0DTE Gambits: Delta Hedging and Endogenous Market

Manipulation at Bitcoin Options Expiry

Robert Gaudiosia,b , Dustin Weiss*c,d , Z. Ivy Zhouc
a
Macquarie University
b
Derivatives Markets Research Centre, Rozetta Institute (formerly CMCRC-SIRCA)
c
School of Business, University of Wollongong
d
Digital Finance Co-operative Research Centre

Abstract
This paper examines the interaction between delta hedging and market manip-
ulation as options near expiration. Using Bitcoin’s fragmented market to distin-
guish futures hedging flows from manipulative spot market trading, we analyse
data from Deribit, which handles over 90% of Bitcoin options volume. We empiri-
cally identify significant return reversals around expiration on high open-interest
days in at-the-money 0DTE options, consistent with patterns of market manipula-
tion that exploits delta hedging price pressure. These dynamics lead to an annual
wealth transfer of $50 million in options markets and a 250% increase in futures
liquidation losses. Generalising these findings in a theoretical model, we provide
two novel insights: (1) directional hedging pressure in 0DTE options amplifies ma-
nipulation incentives, and (2) options ‘charm’ can disentangle manipulation effects
from pure hedging impacts.
Keywords: 0DTE, Bitcoin options, Option Expiration, Manipulation, Hedging

*
Correspondence: Faculty of Business and Law, Building 40, University of Wollongong, North-
fields Avenue, Wollongong, NSW 2522, Australia. E-mail: dsweiss@uow.edu.au.
Dustin Weiss gratefully acknowledges funding from the Digital Finance CRC which is supported by
the Cooperative Research Centres program, an Australian Government initiative. Robert Gaudiosi
acknowledges funding from the Rozetta Institute.
We are grateful to Tālis J Putnin, š, Chanaka Ganepola, Thomas Kokholm, Luke Johnson, Robert
Webb and the participants of the DFCRC Seminar Series and the 2nd Structured Retail Products
and Derivatives Conference for their insightful feedback and constructive comments.

November 6, 2024
1. Introduction

Distortions in financial markets hinder the efficient allocation of capital, with market
manipulation being a significant contributor. Manipulation occurs when the incentives to
engage in such activities outweigh the associated costs. These costs are substantial, includ-
ing penalties imposed if the manipulative actions are detected and the implicit transaction
costs involved in temporarily distorting market prices away from their equilibrium levels.
Despite these costs, market manipulation remains a persistent issue, as evidenced by high-
profile cases such as the manipulation of the WM/R 4pm FX fix and LIBOR, and recent ev-
idence of potential manipulation in academic research (Baltussen et al., 2023; Griffin and
Shams, 2018).
A prevalent form of market manipulation involves investors who hold contractually
linked derivative contracts and subsequently influence the price of the underlying asset
(Jarrow, 1994; Kumar and Seppi, 1992; Putnin, š, 2020). This paper examines the role of
delta hedging in options markets as a factor that may increase incentives for market ma-
nipulation. Previous research indicates that manipulation is particularly common in the
hour preceding options expiration (Chamberlain et al., 1989; Stoll and Whaley, 1987, 1991),
and that delta hedging activity exerts a more pronounced effect on market prices as options
near expiration (Avellaneda and Lipkin, 2003; Ni et al., 2005). The price movements caused
by delta hedging can resemble those of market manipulation, as both create temporary
price pressures that often revert after the expiration. This similarity complicates the em-
pirical disentanglement of delta hedging from manipulative practices as options contracts
approach expiration (Baltussen et al., 2023; Griffin and Shams, 2018; Ni et al., 2005).1
One of the primary objectives of this study is to explore the relationship between delta
hedging and market manipulation, aiming to provide insights that could assist regulators
in distinguishing between the two. The Bitcoin options market offers a particularly suitable
environment for this analysis for several reasons.
First, cryptocurrencies are predominantly traded on self-regulated exchanges, where
the risk of price manipulation is heightened. Similar to the decision problem outlined in
Gneezy (2005), a manipulator must consider the likelihood of detection and the potential
costs of punishment when deciding whether to manipulate settlement prices. Research
by Ni et al. (2005) and Golez and Jackwerth (2012) underscores the importance of market
maker supervision and the detection risk in traditional markets. We focus on Bitcoin op-

1
Griffin and Shams (2018) provide limited evidence for initial position hedging or investors rolling over
hedged positions. However, their analysis does not explore hedging activity induced by changes in delta due
to gamma and charm.

2
tions traded on Deribit, a self-regulated cryptocurrency derivatives exchange that executes
over 90% of the trading volume in Bitcoin options.2 Unlike regulated derivatives exchanges,
where the expiration effect has been extensively tested (e.g., Batrinca et al., 2020; Chiang,
2014), Deribit’s unregulated nature makes it more susceptible to potential manipulation,
particularly during the critical period when the settlement price is determined. Second,
Deribit’s options market makers exclusively use the Deribit perpetual futures contract for
hedging activities (Alexander et al., 2023a). This unique setting allows us to separately
examine hedging activities in futures and potential manipulation on spot exchanges that
Deribit uses to determine the expiry price of the options. Third, the types of investors
involved in cryptocurrency markets exhibit a higher tendency for risk-taking or gambling
(Kim et al., 2020), which is expressed through a high proportion of trading activity in 0DTE
(zero days to expiration) options. These options have changed the frequency of expiration
from months or quarters to daily, and consequently also the frequency of potential settle-
ment price manipulation.
Our analysis covers 1,059 option expiry days from January 1, 2021, to December 31, 2023.
We begin by isolating days that fall within the top decile of at-the-money open interest. We
focus on open interest because it must be sufficiently large for price pressure due to both
hedging and market manipulation to emerge. We show that Bitcoin’s notional open inter-
est in the options market is substantially larger than the turnover on spot exchanges that
contribute to Deribit’s Bitcoin index, which determines the settlement price of the options.
Additionally, we show that hedging activity on Deribit’s perpetual futures is significantly
higher in the hour before expiry on high open interest days, accompanied by unusually
high turnover on the spot exchanges that contribute to the settlement price — suggestive
of potential manipulation.
This increased trading activity before option expiry causes predictable temporary price
pressures. Specifically, we document significant negative Bitcoin spot returns beginning
one hour before option expiration, continuing until the 8 am expiry, followed by a sharp
reversal that persists for the subsequent hour. High open interest is shown to be a signifi-
cant predictor of both the negative return prior to expiration and the subsequent reversal.
This pattern holds economic significance as it enables the formulation of profitable trading
strategies, even after accounting for transaction costs. For instance, a simple spot trading
strategy on high open interest days — shorting Bitcoin at 7 am, buying back and taking a
long position at 8 am, and then selling at 9 am—yields an annual Sharpe ratio of 0.92 after
incorporating transaction costs.

2
See Deribit Sees 17% Growth in Crypto Derivatives Trading Volume in August, Led by Options

3
 Turning to explanations for these anomalies, we develop a theoretical model designed
to separate the effects of delta hedging from those of market manipulation before option
expiry. Our generalized approach ensures that our findings are relevant across different
market settings. This model incorporates delta-hedging option traders alongside a profit-
maximizing trader who holds an open option position. The price of the underlying asset
is determined endogenously, influenced by both hedging pressure and potential manipu-
lation. A direct outcome of the model is that the incentive for manipulation increases as
a function of the traders’ open interest, consistent with earlier work (e.g., Jarrow, 1994;
Kumar and Seppi, 1992). Moreover, we demonstrate that market manipulation becomes
less costly when hedging-induced price pressure is strongest. This finding is closely related
to the option gamma (the rate of change of delta with respect to the price of the under-
lying asset) and, consistent with Golez and Jackwerth (2012), the option charm (the rate
of change of delta as the option approaches expiration). Consequently, we highlight that
a manipulator can exploit hedging activity by "poking" or triggering the hedgers with a
small price move, which is then amplified by the resulting hedging pressure. This means
that a trader who understands the direction of hedging pressure can significantly reduce
the cost of manipulating the settlement price, thereby increasing the incentive to engage
in manipulation.
Using the model, we are able to distinguish between price pressure caused by hedging
and manipulation using the option charm. The intuition for this result is that delta hedg-
ing price pressure resulting from option gamma requires an initial movement in price. In
the absence of manipulation, this movement in price is equally likely to be up or down.
Therefore, systemic unidirectional price pressure at option expiration should be consistent
with pricing pressure due to option charm. If price pressure is consistently in the oppo-
site direction to charm then it can only be due to an external force, which we conjecture is
potential market manipulation.
Using our model predictions, we investigate whether return patterns around Bitcoin op-
tions expiration are driven by delta hedging dynamics or market manipulation. We begin
by exploring the role of gamma hedging pressure in return patterns around Bitcoin option
expirations. Consistent with our model’s expectations, we find that high open interest days
correspond to increased short gamma pressure, where delta hedging trades amplify spot
price movements. To estimate market maker positions, the primary participants involved
in delta hedging (Cho and Engle III, 1999), we use each option’s trading history and assume
market makers are on the opposite side of the aggressor Fournier and Jacobs (2020). This
approach reveals a negative aggregate gamma imbalance on average, with imbalances sig-

4
nificantly larger at higher levels of open interest.3 These results suggest that delta hedging
is a contributing factor in return patterns around expiration. However, within our model
framework, short gamma exposures also heighten incentives for market manipulation.
We further investigate, using the option charm, whether the return patterns can be at-
tributed to pure delta hedging or market manipulation that exploits delta hedging. Specifi-
cally, we highlight that delta hedge rebalancing due to initial option charm does not explain
the return patterns. This points to market manipulation as a driver of return patterns
around expiration. Within our model, manipulative trading exploits delta hedging flows,
specifically when open interest is high, amplifying return patterns in a way that aligns with
manipulation rather than isolated hedging pressure.
Our results are economically meaningful. We ascertain the annual wealth transfer from
call holders to call writers and from put writers to put holders to be approximately $50
million, using hypothetical settlement prices at 7 am. Additionaly, we find a surge in forced
liquidations in Deribit’s perpetual contract just before option expiry. On days with high
open interest, around $50 million worth of assets are liquidated in the hour before expiry.
Further analysis of liquidations on other crypto derivative exchanges reveals that on days
with high open interest, liquidations of bullish Bitcoin positions total $367 million in the
hour before options expire, marking a 250% increase compared to non high open interest
days.
We conduct several robustness checks to rule out alternative explanations, such as fun-
damental news and the well-documented "pinning" effect (Ni et al., 2005). Specifically, our
analysis shows that on days with high open interest and negative gamma exposure, De-
ribit’s Bitcoin settlement prices do not exhibit a tendency to cluster at the nearest strike
price. Instead, on these days, settlement prices are, on average, further away from the near-
est strike. However, we validate the theoretical predictions from (Avellaneda and Lipkin,
2003), who suggest that positive gamma exposure, when combined with high open interest,
significantly increases the likelihood of pinning.
We also show that the observed increase in trading activity and liquidations before ex-
piry is not present in the period before the expiration of Bitcoin options traded on CME.
There are three potential explanations for this: (1) lower open interests and trading ac-
tivity on the CME, (2) greater Bitcoin spot liquidity during the settlement window on the
CME compared to Deribit, and (3) greater regulation on CME. Settlement in a more liquid
time increases the difficulty of price manipulation, as does regulation due to the expected

3
This negative gamma imbalance reflects heightened demand among end-users for long call options and
short put options, supporting the notion that Bitcoin options are predominantly used for bullish speculation
Alexander et al. (2023b).

5
penalty imposed if caught. While the amount of open interest on Deribit is determined in a
free market setting, regulators and market designers can directly influence the latter two
reasons.
Our study contributes to the literature in three important ways. First, we document
that delta hedge rebalancing leading into option expiration can increase the profitability
of market manipulation. While many studies have examined delta hedging around option
expiration, we extend these works by highlighting that market manipulation can emerge
endogenously via delta hedging, reducing the cost function of the manipulator. This helps
explain why researchers continue to find evidence of market manipulation, particularly in
derivative markets (e.g., Griffin and Shams, 2018; Baltussen et al., 2023). Further, it sug-
gests that these option markets are ideal ’hunting grounds’ for manipulators on days with
sufficiently large hedging pressure, and these should be subject to increased regulatory
scrutiny.
In light of this, our second contribution outlines a method to isolate market manipula-
tion from delta hedging, a task that has commonly proved difficult among researchers and
practitioners (Alexander and Cumming, 2022; Cumming et al., 2018), especially in highly
fragmented cryptocurrency markets. Our approach isolates the option charm and claims
that price moves in the opposite direction to charm are consistent with potential market
manipulation. A toolkit that assists in the identification of market manipulation is impor-
tant for effective monitoring and surveillance. From the perspective of the manipulator, a
tool such as this increases the likelihood of being caught, thereby increasing the expected
costs associated with manipulation and reducing overall incentives to manipulate.
Our third and final contribution is to examine our predictions in Bitcoin options, doc-
umenting evidence aligned with delta hedging and manipulation at expiration. In this
sense we contribute towards a set of literature that has examined the pricing of Bitcoin op-
tions (Alexander et al., 2023a) and the informativeness of Bitcoin option trades (Alexander
et al., 2023b).4 Our results highlight the importance of regulation in the cryptocurrency
options market that can reduce the incentives for market manipulation. This issue gains
additional significance given the recent SEC approval of several Bitcoin spot ETFs, with
the major concern relating to potential spot market manipulation.5 The SEC has also ex-
pressed manipulation concerns regarding options on Bitcoin spot ETFs, as reflected in the
recent application re-filed by the Cboe.6

4
We also contribute towards a set of growing literature that examines 0DTE option trading (Beckmeyer
et al., 2023; Brogaard et al., 2023) by demonstrating that these options can be prone to market manipulation.
5
See https://money.usnews.com/investing/articles/new-spot-bitcoin-etfs-to-buy
6
See https://cointelegraph.com/news/movement-on-bitcoin-etf-options-as-cboe-re-files-application

6
 The remainder of this article is organised as follows. Section 2 outlines the model; Sec-
tion 3 introduces the data; Section 4 analyses the returns around Bitcoin option expiration;
Section 5 identifies potential explanations for the return patterns, including hedging and
manipulation; Section 6 overviews the economic implications; and Section 7 concludes.

2. Model

2.1. Setting
The market setting involves a security with tradeable option contracts at n different
strike prices. One call and put option are available at each strike price. Within this frame-
work, there are two types of agents: a risk neutral profit maximising trader and delta
hedgers. The profit-maximizing trader (potential manipulator) holds an open option posi-
tion and chooses to manipulate the price of the underlying if the gain from doing so exceeds
the cost. There are X hedgers that are infinitely risk averse and, as such, perfectly delta
hedge their option positions.7 The profit-maximising trader has knowledge of the net po-
sition of delta hedging traders.8 In the model, delta hedgers and the profit-maximising
trader trade to maximise utility across N equally spaced periods t1 , t2 , ..., tN .

2.2. Price Impact Model

We model the market price within the framework of Obizhaeva and Wang (2013), where
price impact consists of a permanent component θ and temporary component λ that decays
at an exponential rate β . Trades via delta hedging or manipulation incur market impact
which in turn impacts the option settlement price. Hedging trades exhibit a temporary
price impact and a permanent price impact, consistent with Huh et al. (2015) document-
ing options market maker delta hedging trades to be a conduit of private information.9
Manipulator trades exhibit only a temporary impact on prices since these trades only tem-
porarily distort equilibrium prices (e.g., Peng and Röell, 2014). Initial price impact is linear
in trade size, a necessary modelling assumption to avoid quasi-arbitrage (Huberman and
Stanzl, 2004).10 The underyling price unaffected by hedging or manipulator trades follows

7
The infinite risk aversion is equivalent to hedging pressure being exogenous to the model and ensures
that delta hedgers always rebalance their hedge. Our focus is to examine market dynamics given delta hedg-
ing trades where the decision to delta hedge is exogenous.
8
This assumption is not unreasonable since delta hedging positions can be approximated by estimating
market maker option positions (Fournier and Jacobs, 2020). For example, Amberdata provides financial
institutions with gamma exposures of Bitcoin options market markers.
9
In general, option hedging trades impacting spot prices is documented in Basu et al. (2006), Platen and
Schweizer (1998), and Wilmott and Schönbucher (2000).
10
As documented in Huberman and Stanzl (2004), this restriction is only required for permanent price
impact. Several other studies document price impact to exhibit concavity in trade size (Evans and Lyons,
2002; Hasbrouck, 1991). This concavity has no qualitative impact on model conclusions.

7
Brownian motion with volatility σ . Therefore, the market price at time t is given by

t
X t
X t
X
−β(t−ti )
pt = p0 + σ Wi + Ai (θ + λe )+ yi λe−β(t−ti ) (1)
i=1 i=1 i=1
| {z } | {z }
Hedging Trades Manipulator trades

where Ai is the amount traded by hedgers and yi is the amount traded by a manipulator at
time ti . The option settlement price is therefore given as
Z tN
1
S= Pt ∂t (2)
tN − tp tp

where tp is the beginning of the settlement window.

2.3. Delta Hedging Trades

For each strike kj , a hedger Xi has call exposure Cij and put exposure Pij . A negative
value of Cij or Pij indicates a short position. Aggregate hedger position at each strike is
PX
Hj = i=1 Cij + Pij . Each trading decision is characterized by a price and time pair (ti , Pti )
where time is the time to option expiration and the price is a function of time as per Eq.
(1). Hedgers delta-hedge their positions by trading in the underlying an aggregate amount
equal to
n
X  
A(ti+1 ) = −Hj ∗ ∆(t̄i ,pti ),kj − ∆(t̄i−1 ,pti−1 ),kj (3)
j=1

where ∆(t̄i ,pt ),kj is the options delta associated with t̄i = tN − ti (time to expiry) and price
i
pti and strike kj .11 The options delta is the rate of change of a call option price w.r.t the
price of the underlying ∂V
∂pti
which is calculated using the Black-Scholes model (Black and
Scholes, 1973) as

σ2
      
1 pti
∆(t̄i ,pt ),j = N √ ln + r+ (t̄i ) (4)
i σ t̄i kj 2

where N is the cumulative distribution function of the standard normal distribution, σ is

implied volatility, and r is a risk-free interest rate.12 Note that we can express Eq. (3) in
terms of the delta hedge rebalance due to changes in price (gamma) and the delta hedge

11
This specification of delta hedge rebalancing is general and aligns closely with that of Adams et al.
(2024). However, our approach is broader, as Adams et al. (2024) only accounts for delta hedge rebalancing
due to gamma (price changes), while we additionally incorporate rebalancing due to changes in time to expiry
(charm).
12
Implied volatility and the risk-free rate are considered as constant for the purposes of this exercise to
focus on changes in price pti and time to expiration t̄i .

8
rebalance due to changes in time to expiry (charm)

n
X
A(ti+1 ) = P P gamma(j, pti , pti−1 , Cj , Pj ) + P P charm(j, ti , ti−1 , Cj , Pj ) (5)
j=1

+ f (j, pti , pti−1 , ti , ti−1 , Cj , Pj )

| {z }
Interaction Term

where P P gamma(j, pti , pti−1 , Cj , Pj ) is given as

∂∆
P P gamma(j, pti , pti−1 , Cj , Pj ) = −(Cj + Pj ) (pt − pti−1 ) (6)
∂p i

and P P charm(j, t̄i , t̄i−1 , Cj , Pj ) is given as

∂∆
P P charm(j, t̄i , t̄i−1 , Cj , Pj ) = −(Cj + Pj ) (t̄i − t̄i−1 ) (7)
∂ t̄

The interaction of the effects of charm and gamma f (pti , pti−1 , ti , ti−1 , Cj , Pj , j) is zero if
pti = pti−1 or t̄i = t̄i−1 . This is intuitive and simply states that an interaction effect only
exists when both individual effects are present. We also define another variable Γ, related
to P P gamma, which measures the extent to which delta hedge rebalancing in response to
changes in the underlying price either amplifies or dampens price movements.

n
X ∂∆
Γ= (Cj + Pj ) (8)
j=1
∂p

where Γ < 0 indicates delta hedge rebalancing to exacerbate price moves (short gamma)
and Γ > 0 indicates delta hedge rebalancing to contract price moves (long gamma).

2.4. Manipulator Optimisation Problem

The manipulator chooses to trade if their benefit from trading exceeds the cost of doing
so. The functional form for the manipulators optimisation problem depends on the type
of option they hold. We focus our analysis on the manipulator being exposed via (1) short
call options and (2) long put options.13 In this example, the option premium is sunk, so the
profit-maximizing trader who is short a call option determines the amount to trade at each

13
For both of these positions, a manipulator is incentivised to push the price down since that increases
their payoff. A similar exercise can be performed assuming a trader who is long a call option and/or short a
put option to focus on upside manipulation.

9
time t via  
∗
y = arg min x ∗ max E(S(y)) − k̂, 0 + c(y) (9)
y

For a trader who is long a put option, their optimisation problem is

 
y ∗ = arg max x ∗ max k̂ − E(S(y)), 0 − c(y) (10)
y

where x is the number of contracts sold by the manipulator (open interest), and k̂ is the
strike price of the call option. The cost of trading C is equal to the total transaction cost of
all trades
N
X
c(y) = g(yi ) (11)
i=1

where the functional form of the cost function g is proportional to the instantaneous price
impact of a trade
g(w) = vw, v > 0 (12)

where v > 0 represents the liquidity premium of a trade that encompasses instantaneous
temporary price impact (seperate to λ) and the bid-ask spread (Almgren and Chriss, 2001;
Horst and Naujokat, 2011). A risk neutral profit maximising trader will then manipulate
the price when ||y ∗ ||2 > 0 and the expected benefit from doing so outweighs the cost. For a
short call option the condition is
 
x ∗ (max(pt0 − k̂, 0) − max(E(S(y ∗ )) − k̂, 0) − C(y ∗ ) > 0, ||y ∗ ||2 > 0 (13)

and for a long put option14

 
x ∗ max(k̂ − E(S(y ∗ )), 0) − max(k̂ − pt0 , 0) − c(y ∗ ) > 0, ||y ∗ ||2 > 0 (14)

2.5. Model Results

The first set of results relates to the impact of open interest and hedging pressure on
manipulation incentives and are formalised in Lemma 1 and 2. Mathematical proofs are
provided in Appendix 7. Before presenting these results, we introduce a technical assump-
tion regarding hedging pressure’s impact on option payoff maximisation.

14
It is worth noting that these conditions exclude cases where ||y ∗ ||2 > 0 even when S > pt0 . This scenario
can occur if delta hedging trades reduce the profit-maximizing trader’s payoff, making manipulative trading
beneficial to limit losses. In practice, however, a risk-neutral trader aware of hedging pressure could instead
sell the option contract, avoiding such losses. Consequently, we do not focus on this case in our analysis.

10
Assumption 1. In expectation, maximum option payoff for the manipulator is not achieved
in the absence of manipulation. In the case of a short call this requires that E (S(y = 0))−k̂ >
0 and in the case of a long put requires E (S (y = 0)) > 0.

This assumption allows us to restrict attention to cases where manipulation could poten-
tially be profitable by excluding scenarios where hedging pressure alone brings about the
maximum payoff. In these scenarios, the manipulator has no additional incentive to trade,
as natural price impacts from hedging would already optimise their payoff.
Lemma 1. Market manipulation occurs when a manipulator is endowed with open interest
x > x∗ where x∗ is increasing in the liquidity premium (v ).

Lemma 1 holds because a trader’s option payoff scales with their open interest, effectively
amplifying the incentives for manipulation. With sufficiently high open interest, these ben-
efits outweigh the associated liquidity costs (v ) of trading. This result aligns with previous
findings that incentives for market manipulation increase with open interest and market
illiquidity (e.g., Kumar and Seppi, 1992; Jarrow, 1994).
Lemma 2. Hedging pressure (A) increases the net gain from downside market manipula-
tion when for H > 0

∃y < 0
s.t. E (S (y(H = H), H = 0)) − E (S(y(H = H))) > pt0 − E (S(y = 0, H = H)) (15)

and the net gain from market manipulation is increasing in

(E (S(y(H = H), H = 0)) − E (S(y(H = H)))) − (pt0 − E (S(y = 0, H = H))) (16)

Lemma 2 arises from the observation that hedging trades have price impact, influencing
market prices in a way that can enhance a potential manipulator’s option payoff. Eq. (15) in-
dicates that hedging pressure increases the incentives for market manipulation only when
manipulator trades induce additional hedging pressure. Eq. (16) then states that the net
gain from market manipulation is increasing in the amount of additional hedging pres-
sure that manipulator trades induce. This manipulation can occur through two distinct
channels.
The first channel is related to option gamma. For instance, if P P gamma < 0, a manip-
ulator may drive the price closer to an option’s strike price, thereby increasing the hedging
pressure due to gamma. In this scenario, their trading exerts a direct market impact of yλ
that decays exponentially at a rate of β . Additionally, this trading can indirectly trigger
further hedging trades with both permanent price impact θ and temporary price impact λ,
also subject to exponential decay. The manipulator’s trading costs are only proportional to
their own activity (vy ), but the ensuing hedging trades enable a larger option payoff.

11
 The second channel involves option charm. To illustrate this, consider a market situa-
tion where P P charm > 0. In such cases, net hedging trades exert a positive price impact,
which ultimately reduces the option payoff for a potential manipulator. A manipulator can
reverse the direction of hedging pressure through two strategies. The first is applicable
when Γ < 0 (indicating that delta hedgers are short gamma). Here, delta hedge rebalanc-
ing due to option gamma amplifies price movements, allowing a manipulative sell trade to
trigger additional selling pressure via hedge trades. The second strategy entails flipping
P P charm such that P P charm < 0. Since option charm is typically greatest near expira-
tion for at-the-money options, creating price impacts that cause the reference price to fall
below the nearest strike can lead to P P charm < 0.
These insights regarding the conditions when P P charm > 0 culminate in Proposition
1.

Proposition 1. In expectation, a market setting where

Xn
sgn(E(S(y)) − pt0 ) ̸= sgn( P P charm(j, t1 , t0 , Cj , Pj )), Γ < 0 (17)
j=1

is indicative of market manipulation.

Proposition 1 asserts that, in the absence of manipulator trades, option charm dictates the
direction of hedging pressure. Since the random component of price movements average to
zero, any price pressure opposing charm indicates market manipulation (in expectation).
A potential manipulator can instigate hedging trades by creating price impact, thus bene-
fiting from subsequent delta hedge rebalancing.15
Figure 1 illustrates a mock market setting when P P charm > 0 and Γ < 0, highlighting
how open interest and hedging pressure affect the net benefits of market manipulation in
the context of a short call option.16

[Figure 1 here]

Figure 1 encapsulates Lemma 1, Lemma 2, and Proposition 1. Both open interest (Lemma
1) and hedging pressure (Lemma 2) increase the incentive for market manipulation. Both
these effects remain in the case that P P charm > 0, a situation uniquely indicative of ma-
nipulation (Proposition 1). This distinction can be made despite subsequent net selling from

15
Proposition 1 generally holds for Γ > 0, but this is not guaranteed. When Γ > 0, the price may rebound
after charm hedging pressure, provided that delta hedge rebalancing due to gamma outweighs that due to
charm. Nonetheless, significant price movements can still be driven only by market manipulation.
16
The trade-offs are similar for a manipulator holding a long put option.

12
delta hedge rebalancing, which has complicated the differentiation between delta hedging
and market manipulation in prior studies (Baltussen et al., 2023; Ni et al., 2005).

3. Data and summary statistics

We source transaction data on Bitcoin options directly from the Deribit exchange for the
period from 1 January 2021 to 31 December 2023. The exercise-settlement value of options
is calculated using the average of the Deribit Bitcoin index over the last 30 minutes before
expiry. Table 1 presents summary statistics on the Bitcoin index and its daily returns in
panel A and on Deribit’s option data in panel B. The Deribit Bitcoin index’s mean level is
$36,606. During our sample period, Bitcoin traded in a wide range, spanning from a low of
$18,826 to a high of $67,641. Panel B of Table 1 shows that the average daily trading activity
is greatest in out-the-money call and put options, with 3,361 and 2,881 trades, respectively.
There is also a significant fraction of trading activity in the few strikes that have a price
within 2.5% of the spot index price one hour before expiration at 7 am, i.e. at-the-money
options. A consistent pattern emerges in the percentage of 0DTE options traded for these
at-the-money strikes, averaging around 38%.

[Table 1 here]

In Figure 2, we plot the evolution of 0DTE option trading throughout our sample period.
Panel A shows that the proportion of options trades with less than 24 hours to expiry across
all moneyness categories has increased in the second half of the sample. However, as shown
in Panel B, the proportion of 0DTE option trades in at-the-money options has been consis-
tently high. This suggests a regular inclination towards trading very short-term options
with the highest possible gamma exposures. Bitcoin options with shorter maturities and
closer strikes are, therefore, often more liquid, as highlighted by Hoang and Baur (2020).

[Figure 2 here]

An important question is whether open interest in Deribit’s Bitcoin options market is

sufficiently large to have a meaningful impact on Bitcoin price formation. Table 2 provides
summary statistics of open interest, split between calls and puts and option moneyness.
Panel A reports summary statistics for open interest measured using the number of out-
standing contracts. We observe that the normalised open interest imbalance is skewed
towards call options irrespective of the moneyness, suggesting that Bitcoin options are pri-
marily used for bullish speculation. Moreover, the imbalance is greatest for at-the-money
options. Since the price of Bitcoin traded in a wide range during our sample, we also report

13
summary statistics for the notional open interest in Panel B of Table 2. Aggregated across
all strikes and moneyness categories, the average daily expiring notional open interest is
around $ 500 million.

[Table 2 here]

Next, we compare the open interest in the options market to the turnover on several
spot exchanges in the hour before option expiration. Table 3 reports summary statistics
on turnover on major spot exchanges, including Binance, Bitfinex, Bittrex, Coinbase, FTX,
Gemini, HTX Global, Kraken, and OKX in Panel A, on the Bitcoin index constituents, in-
cluding Bitstamp, Coinbase, and Kraken in Panel B,17 and on exchanges not included in
Deribit’s Bitcoin index calculations but used by Deribit as backup markets (Bitfinex and
Gemini) in Panel C. We source this data from various sources. Transaction-level data for
Bitstamp, Kraken, and Bitfinex are sourced directly from the respective exchanges. Hourly
trading volume for the remaining exchanges is sourced from CryptoQuant.

[Table 3 here]

The average about-to-expire open interest of $ 500 million does not seem very large
compared to the aggregate spot turnover of $ 192 million in the hour before expiry. However,
a manipulator would naturally try to influence the price of spot trading pairs that are used
to determine the settlement price of the option. The open interest becomes a lot more
meaningful in comparison to the turnover on these index constituents, as reported in Panel
B of Table 3. This is especially the case in the context of our analysis, which focuses on the
top decile of open interest.

4. Spot returns and trading volume around option expiration

The empirical analysis considers the unfiltered full sample and a subsample of days
in which at-the-money open interest for 0DTE options is in the top decile of our sample
period. The reason to focus on at-the-money open interest is because delta hedging pressure
is stronger due to a larger option gamma and charm. Additionally, there may be greater
incentives for an option writer to manipulate the price in order to avoid paying the option
holder, and for an option holder to manipulate the price to earn a non-zero payoff. The
focus on high open interest is for the same reasons (greater hedging pressure and incentive
for manipulation) and is consistent with the model presented in Section 2 as well as prior

17
We were not able to source data from one of the four index constituents (LMAX Digital).

14
research (Avellaneda and Lipkin, 2003; Golez and Jackwerth, 2012; Jarrow, 1994; Ni et al.,
2005). Open interest is calculated as the sum of open interest for calls and put options at
all eligible strikes.
n
X
OIt = Cit + Pit (18)
i=1

where n is the number of strikes, Cit and Pit are call open interest and put open interest
respectively expiring on day t with strike price i.

4.1. Index returns around expiry

Returns are log returns (reported in % form) calculated using prevailing midquotes for
5 minute intervals from 6am to 10am on each day. Cumulative returns are then computed
as the sum of individual return observations over a given period

n
X Pt,i
CRt = ln( ) (19)
i
Pt,i−1

where Pt,i is the prevailing midquote on day t at time i. Figure 3 plots average cumulative
returns around option expiration for the full sample and for the top decile of open interest.

[Figure 3 here]

A striking downward price move is observed for the top decile of open interest starting about
1 hour before option expiration. The price then reverses and stabilises after about 1 hour
around the initial 7am price level. This price move is not evident for the full sample. Table
4 reports mean and median 20 minute returns across the period from 7am to 9am for the
full sample and high open interest days. Statistical significance tests are conducted on the
mean using a two sided t-test and on the median using a Wilcoxon signed rank test.

[Table 4 here]

For the full sample, all returns are insignificant using both the mean and median. For
high open interest days, returns are negative and statistically significant for the 40 min-
utes before expiration.18 Returns on high open interest days are positive and statistically
significant for the 20 minutes after expiration with less statistical significance obtained for

18
In the 20 minutes directly before expiration, only median returns obtain statistical significance. Other
than the existence of outliers, an alternative explanation for the lack of significance for mean returns is
that hedging pressure and market manipulation is reduced once the settlement window starts. This may
occur because changes in the underlying price have less impact on the settlement price - reducing gamma
hedging pressure. In the context of our model, this has the flow on effect of reducing incentives for market
manipulation.

15
the subsequent 40 minutes. Overall, Table 4 complements the descriptive evidence pre-
sented in Figure 3 of a negative return and subsequent reversal centred around the option
expiration time.
An important question is whether open interest actually has predictive power over re-
turns. To answer this question, we estimate the following two OLS regression models to
understand how open interest impacts pre-expiration returns

ReturnP ret = β0 + β1 OpenInterestt + β2 V olatilityt + β3 V olumet (20)

+ β4 IV ratiot + β5 Surpriset−1 + β6 W eekend + τi + ϵt

and post-expiration returns

ReturnP ostt = β0 + β1 ReturnP ret + β2 OpenInterestt (21)

+ β3 ReturnP ret × OpenInterestt + β4 V olatilityt + β5 V olumet
+ β6 IV ratiot + β7 Surpriset−1 + β8 W eekend + τi + ϵt

where Return Post is the log return (reported in %) from option expiry to 2 hours after option
expiry on day t, Return Pre is the log return (reported in %) from 1 hour before option expiry
to option expiration, and Open Interest is a dummy variable equal to 1 if the option open
interest is in the top decile for the observation window. The remaining variables are control
variables including Volatility computed from 7 am to 10 am as realised volatility using the
Garman and Klass (1980) estimator, spot market Volume (turnover) measured from 7 am to
10 am on Bitstamp in US dollars, and investor IV ratio as the ratio of implied volatility for
the nearest strike call option above the index price at 7 am over the implied volatility for the
nearest strike put option below the index price at 7 am. Surprise is lagged by one day and is
a dummy variable equal to one if the absolute value of the global economic surprise index
is greater than its 90%ile in the sample period19 , Weekend is a dummy variable equal to 1
for weekends capturing changes in volume and return patterns on weekends (Baur et al.,
2019), and τi are fixed effects for year i. The coefficients and corresponding t-statistics of
the regression specifications in Eq. (20) and (21) are reported in Table 5 with and without
control variables.

[Table 5 here]

For Eq. (20), OpenInterest is significantly negative at a 5% level. The only significant
control variable is the dummy representing weekends which is negative, implying that the

19
Using Surprise at levels does not have any significant impact on results.

16
return prior to option expiration is more negative on weekends. The lack of significance of
Surprise in any regression specification indicates that fundamental news is not a primary
driver of the return reversal. For Eq. (21), the interaction between ReturnPre and Open-
Interest is negative and statistically significant at a 1% level. This variable captures the
reversal effect, a negative coefficient implying that returns tend to reverse on days when
open interest is higher. These results support the hypothesis that high open interest can
predict returns at option expirations. More specifically, high open interest in at-the-money
options is associated with a negative return in the hour before expiration followed by a
positive return in the subsequent hour.

4.2. Trading strategy

Turning to the economic significance of these results, we consider a simple trading strat-
egy on high open interest days that shorts Bitcoin at 7am, buys back and takes a long po-
sition at 8am, and closes out at 9am. Trading strategy returns are computed with and
without transaction costs, where transaction costs assume purchases are executed at the
prevailing ask and sales at the prevailing bid. Table 6 reports trading strategy results con-
sidering a switch from a short position to a long position at 7:30am, 8:00am, and 8:30am
to assess whether the option expiration at 8:00am is the optimal switching time. We also
calculate a bootstrapped p-value by resampling with replacement from the original dataset
of high open interest days. The p-value (P ) is then computed as

500
X
P = pi (22)
i=1

where 
1 if p > 0.05
i
pi = (23)
0 otherwise

and pi is the p-value from the ith resampled dataset. Table 6 also documents the percent-
age of profitable days using the strategy and an annual Sharpe ratio that is computed as
expected return scaled by standard deviation multiplied by the square root of yearly obser-
vations, as in Lucca and Moench (2015).

[Table 6 here]

Expected returns (in %) from the strategy are highest using an 8am switching point with
and without transaction costs, implying that option expiration is an economically mean-
ingful event. Annualised Sharpe ratios are 1.09 without transaction costs and 0.92 with
transaction costs, around 60% of days enabling a profit. Bootstrapped p-values are 0.03

17
without transaction costs and 0.05 with transaction costs, enabling the robustness of re-
sults.

5. Potential explanations

5.1. Hedging
5.1.1. Gamma hedging pressure
In this section, we investigate whether hedging trades in spot markets impact the price
of Bitcoin before option expiration. Delta hedging by risk averse option market makers
is a well documented practice where traders protect their position against movements in
the underlying (Huh et al., 2015; Ni et al., 2021). For hedging to explain directional price
changes before option expiration, delta hedge rebalancing should excacerbate price moves.
This depends on the option positions of delta hedging market participants, the most likely
candidate being option market makers (Cho and Engle III, 1999). Following Fournier and
Jacobs (2020), we proxy market maker positions for each contract by summing up the neg-
ative of daily end users’ net demand over the life of each contract.
We determine whether delta hedge rebalancing in response to a change in price exac-
erbates or dampens spot price movements by empirically estimating Γ (Eq. 8). If Γ > 0
then delta hedging dapmpens price movements by causing hedging trades in the opposite
direction. On the contrary, if Γ < 0 then delta hedging trades exacerbate price moves.
The intuition is that delta hedging a long call or put position involves trading against price
moves, whereas delta hedging a short call or put position trades involves trading in the
same direction as price moves.
Gamma hedging holds up as an explanation for the documented return reversal around
option expiration if Γ is negative and if Γ is more negative on high open interest days. Table
7 reports average net gamma (Eq. 8), and proxied at-the-money market maker positions for
the full sample and high open interest days.

[Table 7 here]

Table 7 demonstrates that Bitcoin option market makers on Deribit are primarily short call
options. Some evidence also exists, albeit not statistically significant, that options market
makers are long put options. Importantly, Γ is significantly negative and more negative on
high open interest days, following from option market makers holding short call options.
We next investigate how hedging pressure due to gamma varies throughout the day.
Figure 4 documents price pressure to gamma to peak at 7 am before reducing after options
expiration. This follows because option gamma is largest for at-the-money options when
the option approaches expiration. The reason for the higher option gamma is that as an

18
at-the-money option approaches expiration small changes in price have a more significant
impact on the probability of an option expiring in-the-money, represented by delta. After
expiration, the exposure to these options disappears, and gamma exposure is reduced. This
reduction in gamma exposure follows because the delta of options that expire on later days
is less sensitive to changes in price.

[Figure 4 here]

In summary, delta hedging by option market makers is a plausible explanation for the
spot Bitcoin return reversal around option expiration, as documented in this paper. Price
pressures can be exacerbated by market makers holding negative gamma exposure, which is
more pronounced on high open interest days. However, as highlighted by Lemma 2, hedging
pressure increases the incentive for a trader to manipulate the price as it can amplify price
moves. Therefore, delta hedging does not necessarily rule out market manipulation.

5.1.2. Hedging with perpetual contracts

Having established the potential hedging pressure, we now examine how it affects trad-
ing activity around option expirations. We focus on trading volume in Deribit’s perpetual
contract. Alexander and Imeraj (2023) provide evidence that perpetuals are the most effec-
tive hedging instruments and suggest that Deribit options market makers solely use the
Deribit perpetual for hedging activities. Hedging on the same exchange has at least two
advantages: First, it results in lower margin requirements due to cross-margining between
option exposures and offsetting positions in the perpetual. Second, if an exchange is offline,
both the perpetual position and the option prices won’t update so that the hedger is not
exposed to one-sided mark-to-market losses.
Panel A of Figure 5 plots the fraction of daily turnover measured in half-hour intervals
for both the entire sample and for days in the top decile of at-the-money open interest. On
days with high open interest, we observe a spike in trading activity around option expiration
at 8 am. Specifically, the fraction of daily turnover in the 30-minute window before expiry is
approximately three times as high as it is on an average day. This highlights that hedging
needs are particularly high around expiry. In Panel B of Figure 5, we examine the trading
activity in the hour before expiry in relation to at-the-money open interest. We observe a
strikingly positive correlation between open interest and trading activity before the expiry
of Deribit options.

[Figure 5 here]

19
 Next, we examine whether these hedging flows are aligned with our estimated aggregate
gamma exposure. To answer this question, we split the sample into days with positive and
negative gamma exposure and estimate the following OLS regression model:

T urnovert = β0 + β1 OpenInterestt + β2 V olatilityt (24)

+ β3 IV ratiot + β4 Surpriset−1 + β5 W eekend + τi + ϵt

where Turnover is either the dollar volume 30 minutes before option expiry or the dollar
buying dollar volume
volume imbalance, defined as selling dollar volume
. The control variables Volatility, IV ratio,
Surprise, and Weekend are as defined in Eq. (20) and Eq. (21). We also include year fixed
effects, τi . The variable of interest is OpenInterest. Panel A of Table 8 reports the results for
days with aggregate positive gamma exposure. While there is significantly higher turnover
on days with high open interest, we do not find a significant order imbalance. However,
we find significant selling pressure on days with negative gamma exposure, as reported
in Panel B of Table 8. These results provide evidence for the predictions from Avellaneda
and Lipkin (2003), who argue that long gamma exposures lead to pinning due to buying
(selling) pressure from below (above) the strike price. Conversely, if market makers are net
short gamma, pinning decreases and hedging flows exacerbate price moves. The negative
turnover imbalance on days with negative gamma and high open interest is consistent with
the negative returns observed prior to expiration.

[Table 8 here]

5.2. Manipulation
5.2.1. Charm price pressure
Spot market manipulation by traders holding an open derivative position is well doc-
umented as a theoretical concept (Jarrow, 1994). However, there is limited empirical evi-
dence, partly due to the difficulty in identifying market manipulation with certainty. No-
table papers that provide evidence consistent with spot market manipulation around option
expiration include Baltussen et al. (2023), Griffin and Shams (2018), and Ni et al. (2005).
A limitation of these studies is they are unable to disentangle the effect of delta hedging
by option market makers. A key contribution of this paper is to present a novel method to
isolate price behaviour around option expiration that is, in theory, only related to market
manipulation and not delta hedging practices (Proposition 1).
If negative returns are driven purely by hedging pressure then we would expect the
return patterns documented in Table 5 to be driven by negative charm price pressure. Pos-

20
itive charm price pressure in the hour before expiration implies a downward price move-
ment only occurs if an initial price move triggers additional hedging pressure (Lemma 2,
Proposition 1). This type of market movement can occur due to random market movements
(e.g., large order execution, news etc.), however, these events are just as probable to move
prices in the opposite direction and, therefore, do not explain uni-directional price moves
on days with high open interest. The intuition of our approach using the option charm is
illustrated in Figure 6.

[Figure 6 here]

As an option approaches expiration, its delta tends towards 0 or 1 (or -1 for a put option),
depending on whether it is in-the-money or out-the-money. The speed and direction of this
change is measured by charm. Charm is most significant for an at-the-money option, and
increases into expiry, as documented in Baltussen et al. (2023). Our approach determines
the price pressure due to charm (Eq. 7) calculated at 7 am using the aggregate position of
option market makers (see Table 7).
Our formal approach to test the impact of charm on the return patterns around expi-
ration is to extend Eq. (20,21) by including charm as an independent variable that also
interacts with open interest and returns. Our aim is to ascertain whether the return pat-
terns that we document in Table 5 are driven by price pressure related to the option charm.
The OLS regression model estimated for returns 1-hour pre expiration is given as

ReturnP ret = β0 + β1 OpenInterestt + β2 Charmt + β3 OpenInterestt × Charmt (25)

X
+ βi Controlsit + τi + ϵt
i

and for returns in the 2-hours post expiration

ReturnP ostt = β0 + β1 OpenInterestt + β2 Charmt + β3 ReturnP re (26)

+ β4 OpenInterestt × Charmt + β5 OpenInterestt × P reReturnt
+ β6 P reReturnt × Charmt + β7 OpenInterestt × P reReturnt × Charmt
X
+ βi Controlsit + τi + ϵt
i

where Charm represents charm price pressure (Eq. 7). Other variables are as defined in
Eq. (20) and Eq. (21) with the same controls.

[Table 9 here]

21
 Table 9 documents that charm price pressure is positively related to pre option expi-
ration returns and negatively related to post option expiration returns, however, this re-
lationship is not statistically significant. Most importantly, the interaction terms between
OpenInterest and Charm (Eq. 25) and between OpenInterest, PreReturn, and Charm (Eq.
26) are not statistically signficant in the model. This implies that option charm is not driv-
ing the negative pre expiration return and subsequent reversal documented in Table 5. We
further document that both the OpenInterest variable in Eq. (25) and the interaction be-
tween OpenInterest and PreReturn in Eq. (26) remain negative and statistically significant.
A two-sided Wald test on the sum of coefficients for Charm and OpenInterest × Charm (Eq.
25) and ReturnPre × Charm and OpenInterest × ReturnPre × Charm is also statistically
insignificant.
In summary, these results confirm that delta hedge rebalancing due to option charm
calculated at 7am is not able to explain the return patterns around option expiration. In
the context of our model, this suggests that market manipulation is a plausible alternative
explanation (Proposition 1).

5.2.2. Manipulation in the spot market

In the previous sections, we demonstrated that high open interest combined with neg-
ative gamma exposure (Table 8) can potentially exacerbate price movements. However, it
remains unclear whether these hedging flows alone are causing the observed negative re-
turns before option expiry. While hedging primarily occurs in Deribit’s perpetual contract
(Alexander and Imeraj, 2023), price manipulation is more likely to occur in the constituents
of Deribit’s Bitcoin index, which is used to calculate option payoffs. When manipulators at-
tempt to influence the price of the underlying asset to increase their profit from the options
position, they ultimately generate abnormal trading volume.
To identify potential manipulation, we examine trading activity on several spot ex-
changes, including Deribit’s index constituents and the spot markets that Deribit uses as
backups. As a control group, we examine other spot markets that do not contribute to the
determination of the settlement price. Specifically, we estimate the following OLS regres-
sion model:

T urnovert = β0 + β1 OpenInterestt + β2 V olatilityt + β3 Surpriset−1 + β4 W eekend + ϵt

(27)

where Turnover is the dollar volume during the specific hour on the respective spot
exchange. Open Interest is the variable of interest and is a binary variable equal to 1 if

22
the at-the-money option open interest is in the top decile for the observation period. The
control variables Volatility, Surprise, and Weekend are as defined in Eq. (20) and Eq. (21).
Table 10 reports the coefficients and t statistics for Open Interest. Panel A reports the
results for the entire sample period. On days with high open interest, spot turnover on the
index constituents significantly increases in the hour prior to expiration. We do not observe
similar effects on the other spot markets, including Deribit’s backup markets. Neither do
we observe an increase in trading activity prior to the settlement of CME Bitcoin options
in the hour from 2 pm to 3 pm UTC or during any hour of the trading day. Therefore, the
results suggest potential manipulation in Deribit’s index constituents.

[Table 10 here]

Next, we examine days with positive and negative gamma exposure. An important ob-
servation from the model is that manipulation is more likely to occur if the desired price
move aligns with anticipated hedging pressure. In Table 12, we show that positive gamma
exposures significantly increase the probability of pinning, making manipulative efforts to
affect the settlement price increasingly expensive. Consequently, we expect manipulation
to occur on days with negative gamma exposure. We test this prediction by splitting the
sample into days with positive and negative gamma exposures. Panels B and C of Table 10
report the results. On days where hedging pressure increases the cost of manipulation, we
find only weak evidence of abnormal trading volume in Deribit’s index constituents. How-
ever, on days where hedging flows exacerbate price moves, we observe significantly higher
volume.
In conclusion, our findings provide suggestive evidence that manipulation in Deribit’s
index constituents is more likely to occur on days with negative gamma exposure, where
hedging flows amplify price movements, resulting in increased trading activity and poten-
tial price manipulation.

5.2.3. Behaviour of market participants

An important question that arises from our analysis is the identification of market par-
ticipants who could be engaging in manipulation around option expiration times. Table 11
provides insights into this issue by examining the relationship between miner outflows and
open interest.
Table 11 presents the results of both logistic and ordinary least squares (OLS) regression
models, where the dependent variable is the open interest on miner outflows (scaled by 100
for the logistic regression). The table includes two sets of regressions: the first specifies
open interest as a binary indicator variable (equal to 1 if open interest is in the top decile

23
across the sample period), while the second specifies open interest as a continuous variable.
Miner outflows are lagged by one day and serve as the key independent variable.
The results indicate a strong relationship between miner outflows and open interest.
Specifically, in the logistic regression (Columns 1 and 2), miner outflows from the previous
day are positively associated with the probability of high open interest. This suggests that
higher miner outflows are a significant predictor of elevated open interest. Similarly, in the
OLS regressions (Columns 3 and 4), miner outflows are positively associated with the level
of open interest.
The strong positive relationship between miner outflows and open interest suggests that
miners, who are significant holders of cryptocurrency, could be actively participating in the
options market in a way that influences settlement prices. This raises the possibility that
miners may be engaging in strategic trading behaviour around option expiration times,
potentially with the intention of influencing market prices to their advantage.

[Table 11 here]

5.3. Pinning and (non)-fundamental shocks

The phenomenon of stock prices being close to the nearest strike at option expiration has
been well-documented (Ni et al., 2005). Golez and Jackwerth (2012) studied this occurrence
for S&P 500 futures and found a strong tendency to finish close to the at-the-money strike
on option expiration days. To investigate whether the pinning effect exists in the Bitcoin
options market, we examine the distance between Bitcoin option settlement prices and the
nearest strike price. Since every day in our sample is an option expiration day, we focus on
days that are in the top decile of at-the-money open interest.20
Figure 7 plots the mean and median absolute distances between option settlement prices
and the nearest strike price surrounding dates with high open interest. We do not find a
tendency for Bitcoin settlement prices to pin on days with high open interest. Instead,
Panel A of Figure 7 shows that settlement prices tend to be, on average, further away from
the nearest strike.

[Figure 7 here]

We next turn our attention to the impact of hedging pressure on the tendency of settle-
ment prices to be pinned to the nearest strike at expiration. This approach is different to the
analysis in the previous sections because it only considers settlement prices at 8 am UTC
instead of the index return in the hour to expiry. Predictions from Avellaneda and Lipkin

20
Golez and Jackwerth (2012) suggest that at-the-money open interest increases the tendency for pinning.

24
(2003) suggest that market maker hedging activities affect the likelihood of pinning as well
as anti-pinning. Therefore, we focus on extreme positive and negative values of gamma and
charm in our sample period. Specifically, we use the top and bottom deciles of gamma and
charm-related hedging pressure to examine their effect on pinning. We specify pinning as
a binary variable that takes the value of 1 if settlement prices are less than 0.20% away
from the nearest strike price at settlement. We choose a threshold of 0.20% because it is
the median absolute distance to the nearest strike, as shown in Panel B of Figure 7.
The results are reported in Table 12. Dummy variables for high open interest, extremely
negative gamma, and extremely positive or negative charm do not affect the probability of
pinning. However, extremely positive gamma exposures significantly increase the proba-
bility of pinning, supporting the predictions from Avellaneda and Lipkin (2003) and con-
firming that pinning behaviour in Bitcoin options is consistent with economic theory.

[Table 12 here]

Another explanation for the observed return patterns relates to fundamental and non-
fundamental price shocks in spot markets that can create temporary price pressures that
subsequently revert. As shown in Table 5, surprises in the Global economic surprise index,
i.e. fundamental shocks, fail to explain the return reversal.
Non-fundamental shocks arise when traders with a strong demand for immediacy are
supplied liquidity by market makers who take on substantial inventory risk (Grossman
and Miller, 1988). Market makers require a premium for holding any price risk and are
compensated by prices reverting following demand shocks. This phenomenon can explain
the return reversal at option expiration if returns are persistently negative in the hours
leading up to 8 am. Therefore, lagged returns from 5 am to 6 am and 6 am to 7 am should
be positively related to returns in the hour before expiration and negatively related to post-
expiration returns. We test this hypothesis by estimating the following OLS regression
model on returns in the hour before and 2 hours post option expiration

Returnt = β0 + β1 Return[6,7],t + β2 Return[5,6],t + β3 OpenInterestt (28)

+ β4 Return[6,7],t × OpenInterestt + β5 Return[5,6],t × OpenInterestt + ϵt

where Return[6,7] is the index return from 6 am to 7 am and Return[5,6] is the index
return from 5 am to 6 am. Table 13 documents the coefficient on Return[5,6] and Return[6,7]
to be insignificant in both regression specifications with and without control variables.21

21
A two-sided Wald test confirms that the sum of coefficients on Return[6,7] (Return [5,6] ) and the interaction
of Return[6,7] (Return[5,6] ) with OpenInterest is also not statistically significant.

25
In summary, we find limited support that non-fundamental shocks are responsible for any
return reversal around Bitcoin option expirations.

[Table 13 here]

6. Economic Implications

6.1. Wealth transfer

An important implication of the return reversal around option expiration is the impact
on wealth allocation. A primary function of financial markets is to ensure efficient resource
allocation, which is compromised by artificial market prices. To understand the economic
significance of our results, we compute the wealth transfer caused by negative returns in
the hour before option expiration on days comprising the top decile of open interest.
Panel A of Figure 8 shows that the notional value of 0DTE in-the-money call options
decreases significantly in the 1 hour before option expiration while the notional value of
in-the-money 0DTE put options increases.

[Figure 8 here]

This pattern is consistent with a negative return prior to expiration, causing the Bitcoin
spot price to cross under the nearest strike price. Panel B of Figure 8 plots the transfer of
wealth from call writers to holders and from put writers to holders using various open in-
terest thresholds. On days when option open interest is low, the wealth transfer is smaller;
however, it increases significantly for high open interest days. This result underscores the
economic importance of option open interest, not only in predicting spot returns, but in the
allocation of capital.
To formalise the wealth transfer results, we compare actual option payoffs with the
payoff of a hypothetical option that settles at 7 am. Option payoffs for calls are calculated
as
X
V alueCalls = max(0, S − Ki ) ∗ OpenInteresti (29)
i

and for puts as

X
V alueP uts = max(0, Ki − S) ∗ OpenInteresti (30)
i

and expressed in dollar terms.

[Table 14 here]

26
 Table 14 shows that the price move from 7 am to 8 am results in a statistically significant
wealth transfer from call holders to call writers of $0.5 million and from put writers to
holders of another $0.5 million on days comprising the top decile of open interest. This
equates to an annual wealth transfer of more than $50 million for the most recent year in
our sample.

6.2. Liquidations
In this subsection, we present an analysis of the forced liquidations of perpetual futures
contracts, with a focus on how these liquidations are distributed throughout the trading
day. The data are summarized in Figure 9. Panel A focuses on Deribit’s futures contracts
and on days where open interest in at-the-money options is in the top decile. We observe
substantial spikes in liquidations before option expiry at 8 am, supporting the notion of
abnormal trading activity around the options expiry time, especially on days with high
open interest.
Next, we extend our analysis to examine forced long liquidations across a broader set of
cryptocurrency derivative exchanges, including Bitfinex, Gate.io, Binance, BitMEX, Bybit,
Deribit, FTX, HTX Global, and OKX. In unreported results, we find that cumulative long
liquidations affected by the return pattern exceed $350 million during our sample period,
indicating that the impact of high open interest on liquidation activity is not confined to a
single market on which options market makers are primarily active. Panel B of Figure 9
contrasts the liquidation patterns observed on high open interest days with those across
the full sample period. The ratio of median daily liquidations on high open interest days
relative to the entire sample period shows that liquidation activity is significantly elevated
on high open interest days. Specifically, the ratio spikes before option expiry at 8 am.

[Figure 9 here]

7. Conclusion

The central finding of this paper is the identification of a statistically and economically
significant return reversal occurring within two hours of 0DTE Bitcoin option expirations
on Deribit, driven primarily by option open interest. To explore the underlying causes of
this market inefficiency, we develop a model that incorporates delta hedgers and a potential
price manipulator. Our analysis demonstrates that the incentives for market manipulation
increase with the level of hedging pressure, particularly when a manipulator can influence
hedging trades that increase their option payoff. Crucially, we establish that option charm
serves as a key differentiator between price movements resulting from hedging and those

27
stemming from manipulation. Specifically, if market prices move contrary to charm pres-
sure, it indicates that hedging alone cannot fully explain the observed price dynamics.
Our empirical analysis reveals that, on average, Deribit option market makers are short
gamma, which aligns with the observed return patterns through delta hedge rebalancing.
This finding is further supported by an increase in net selling of Deribit perpetual contracts
in the hour leading up to option expiration. Notably, the negative return pattern persists
even when hedging driven by charm pressure would typically lead to positive returns. The
potential for manipulation is underscored by net selling of bitcoin on spot exchanges that
are used to determine the option settlement price. Additionally, we observe a flow of funds
out of wallets belonging to market participants documented to hold short call options in the
hour preceding option expiration.
Our findings carry significant implications for regulators and policymakers. The return
patterns surrounding Bitcoin option expirations result in a wealth transfer of $50 million
from call holders to call writers, and from put writers to put holders. We propose two po-
tential enhancements to market design that could mitigate this inefficiency: (1) scheduling
option settlements between 14:00 and 15:00 UTC, when spot trading volume is highest,
and (2) implementing enhanced regulatory oversight of bitcoin spot markets, particularly
when hedging flows are significant. The first recommendation is based on the premise that
trade price impact diminishes in more liquid markets (Chan and Lakonishok, 1995). The
second recommendation aims to increase the costs associated with market manipulation
(Gneezy, 2005), thereby deterring illegitimate activities that affect prices. Lastly, the abil-
ity to distinguish between return patterns resulting from hedging and those due to market
manipulation using option charm can serve as a valuable tool for regulators in identifying
manipulation across markets beyond cryptocurrencies.

28
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32
Figures
Open interest, hedging pressure and manipulator net benefit with positive charm price pressure
This figure plots hedging pressure ( H = (C + P )) and open interest (x) against a manipulators net benefit. Market manipulation
P P

is incentive compatible when the net benefit is positive. A net benefit of zero implies no market manipulation. The market setting
includes three option strikes at k1 = 0.95, k2 = 1, and k3 = 1.05 with equal hedging pressure at each strike h1 = h2 = h3 . Other
parameters are the initial underlying price p0 = 1.1, permanent price impact θ = 0.01, temporary price impact λ = 0.02, temporary
impact exponential decay β = 20, 000 (half-life ≈ 12.5 minutes), liquidity premium v = 0.2, volatility σ = 1 (100% annual volatility).
The model is solved for N = 6 time periods from 1-hour before expiration with the option settlement price computed as the average in
the last 3 time periods (Tp = 3, TN = 6).

Figure 1: Open interest, hedging pressure and manipulator net benefit with positive charm price pressure

33
Option trading and 0DTE option trading
This figure depicts a time series of the proportion of listed options that have less than 1 day to expiration and the proportion of trades for
options that have less than 1 day to expiration. Panel A considers options of any moneyness and panel B plots only at-the-money options
where an option is considered at-the-money if its strike price is within 2.5% of the spot index price at 7am.

Panel A. All options

Panel B. At-the-money options

Figure 2: Option trading and 0DTE option trading

34
Cumulative returns of spot Bitcoin around option expiration for full sample and top decile of open interest days
This figure depicts cumulative log returns of the Bitcoin spot index around the option expiration time at 8am. The figure plots mean
and median returns for the full sample and for days in the top decile of at the money open interest. Options whose strike price is within
2.5% of the underyling price at 7am are labelled at-the-money.

Panel A. Full Sample vs High Open Interest Mean

Panel B. Full Sample vs High Open Interest Median

Figure 3: Cumulative returns of spot Bitcoin around option expiration for full sample and top decile of open
interest days

35
Bitcoin option gamma exposure and time of day
This figure depicts the mean and median gamma price pressure for each hour from 3am to 11 am for the full sample and for the top
PN Pn
decile of open interest. Gamma price pressure is computed as P Pgamma = i=1 j=1 (Cij + Pij ) ∂∆
∂p
where Cji is the aggregate
market maker position in call options for thej th strike with i days to expiry, Pji is the aggregate market maker position in put options
for the j th strike with i days to expiry, and ∂∆
∂p
is the option gamma (sensitivity of delta to change in the underlying price). Gamma
price pressure is the dollar hedge rebalance for a $1 move in the underlying. Panel A plots the mean, and Panel B plots the median.

Panel A. Full Sample vs High Open Interest Mean

Panel B. Full Sample vs High Open Interest Median

Figure 4: Bitcoin option gamma exposure and time of day

36
Deribit turnover
This figure depicts the trading activity in Deribit’s perpetual contract as a fraction of daily turnover. Panel A displays the average
fraction of daily turnover measured in half-hour intervals for both the entire sample and for days in the top decile of at-the-money open
interest. Panel B shows the relationship between the fraction of daily turnover one hour prior to option settlement and at-the-money
open interest.

Panel A. Fraction of Daily Turnover: Full Sample vs High Open Interest Mean

Panel B. Fraction of Daily Turnover (1 Hour Before Settlement) vs Open Interest

Figure 5: Deribit perpetual turnover

37
Sensitivity of delta to time to expiry
This figure depicts call option delta ∆ as time t approaches expiry for different underyling prices p 1 hour before expiration that corre-
spond to different values of charm ( ∂∆
∂ t̄
) where t̄ is the time to option expiry. Other parameters include the risk free rate set to 0.05 (5%),
the option strike price set to 50, and annuallised implied volatility set to 1 (100%).

Figure 6: Sensitivity of delta to time to expiry

38
Pinning
This figure depicts the mean and median absolute distances between option settlement prices and the nearest strike price surrounding
dates that fall within the top decile of at-the-money open interest. These high open interest days are labelled as trade date ’0’, with the
day before labelled as date ’-1’ relative to the option expiration date, and so on.

Panel A. Mean

Panel B. Median

Figure 7: Pinning

39
Wealth transfer
Panel A plots the notional value of in the money call and put options from 5am to option expiration at 8am. Panel B plots the sum of
actual option payoffs minus a hypothetical option payoff if the settlement price was the prevailing price at 7am against the open interest
percentile. The calculated wealth transfer is for all days when open interest is lower than the xth percentile of open interest across the
sample period.

Panel A. Notional Value of ITM Calls and Puts

Panel B. Wealth Transfer vs Open Interest

Figure 8: Wealth transfer

40
Perpetual futures liquidations
This figure depicts the total forced liquidations in perpetual contracts during the trading day. Panel A displays cumulative liquidations
of Deribit’s perpetual at half-hour intervals on days with high open interest in at-the-money options. Panel B depicts the total forced
liquidations of long positions in the perpetual contracts from Bitfinex, Gate.io, Binance, BitMEX, Bybit, Deribit, FTX, HTX Global, and
OKX at one-hour intervals during the trading day. We plot the ratio between the median daily liquidations on days with high open
interest relative to the entire sample period (2021-2023).

Panel A. Deribit cumulative liquidations on days with high open interest (2021-2023)

Panel B. Long liquidations (multiple exchanges)

Figure 9: Perpetual futures liquidations by time of day

41
Tables
Summary statistics
Panel A of this table reports daily descriptive statistics on the price level of the Bitcoin index used to construct the settlement value of
Deribit options. This index is comprised using price data from spot Bitcoin traded on coinbase, bitstamp, gemini, itbit, kraken, bitfinex,
bittrex, and LMAX digital. Panel B reports descriptive statistics seperately for options categorised by moneyness. Complex trades are
trades with more than one leg. At the money options are options whose strike price are within 2.5% of the index price at 7am. In-the
money options are all other in-the money options and out-the money options are all other out-the money options.

Table 1: Summary statistics

Mean SD Min Median Max

Panel A. Deribit Bitcoin index

Index level ($) 34,764 12,506 15,915 31,872 67,641
Daily return 0.001 0.034 -0.155 -0.000 0.193

Panel B. Deribit Bitcoin options

Call IV (%) 68.84 22.90 26.82 66.60 166.88
Put IV (%) 73.77 25.88 27.11 73.00 176.14
Days to expiration 25.76 10.67 6.45 24.02 120.20
Complex trades (%) 9.54 9.54 3.39 9.43 21.74

In-the-Money
# Calls traded per day 2,554 2,148 183 1,954.5 14,782
% 0DTE 0.11 0.07 0.00 0.11 0.77
# Puts traded per day 2,321 2,080 98 1,773 21,538
% 0DTE 0.13 0.08 0.00 0.12 0.86
# Strikes traded 17.64 7.77 4 16.0 52
# Strikes traded 0DTE 6.74 7.14 0 4.00 45

At-the-Money
# Calls traded per day 1,311 845 149 1,095 5,930
% 0DTE 0.38 0.09 0.13 0.38 0.76
# Puts traded per day 1,160 779 78 943 5,711
% 0DTE 0.38 0.11 0.06 0.38 0.69
# Strikes traded 3.76 1.95 1 3.00 12
# Strikes traded 0DTE 3.63 1.84 1 3.00 11

Out-the-Money
# Calls traded per day 3,361 2,288 528 2,776 15,141
% 0DTE 0.09 0.05 0.00 0.08 0.68
# Puts traded per day 2,881 2,229 307 2,296 21,889
% 0DTE 0.10 0.06 0.00 0.09 0.78
# Strikes traded 49.43 10.29 29 49.00 94
# Strikes traded 0DTE 15.30 9.26 1 12.00 65

42
Open interest
Panel A of this table reports decsriptive statistics on option open interest split between calls and puts and option moneyness, and option
open interest imbalance. At the money options are options whose strike price are within 2.5% of the index price at 7am. In-the money
options are all other in-the money options and out-the money options are all other out-the money options. Open interest imbalance is
call open interest minus put open interest. Normalised open interest imbalance is the open interest imbalance scaled by the total call
and put open interest. Panel B reports open interest by the notional option value.

Table 2: Open interest

Open Interest Open Interest Open Interest

Call Put Imbalance Normalised Imbalance
Panel A. OI (Contracts)
OTM 3,785 3,117 668 0.014
ATM 891 715 176 0.067
ITM 3,150 2,586 564 0.032
Panel B. Notional OI ($ millions)
OTM 131.6 109.5 22.1 -
ATM 30.4 24.5 5.8 -
ITM 107.9 92.3 15.5 -

43
Spot turnover
This table reports summary statistics on the dollar volume (in millions) across various cryptocurrency exchanges from 7 am to 8 am UTC,
i.e. one hour before Deribit options expiration. Panel A reports aggregate turnover from major spot exchanges, specifically Binance,
Bitfinex, Bittrex, Coinbase, FTX, Gemini, HTX Global, Kraken, and OKX. Panel B provides dollar volume data for Deribit’s index
constituents, including Bitstamp, Coinbase, and Kraken. Panel C reports dollar volume from exchanges not included in Deribit’s Bitcoin
index calculations but used by Deribit as backup markets.

Table 3: Spot turnover ($ millions)

Mean SD Min Median Max

Panel A. All exchanges
All exchanges 192.61 169.95 4.34 151.75 1271.31
Panel B. Deribit Index Constituents
Bitstamp 4.03 6.21 0.08 1.85 75.79
Coinbase 16.95 15.86 0.79 12.73 163.68
Kraken 4.16 5.31 0.04 2.33 54.39
Panel C. Backup Markets
Bitfinex 7.17 12.9 0.0 3.01 163.11
Gemini 1.22 2.17 0.0 0.48 25.54

44
Spot Returns around expiry
This table reports percentage index log returns (reported in % form) around option expiry over 20-minute intervals around option
expiration where T is from option expiration (8 am) until 08:20:00. Mean and median returns are reported for the full sample (n = 1, 058)
and for the top decile of open interest (calls and puts). Statistical significance around the mean is determined using a t-test around the
mean and around the median using a non-parametric Wilcoxon signed rank test. ***, **, and * denote significance at a 1%, 5%, and 10%
level respectively.

Table 4: Spot returns around expiry

Full Sample High Open Interest

Mean Median Mean Median
(1) (2) (3) (4)

T −3 0.006 0.006 0.028 -0.019

T −2 0.007 -0.001 -0.112∗∗∗ -0.025∗∗
T −1 -0.001 0.000 -0.003 -0.034∗∗
T 0.013 0.009 0.061∗∗ 0.038∗∗
T +1 0.007 -0.010 -0.013 0.012
T +2 -0.005 0.003 0.060∗ -0.003

45
Regression analysis on index returns around option expiration
This table reports coefficients and t statistics for two Ordinary Least Squares (OLS) regressions. Specifically column (4) reports coeffi-
cients and t-statistics of the OLS regression

ReturnP ostt = β0 + β1 ReturnP ret + β2 OpenInterestt + β3 ReturnP ret × OpenInterestt + β4 V olatilityt

+ β5 V olumet + β6 IV ratiot + β7 Surpriset−1 + β8 W eekendt + τi + ϵt

where Return Post is the log return (reported in % form) from option expiry to 2 hours after option expiry, Return Pre is the log return
(reported in % form) from 1 hour periods before option expiry to option expiration, and Open Interest is a dummy variable equal to 1 if the
option open interest is in the top decile for the observation period. The remaining variables are control variables, including Garman and
Klass (1980) realised Volatility measured from 7am to 10am, spot market Volume (turnover) from 7am to 10am at major spot exchanges:
Binance, Bitfinex, Bittrex, Coinbase, FTX, Gemini, HTX Global, Kraken, and OKX, and IV ratio as the ratio of implied volatility for the
nearest strike call option above the index price at 7 am over the implied volatility for the nearest strike put option below the index price
at 7 am. Surprise is lagged by one day and is a dummy variable equal to one if the absolute value of the global economic surprise index is
greater than its 90%ile in the sample period, Weekend is a dummy variable equal to 1 if the day is a weekend, and τi are fixed effects for
year i. Column (3) drops the control variables. Columns (1) regresses Return Pre on Open Interest and column (2) adds control variables.
***, **, and * denote significance at a 1%, 5%, and 10% level respectively.

Table 5: Regression analysis on index returns around option expiration

Pre Return Pre Return Post Return Post Return

(1) (2) (3) (4)

Return Pre 0.080 0.084

(0.851) (0.887)
Open Interest -0.121* -0.146** 0.163 0.163
(-1.817) (-2.10) (1.136) (1.065)
Return Pre × Open Interest -0.674*** -0.688***
(-2.940) (-2.992)
Volatility 0.023 -0.043
(1.157) (-0.903)
Volume -0.031 -0.049
(-1.430) (-1.538)
IV ratio 0.027 -0.048
(0.743) (-0.958)
Surpriset−1 0.245 -0.176
(1.744) (-0.691)
Weekend -0.101** 0.015
(-2.327) (0.161)
Intercept 0.024 1.014 -0.025 -1.897
(1.19) (1.400) (-0.566) (-1.223)

Year FE No Yes No Yes

Adjusted R2 0.002 0.012 0.029 0.027
Observations 1,058 1,058 1,058 1,058

46
Results after a bootstrap with resampling + Sharpe ratios
This table reports results that characterise the economic significance of the return reversal around option expiration. The trading
strategy shorts Bitcoin on bitstamp at 7 am and buys back at 8 am. At 8 am, a long position is taken and then liquidated at 10 am. Daily
expected returns and standard deviations are based on a strategy that trades only if the decision rule is met and are reported in % form.
Results are considered with and without transaction costs. Bootstrapped p-values are constructed by creating 500 artificial samples
by resampling the original sample with replacement, then assessing the statistical significance of returns in each new sample. Sharpe
ratios are calculated as the daily expected return scaled by the standard deviation of daily returns. Annual Sharpe ratios are computed
as per Lucca and Moench (2014) by multiplying by the square root of the number of yearly observations.

Table 6: Results after a bootstrap with resampling + Sharpe ratios

Switch 07:30:00 Switch 08:00:00 Switch 08:30:00

Mid With TC Mid With TC Mid With TC

(1) (2) (3) (4) (5) (6)
Daily Expected return 0.25 0.19 0.36 0.29 0.27 0.20
Daily Standard deviation 1.68 1.70 1.96 1.86 1.82 1.80
Bootstrapped P-value 0.09* 0.11 0.03** 0.05** 0.09* 0.13
Annual Sharpe ratio 0.88 0.67 1.09 0.92 0.87 0.67
% Profitable days 59.43 56.60 59.43 59.43 58.49 56.60

47
Market maker positions and hedging pressure around option expiry
This table reports mean and median values of market maker call and put positions for 0DTE at the money options as well as net option
gamma. Results are reported for the full sample and for the top decile of open interest. An option is classified as at the money if the
underlying price at 7 am is within 2.5% of its strike price. Market maker positions are proxied as the negative of the notional directional
trade amount for each option. The position values reported are for at-the-money options only. Net gamma is computed as

n
X ∂∆
Γ= (Cj + Pk )
j=1
∂p

where Cj is the aggregate market maker position in call options for the j th strike, Pj is the aggregate market maker position in put
∂∆
options for the j th strike, and ∂p
is the option gamma (sensitivity of delta to change in the underlying price) calculated using at the
money 0DTE options. Significance tests are conducted on the full sample, top decile of open interest and on the difference between the
full sample and the top decile of open interest. Statistical significance around the mean is determined using using a t-test around the
mean and around the median using a non-parametric Wilcoxon signed rank test. ***, **, and * denote significance at a 1%, 5%, and 10%
level respectively.

Table 7: Market maker positions and hedging pressure around option expiry

Full Sample High Open Interest Difference

Mean Median Mean Median Mean Median
(1) (2) (3) (4) (5) (6)
∗∗∗ ∗∗∗ ∗∗ ∗∗∗
Call Position (thousands) -1,488.9 -38.8 -13,916.6 -4,589.5 12,427.7 4,550.8∗∗
Put Position (thousands) 286.1 228.6 1,316.6 -563.6 -1,030.5 792.3
Gamma Price Pressure -2,836.8∗∗∗ -460.23∗∗∗ -5,704.3∗∗∗ -2,520.6∗∗∗ 3,317.4∗∗ 2,060.3∗∗∗

48
Deribit turnover around expiration
This table reports coefficients and t statistics for two Ordinary Least Squares (OLS) regressions. Specifically column (2) reports coeffi-
cients and t-statistics of the OLS regression:

T urnovert = β0 + β1 OpenInterestt + β2 V olatilityt + β3 IV ratiot + β4 Surpriset−1 + β5 W eekend + τi + ϵt

where Turnover is the dollar volume 30 minutes before option expiry and Open Interest is a dummy variable equal to 1 if the option open
interest is in the top decile for the observation period. The remaining variables are control variables, including Garman and Klass (1980)
realised Volatility and IV ratio as the ratio of implied volatility for the nearest strike call option above the index price at 7 am over the
implied volatility for the nearest strike put option below the index price at 7 am. Surprise is lagged by one day and is a dummy variable
equal to one if the absolute value of the global economic surprise index is greater than its 90%ile in the sample period, Weekend is a
dummy variable equal to 1 if the day is a weekend, and τi are fixed effects for year i. Column (1) drops the control variables. Column (3)
buying dollar volume
regresses T urnoverImbalance, defined as selling dollar volume
on OpenInterest and column (4) adds control variables.

Table 8: Regression analysis on Deribit perpetual turnover before options expiration

Turnover Turnover Turnover Imbalance Turnover Imbalance

(1) (2) (3) (4)
Panel A. Days with positive gamma
Open Interest 1.35*** 1.27*** -0.02 0.04
(9.88) (9.68) (-0.13) (0.33)
Volatility 0.37*** -0.04
(15.49) (-0.68)
IV ratio 0.03 0.04
(0.27) (0.26)
Surpriset−1 0.14 0.11
(1.25) (0.86)
Weekend -0.27*** 0.18
(-3.50) (1.64)
Intercept 16.01*** 19.28*** 1.17*** 0.82
(357.76) (76.66) (24.80) (1.59)

Year FE No Yes No Yes

Adjusted R2 0.123 0.510 -0.002 0.007
Observations 483 483 483 483

Panel B. Days with negative gamma

Open Interest 1.25*** 1.33*** -0.22*** -0.17**
(11.04) (14.30) (-2.94) (-2.00)
Volatility 0.33*** -0.07
(14.63) (-1.38)
IV ratio 0.06 -0.07
(1.37) (-0.93)
Surpriset−1 0.24* -0.04
(1.77) (-0.28)
Weekend -0.15** 0.29***
(-2.30) (2.83)
Intercept 16.03*** 19.12*** 1.26*** 1.17***
(379.28) (75.57) (26.38) (3.67)

Year FE No Yes No Yes

Adjusted R2 0.158 0.514 0.003 0.066
Observations 575 575 575 575

49
Option charm price pressure
This table reports regression coefficients and t-statistics for the following two regression specifications on Bitcoin spot index log returns
(reported in % form) one hour before expiration (Pre Return) and two hours after expiration (Post Return) over the full sample (n = 1058)

X
ReturnP ret = β0 + β1 OpenInterestt + β2 Charmt + β3 β2 OpenInterestt × Charmt + βi Controlsit + ϵt
i

ReturnP ostt = β0 + β1 OpenInterestt + β2 Charmt + β3 ReturnP re + β4 OpenInterestt × Charmt

+ β5 OpenInterestt × P reReturnt + β6 P reReturnt × Charmt
X
+ β7 OpenInterestt × P reReturnt × Charmt + βi Controlsit + ϵt
i

Pn ∂∆
where Charm is charm price pressure computed as P P charm = − j=1 (Cj + Pj ) ∂ t̄j where Cj is the aggregate market maker
position in call options (notional value) for the
j th strike, Pj is the aggregate market maker position in put options (notional value) for
∂∆
the j th strike, and ∂ t̄j is call option charm (sensitivity of delta to option time to expiry). OpenInterest is a dummy variable equal to
1 if a day is in the top decile at-the-money option open interest, and control variables are as per Table 5 (not reported).

Table 9: Option charm price pressure

Pre Return Post Return

(1) (2)

Open Interest -0.127∗∗ 0.141

(-1.992) (1.026)
Charm 0.004 -0.004
(1.503) (-0.714)
Return Pre 0.091
(1.043)
Open Interest × Charm -0.002 -0.002
(-0.730) (-0.249)
Open Interest × Return Pre -0.506∗∗
(-2.331)
Return Pre × Charm 0.002
(0.197)
Open Interest × Pre Return × Charm 0.005
(0.370)

50
Spot turnover around expiration
This table reports coefficients and t statistics for Open Interest on several spot exchanges in the following OLS regression:

T urnovert = β0 + β1 OpenInterestt + β2 V olatilityt + β3 Sentimentt + β4 Surpriset−1 + β5 W eekend + ϵt

where Turnover is the dollar volume during the specific hour on the respective spot exchange. Open Interest is the variable of interest and
is a dummy variable equal to 1 if the ATM option open interest is in the top decile for the observation period. The remaining variables
are control variables, including Garman and Klass (1980) realised Volatility, IV ratio as the ratio of implied volatility for the nearest
strike call option above the index price over the implied volatility for the nearest strike put option below the index price. Surprise is
lagged by one day and is a dummy variable equal to one if the absolute value of the global economic surprise index is greater than its
90%ile in the sample period, and Weekend is a dummy variable equal to 1 if the day is a weekend. ***, **, and * denote significance at
a 1%, 5%, and 10% level respectively. Column 1 reports results for aggregate turnover from major spot exchanges, including Binance,
Bitfinex, Bittrex, Coinbase, FTX, Gemini, HTX Global, Kraken, and OKX.

Table 10: Spot turnover around expiry

Index Constituents Backup Markets

All Exchanges Bitstamp Coinbase Kraken Bitfinex Gemini
(1) (2) (3) (4) (5) (6)
Panel A. Full Sample
05:00 - 06:00 -0.12 -0.11 -0.13** -0.07 -0.29** -0.30***
(-1.27) (-1.11) (-2.01) (-0.83) (-1.97) (-2.59)
06:00 - 07:00 -0.14 -0.03 -0.12* -0.04 -0.21 -0.40**
(-1.48) (-0.32) (-1.89) (-0.40) (-1.58) (-2.48)
07:00 - 08:00 -0.10 0.31*** 0.28*** 0.29*** 0.21 -0.06
(-0.99) (3.89) (4.82) (3.31) (1.54) (-0.38)
08:00 - 09:00 -0.16* 0.03 0.07 0.07 -0.06 -0.25**
(-1.67) (0.36) (1.25) (0.93) (-0.47) (-2.29)
09:00 - 10:00 -0.14 0.04 0.02 0.01 -0.16 -0.29**
(-1.38) (0.43) (0.43) (0.17) (-1.19) (-2.46)
10:00 - 11:00 -0.00 -0.00 0.01 0.11 -0.13 -0.20**
(-0.01) (-0.06) (0.12) (1.40) (-0.97) (-1.99)
...

14:00 - 15:00 0.01 0.08 0.07 0.09 -0.03 -0.16*

(0.06) (0.88) (1.29) (1.12) (-0.27) (-1.85)
Panel B. Days with positive gamma
07:00 - 08:00 -0.10 0.16 0.20** 0.13 0.04 -0.54
(-0.62) (1.42) (2.10) (0.86) (0.23) (-1.44)
14:00 - 15:00 -0.04 0.14 0.13 0.14 0.15 -0.12
(-0.26) (1.18) (1.59) (1.32) (0.79) (-0.91)
Panel C. Days with negative gamma
07:00 - 08:00 -0.09 0.37*** 0.32*** 0.37*** 0.31* 0.19
(-0.68) (3.48) (4.29) (3.33) (1.72) (1.20)
14:00 - 15:00 0.03 0.03 0.04 0.06 -0.10 -0.15
(0.32) (0.22) (0.60) (0.54) (-0.61) (-1.31)

51
Miner outflows
This table reports coefficients and t (z) statistics for an OLS and logistic regression model of open interest on miner outflows (scaled by
100 for the logistic regression). In the OLS regression open interest is specified as continous whereas for the logistic regression open in-
terest is specified as a dummy variable equal to 1 if open interest is in the top decile across the sample period. Miner outflows are one day
lagged daily outflows. The remaining variables are control variables including Garman and Klass (1980) realised Volatility, spot market
Volume (turnover) spot market from major spot exchanges: Binance, Bitfinex, Bittrex, Coinbase, FTX, Gemini, HTX Global, Kraken, and
OKX. Surprise is lagged by one day and is a dummy variable equal to one if the absolute value of the global economic surprise index is
greater than its 90%ile in the sample period, Weekend is a dummy variable equal to 1 if the day is a weekend. Year fixed effects are also in-
cluded, and standard errors are corrected for heteroscedasticity. ***, **, and * denote significance at a 1%, 5%, and 10% level respectively.

Table 11: Logistic and OLS regression on OI

OI Indicator OI Indicator OI OI
(1) (2) (3) (4)

Miner outflowst−1 1.10*** 0.007** 0.093*** 0.049**

(4.32) (2.24) (4.53) (2.44)
Volatility -0.277* -136.654**
(-1.90) (-2.11)
Volume 0.180 66.632
(1.29) (0.83)
Surpriset−1 0.361 67.372
(0.77) (0.27)
Weekend -2.388*** -451.226***
(-4.48) (-4.74)
Intercept -2.681*** -10.713* 1229.598*** -3143.369
(-16.46) (-1.94) (14.40) (-1.27)

Year FE No Yes No Yes

Adjusted R2 0.027 0.071
Pseudo R2 0.024 0.103

52
Hedging pressure and pinning
This table reports coefficients and z statistics for logistic regressions on pinning at option expiration. Pinning is a dummy variable equal
to 1 if the absolute distance to the nearest strike at expiration is below 0.20%. Open Interest is a dummy variable equal to 1 if the option
open interest is in the top decile for the observation period. Positive (negative) Gamma is a dummy variable equal to 1 if the aggregate
market maker gamma exposure is in the top (bottom) decile for the observation period. Similarly, Positive (negative) Charm is a dummy
variable equal to 1 if the aggregate market maker charm-related hedging pressure is in the top (bottom) decile for the observation period.
***, **, and * denote significance at a 1%, 5%, and 10% level respectively.

Table 12: Hedging pressure and pinning

Pinning Pinning Pinning Pinning Pinning

(1) (2) (3) (4) (5)
Open Interest 0.2638
(1.25)
Negative Gamma 0.1357
(0.64)
Positive Gamma 0.6299***
(2.93)
Negative Charm 0.1227
(0.58)
Positive Charm -0.2381
(-1.14)
Intercept 0.0479 0.0581 -0.0011 0.0563 0.0964
(0.22) (0.27) (-0.01) (0.26) (0.45)

Pseudo R2 0.017 0.016 0.021 0.016 0.016

Observations 1,058 1,058 1,058 1,058 1,058

53
Non-fundamental price shocks
This table reports coefficients and t statistics for two Ordinary Least Squares (OLS) regressions of the form

Returnt = β0 + β1 Return[6,7],t + β2 Return[5,6],t + β3 OpenInterestt + β4 Return[6,7],t × OpenInterestt (31)

+ β5 Return[5,6],t × OpenInterestt + τi + ϵt

where Return is the return from 1 hour before option expiry to option expiration in columns (1) and (2) and the return from option expiry
to 2 hours after option expiry in columns (3) and (4). Return[6,7] is the log return from 6am to 7am, Return[5,6] is the log return from
5am to 6am hour periods before option expiry to option expiration, and Open Interest is a dummy variable equal to 1 if the option open
interest is in the top decile for the observation period. Control variables in columns (2) and (4) are Volatility as Garman and Klass (1980)
realised volatility measured from 7am to 10am, and IV ratio as the ratio of implied volatility for the nearest strike call option above the
index price at 7 am over the implied volatility for the nearest strike put option below the index price at 7 am. Surprise is lagged by one
day and is a dummy variable equal to one if the absolute value of the global economic surprise index is greater than its 90%ile in the
sample period, Weekend is a dummy variable equal to 1 if the day is a weekend, and τi are fixed effects for year i. ***, **, and * denote
significance at a 1%, 5%, and 10% level respectively.

Table 13: Non-fundamental price shocks

Pre Return Pre Return Post Return Post Return

(1) (2) (3) (4)

Return[6,7] -0.070 -0.067 -0.075 -0.079

(-0.79) (-0.80) (-0.94) (-0.99)
Return[5,6] -0.052 -0.057 -0.046 -0.048
(-0.58) (-0.65) (-0.67) (-0.68)
Open Interest -0.116 -0.139 0.160 0.165
(-1.84) (-2.03) (1.07) (1.09)
Return[6,7] × Open Interest -0.101 -0.147 -0.623 -0.621
(-0.50) (-0.73) (-0.77) (-0.76)
Return[5,6] × Open Interest 0.360 0.391 -0.550 -0.553
(0.91) (0.95) (-1.15) (-1.15)
Intercept 0.014 0.059 0.015 -0.119
(0.71) (0.83) (0.58) (-0.34)

Controls No Yes No Yes

Adjusted R2 0.015 0.029 0.028 0.026
Observations 1,058 1,058 1,058 1,058

54
Wealth transfer
This table reports values for actual and hypothetical Bitcoin option settlement values. Actual settlement values are the cumulative
payoff of Bitcoin options at settlement where the settlement price is constructed as a weighted average of an index price between 730 am
and 8 am comprising of Bitcoin spot data from 4 exchanges: Bitstamp, Coinbase, Kraken, and LMAX digital. Hypothetical settlement
considers the option payoff at 7 am. Option payoffs for calls are given by V alueCalls = i max(0, S − Ki ) ∗ OpenInteresti and for
P

puts as V alueP uts = . Results are reported for the full sample (n = 1, 058) and separately for
P
i max(0, K i − S) ∗ OpenInterest i
the top decile sorted by open interest. The difference is actual settlement value minus hypothetical settlement value and represents the
transfer from option writers to option holders. A t test is conducted to ascertain statistical significance for the differences. ***, **, and
* represent 10%, 5%, and 1% significance levels respectively.

Table 14: Wealth transfer

Calls Puts
Full Sample High OI Full Sample High OI
(1) (2) (3) (4)

Actual Settlement 4.7 38.4 3.1 16.5

7am Settlement 4.7 38.9 3.0 16.0
Difference -0.0 -0.5∗∗ 0.1 0.5∗∗

55
A. Mathematical Proofs

Lemma 1
For a short call option, y ̸= 0 occurs when

x(max(S(y) − k̂, 0)) + C(y) < x(max(S(y = 0) − k̂)) (A-1)

subject to
max(S(y) − k̂, 0) < max(S(y = 0) − k̂) (A-2)

Rearranging Eq. (A-1) in the form x > x∗ gives x∗ as

C(y)
x∗ = (A-3)
max(S(y = 0) − k̂, 0) − max(S(y) − k̂, 0)

Assumption 1 ensures that

E(max(S(y = 0) − k̂, 0)) = S(y = 0) − k̂ (A-4)
∗ ∂C(y)
∂x
which guarantees the existence of an x > x∗ that satisfies the constraint in Eq. (A-2). It is trivial to verify ∂C(y)
> 0 and ∂k
> 0.
The proof for a long put option follows the same process.

Lemma 2
For a short call option the net gain from manipulation is given by Eq. (13) and is decreasing in (max(E(S(y)) − k̂, 0)) (weakly)
and c(y) (strictly). It is trivial to verify that

∂c(y)
= −v < 0, y < 0 (A-5)
∂y
tN
∂E(S(y)) X
> 0, Ω = G(A(i, y)) ≤ 0 (A-6)
∂y i=t p

where G(A(t, y)) (total hedging price impact at time t) is

t
X t X
X i
G(A(t, y)) = θ A(i, y) + A(j, y)λe−β(ti −tj ) (A-7)
i=1 i=1 j=1

In the case where Ω < 0, it follows that hedging pressure increases manipulation incentives whenever

E (S(y(H = H), H = 0)) − E (S(y(H = H))) > pt0 − E (S(y = 0, H = H)) (A-8)

since for a fixed y , C(y) is fixed but the expected settlement price is lower than that caused soley by manipulator trades and hedging
pressure in the absence of manipulator trades

 
tN i tN
1 X X
−β(i−j)
X
E(S(y)) < p0 +  yj λe + G (A(||y||2 = 0, t)) (A-9)
tN − tp i=t j=1 i=t
p p

We can then re-write the condition for y ̸= 0 as

 
x max(S(y = 0) − k̂, 0) − max(S(y) − k̂, 0) − C(y) > 0 (A-10)

Eq. (A-10) is weakly increasing in max(S(y = 0)−k̂, 0)−max(S(y)−k̂, 0) which is increasing in (E (S(y(H = H), h = 0)) − E (S(y(H = H))))−
(pt0 − E (S(y = 0, H = H))).

56
Proposition 1
Consider the case of no manipulation where ||y||2 = 0. In expectation the market price at time t is

t t
!
X X  
E(pt ) = p0 + σE Wt + Ai θ + λe−β(t−ti ) (A-11)
i=1 i=1
t
X  
= p0 + Ai θ + λe−β(t−ti ) (A-12)
i=1

It follows that E(p0 ) = E(p1 ) and E(p2 ) can be expressed as

n
X
E(p2 ) = E(p1 ) + P P charm(j, t1 , t0 , Cj , Pj ) (A-13)
j=1

and therefore for Γ < 0, a general E(p3 ) is

n
X
E(p3 ) = E(p2 ) + P P charm(j, t2 , t1 , Cj , Pj ) + αsgn(E(p2 ) − E(p1 )) (A-14)
j=1
| {z }
Gamma Price Impact

where α > 0. We can then derive a general form for E(pt ) as

n
X
E(pt ) = E(pt−1 ) + P P charm(j, tt−1 , tt−2 , Cj , Pj ) + αsgn(E(pt−1 ) − E(pt−2 )) (A-15)
j=1
| {z }
Gamma Price Impact

In the absence of the market price crossing a strike price, this demonstrates

 
n
X
sgn (E(pT ) − E(pt )) = sgn  P P charm(j, t1 , t0 , Cj , Pj ) , ∀ T >t>0 (A-16)
j=1

In the case of the market price crossing a strike price, sgn(P P charm) flips with positive probability. In this case we can use Eq. (A-15),
and observe that the price then reverts direction until it crosses back over the same strike price before reverting
 again. Ultimately, this 
Pn
causes the market price to cluster at the nearest strike price, but, still consistent with the direction of sgn j=1 P P charm(j, t1 , t0 , Cj , Pj ) .
Therefore, Eq. (A-16) implies

 
n
X
sgn (E(S(y = 0)) − E(pt0 )) = sgn  P P charm(j, t1 , t0 , Cj , Pj ) (A-17)
j=1

P 
n
such that sgn (E(S(y = 0)) − E(pt0 )) ̸= sgn j=1 P P charm(j, t1 , t0 , Cj , Pj ) can only occur due to market manipulation.

57