While Stability Lasts: A Stochastic Model of Stablecoins
aa r X i v : . [ q -f i n . T R ] A p r While Stability Lasts: A Stochastic Model ofStablecoins ∗ Ariah Klages-Mundt † Andreea Minca ‡ April 6, 2020
Abstract
The ‘Black Thursday’ crisis in cryptocurrency markets demonstrateddeleveraging risks in over-collateralized lending and stablecoins. We de-velop a stochastic model of over-collateralized stablecoins that helps ex-plain such crises. In our model, the stablecoin supply is decided by spec-ulators who optimize the profitability of a leveraged position while incor-porating the forward-looking cost of collateral liquidations, which involvesthe endogenous price of the stablecoin. We formally characterize stableand unstable domains for the stablecoin. We prove bounds on the prob-abilities of large deviations and quadratic variation in the stable domainand distinctly greater price variance in the unstable domain. The unsta-ble domain can be triggered by large deviations, collapsed expectations,and liquidity problems from deleveraging. We formally characterize a de-flationary deleveraging spiral as a submartingale that can cause such liq-uidity problems in a crisis. We also demonstrate ‘perfect’ stability resultsin idealized settings and discuss mechanisms which could bring realisticsettings closer to the idealized stable settings.
On March 12, 2020, called ‘Black Thursday’ during the COVID-19 market panic,cryptocurrency prices dropped ∼
50% in the day. This was accompanied by cas-cading liquidations on cryptocurrency leverage platforms, including both cen-tralized platforms like exchanges and new decentralized finance (DeFi) platformsthat facilitate on-chain over-collateralized lending. Among many events fromthis day, the story of Maker’s stablecoin Dai stands out, which entered a defla-tionary deleveraging spiral. This triggered high volatility of the ‘stable’ assetand a breakdown of the collateral liquidation process. Due to market illiquidityexacerbated by network congestion, some collateral liquidations were performed ∗ This paper is based on work supported by NSF CAREER award † Cornell University, Center for Applied Mathematics. ‡ Cornell University, Operations Research & Information Engineering.
1t near-zero prices. As a result, the system developed a collateral shortfall,which prompted an emergency response and had to be made up by selling newequity-like tokens to recapitalize [18].During this time, there was a huge demand for Dai. It became a much riskierand more volatile asset, yet traded at a high premium and fetched lending ratesin the mid double digits. Leveraged speculators, who must repurchase Dai inorder to deleverage their positions, were exhausting Dai liquidity, driving upthe price of Dai and subsequently increasing the cost of future deleveraging(we discuss some further causes that led to market illiquidity in developingthe model in the next section). These speculators began to realize that, inthese conditions, they face concrete risk that a debt reduction of $1 could costa significant premium. Eventually, a new exogenously stable asset–the USD-backed custodial stablecoin USDC–had to be brought in as a new collateraltype to stabilize the system [10].
A stablecoin is a cryptocurrency with added economic structure that aims tostabilize price/purchasing power. For a recent overview of stablecoins, see e.g.[4] and the references therein. Stablecoins are meant to bootstrap price stabilityinto cryptocurrencies as a stop-gap measure for adoption. Current projects areeither custodial and rely on custodians to hold reserve assets off-chain (e.g.,$1 per coin) or non-custodial and set up a risk transfer market through on-chain contracts. Non-custodial stablecoins aim to retain the property of reducedcounterparty/censorship risk.Non-custodial stablecoins transfer risk from stablecoin holders to specula-tors, who hold leveraged collateralized positions in cryptocurrencies. A dy-namic deleveraging process balances positions if collateral value deviates toomuch, as determined by a price feed. This is similar to a tranche structure,in which stablecoins act like senior debt, with the addition of dynamic delever-aging. Two major risks in these stablecoins emerge around market structurecollapse and price feed and governance manipulation. In this paper, we focuscompletely on the market structure risk, assuming that price feeds, governance,and the underlying blockchain perform as expected. In addition to the COVID-19 panic, the effects of these risks are also wit-nessed in bitUSD, Steem Dollars, and NuBits, which have suffered serious de-pegging events in 2018 [16], and Terra and Synthetix, which suffered price feedmanipulation attacks in 2019 ([27], [28], [30]) and similar manipulations on thebZx lending protocol in 2020 ([22], [23]). Stablecoins currently serve a centralrole in an increasingly complex decentralized finance environment, involvingcomposability with other DeFi platforms. In addition, many other blockchainassets, such as synthetic and cross-chain assets, rely on the basic mechanismbehind stablecoins, which we discuss further in the discussion section. ‘Leverage’ means that speculators holds > × initial assets but face new liabilities. Note, however, that blockchain congestion can serve to decrease elasticity in the marketstructure, which we discuss in the model construction. .2 This paper In this paper, we construct a stochastic model of over-collateralized stable assets,including non-custodial stablecoins, with an endogenous price (Section 2). Thesystem is based on a speculator who solves an optimization problem accountingfor potential returns from leverage as well as potential liquidation costs. Thespeculator decides the supply of stablecoins secured by its collateral positionwhile considering demand for the stablecoin.We derive fundamental results about the model, including economic limitsto the speculator’s behavior, in Section 3. In Section 4 we develop the primaryresults of the paper: we analytically characterize regions in which the stablecoinexhibits stability (Theorems 1 and 2) and relative instability (Theorems 4 and5), and a region in which a deleveraging spiral occurs that can cause liquidityproblems in a crisis (Theorem 3).The context for our analytical results is a model with a single speculator fac-ing imperfectly elastic demand for the stablecoin; however, many of the meth-ods can extend to generalized settings. We consider such generalizations ofthe model in Section 5 and consider how the stability results will differ givendifferent model structure, including ‘perfect’ stability in idealized settings andpractical applications to realistic settings.We discuss in Section 6 a seeming contradiction that arises: while the goalis to make decentralized non-custodial stablecoins, these can only be fully sta-bilized from deleveraging effects by adding uncorrelated assets, which are cur-rently centralized/custodial. This is a consequence of our instability results inSection 4 and, as introduced in Section 5, the absence of a stable region inidealized settings when underlying asset markets deviate from a submartingalesetting. We suggest an alternative: a buffer to dampen deleveraging effectswithout directly incorporating custodial assets. This buffer works by separatingthose who are willing to have stablecoins swapped to custodial assets in a crisis(in return for an ongoing yield from option buyers) from those who require fulldecentralization.
While there is a rich literature on related financial instruments, there is limitedresearch directly applicable to stablecoins.We develop a simple stable asset model in [15] and introduce the concept ofdeleveraging spirals. This paper supersedes that model and its results. Whereasthe model in [15] doesn’t directly account for the actual repurchase price indeleveraging–instead delegating to a risk constraint in the optimization–we setup a stochastic process model in this paper that includes forward-looking liq-uidation prices in the speculator’s optimization. Our analytical results in thispaper supersede [15] in the following ways: • We formally characterize a deleveraging spiral as a submartingale, whereas[15] lacks a formal treatment. 3
Stability results in [15] are based on a volatility estimator. We prove sta-bility in terms of probabilities of large deviations and quadratic variation. • An unstable region is conjectured in [15], backed by simulation. We for-mally prove distinct price variances in stable and unstable regions.Option pricing theory is applied in [6] to value tranches in a proposed sta-blecoin using PDE methods. In doing so, they need the simplifying assumptionthat payouts (e.g., from liquidations) are exogenously stable, whereas they areactually made in ETH and can cause price feedback effects in the stable asset.In particular, stablecoin holders either hold market risk or are required to re-buy into a reduced stablecoin market following liquidations. This motivates ourmodel to understand stablecoin feedback effects.Stability in proposed stablecoins is simulated in [8] and [24], but under verystrict assumptions.[9] discusses stablecoin concepts based on monetary policy and hedgingstrategies and introduces methods for enhancing liquidity using combinatorialauctions and automated market makers. [17] studied custodial stablecoins andconsiders the use of hedging techniques to build an asset-backed cryptocurrency.[11] explores the robustness of decentralized lending protocols to shocks and liq-uidations. [7] explores competition between decentralized lending yields andstaking yields in proof-of-stake blockchains. However, these do not model astablecoin mechanism with endogenous price behavior.[13] designs a reputation system for crypto-economic protocols to reduce col-lateral requirements. This does not readily apply to understanding stablecoincollaterals, however, as it requires identification of ‘good’ behavior and, addi-tionally, stablecoin speculators face leveraged exchange rate bets and will havereason to provide greater than minimal collateral. This additionally motivatesour model to understand how liquidation effects affect speculator decisions.Stablecoins share similarities with currency peg models, e.g., [20] and [12]. Inthese models, the government plays a mechanical market making role to seek sta-bility and is not a player in the game. In contrast, in non-custodial stablecoins,decentralized speculators take the market making role. They issue/withdrawstablecoins to optimize profits and are not committed to maintaining a peg. Ina stablecoin, the best we can hope is that the protocol is well-designed and thatthe peg is maintained with high probability through incentives. A fully strategicmodel would be a complicated (and likely intractable) dynamic game.There are also similarities with collateral and debt security markets andrepurchase agreements. These have also experienced unprecedented stress inthe COVID-19 market panic, during which even 30-year US government bonds–normally highly liquid–have been difficult to trade [25]. Such debt securitiesdiffer from stablecoins in that dollars are borrowed against the collateral asopposed to a new instrument, like a stablecoin, with an endogenous price. Thesedebt security markets do, however, demonstrate that liquidity in the underlyingmarkets can dry up in crises even in highly liquid markets. Stablecoins face thisliquidity risk in the underlying market as well as an endogenous price effect onthe stable asset. 4he problem resembles classical market microstructure models (e.g., [21]);it is a multi-period system with agents subject to leverage constraints thattake recurring actions according to their objectives. In contrast, the stablecoinsetting has no exogenously stable asset that is efficiently and instantly available.Instead, agents make decisions that endogenously affect the price of the ‘stable’asset and affect futures incentives.
Our model is very closely related to Maker’s stablecoin Dai [19]. The model con-tains a stablecoin market and two assets: a risky asset (ETH) with exogenousprice X t and an ETH-collateralized stablecoin STBL with endogenous price Z t .The stablecoin market connects stablecoin holders, who seek stability, and spec-ulators, who make leveraged bets backing STBL. The STBL protocol requiresthe STBL supply to be over-collateralized in ETH by collateral factor β .In order to focus on the effects of speculator decisions in this paper, wesimplify the stablecoin holder demand as exogenous with constant unit price-elasticity. This is equivalent to a fixed STBL demand D in dollar terms, thoughnot quantity. We relax this to arbitrary elasticity in Section 5, including theperfectly elastic case. Note that there is no direct redemption process for sta-blecoin holders aside from a global settlement/shutdown of the system at parvalue, which can be triggered by a governance process (see [19]).From a practical perspective, STBL demand is not elastic, at least short-term, even if it were in principle elastic longer-term. A significant portionof stablecoin supplies are locked in other applications, like lending protocolsand lotteries. These applications promise (in some sense) value safety in over-collateralization, but don’t guarantee liquidity to withdraw. Additionally, Ethereumtransactions cannot be executed in parallel; during volatile times, transactionscan be delayed due to congestion, causing timely trades (especially involvingtransfer to/from centralized exchanges) to fail. This occurs even if, in principle,there is liquidity in these markets. On the other hand, longer-term demand elas-ticity will naturally depend on the presence of good uncorrelated alternatives. We focus on the case of a single speculator, though we consider general-izations that accommodate many speculators in Section 5. The speculator hasETH locked in the system and decides the STBL supply, which represents a li-ability against its locked collateral. At the start of step t , there are L t − STBLcoins in supply. The speculator holds N t − ETH and chooses to change theSTBL supply by ∆ t = L t − L t − . If ∆ t >
0, the speculator sells new STBL onthe market for ETH at the market clearing price Z t , which is added to N t . If∆ t <
0, the speculator buys STBL on the market, reducing N t . The specula- We designate the risky asset as ETH for simplicity. In principle, it could be anothercryptoasset or even outside of a cryptocurrency setting. From another perspective, a strategic stablecoin holder would take into account expec-tations about speculator issuance and ability to maintain the price target and expectationsabout a global settlement. This is outside of our model as formulated. N t and may or may not be equivalent to N t . Informedby limitations of actual implementations, we develop a particular formulationfor the process ( ¯ N t ) based on ( N t ) in this section, though we discuss ways thatthis can be generalized in Section 5. The speculator decides L t by optimiz-ing expected profitability in the next period based on expectations about ETHreturns and the cost of collateral liquidation if the collateral factor is breached.Given supply and demand, the STBL market clears by setting demand equalto supply in dollar terms. This yields the clearing price Z t = DL t . This clearingequation is related to the quantity theory of money and is similar to the clearingin automated market makers [1] but processed in batch. We formalize the model as follows. We define the following parameters : • D = STBL demand in dollar value (equivalent to constant unit price-elasticity) • β = STBL collateral factor • α ≥ processes : • ( X t ) t ≥ = exogenous ETH price process • L t = stablecoin supply at time t that obeys L t = ζ + L t − + ∆ t where L t − > t is the speculator’s change in liabilities at time t (such that L t = L t − + ∆ t ), and ζ ≥ • N t = speculator’s ETH position at time t , including collateral • ¯ N t = speculator’s locked ETH collateral at time t (and start of time t + 1) • ( Y t ) t ≥ = speculator’s value process • Z t = DL t defines the STBL price processWe take ( F t ) t ≥ to be the natural filtration where F t = σ ( X , . . . , X t , L , . . . , L t ).The system is driven by the process ( X t ) subject to the speculator’s decisions∆ t (equivalently L t given L t − ). In principle, the speculator’s decision could be extended to deciding ¯ N t in addition to L t .Note though that this would make most sense if the speculator’s position is further extendedto include multiple assets.
6o simplify the exposition of analytical results going forward, we simplify tothe case that β = (the collateral factor used in Maker’s Dai stablecoin) and ζ = 0. Note that under these conditions, and in the remainder of the paper, weuse L t and L t interchangeably . However, similar analytical results will extendto the general setting, as we discuss in Section 5. We note that a non-zero ζ may make numerical results more applicable to real settings as it removesthe singularity effect if L t approaches 0, which is why we leave it in the modelexposition. The collateral constraint requires the collateral locked in the system to be ≥ a factor of β times by liabilities. It applies in both a pre-decision and post-decision sense. The pre-decision version determines when a liquidation occurs:a liquidation is triggered at the start of time t if¯ N t − X t < βL t − . The post-decision version constrains the speculator’s decision-making, limiting L t such that ¯ N t X t < βL t . ∆ t We assume the speculator is risk-neutral and optimizes its next-period expectedvalue, taking into account expectations around liquidations. Note that theassumption of risk neutrality can be removed by instead applying an appropriateutility function–in some reasonable cases the results will still hold. Its value attime t is its equity at the start of period (pre-decision), given by Y t = N t − X t − L t − − liquidation effectA liquidation effect is triggered as outlined in a following subsection.Note that N t is a function of the decision variable ∆ t , and recall L t = L t − + ∆ t . The speculator decides ∆ t (equivalently L t given L t − ) to optimizenext-period expected value subject to the post-decision collateral constraint inthe current period: max ∆ t E [ Y t +1 |F t ]s.t. ¯ N t X t ≥ βL t We take the speculator’s collateral at stake at the start of time t + 1 to be¯ N t = N t − minus any collateral liquidation that happens at time t . This isconsistent with the speculator’s collateral being blocked: it cannot be used torepurchase STBL in the same step. This means that the speculator (1) has an7utside amount (or is able to borrow) to repurchase STBL if ∆ t < t >
0) as collateral within the same step.While there are settings in which we could alternatively use N t as the col-lateral at stake at the start of t + 1 (e.g., if flash loans are used), the choiceof N t − additionally leads to a simpler exposition of results as it decouples thecollateral from the decision variable. This said, the general methods and resultswould extend into the setting with N t collateral, as we discuss in the Appendix[refer to section]. In time t + 1, the pre-decision collateral constraint is ¯ N t X t +1 ≥ βL t . If this isbreached, then the speculator’s collateral is partially liquidated, if possible, torepurchase an amount ℓ t +1 > N t X t +1 − ℓ t +1 = β ( L t − ℓ t +1 ). If a liquidation occurs, this is ℓ t +1 = βL t − ¯ N t X t +1 β − L t − N t X t +1 with β = . In a time step with a liquidation, the liquidation forces an up-per bound ∆ t +1 ≤ − ℓ t +1 , but the speculator could choose to repurchase moreSTBL to further reduce leverage. The repurchase of ℓ t +1 through the liquida-tion mechanism is subject to a liquidation fee multiple α ≥ α × the STBL market price. The purpose of this fee is that,in real stablecoin systems, these liquidations are performed by arbitrageurs whocapture this fee. Notice that the STBL market price will itself be affected by theliquidation. The liquidation does not guarantee that the collateral constraint ismet post-liquidation; whether this occurs will depend on the repurchase pricein the market.Two thresholds are relevant at time t for calculating expectations of a liq-uidation effect at time t + 1. These are non-time-dependent functions of therandom variable L t : b ( L t ) := βL t ¯ N t c ( L t ) := 12 ¯ N t (cid:16)q α D + 4 α D L t + L t − α D + L t (cid:17) The threshold b ( L t ) gives the highest t +1 ETH price that breaches the collateralconstraint while the threshold c ( L t ) gives the t + 1 ETH price that consumesthe entirety of the speculator’s locked collateral in a liquidation repurchase dueto the effect on STBL repurchase price. Below this level, the speculator cannotmeet the collateral demand even by liquidating everything. The formulation of b ( L t ) follows directly from the collateral constraint; the formulation of c ( L t ) Note that the probability of a large deviation like this is not zero. For instance, it couldrepresent the possibility of a contentious hard fork that splits ETH value. ℓ t +1 to ¯ N t X t +1 andsolving for X t +1 .If c ( L t ) ≤ X t +1 ≤ b ( L t ), then the liquidation effect is ℓ t +1 − ℓ t +1 DL t − ℓ t +1 α .This represents a repurchase of ℓ t +1 STBL (reducing collateral by the repurchaseprice DL t − ℓ t +1 with liquidation fee factor α ) and subsequent reduction of thespeculator’s liabilities by the ℓ t +1 . The variables L t +1 and N t are affectedsimilarly. If X t +1 < c ( L t ), then the speculator’s collateral position is zeroedout in the liquidation. We define the corresponding events A t = { X t +1 ≥ b ( L t ) } B t = { c ( L t ) ≤ X t +1 < b ( L t ) } Putting all the pieces together, we have the following system of random variablesdriven by the random process ( X t ): X t Y t +1 = ∆ t D X t +1 L t X t + ( ¯ N t X t +1 − L t ) A t ∪ B t + B t (3 L t − N t X t +1 ) (cid:16) − α D N t X t +1 − L t (cid:17) ∆ ∗ t = min (cid:16) arg max ∆ t E [ Y t +1 |F t ] , ¯ N t − X t β − L t − (cid:17) if X t ≥ βL t − ¯ N t − min (cid:16) arg max ∆ t E [ Y t +1 |F t ] , − (3 L t − − N t − X t ) (cid:17) if X t < βL t − ¯ N t − L t = L t − + ∆ ∗ t N t = ( N t − + ∆ ∗ t Z t X t if X t ≥ βL t − ¯ N t − N t − + Z t X t (∆ t + (1 − α )(3 L t − − N t − X t )) if X t < βL t − ¯ N t − ¯ N t = ( N t − if X t ≥ βL t − ¯ N t − N t − − α (3 L t − − N t − X t ) if X t < βL t − ¯ N t − Z t = DL t In the above, the first case for ∆ ∗ t comes from maximizing expected valuesubject to the post-decision collateral constraint while the second cases for ∆ ∗ t , N t , and ¯ N t apply the liquidation effects that occur during time t . In this section, we derive foundational results about the model that we will useto prove the primary results of the paper in the next section. Note that N t is affected because this is the locked collateral at time t + 1. Alternatively,working with N t +1 as locked collateral, we would update N t +1 . .1 Assumptions We begin by defining the assumptions we will use in the rest of the paper.
Assumption 1. ( X t ) is a submartingale with respect to ( F t ) and is independentfrom ( L t ) and ( N t ) . Note that the submartingale assumption can be relaxed somewhat while pre-serving some results. It is useful, though not necessarily critical, in our proof ofproblem concavity. However, the results are most meaningful in a setting like asubmartingale, which always provides a fundamental reason that a speculatormight desire leverage. In such a setting, it is conceivable that the stablecoincould maintain a dollar peg, whereas in long periods of negative expected re-turns, the stablecoin concept falls apart as no speculators will want to partic-ipate. As noted in the introduciton, such a deviation from the submartingalesetting appears to have occurred in March 2020. In Section 5, we elaborate howthe concept falls apart in such negative settings, even given otherwise perfectmarket structure.Also note that the submartingale differences need not be independent formost results. In the Appendix, we further consider ways in which independenceof ( X t ) and ( L t ) can be relaxed. Assumption 2.
Each X t +1 has a conditional probability distribution given F t ,which admits a density function f t that is a.s. continuous. Equivalently, we consider the process in terms of returns R t , where X t +1 = X t R t +1 . Conditioned on F t , then R t +1 admits density function g t . In the i.i.d.setting for ( R t ), the time dependence can be dropped. As noted above, for mostresults, we do not need to assume i.i.d. Assumption 3.
There is some upper bound r ≥ sup n E [ R n |F n − ] . The next assumption is needed to interchange derivative and integrationoperators in the improper setting. Note that it also translates to an upperbound on L t and a lower bound on N t − . Assumption 4.
There is some upper bound u ≥ c ( L t ) for all L t . The next assumption bounds the STBL price away from singularity. Asdiscussed previously, it can be avoided under an alternative formulation of themodel.
Assumption 5. L t ≥ v > for some v . The next assumption simplifies repurchase considerations. It is reasonablegiven a reasonable bound r on expected returns. Assumption 6.
The liquidation premium factor α is sufficiently high that therepurchase price in a liquidation is a.s. > . X distributionsconsidering b ( L t ) is increasingly linearly whereas c ( L t ) decreases with L t . Assumption 7. P ( B t |F t ) = P (cid:16) c ( L t ) ≤ X t +1 ≤ b ( L t ) |F t (cid:17) is increasing in L t . Define ψ ( L t ) := E [ Y t +1 |F t ]. Note that ψ could have a subscript t , or equiv-alently other time t inputs ( ¯ N t , X t , g t ), but we relax notation as we only use itin the context of time t . The next assumption ensures that ψ is concave in L t ,a result that we prove in Prop. 1. Assumption 8. α D Nc t Nc t −L t ) ≤ (note L t ≥ α D is sufficient). Additionally, the next assumption ensures that ψ is strictly concave in L t ,which we also prove in Prop. 1. Notice that this means that either the sub-martingale inequality is strict at time t or there is non-zero probability thata liquidation is triggered in the next step. Given that the latter is certainlyreasonable, this assumption is not much stronger than the basic submartingaleassumption. Assumption 9.
Either E [ R t +1 |F t ] > or P ( B t |F t ) = P (cid:16) c ( L t ) ≤ X t +1 ≤ b ( L t ) |F t (cid:17) > . While strict concavity of ψ is not necessary for all results, it does simplifythe analysis considerably. More generally, concavity of ψ could reasonably beexpected in many settings, and so the assumptions can probably be relaxed.Informally, reasonable distributions for X t will have concentration about thecenter. In this case, moving ∆ t in the positive direction, expected liabilitiesincrease faster than revenue from new STBL issuance. Moving ∆ t in the neg-ative direction, the cost to buyback grows faster than the decrease in expectedliabilities. Our first result is to prove that ψ ( L t ) is concave in L t . Prop. 1.
Given Assumptions 1-8, ψ ( L t ) := E [ Y t +1 |F t ] is concave in L t .Further, given additional Assumption 9, ψ ( L t ) is strictly concave in L t . [Link to Proof] In deriving some results, it will be useful to make assumptions about the scaleof the system. The next result shows that results about Z t should translate todifferently scaled systems, validating that such results will describe the STBLprice process more generally. In the following, we define h to output L t as afunction of the system state. 11 rop. 2. Consider a system setup ( L t − , D , N t − ) with ETH price process ( X t ) . For γ > , h ( γL t − , γ D , γN t − , X t ) = γh ( L t − , D , N t − , X t ) h ( L t − , D , γ N t − , γX t ) = h ( L t − , D , N t − , X t ) As a result, the STBL price process ( Z t ) is equivalent across these system rescal-ings. [Link to Proof] Under these condtions, we can interchange derivative and integration op-erators in ∂ψ∂ L t according to Leibniz integral rules (a variation of dominatedconvergence theorems). The speculator’s choice of L t will fulfill the first ordercondition of ∂ψ∂ L t = 0. From concavity, we can then conclude that the speculatorchooses to increase the STBL supply when ∂ψ∂ L t ( L t − ) > ∂ψ∂ L t ( L t − ) < We now present some fundamental results that bound the speculator’s decision-making. These results will be useful in developing the primary results of thepaper in the next section. The next result introduces a lower bound to the spec-ulator’s STBL supply decision that arises from the fundamental price impact ofrepurchasing STBL.
Prop. 3.
Suppose the pre-decision collateral constraint is met at time t . Thereis a computable lower bound to ∆ t . We can interpret the lower bound in terms of a balance sheet constraintdescribing when the speculator’s ETH position is exhausted in a repurchase.We give the specific bound in the proof but note that it is not especially usefulon its own. Given information about the returns distribution and the level ofcurrent collateral and considering ∂ψ∂ L t , much better bounds are possible. [Link to Proof] The next result provides a useful upper bound to the speculator decision L t .The result is derived from incentives to issue STBL. Intuitively, it says that ifsupply is below this bound, then in some sense a marginal speculator may seea profitable opportunity to expand supply. It’s simply not profitable to issue12ore STBL than this bound. This doesn’t mean that the speculator decides toachieve the bound, however, as it underestimates the liquidation costs that thespeculator might face. Notice that the bound is strongest when we have κ ∼ Prop. 4.
Suppose either of the following hold for given κ : • R b ( Lt ) Xtc ( Lt ) Xt (cid:16) − α D ¯ N t X t z
2( ¯ N t X t z −L t ) (cid:17) g t ( z ) dz ≤ and P ( A t ∪ B t |F t ) ≥ κ − > • ≥ P ( A t |F t ) − P ( B t |F t ) ≥ κ − > Then L t ≤ p κ L t − D E [ X t +1 |F t ] /X t [Link to Proof] .The first condition comes from the derivative of the expected liquidationeffect with respect to L t taking β = . The integrand can be interpreted asthe effective leverage change in a given liquidation. Notice that this is < b ( L t ) (small liquidations effectively reduce leverage) whereas it is > c ( L t ) (in very large liquidations, leverage reduction may notbe effective due to effect on repurchase price). The integral condition then saysthat, in expectation, liquidations effectively reduce leverage. This is a generallyreasonable assumption given a starting state of sufficient over-collateralization,since reasonable distributions of X t +1 will place most mass in the integral around b ( L t ) as opposed to c ( L t ), which is a tail event.The second (alternative) condition says that the probability of having aliquidation is sufficiently smaller than not having a liquidation.This result holds if either of the two conditions hold, both of which could bechecked in data-driven modeling. We will formalize an assumption like the firstcondition in the next section. Note, however, that similar results going forwardcould be derived instead using a variation on the second condition. The primary results of the paper characterize regions in which the stablecoinprice process can be interpreted as ‘stable’ and ‘unstable’. In this section, wederive these results for the given model of a single speculator facing imperfectlyelastic demand for STBL. In the next section, we consider generalizations of themodel and how the stability results will differ given different model structure. Note that the model as formulated does not incorporate an interest rate paid by thespeculator on issued STBL (the ‘stability fee’ in Dai). Additionally, it does not incorporatea possible yield if the speculator creates STBL to lend on a lending platform as opposedto selling on the market. Under either of these extensions, Prop. 4 would change by anappropriate factor. .1 Domain barriers/Stopped processes We establish stability results in terms of barriers. While the stablecoin processis within certain barriers, we prove that it behaves in ways that are interpretableas ‘stable’ and ‘unstable’. These barriers are generally stopping times, and weproceed by considering the stopped processes.Assume that in the initial condition, E h L |F i ≤ L . We define the follow-ing stopping times: • τ is the hitting time of E h L t +1 |F t i > L t , • T m is the hitting time of Z t > m , for m ≥ Z , • S is the hitting time of E [ L t +1 |F t ] < L t , • S is the hitting time of E [ L t +1 |F t ] ≥ L t such that S > S .As we will see, while the stablecoin mechanism is working as intended, wegenerally expect the STBL supply to increase (equivalently in this setting, theSTBL price to decrease, though in slow and bounded way). With this context inmind, τ represents the first time we expect the STBL price to increase. Noticethat this is an expectation of reciprocal of supply, a convex function, and sothrough Jensen’s inequality, this is weaker than expecting the speculator todeleverage/reduce supply. It will be influenced heavily by the tails possibilities.In particular, we have τ ≤ S .Note that the expectations of the process are not necessarily the same as theactual movements of the process: τ does not necessarily correspond to the firsttime the process actually increases in price. We track this with T m , the timethe STBL price breaches a given level above Z , which may be before or after τ . The stopping times S and S track when expectations about STBL sup-ply change. These can be equivalently stated (and calculated in a data-drivenmodel) based on expectations about the derivative of E [ Y t +2 |F t ] with respectto L t +1 evaluated at L t , similarly to the discussion from the previous sectionon concavity.Before proceeding, we formalize stopped versions of assumptions in Prop. 4.The interpretation of these assumptions is the same as discussed in the previoussection. Note that the results going forward could also apply more generallysubject to additional stopping times embedding these assumptions. For no-tational simplicity, we just present the results subject to the stopping timesalready defined with the assumptions given. Assumption 10.
For t ≤ τ , P ( A t ∪ B t |F t ) = P ( X t +1 ≥ c ( L t ) |F t ) ≥ κ − > . Assumption 11.
For t ≤ τ , R b ( Lt ) Xtc ( Lt ) Xt (cid:16) − α D ¯ N t X t z
2( ¯ N t X t z −L t ) (cid:17) g t ( z ) dz ≤ κ will be > ∼ X < c ( L t ) is a low probability event.Recall that the STBL price Z t is a function of collateral value, expectationsabout ETH returns, and expectations of liquidation costs (related to tail risks).These factors go into the speculator’s supply decision, which goes into Z t . Goingforward, we will explore how changes in these affect the STBL price process. We characterize stability of the stablecoin process subject to the barriers τ and T m . In this domain, we derive bounds relating to large price movements andquadratic variation.Our first stability result bounds Z t under the condition T Z > τ . Con-ditioned on this, the price is contained within small variation–e.g., consider Z = 1 and consider κr ∼ Prop. 5.
Let r := sup t E [ X t +1 ] X t . If T Z > τ , then Z ≥ Z t ∧ τ ≥ s D κ L t ∧ τ − r ≥ D ( κ D r ) t − t L t And for any t , L t ∧ τ ≤ κ D r and Z t ∧ τ ≥ κr . [Link to Proof] Notice, however, that the condition T Z > τ introduces dependence on futureevents. As such, we can’t conclude with the information at time t that the t + 1price is bounded in this way.However, we can bound our expectations on the t + 1 price given the infor-mation at time t ( F t ). This approach relies on the fact that the versions of theprocess behaves nicely as submartingales in the stopped setting. Prop. 6. ( L t ∧ τ ) is a submartingale bounded above and ( Z t ∧ τ ) is a supermartin-gale bounded below. Thus they converge a.s. [Link to Proof] An immediate bound on expected price comes from the fact that stoppedversion of Z t is a supermartingale. This is the first result of the next proposition.Additionally, with a stronger assumption on ( X t ) that conditional expectation ofreturns is non-decreasing within the domain barriers, we can bound the expectedprice further. Prop. 7.
The process ( Z t ∧ τ ∧ T Z ) is bounded in expectation by Z ≥ E [ Z t ∧ τ ∧ T Z ] ≥ κr . Further, assuming that for t < τ , ( E [ R t +1 |F t ]) is non-decreasing, then for t ≤ τ , Z t − ≥ E [ Z t ∧ τ |F t − ] ≥ s D κ L t − E [ R t |F t − ]15 Link to Proof]
Going forward, we will work with a variation on the price process Z ′ t := | m − Z t | for given m ≥ Z . Using m = 1, this has concrete interpretation as the absolute price deviationfrom the stablecoin peg. The stopped version of this process has the usefulproperty of being a non-negative submartingale. In addition, ( Z ′ t ) shares similarlarge deviation and quadratic variation properties with ( Z t ), which we explorein the remainder of this subsection. Lemma 1.
The stopped process ( Z ′ t ∧ τ ∧ T m ) is a non-negative submartingale. [Link to Proof] We define the maximum process over some process ( θ t ) as θ ∗ N = max t ≤ N | Θ t | .The next result bounds the expected maximum of the deviation process ( Z t ). Prop. 8.
Suppose m ≥ Z . Denote E := E [ Z τ ∧ T m − m | Z τ ∧ T m > m ] . Supposeany one of the following conditions holds: • κr > m and E > κr − m • κr = m and E > • κr < m and E ≥ Then E [ Z ′∗ τ ∧ T m ] ≤ (cid:0) m − κr (cid:1) . [Link to Proof] The value ( m − κr ) describes the range of the domain considered. Prior to T m , we know that the price falls in this range. The nontrivial part is describingwhat happens at the stopping time as it exceeds this range if the stop is triggeredby T m . The value E is the expected deviation at the stopping time given that T m triggers the stop. By definition, E >
0. Given reasonable κ , r , and m ,the condition for Prop. 8 is satisfied quite broadly. For instance, the concreteinstance with m = 1 is satisfied since κr < κ .Notice that the analysis for the proof can lead to better bounds if we havemore information about E or p := P ( Z τ ∧ T m ≤ m ), e.g., by incorporating in-formation from other results above or from knowledge about the distributionsof ( X t ). Additionally, the analysis can be used to bound either E or p givenbounds on the other.We now state the first main results of the paper. Our next result appliesDoob’s inequality to bound the probability of large deviations in the stoppedprocess. 16 heorem 1. For m ≥ Z and ǫ > , P (cid:18) max n ≤ τ ∧ T m Z ′ n > ǫ (cid:19) ≤ ǫ − (cid:18) m − κr (cid:19) . [Link to Proof] The result can be pretty powerful. Consider the concrete case of m = 1, inwhich case Z ′ t describes the deviation from the peg, and take (arguably reason-able) κ − = 0 .
999 (99 .
9% chance X t won’t drop below c ( L t )) and r annualizedas 1.5 (daily r = 1 . P ( Z ′∗ τ ∧ T > . ≤ . Z ′ t ) by[ Z ′ ] t := t X k =1 ( Z ′ k − Z ′ k − ) . The quadratic variation is a stochastic process that measures how spread outthe underlying process is. Its expectation at time t is related to the varianceat that time, supposing variance is defined–in particular, they are equal if theunderlying process is a martingale. The result bounds the probability of largequadratic variation in the stopped process. In essence, with high probability,the quadratic variation can’t be too far away from the expected maximum. Theorem 2.
Suppose m ≥ Z and ǫ > . Then P (cid:16)p [ Z ′ ] τ ∧ T m > ǫ (cid:17) ≤ ǫ − (cid:18) m − κr (cid:19) [Link to Proof] This result is also pretty powerful. Considering the same setting as above,we have P ( p [ Z ′ ] τ ∧ T > . ≤ .
127 in the stable domain.Bounds on the expectation of quadratic variation can also be obtained usinga more classical form of Burkholder’s inequality, albeit with stronger assump-tions. We develop this idea in the next remark.
Remark 1.
There is an additional form of Burkholder’s inequality that extendsto non-negative submartingales. If we are additionally given a useful bound on E h(cid:0) Z ′ τ ∧ T m (cid:1) p i for some < p < ∞ (for instance, if we have some distribu-tion assumptions on ( X t ) ), then we can apply Lemma 3.1 in [5] to derive thefollowing bound on quadratic variation expectations: E h(cid:0) [ Z ′ ] τ ∧ T m (cid:1) p i p ≤ p − p − E h(cid:0) Z ′ τ ∧ T m (cid:1) p i p . There is a lot of research on obtaining the best constants/bounds in Burkholder’sinequality, which may be able to tighten the bound. ote that the classical two-sided Burkholder inequailty may not extend tonon-negative submartinagales. In general, only the first half of the Burkholderinequality (bounding expectations about quadratic variation by the maximum)extends to this setting and only for < p < ∞ . This contrasts with Prop. 2,where we can derive results about probability of large quadratic variation of non-negative submartingales for the p = 1 case. From a practical point of view, thismay be good enough. Notice that with an effective bound on the expectation of quadratic variation(QV) of the entire stable process, we have by law of large numbers
QVn → n → ∞ . So the longer the process is stable, the smaller the variability.As we’ve characterized the stable domain based on τ and T m , an exit fromthis region corresponds to either a change in expectations ( τ ) or a large deviationevent ( T m ). We now characterize how the stablecoin can be unstable outside of the stableregion barriers described above. The intuition for instability is that, in unstableregions, the speculator’s position is nearer to c ( L t ) and b ( L t ), and so expectedcosts of liquidation increase and are more sensitive to the threshold proximity, inaddition to being driven by the volatile process ( X t ). The remaining results inthis section characterize a deflationary region and instability in terms of varianceof stablecoin prices corresponding to large deviations into such regions.Our next result characterizes a deflationary region defined by stopping times S and S . In such a setting, an opposite behavior occurs compared to the stableregion: ( Z t ) behaves as a submartingale, depicting a deleveraging spiral. Theorem 3.
Restarting the process at S , we have ( L t ∧ S ) is a supermartingaleand ( Z t ∧ S ) is a submartingale. [Link to Proof] The previous result guarantees that the process, after crossing S , enters adeflationary region in a precise sense. Such a deflationary region will correspondto depressed ETH expectations or collateral levels or increased expectationsaround liquidation or deleveraging costs.We now derive more practical tools to characterize instability without havingto detect if S has occurred. With these tools, we will establish more measurableresults in terms of forward-looking price variance of the stablecoin. We beginin the next remark by setting up a variance estimation idea based on Taylorapproximation. 18 emark 2. (Estimating variances) Taylor approximations can be applied toestimate the variances of the stablecoin process. Consider X t = X t − R t forreturn R t ≥ . For notational clarity, define h ( ρ, n ) := arg max L t E [ Y t +1 |F t ] = L t , where ρ, n are realizations of R t , ¯ N t . Variance in stablecoin supply followsVar ( L t |F t − ) ≈ h ′ (cid:0) E [ R t |F t − ] , ¯ N t (cid:1) Var ( R t |F t − ) And the stablecoin price variance approximation isVar ( Z t |F t − ) ≈ D h ′ ( E [ R t |F t − ] , ¯ N t ) E [ L t |F t − ] Var ( R t |F t − ) (1) This is given informally, but could in principle be formalized using two steps ofcompounded Taylor approximation error. The approximation error is arguablymoderate considering that our domain is bounded away from singularities (e.g.,our lower bound results on L ). This variance approximation (Eq. 1 in Remark 2) is low in the stable do-main and can be high in the unstable domain, as formalized in the followingTheorem 4. We introduce a few more assumptions that we use only in derivingthe remaining results in this section. Note that all of these assumptions comedown to assumed properties of the R t distribution. Assumption 12.
The post-decision collateral constraint at time t is not bindingin the speculator’s maximization. This first assumption means that the speculator’s objective fully accountsfor the post-decision collateral constraint (i.e., by maximizing the objective, thespeculator by extension also satisfies the constraint), which is reasonable unlessexpected returns are excessively high.
Assumption 13.
Returns R t − and R t are independent. Assumption 14. ψ is twice continuously differentiable. This last assumption restricts the density g t . We now present the result,which applies the implicit function theorem to derive the derivatives of h , whichdescribe the sensitivity of h to price and collateral level. Theorem 4.
Under the above assumptions, the following hold:1. ∂∂ρ h ( ρ, n ) ∂∂n h ( ρ, n ) exist.2. ∂∂ρ h ( ρ, n ) ≥ and is increasing in − ρ by order of ρ for ρ ≥ b t − X t − , L t > . As in the case of ψ , h could have a subscript t (or equivalently other time t inputs), butwe relax notation as we only use in the context of time t . . ∂∂n h ( ρ, n ) ≥ and is increasing in − n by order of n for n ≥ b t − X t , L t > .4. ∃ ε with < ε < , s.t. ∂∂ρ h ( ρ, n ) > if ρ < ε , L t > α D , and c t > .As a result, the variance approximation in Eq. 1 increases by order of R t in − R t and N t in − ¯ N t . [Link to Proof] Theorem 4 shows that the variance approximation in Eq. 1 in Remark 2increases by order of R t during an ETH return shock (result 2). Similarly,settings with lower collateralization in the initial conditions have higher varianceapproximation by order of N t (result 3). Such differences in initial conditions ofcollateral could result from, for example, different realizations of liquidations orthe speculator abandoning its collateral position (and so extracting any excesscollateral it can). Result 4 shows that there are cases where the h ′ factor in thevariance approximation is >
1, meaning that the variance of R t , the inherentlyvolatile process, will carry through directly to Z t , the ‘stable’ process.Note that the extra conditions on the scale of L t and c t in Theorem 4 results2-4 may seem strange at first sight. Since the ( Z t ) process is scale-invariant,as proven in Prop. 2, the results about Z t variance hold more generally. Inparticular, recall that a term of ∼ L t shows up in the variance approximationin Remark 2, which will cancel out the conditions on scale.Up to this point, we have only been able to say things about variance estima-tions. We will now show that the ‘stable’ and ‘unstable’ regimes are well-definedin the following sense: given different initial conditions of the same process, theforward-looking stablecoin price variances are indeed distinct. This is whatjustifies calling them stable as opposed to unstable. If we start in the unstableregime, we will always have variance higher than if we start in the stable regime.The next result formalizes this. Theorem 5.
In addition to the previous assumptions, suppose X t ≥ b ( L t − )+ ǫ for some ǫ > (the pre-decision collateral constraint is exceeded by ǫ , whichrestricts the ranges of both X t and ¯ N t − ). Consider two possible states s and u of the stablecoin at time t that differ only in collateral amounts ¯ N st − > N ut − and evolve driven by the common price process ( X t ) . Then the forward-lookingprice variances satisfy Var ( Z st |F t − ) < Var ( Z ut |F t − ) . [Link to Proof] Notice that special care should be given to the treatment of Z t under thecondition X t ≤ c ( L t − ), as the STBL price may no longer be well-defined with-out ζ > X t in20he above result is partly to keep things well-defined and partly because therecan be a non-smooth point in h at X t = b ( L t − ).Similar variance difference results can be derived for varying initial condi-tions of X t − and L t − as opposed to ¯ N t − . In some sense, these are all similaras they change the initial collateralization level, though there will be some dif-ference in price effect.The above analytical results describe domains of relative stability and insta-bility for the stablecoin. They may also be adaptable into data-driven risk tools,for instance to estimate probabilities of peg deviations and to infer about whenbarriers are crossed exiting the stable domain. However, an obvious cautionapplies that specific numerical results are highly model-specific and sensitive tomarket structure and distributions of underlying assets. We now explore how the analytical results will extend to different settings.We first demonstrate good stability results in settings with ‘perfect’ marketstructure. We then consider how analytical results from the previous sectionwill extend to generalized model settings.
In the previous section, we considered the given model of a single speculatorfacing imperfectly elastic demand for STBL. We now consider idealized settings,in which STBL demand is perfectly elastic and/or unlimited speculator supplyexists.
Perfectly elastic demand
Under perfectly elastic demand, STBL demandis time-dependent D t , which adapts in each time period to match STBL supply.This results in Z t = 1. In this case, the speculator’s issue and repurchase priceis always $1 and $ α in a liquidation. The problem simplifies to evaluating E [ Y t +1 |F t ] = ∆ t E [ R t +1 |F t ] + Z ∞ ctXt ( ¯ N t X t z − L t ) g ( z ) dz + (1 − α ) Z btXtctXt (3 L t − N t X t z ) g ( z ) dz, where the liquidation effect is now ℓ t +1 (1 − α ) where ℓ t +1 = 3 L t − N t X t +1 and c ( L t ) = 3 α L t ¯ N t (2 α + 1) . In this setting, we have ∂ψ∂ L t = E [ R t +1 |F t ] − P ( A t ∪ B t ) − α − P ( B t ).Recalling P ( A t ) and P ( B t ) are functions of L t and supposing a non-binding21ollateral constraint, the speculator chooses L t such that E [ R t +1 |F t ] = P ( A t ∪ B t ) + 3( α − P ( B t ) . Noting that E [ R t +1 ] ≥ P ( A t ∪ B t ) is decreasing in L t but generally ∼ P ( B t ) is increasing in L t , this is interpretable as the speculator balancingexpected return against 3 × the expected (constant) liquidation cost in decidingwhether to issue a new unit of STBL.In this setting, the STBL price is identically $1 and the speculator only facesthe risk of leveraged ETH declines subject to a fixed liquidation fee. Liquidationsgenerally work well to keep the system over-collateralized, and the only real riskto STBL holders is from extreme single period declines in ETH price. Unlimited speculator supply
Suppose there is an infinite depth of specu-lators (with capital) ready to enter the STBL market given what they see as aprofitable opportunity subject to STBL demand D . A marginal speculator insuch a market would choose to deposit collateral and issue new STBL at time t if DL t − L t E [ R t +1 |F t ] − γ >
0, where γ represents the marginal speculator’s ex-pected liability and liquidation cost after entering the market. Arguably, γ ∼ L t = p γ DL t − E [ R t +1 |F t ] . Notice the similarity with the upper bound in Prop. 4. In this case, we attainthe upper bound on supply because either the initial speculators act to increasesupply or a marginal speculator will see a profitable opportunity and bring usto the upper bound.Further using that ( X t ) is a submartingale, in which case E [ R t +1 |F t ] ≥ Z ≥ Z t ≥ γr . Thisresembles the perfectly elastic demand case as existing speculators are ableto liquidate positions without influencing STBL price, in this case because newmarginal speculators are always willing to issue new STBL to offset a liquidation. No stable region if ( X t ) is not a submartingale Notice that the mech-anisms that give the idealized settings perfect stability break down when theETH price process ( X t ) is not a submartingale. This stresses how fragile thestablecoin market is to negative expectations in the primary ETH market, evenunder these idealized settings. In the unlimited speculator case, marginal spec-ulators no longer enter the market if expectations are negative, and so we don’tachieve the supply bound developed above. Instead, we return to the main set-ting of the paper, which is unstable under negative expectations as it leads todeleveraging effects. In the perfectly elastic demand setting, the STBL supplygoes to zero as the speculator chooses not to participate.22 .2 Generalizing the model We can generalize the single speculator model while retaining similar analyticalresults. We briefly sketch out what these generalized models can look like anddiscuss how these relate to realistic settings.
Concurrent collateral
In the main analysis, we considered the speculator’scollateral at stake at time t to be N t − (minus any applicable liquidation).Alternatively, we now consider the collateral at stake at time t to be N t = N t − + ( L t − L t − ) DL t X t where the second term represents the value in ETH obtained from issuing newstablecoins at time t (negative if redeeming).For notational simplicity drop subscripts as follows: N t N , X t X , L t
7→ L , c ( L t ) c , b ( L t ) b , R t +1 R . And define ψ := E [ Y t +1 |F t ]. Noticethat N is now a function of L . Both b and c , which were previously functionsof L with parameter N , are now further complicated by N ’s dependency on L .In this setting we have ψ ( L ) = Z ∞ c/X ( N Xz − L ) g ( z ) dz + Z b/Xc/X (cid:18) L − α DL N Xz − L − N Xz (cid:19) g ( z ) dz. The partials become more complicated algebraically since N is now a func-tion of L , e.g., partial with respect to L is ∂ψ∂ L = Z ∞ c/X (cid:18) DL t − z L − (cid:19) g ( z ) dz + Z b/Xc/X α DL (cid:16) DL t − z L − (cid:17) (2 N Xz − L ) − α D N Xz − L − DL t − z L + 3 g ( z ) dz. Second partials of ψ are further complicated, as are the partials of b and c .Similar results can in principle be derived in this setting, although we needfurther conditions, for instance to extend the ψ concavity proof (e.g., to ensure ∂b∂ L ≥ Generalized STBL demand
We can consider more general STBL demandfunctions that depend on Z t . For instance, consider a constant elasticity market.Let q be the quantity demanded of STBL at $1 price and suppose the quantitydemanded changes with price subject to a constant price elasticity − γ < Q ( Z t ) = q (1 − γ (1 − Z t )) . D ( Z t ) = Z t Q ( Z t ) = Z t q (1 − γ (1 − Z t )) . Our analysis in the previous section becomes the simplified case where γ = 1,in which case dollar-denominated demand (but not quantity demanded) is con-stant.In clearing the market, the generalized price process becomes Z t = 1 γ (cid:18) L t q − (cid:19) + 1and the collateral process becomes N t = N t − + ( L t − L t − ) (cid:18) γ (cid:18) L t q − (cid:19) + 1 (cid:19) X t which we can apply to get the generalized version of ψ . The general methodsused above can again apply to this formulation, though additional assumptionsmay again be needed, for instance to extend the ψ concavity proof. Generalized collateral factors and supply depth
For a simple exposition,we made the simplifications that the STBL supply is composed solely of coinsissued by the speculator (i.e., L t = L t with outside supply ζ = 0) and thecollateral factor β = 3 /
2. The results will apply for more general ζ ≥ β >
1. The equations for this setting are as follows.Drop subscripts: ¯ N t N , X t X , L t L , c t c , b t b , g t g , R t +1 R . And recall L t = ζ + L t . Then ψ ( L ) = ( L − L t − ) D L + ζ E [ R |F t ] + Z ∞ c/X ( N Xz − L ) g ( z ) dz + Z b/Xc/X βL − N Xzβ − − α D L + ζ − βL − NXzβ − ! g ( z ) dz with c ( L ) and b ( L ) similarly redefined. Endogeneity of ETH prices
We can also extend the model to considercertain endogenizations of ETH prices. The intuition here is that a large enoughliquidation of collateral could have a market impact on ETH prices. One possibleway to endogenize ETH prices is to replace X t +1 f ( X t +1 , ¯ N t , L t ), where f describes the market impact of a collateral liquidation at t + 1 on t + 1 ETHprice and X t +1 describes the (exogenous) price of ETH absent any liquidation.We expect that the general methods used in this paper can be applied topartial equilibrium settings such as this. Naturally, this would necessitate condi-tions on f . Notice that the transformed ETH price process (cid:16) f ( X t +1 , ¯ N t , L t ) (cid:17) t ≥ may no longer be a submartingale. In this case, we would need further condi-tions on f that ensure ψ remains concave.24 ormulating as a multi-period control problem So far, we’ve specifiedthe speculator’s decision-making in terms of a sequence of one-period optimiza-tion problems. However, there could be better long-term strategies. Alterna-tively, the speculator could strategically coordinate the sequence of decisionsfurther into the future.This can be formulated using an exit time for the speculator based on arandom clock, possibly exponential. If this terminal time is deterministic, theproblem can be formulated as a dynamic program, in which the terminal decisionis the one-period optimization, intermediate decisions solve a Bellman equationconditioned on the information revealed up to that point, and random returnsare independent. It is possible to extend these results to a random exit time, ifthat exit time is a ‘nice’ stopping time. For instance, [2] sets up a supermodulargame, for which this works.For this to make sense conceptually, we need to assume the speculator cancash out of its position by selling to someone else at par at the exit time. Thiscan include a noise factor of when this is possible. We expect that this noisefactor would need to be independent of the state of the system in order for theproblem to be tractable. A main challenge is that this will not in reality beindependent.
Realistic settings are likely to be somewhere in between the idealized settingsdescribed in the previous subsection and the single speculator with imperfectlyelastic demand setting explored in this paper. As discussed in the model setup,demand will be imperfectly elastic, at least in the short term. A reasonableinelastic setting can be set by choosing an appropriate elasticity parameter forthe model.A realistic setting will have multiple speculators, including some marginalspeculators, but the depth of the speculators will not be infinite. Further com-plications will come when different speculators maintain positions with differentleverage points and/or ETH expectations. This can lead to a sequential sched-ule of liquidation points at a given time throughout the system, which will bereflected in a speculator’s expected liquidation costs. In particular, a givenspeculator will take into account price effects from the potential liquidations ofother speculators’ positions in addition to their own when evaluating expectedliquidation costs. Additionally, expected liquidation costs will reflect expecta-tions of marginal speculators stepping in to expand the supply. Of course, givenfinite depth, the speculator market can dry up. For instance, the number ofpeople who expect positive ETH returns in an extended bear market may bequite limited.From the perspective of data-driven applications, we would use a comple-mentary version of the model that incorporates an estimation function that thespeculator uses to estimate liquidation costs and price effect since the exactmarket structure is not generally known in a real setting. See [1] and [14] forexamples developing agent-based models in this direction.25
Discussion
This paper presents a new stochastic model of over-collateralized stablecoinswith an endogenous price. In this model, we formally characterize stable andunstable domains for the stablecoin and a deflationary deleveraging spiral as asubmartingale, which can cause liquidity problems in a crisis. We prove thatthe stablecoin behaves in a stable way by bounding the probabilities of largedeviations and quadratic variation, restricted to a certain region. We also provethat price variance is distinctly greater in a separate region, which can be trig-gered by large deviations, collapsed expectations, and liquidity problems fromdeleveraging.An observation from the model is that the speculator chooses a collaterallevel above the required collateral factor. This is because the expected liqui-dation cost is greater than the $1 face value. The speculator will desire toincrease the collateralization during times when the expected liquidation costis higher, which can occur after a shock to collateral value or if the speculatorexpects the collateral to be more volatile. This generally explains the high levelof over-collateralization seen in Dai, which typically ranges 2 . − × althoughthe collateral factor is 1 . × .The presence of deleveraging effects poses fundamental trade-offs in decen-tralized design. One way to bring the stablecoin closer to the ‘perfect’ stabilitycases is to increase elasticity of demand. This relies on the presence of good un-correlated alternatives to the stablecoin. As all non-custodial stablecoins likelyface similar deleveraging risks, greater elasticity relies on custodial stablecoinsor greater exchangeability to fiat currencies. Another way to bring the stable-coin closer ‘perfect’ stability is to increase the supply of marginal speculators.As there will not be unlimited supply of speculators with positive ETH expecta-tions (especially during an extended bear market), this relies on having anotheruncorrelated collateral asset. As all decentralized assets are very correlated, thisagain largely relies on including custodial collateral assets, like Maker’s recentaddition of USDC. While these measures strengthen the stability results, it’sat the expense of greater centralization and moves the system away from being‘non-custodial’.We suggest a buffering idea toward damping deleveraging effects withoutgreater centralization. The Maker system charges fees to speculators, part ofwhich it passes on to Dai holders as an interest rate if the holder locks the Daiinto a savings pool. With modified mechanics, this savings pool can providea buffer to deleveraging effects. For instance, if we allow Dai in the savingspool to be bought out at a reasonable premium by a speculator who uses it todeleverage, then deleveraging effects are bounded by the premium amount upto the size of the savings buffer. The Dai holders who participate in this savingspool are then compensated for providing a repurchase option to the speculator.The Dai holder could elect to have the repurchase fulfilled in the collateral asset, Recall that custodial assets face their own risks, however, which may not be uncorrelatedin extreme crises. This includes counterparty risk, bank run risks, asset seizure risk, andeffects from negative interest rates.
26r something else, like a custodial stablecoin. In this way, this mechanism canprovide some of the benefits of the ‘perfect’ stability settings while enabling Daiholders to choose how decentralized they want to be. A Dai holder who does notrequire high decentralization would elect to receive the compensation from thesavings pool whereas a Dai holder who requires higher decentralization wouldchoose not to use the savings pool. Our model is easily extended to considermechanisms like this.Our model and results can also apply more broadly to synthetic and cross-chain assets and over-collateralized lending protocols that allow borrowing ofilliquid and/or inelastic assets– whenever the mechanism is based on leveragedpositions and leads to an endogenous price of the created or borrowed asset. Syn-thetic assets generally use a similar mechanism just with a different target peg.Cross-chain assets that port an asset from a blockchain without smart contractcapability (e.g., Bitcoin) to a blockchain with smart contracts (e.g., Ethereum)also tend to rely on a similar mechanism. In non-custodial constructions suchas [31] and [29], vault operators are required to lock ETH collateral in additionto the deliverable BTC asset. They bear a leveraged ETH/BTC exchange raterisk and face similar deleveraging risk. In particular, to reduce exposure, theyneed to repurchase the version of the cross-chain asset on Ethereum.
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The Annalsof Probability , 1(1):19-42.[6] Chao, Y., Dai, M., Kou, S., Li, L., Yang, C. (2018). Designing stable coins. Duo NetworkAcademic Whitepaper. https://duo.network/papers/duo_academic_white_paper.pdf .[7] Chitra, T. (2020) Competitive equilibria between staking and on-chain lending. CryptoEconomic Systems ’20.[8] cLabs Team. An analysis of the stability characteristics of Celo. https://celo.org/papers/Celo_Stability_Analysis.pdf , Sep 2019.[9] Detrio, C. (2015). Smart markets for stablecoins. Green paper. http://cdetr.io/smart-markets/ .[10] Foxley, W. MakerDAO adds USDC as DeFi col-lateral following ‘Black Thursday’ chaos. Coindesk, ,17 Mar 2020[11] Gudgeon, L., Perez, D., Harz, D., Gervais, A., Livshits B. (2020). The decentralizedfinancial crisis: attacking DeFi. Available at arXiv.[12] Guimaraes, B., Morris, S. Risk and Wealth in a Model of Self-Fulfilling Currency Attacks.
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Proceedings of the 2019 ACM SIGSAC Conference onComputer and Communications Security , 1485-1502.[14] Kao, H.T., Chitra, T., Chiang. R., Morrow, J. (2020). Market risk assess-ment: an analysis of the financial risk to participants in the Compound protocol. https://gauntlet.network/reports/CompoundMarketRiskAssessment.pdf .[15] Klages-Mundt, A., Minca, A. (2019) (In)stability for the blockchain: deleveraging spiralsand stablecoin attacks. Presented at Crypto Valley Conference on Blockchain Technology’19. Available at arXiv.[16] Klages-Mundt, A. The state of stablecoins–update 2018. https://medium.com/coinmonks/the-state-of-stablecoins-update-2018-56fb82efe6de ,14 Dec 2018.[17] Lipton, A., Hardjono, T., Pentland, A. (2018). Digital Trade Coin (DTC): Towards amore stable digital currency.
Journal of the Royal Society Open Science–Special Issue onBlockchain Technology .[18] MakerDAO. Black Thursday response thread. https://forum.makerdao.com/t/black-thursday-response-thread/1433 ,12 Mar 2020.[19] MakerDAO. The Dai stablecoin system Whitepaper. https://makerdao.com/whitepaper/DaiDec17WP.pdf , 2017.[20] Morris, S., Shin, H.S. Unique equilibrium in a model of self-fulfilling currency attacks.
American Economic Review , 88(3):587–597 (1998).[21] O’Hara, M. Market Microstructure Theory, Basil Blackwell, Cambridge, MA (1995).[22] Peckshield. bZx hack full disclosure (with detailed profit analysis). https://medium.com/@peckshield/bzx-hack-full-disclosure-with-detailed-profit-analysis-e6b1fa9b18fc ,17 Feb 2020.[23] Peckshield. bZx hack II full disclosure (with detailed profit analysis). https://medium.com/@peckshield/bzx-hack-ii-full-disclosure-with-detailed-profit-analysis-8126eecc1360 ,18 Feb 2020.[24] Platias, N., Di Maggio, M. Terra money: stability stress test. https://agora.terra.money/t/stability-stress-test/55 , Jun 2019.[25] Rennison, J., Stafford, P., Smith, C., Wigglesworth, R. ‘Great liq-uidity crisis’ grips system as banks step back. Financial Times, , 23 Mar 2020,[26] See, C.T., Chen, J. (2008). Inequalities on the variances of convex functions of randomvariables.
Journal of inequalities in pure and applied mathematics , 9(3):5.[27] Synthetix. Synthetix response to oracle incident. https://blog.synthetix.io/response-to-oracle-incident/ , 25 Jun 2019.[28] Synthetix. Addressing claims of deleted balances. https://blog.synthetix.io/addressing-claims-of-deleted-balances/ , 16 Sep 2019.[29] tBTC: a decentralized redeemable BTC-backed ERC-20 token. https://docs.keep.network/tbtc , Mar 2020.[30] Terra. Increasing robustness of the Terra oracle. https://agora.terra.money/t/increasing-robustness-of-the-terra-oracle/82 , 1Jul 2019.[31] Zamyatin, A., Harz, D., Lind, J., Panayioutou, P., Gervais, A., Knottenbelt, W. (2019).XClaim: trustless, interoperable cryptocurrency-backed assets. IEEE Symposium on Secu-rity and Privacy ’19. Proofs
In the proofs, we often use the following elementary result
Lemma 2.
For α, D , L ≥ , α D + L ≤ p α D + 4 α D L + L ≤ min (cid:16) α D + L, α D + L + √ α D L (cid:17) Proof.
Define ε := √ α D + 4 α D L + L . We have ε ≤ α D + L as long as α D ≥ L ( √ − α, D , L ≥
0. Next, notice that ε = p ( α D + L ) + 2 α D L . Thus ε > α D + L since 2 α D L ≥
0. Lastly, by concavity, ε ≤ α D + L + √ α D L . Prop. 1
Proof.
Consider X t +1 = X t R t +1 . For notational simplicity, drop subscripts as follows: ¯ N t N , X t X , L t
7→ L , ∆ = L t − L t − , c ( L t ) c , b ( L t ) b , g t g , R t +1 R . Define ψ := E [ Y t +1 |F t ]. Then ψ ( L ) = ∆ · DL E [ R |F t ] + Z ∞ c/X ( NXz − L ) g ( z ) dz + Z b/Xc/X (cid:18) L − α DL NXz − L − NXz (cid:19) g ( z ) dz Recall that the integrand factor (cid:16) L − α DL NXz −L − NXz (cid:17) evaluated at Xz = c is L − Nc (the liquidation zeros out the speculator’s collateral position), and evaluated at Xz = b is 0(on the threshold of liquidation).Taking derivatives using Leibniz integral rule: ∂ψ∂ L = DL t − L E [ R |F t ] − (cid:16) NX cX − L (cid:17) g (cid:16) cX (cid:17) ∂c∂ L X − Z ∞ cX g ( z ) dz − (cid:16) L −
NX cX (cid:17) g (cid:16) cX (cid:17) ∂c∂ L X + Z bXcX (cid:18) − α D NXz NXz − L ) (cid:19) g ( z ) dz = DL t − L E [ R |F t ] − Z ∞ cX g ( z ) dz + Z bXcX (cid:18) − α D NXz NXz − L ) (cid:19) g ( z ) dz∂ ψ∂ L = − DL t − L E [ R |F t ] + g (cid:18) bX (cid:19) ∂b∂L X (cid:18) − α D Nb Nb − L ) (cid:19) − g (cid:16) cX (cid:17) ∂c∂L X (cid:18) − α D Nc Nc − L ) (cid:19) − Z bXcX α D NXz ( NXz − L ) g ( z ) dz Notice that ∂b∂L > ∂c∂L > g ≥
0, and3 − α D Nb Nb − L ) = 3 − α D β L L ( β − = 3 − α DL < ≥
1. Additionally, the remainingintegral is always positive as the integrand is positive between the limits and and g ≥ E [ R |F t ] ≥ X t ) is a submartingale. Thus under the given conditions, ∂ ψ∂ L ≤ ≤ E [ R |F t ] > P (cid:16) c ( L ) < XR < b ( L ) (cid:17) = R b/Xc/X g ( z ) dz > ∂ ψ∂ L < in the bound is related to the choice β = . rop. 2 Proof.
Easily verifiable by substitution, noting that factors of γ cancel in the integral limits. Prop. 3
Proof.
The speculator can at most buy back using all its ETH. At time t , this amount is thesolution ∆ t to the following ∆ t D L t − + ∆ t + N t − X t − L t − − ∆ t = 0supposing there is no liquidation at time t . It is straightforward to verify the solution, givingthe lower bound:∆ t ≥ (cid:16) − q D − DL t − + 2 D N t − X t + N t − X t + D − L t − + N t − X t (cid:17) . Note that if the speculator is not soluble at time t , then there is no real solution. Prop. 4
Proof.
As above, consider X t +1 = X t R t +1 . And for notational simplicity, drop subscriptsas follows: ¯ N t N , X t X , L t
7→ L , ∆ = L t − L t − , c ( L t ) c , b ( L t ) b , g t g , R t +1 R , P ( A t |F t ) P ( A ), P ( B t |F t ) P ( B ).Suppose the first condition is true. We have ∂ψ∂ L = DL t − L E [ R |F t ] − Z ∞ cX g ( z ) dz + Z bXcX (cid:18) − α D NXz NXz − L ) (cid:19) g ( z ) dz ≤ DL t − L E [ R |F t ] − P ( A ∪ B ) ≤ DL t − L E [ R |F t ] − κ − Notice this is monotonic decreasing in L over the domain, so the critical point will be a boundfor the optimal value of L ∗ . Setting equal to 0, we have L ∗ ≤ p κ DL t − E [ R |F t ]Now suppose the second condition is true instead. We have ∂ψ∂ L = DL t − L E [ R |F t ] − Z ∞ bX g ( z ) dz + 2 Z bXcX g ( z ) dz − Z bXcX α D NXz NXz − L ) g ( z ) dz ≤ DL t − L E [ R |F t ] − (cid:16) P ( A ) − P ( B ) (cid:17) ≤ DL t − L E [ R |F t ] − κ − which delivers the desired result as above. rop. 5 Proof.
By assuming T Z > τ , we have Z ≥ Z t ∧ τ . Applying Proposition 4 to Z t = DL t provides Z t ∧ τ ≥ q D κ L t ∧ τ − r . Notice that the upper bound on L t and the lower bound on Z t can be written respectively as increasing and decreasing sequences in t starting from initialstate as follows: L t = ( κ D r ) t − t L t Z t = D ( κ D r ) t − t L t These have limits L ∞ = κ D r and Z ∞ = κr that also bound L t and Z t respectively. Prop. 6
Proof.
For t − < τ , D E [ L t |F t − ] ≤ E (cid:20) DL t |F t − (cid:21) ≤ DL t − by Jensen’s inequality and condition for τ > t −
1. Thus we have E [ L t ∧ τ |F t − ] ≥ L t ∧ τ − and ( L t ∧ τ ) is a submartingale. ( Z t ∧ τ ) is a supermartingale by condition of τ .Applying Proposition 5, L t ∧ τ is bounded above and Z t ∧ τ is bounded below. Thus theyconverge a.s. by Doob’s martingale convergence theorem. Prop. 7
Proof.
The first inequality follows from Prop. 5 and supermartingale properties.Since Z t ∧ τ is supermartingale, we have Z t − ≥ E [ Z t |F t − ]. Assume ( E [ R t +1 |F t ]) isnon-decreasing for t < τ . Then subject to the stopping time τ , E [ Z t |F t − ] ≥ E "s D κ L t − E [ R t +1 |F t ] |F t − (Apply Prop. 4) ≥ vuut D κ L t − E h E [ R t +1 |F t ] |F t − i (Jensen’s inequality)= s D κ L t − E [ R t +1 |F t − ] (Tower property) ≥ s D κ L t − E [ R t |F t − ]since E [ R t +1 |F t ] ≥ E [ R t |F t − ]. emma 1 Proof.
For t − < τ ∧ T m , E [ | m − Z t ||F t − ] ≥ | E [ m − Z t |F t − ] |≥ | m − Z t − | by Jensen’s inequality and the condition for t − < T m that m − Z t − ≥
0. Thus (cid:16) Z ′ t ∧ τ ∧ T m (cid:17) is a non-negative submartingale. Prop. 8
Proof.
Note for t < τ ∧ T m , have Z ∗ t ≤ m , and so Z ′∗ τ ∧ T m − ≤ m − κr . Thus Z ′∗ τ ∧ T m ≤ max (cid:16) m − κr , Z ′ τ ∧ T m (cid:17) .Consider time t = τ ∧ T m and note that optional stopping applies since Z is bounded.Denote W := m − Z t , E := E [ − W | Z t > m ], and p := P ( Z t ≤ m ). From optional stopping,we recall that m ≥ E [ Z t ] ≥ κr , and so 0 ≤ E [ W ] ≤ m − κr . Then E [ W ] = E [ W Z t ≤ m ] − E [ − W Z t >m ] ≤ p (cid:18) m − κr (cid:19) − (1 − p ) E Combining with 0 ≤ E [ W ], we have 0 ≤ p ( m − κr ) − (1 − p ) E , which gives p ≥ Em − κr + E Then noting that (1 − p ) E ≤ E (1 − Em − κr + E ), p ≤
1, and E [ Z ′ t ] = E [ W Z t ≤ m ] + E [ − W Z t >m ], we have E [ Z ′∗ t ] ≤ p E [ Z ′∗ t − ] + (1 − p ) E ≤ m − κr + E − Em − κr + E ! = m − κr + E ( m − κr ) m − κr + E Notice further that given either of the following conditions • κr > m and E > κr − m • κr = m and E > • κr < m ad E ≥ ≤ (1 − p ) E ≤ E ( m − κr ) m − κr + E ≤ m − κr Thus, recalling we used t = τ ∧ T m , we get the following result E [ Z ′∗ τ ∧ T m ] ≤ (cid:18) m − κr (cid:19) heorem 1 Proof.
Given Lemma 1 and Prop. 8 and noting E [ Z ′ τ ∧ T m ] ≤ E [ Z ′∗ τ ∧ T m ], apply Doob’s maximalinequality. Theorem 2
Proof.
Apply Theorem 3.1 in [5], noting that sup n E [ Z ′ n ∧ τ ∧ T m ] ≤ E [ Z ′∗ τ ∧ T m ] by Jensen’sinequality. Theorem 3
Proof.
For S ≤ t < S , we have E (cid:20) DL t |F t − (cid:21) ≥ D E [ L t |F t − ] ≥ DL t − by Jensen’s inequality and the S condition E [ L t |F t − ] ≤ L t − . Thus ( Z S ∨ t ∧ S ) is a sub-martingale (though note that it can be a submartingale for more general stopping times thanthis). L started at S and stopped S is a supermartingale (by def). Theorem 4
Proof.
As above, consider X t +1 = X t R t +1 . And for notational simplicity, drop subscriptsas follows: ¯ N t N , X t − X (notice this is different from previous usage), L t
7→ L ,∆ = L t − L t − , c ( L t ) c , b ( L t ) b , and g t g .Let ρ be (deterministic) variable representing the outcome of R t , such that now we havethe outcome X t = Xρ . And define h ( ρ ) = arg max L ψ ( ρ, L ) = E [ Y t +1 |F t ]. By first ordercondition, ∂∂L ψ ( ρ, h ( ρ )) = 0. The assumptions on ψ provide unique maximum and fulfillconditions of the implicit function theorem, which gives us ∂h∂ρ ( ρ ) exists and ∂h∂ρ ( ρ ) = − ∂ ∂ρ∂L ψ ( ρ, h ( ρ )) ∂ ∂L ψ ( ρ, h ( ρ ))Calculating derivatives using the Leibniz integral rule (recalling c, b are functions of L ), ∂ ψ∂ρ∂L = g (cid:18) cXρ (cid:19) cXρ (cid:18) − α D Nc Nc − L ) (cid:19) − g (cid:18) bXρ (cid:19) bXρ (cid:18) − α D Nb Nb − L ) (cid:19) + Z bXρcXρ α D NXz ( NXρz + L )2( NXρz − L ) g ( z ) dz∂ ψ∂L = − DL t − E [ R t +1 ] L + g (cid:18) bXρ (cid:19) ∂b∂L Xρ (cid:18) − α D Nb Nb − L ) (cid:19) − g (cid:18) cXρ (cid:19) ∂c∂L Xρ (cid:18) − α D Nc Nc − L ) (cid:19) − Z bXρcXρ α D NXρz ( NXρz − L ) g ( z ) dz Notice that (and continuing with β = 3 / − α D Nb Nb − L ) = 3 − α D β L L ( β − = 3 − α DL < y assumption that liquidation repurchase price always ≥
1. And α D Nc Nc − L ) ≤ α D (2 α D + L − α D + L ) − α D (2 α D + L ) + 2 L ( α D + L ) + 2 α D + 2 α D L + 2 L = α D ( α D + 2 L )4( α D + L )(2 L − α D )= α D α D + L ) + α D L − α D ) ≤
112 + α D L − α D )This is ≤ L ≥ α D . Thus under this condition4 − α D Nc Nc − L ) > − α D Nc Nc − L ) ≥ ∂ ψ∂ρ∂L are non-negative and all terms of ∂ ψ∂L are non-positive. Given ρ ≥ b/X , we have g (cid:16) cXρ (cid:17) and g (cid:16) bXρ (cid:17) are increasing in 1 /ρ . Note also that ∂b∂L , ∂c∂L , and DL t − E [ R t +1 ] L are constant in ρ . Lastly, the numerator and denominator integrals can berewritten respectively as1 ρ Z bc α D Nz ( Nz + L )2( Nz − L ) g (cid:18) zXρ (cid:19) dz and Z bc α D Nz ( Nz − L ) g (cid:18) zXρ (cid:19) dz and α D Nz ( Nz + L )2( Nz −L ) ≥ α D Nz ( Nz −L ) given Nz + L ≥ Nc + L >
2, for which L > | h ′ ( ρ ) | are growing by a factor 1 /ρ faster than the terms inthe denominator as ρ decreases, proving (2).Next, note that under the condition 0 < ρ < bXρ = βLNXρ = dbdL LXρ ≥ dbdL XρcXρ ≥ dcdL Xρ ≥ dcdL Xρ The last relation uses the fact that dcdL ≤ α D + L α D + L ) + 1 <
2, and so c > dcdL under the problemsetup.Next note that for ρ ≤ L and c ≤ Xρz ≤ b , we have α D NXz ( NXρz + L )2( NXρz − L ) ≥ αDNXρz ( NXρz − L ) This is because the expression (1) simplifies to
NXρz + L ≥ ρ , (2) to be true over the wholerange of z , we need Nc + L ≥ ρ , and (3) ρ ≤ L is sufficient for this. Thus Z bXρcXρ α D NXz ( NXρz + L )2( NXρz − L ) g ( z ) dz ≥ Z bXρcXρ α D NXρz ( NXρz − L ) g ( z ) dz under these conditions.Then note that all terms in the numerator of h ′ ( ρ ) are greater than and grow faster in 1 /ρ than the comparable terms in the denominator. This leaves the first term in the numerator,which is constant in ρ . To get (3), then note that ε can be chosen such that for ρ = ε , thenumerator and denominator are equal.We can derive the results for ∂h∂n in essentially the same way. Alter the above dropping ofsubscripts with X t X , let n be a variable representing the realization of ¯ N t , and consider h as a function of n . Note the following relevant derivatives. ∂b∂n = − βLn = − bn c∂n = − n (cid:16)p α D + 4 α D L + L − α D + L (cid:17) = − cn∂ ψ∂n∂L = g (cid:16) cX (cid:17) cn (cid:18) − α D nc nc − L ) (cid:19) − g (cid:18) bX (cid:19) bn (cid:18) − α D nb nb − L ) (cid:19) + Z bXcX α D nXz ( nXz + L )2( nXz − L ) g ( z ) dz And translating the following to the new notation ∂ ψ∂L = − DL t − E [ R t +1 ] L + g (cid:18) bX (cid:19) ∂b∂L X (cid:18) − α D nb nb − L ) (cid:19) − g (cid:16) cX (cid:17) ∂c∂L X (cid:18) − α D nc nc − L ) (cid:19) − Z bXcX α D nXz ( nXz − L ) g ( z ) dz And by applying implicit function theorem, we get ∂h∂n ( n ) = − ∂ ∂n∂L ψ ( n, h ( n )) ∂ ∂L ψ ( n, h ( n )) . From here we can proceed with the same analysis using factors of n instead of ρ . Theorem 5
Proof.
For notational simplicity, drop subscripts X t X , ¯ N t − N , L t −
7→ L . Andconsider x a realization of X as variable in h . Define the function f ( X, n ) = h ( X,n ) where n represents the realization of N . With probability 1, the following are true: • h is concave in x and n because h ′ is decreasing, as shown in the previous result. • f is differentiable (wrt n and x ) over domain using chain rule and implicit functiontheorem. • f is convex: it’s the composition of 1 /x and h , and since 1 /x is convex and non-increasing and h is concave, so is f (see [3] 3.2.4). • f is (strictly) decreasing (in n and x ) since h is increasing. • By assumption, we’ve restricted NX . The derivative of f at the minimum value existsand is bounded. • f is non-negative since h is non-negative. • ∂f∂n is (strictly) increasing in n . We have f ′ ( x, n ) = − h ( x, n ) h ′ ( x, n ) , where h ′ ( x, n ) is derived in the previous proof using the implicit function theorem. h is increasing in n and h ′ is non-negative and decreasing in n . Thus h ′ h is decreasing in n , and so − h ′ h is increasing. • ∂h∂n is increasing in x . This can be seen using the formulation at the end of the prooffor the previous result as terms in ∂ ψ∂L grow slower in x (in magnitude) than termsin ∂ ψ∂n∂L . In particular, the first term of ∂ ψ∂L is decreasing in magnitude since L isincreasing in x . And the integral in ∂ ψ∂n∂L increases faster in x than the integral in ∂ ψ∂L , as can be seen by comparing the integrand numerators (a factor of x in ∂ ψ∂n∂L vs. a factor of x in ∂ ψ∂L ). ∂f∂n is (strictly) increasing in x This is because h is increasing in x and ∂h∂n is non-negativeand increasing in x (previous bullet).Note additionally that, from the system setup assumptions, all of the functions are ap-propriately bounded.Thus we can apply Theorem 3.1 in [26] to getVar (cid:16) f ( X, N s ) |F t − (cid:17) < Var (cid:16) f ( X, N u ) |F t − (cid:17) . Note that the variances exist because h = L t is bounded, as shown in previous results. Thevariances of Z st and Z ut are then obtained by multiplying the above inequality by D ..