\text{last}}$. Let $O_k(S_{\text{last}}, K_k)$ be the market value of option $k$ (per unit) at price $S_{\text{last}}$ when the hedge $H_{\text{last}}$ was established. The P&L from this point until expiration is:

$$\text{P\&L}_{S_{\text{last}} \to S_T}(S_T) = \sum_k Q_k \left[O_k(S_{\text{last}}, K_k) - \text{Payoff}_k(S_T, K_k)\right] + H_{\text{last}} (S_T - S_{\text{last}})$$

Here, $Q_k$ represents the quantity of option $k$ sold by the MM (so $Q_k > 0$ for short positions). $\text{Payoff}_k(S_T, K_k)$ is the final payoff of option $k$ at expiration if the underlying price is $S_T$. For example, for a call option, $\text{Payoff}_k(S_T, K_k) = \max(0, S_T - K_k)$. This P&L can be rewritten to separate terms dependent on $S_T$:

$$\text{P\&L}_{S_{\text{last}} \to S_T}(S_T) = \left( \sum_k Q_k O_k(S_{\text{last}}, K_k) - H_{\text{last}} S_{\text{last}} \right) + H_{\text{last}} S_T - \sum_k Q_k \text{Payoff}_k(S_T, K_k)$$

The term in parentheses, $\left( \sum_k Q_k O_k(S_{\text{last}}, K_k) - H_{\text{last}} S_{\text{last}} \right)$, is constant with respect to the final expiration price $S_T$. The payoff functions $\text{Payoff}_k(S_T, K_k)$ are piecewise linear with kinks at their respective strikes $K_k$. Therefore, $\text{P\&L}_{S_{\text{last}} \to S_T}(S_T)$ is also piecewise linear in $S_T$.

Optimal Expiration Price ($S_{\text{MM}}^*$) for the MM

The MM, having established the hedge $H_{\text{last}}$, seeks to maximize $\text{P\&L}_{S_{\text{last}} \to S_T}(S_T)$ with respect to the unknown expiration price $S_T$. The optimal expiration price for the MM, $S_{\text{MM}}^*$, can be found by examining the derivative of this P&L with respect to $S_T$:

$$m(S_T) = \frac{d \text{P\&L}_{S_{\text{last}} \to S_T}(S_T)}{d S_T} = H_{\text{last}} - \sum_k Q_k \frac{d \text{Payoff}_k(S_T, K_k)}{d S_T}$$

Let $\Delta_k^{\text{exp}}(S_T, K_k) = \frac{d \text{Payoff}_k(S_T, K_k)}{d S_T}$ be the option’s delta at expiration. This delta is:

• For a call option: $1$ if $S_T > K_k$ (ITM), $0$ if $S_T < K_k$ (OTM), and can be considered $0.5$ or undefined if $S_T = K_k$.
• For a put option: $-1$ if $S_T < K_k$ (ITM), $0$ if $S_T > K_k$ (OTM).

The sum $\sum_k Q_k \Delta_k^{\text{exp}}(S_T, K_k)$ represents the Net Delta Obligation at Expiration, $\text{NDO}(S_T)$, if the price settles at $S_T$. This is the total number of shares the MM will have to deliver (for short calls) or receive (for short puts that are exercised against them). So, the slope of the P&L function is:

$$m(S_T) = H_{\text{last}} - \text{NDO}(S_T)$$

Since $H_{\text{last}} = \sum_k Q_k \Delta_k(S_{\text{last}}, K_k)$ (the aggregate delta of the options book valued at $S_{\text{last}}$, the price at which the last hedge was made), we have:

$$m(S_T) = \sum_k Q_k \Delta_k(S_{\text{last}}, K_k) - \text{NDO}(S_T)$$

The function $m(S_T)$ is piecewise constant and changes values only at the strike prices $K_k$. The optimal expiration price $S_{\text{MM}}^*$ for the MM is typically a strike price $K_k$ where $m(S_{\text{MM}}^*) \approx 0$, or more precisely, where $m(S_T)$ changes sign from positive (for $S_T < S_{\text{MM}}^*$) to negative (for $S_T > S_{\text{MM}}^*$) for a local maximum. This condition $H_{\text{last}} \approx \text{NDO}(S_{\text{MM}}^*)$ means that the MM’s hedge $H_{\text{last}}$ (taken at $S_{\text{last}}$) most closely matches the net shares needed for settlement at expiration if $S_T = S_{\text{MM}}^*$.

5. Potential for Price Manipulation by the MM

The breakdown of perfect hedging near expiration, as detailed in Section 2.2, presents the market maker (MM) with a critical challenge. Continuous, high-frequency re-hedging often becomes prohibitively costly, operationally complex, and ultimately ineffective against the onslaught of extreme gamma. As a result, the MM is frequently left holding a share hedge, $H_{\text{last}}$, that was established at a prior price $S_{\text{last}}$ and is now ‘stuck’ or significantly misaligned with the true delta of their options book at the current market price $S_c$.

In these final, turbulent moments before expiration, rather than persisting with numerous, potentially futile, and very expensive micro-adjustments to their stock hedge, the MM might strategically evaluate a more decisive alternative: influencing the final settlement price $S_T$. This approach represents a shift from a strategy of continuous risk mitigation (which is failing) to a ‘one-shot’ attempt to optimize the P&L outcome of their substantial, existing options portfolio, given their now-fixed hedge $H_{\text{last}}$. The goal would be to steer the final settlement price $S_T$ away from its anticipated “natural” level $S_T^n$ (which could be proxied by the current market price $S_c$) towards a manipulated price $S_T^m$ that better aligns with their P&L objectives defined by their options book and the fixed hedge $H_{\text{last}}$.

Naturally, such an attempt to influence the market price is not without significant cost. Let $a$ represent the marginal cost for the MM to move the price by one dollar. This cost, $a$, arises from the very same market frictions that complicate continuous hedging: transaction fees, the bid-ask spread, and the market impact of their own large trades (which can move the price adversely as they execute substantial orders). Liquidity constraints, especially for the volumes potentially needed to influence the price as expiration looms, further contribute to this cost. Thus, the factors that make perfect hedging an elusive ideal (Section 2.2) also render price manipulation an inherently costly endeavor. It’s important to note that $a$ is a simplification, as actual manipulation costs are complex, nonlinear, and highly dependent on prevailing market conditions, including volatility and depth.

Crucially, this manipulation cost, $a$, is likely not static; it may decrease as the moment of expiration draws very near. Influencing the closing auction or the price in the final minutes or seconds often requires less capital and sustained market effort than maintaining an artificial price level for an extended period. The market’s natural mean-reverting tendencies have less time to counteract a short-term price push at the very end. This dynamic can make manipulation potentially more effective and less costly per dollar of price movement as expiry approaches, increasing the attractiveness of this ‘one-shot’ strategy at the precise time when continuous hedging is most challenged.

Despite these costs, price manipulation might become an economically rational choice if the potential P&L benefit from steering the settlement price for their entire existing options book outweighs the costs of this single, larger intervention. The marginal P&L gain for the MM from moving $S_T$ is given by the slope of their P&L function, $m(S_T) = H_{\text{last}} - \text{NDO}(S_T)$, as derived in Section 4. Manipulation becomes an attractive proposition if the absolute value of this marginal P&L gain at the ‘natural’ expiration price $S_T^n$ exceeds the marginal cost $a$ of effecting that price change. The condition for initiating manipulation is then:

$$\vert m(S_T^n) \vert > a$$

Substituting the expression for $m(S_T^n)$:

$$\vert H_{\text{last}} - \text{NDO}(S_T^n) \vert > a$$

The term $H_{\text{last}} - \text{NDO}(S_T^n)$ represents the MM’s “excess hedge” (if positive) or “hedge deficit” (if negative) relative to what would be needed if the price expires naturally at $S_T^n$.

• If $H_{\text{last}} - \text{NDO}(S_T^n) > a$: The MM is significantly over-hedged for an expiration at $S_T^n$ (i.e., they hold substantially more shares than needed if the price settles at $S_T^n$) and thus benefits if $S_T$ rises. They have an incentive to push the price up towards their optimal price $S_{\text{MM}}^*$.
• If $H_{\text{last}} - \text{NDO}(S_T^n) < -a$: The MM is significantly under-hedged for $S_T^n$ (i.e., they hold substantially fewer shares than needed, or are even net short shares relative to their needs) and thus benefits if $S_T$ falls. They have an incentive to push the price down towards $S_{\text{MM}}^*$.

The MM would theoretically attempt to move $S_T$ from $S_T^n$ towards $S_{\text{MM}}^*$ as long as the marginal benefit from doing so, $\vert m(S_T) \vert$, continues to exceed the marginal cost $a$. This framework suggests that under specific conditions, particularly near expiration with a ‘stuck’ hedge, influencing the settlement price can become a calculated P&L optimization strategy for the market maker.

6. Deriving a Market Observable for $S_{\text{MM}}^*$ and Manipulation Indicator

Can we leverage this behavior somewhat using only observable data in the market? To make this operational, we assume a single (monopolist) MM whose sold quantities $Q_k$ correspond to the total market open interest (OI) for each option strike $K_k$.

• $Q_k^{\text{call}} = \text{OI}_{\text{C}}(K_k)$ (Open Interest for calls at strike $K_k$)
• $Q_k^{\text{put}} = \text{OI}_{\text{P}}(K_k)$ (Open Interest for puts at strike $K_k$)

The MM’s optimal expiration price $S_{\text{MM}}^*$ is where their hedge $H_{\text{last}}$ approximately equals their Net Delta Obligation $\text{NDO}(S_{\text{MM}}^*)$. For practical observability, we estimate the MM’s current hedge quantity based on current market conditions.

MM’s Estimated Current Delta Hedge ($H_{\text{est}}(S_c, \sigma)$)

We proxy $S_{\text{last}}$ with the current underlying price $S_c$ (or can be picked by the trader in real applications). The time to expiration $\tau_c$ is close to zero for expiring options. The MM’s current total delta hedge, for the purpose of creating an observable indicator, is estimated as $H_{\text{est}}$, using current option open interests and Black-Scholes deltas calculated at $S_c$ and an estimated 0DTE implied volatility $\sigma$:

$$H_{\text{est}}(S_c, \sigma) = \sum_{K_k} \text{OI}_{\text{C}}(K_k) \Delta_{\text{C}}(S_c, K_k, \sigma, \tau_c) + \sum_{K_k} \text{OI}_{\text{P}}(K_k) \Delta_{\text{P}}(S_c, K_k, \sigma, \tau_c)$$

Where $\Delta_{\text{C}}(S_c, K_k, \sigma, \tau_c)$ and $\Delta_{\text{P}}(S_c, K_k, \sigma, \tau_c)$ are the deltas of call and put options, respectively, at strike $K_k$, given current price $S_c$, volatility $\sigma$, and time to expiry $\tau_c$. This $H_{\text{est}}(S_c, \sigma)$ serves as a practical proxy for the theoretically “stuck” hedge $H_{\text{last}}$ when evaluating the situation from the current price $S_c$.

Observable $S_{\text{MM}}^*$ Calculation

An observable estimate for $S_{\text{MM}}^*$ can be calculated as follows:

1. Collect current open interest data: $\text{OI}_{\text{C}}(K_k)$ and $\text{OI}_{\text{P}}(K_k)$ for all relevant strikes $K_k$.
2. Use the current underlying price for $S_c$.
3. Calculate the MM’s estimated current total delta hedge $H_{\text{est}}(S_c, \sigma)$.
4. For each strike price $K_j$ in the option chain, calculate the Net Delta Obligation $\text{NDO}(K_j)$ that would occur if $S_T = K_j$.
5. $S_{\text{MM}}^*$ is the strike $K_j$ for which $\text{NDO}(K_j)$ is closest to $H_{\text{est}}(S_c, \sigma)$. That is, $S_{\text{MM}}^*$ is the $K_j$ that minimizes $\vert H_{\text{est}}(S_c, \sigma) - \text{NDO}(K_j) \vert$. This is the strike where the MM’s P&L slope $m(K_j)$, using $H_{\text{est}}$ as the hedge, is closest to zero.

6.1. An Indicator for Predicting Price Manipulation

Let $S_c$ be the current market price, taken as a proxy for $S_T^n$ (the “natural” expiration price if no manipulation occurs). The MM’s current effective hedge is proxied by $H_{\text{est}}(S_c, \sigma)$. The Current Delta Imbalance (CDI) at $S_c$ is defined as:

$$\text{CDI}(S_c) = H_{\text{est}}(S_c, \sigma) - \text{NDO}(S_c)$$

This $\text{CDI}(S_c)$ represents the slope $m(S_c)$ of the MM’s P&L function $\text{P\&L}_{S_{\text{last}} \to S_T}(S_T)$ evaluated at the current price $S_c$, assuming their hedge is $H_{\text{est}}(S_c, \sigma)$. It indicates the MM’s exposure per dollar change in $S_T$ from $S_c$.

A Manipulation Indicator can be formulated: Price manipulation by the MM is predicted if $\vert \text{CDI}(S_c) \vert > a$, where $a$ is some threshold value we choose that represents the marginal cost of manipulation.

• If $\text{CDI}(S_c) > a$: The MM is (effectively) over-hedged at $S_c$ and has an incentive to push the price upward, towards $S_{\text{MM}}^*$ (which would likely be greater than $S_c$).
• If $\text{CDI}(S_c) < -a$: The MM is (effectively) under-hedged at $S_c$ and has an incentive to push the price downward, towards $S_{\text{MM}}^*$ (which would likely be less than $S_c$).

This theoretical framework outlines how a market maker’s hedging activities, particularly in the high-gamma environment of 0DTE options, can lead to specific P&L profiles and potential incentives to influence the market’s expiration price. The increasing difficulty of perfect hedging as expiration approaches can lead to an effectively “stuck” hedge ($H_{\text{last}}$). This “stuck” hedge, when evaluated against the potential share obligations at different expiration prices ($S_T$), creates a P&L landscape where the MM may have a preferred expiration point ($S_{\text{MM}}^*$). The concept of $S_{\text{MM}}^*$, the MM’s optimal expiration price given their “stuck” hedge, and the Current Delta Imbalance (CDI) (using $H_{\text{est}}$ as a proxy for $H_{\text{last}}$) provide a basis for understanding potential “pinning” pressures around certain strikes. While simplified, particularly in its assumptions about a monopolist MM and manipulation costs, this model offers insights into the complex dynamics market makers face and potentially induce in 0DTE markets when their ability to continuously and cost-effectively re-hedge is compromised by the extreme nature of near-expiry options.