Liquidity Provider Returns in Geometric Mean Markets
LLiquidity Provider Returns in Geometric Mean Markets
Alex Evans ∗ June 2020
Abstract
Geometric mean market makers (G3Ms), such as Uniswap and Balancer, comprise a popular classof automated market makers (AMMs) defined by the following rule: the reserves of the AMM beforeand after each trade must have the same (weighted) geometric mean. This paper extends several resultsknown for constant-weight G3Ms to the general case of G3Ms with time-varying and potentially stochasticweights. These results include the returns and no-arbitrage prices of liquidity pool (LP) shares thatinvestors receive for supplying liquidity to G3Ms. Using these expressions, we show how to create G3Mswhose LP shares replicate the payoffs of financial derivatives. The resulting hedges are model-independentand exact for derivative contracts whose payoff functions satisfy an elasticity constraint. These strategiesallow LP shares to replicate various trading strategies and financial contracts, including standard options.G3Ms are thus shown to be capable of recreating a variety of active trading strategies through passivepositions in LP shares.
Decentralized Finance (DeFi) consists of a set of protocols and applications that provide automated financialservices through smart contracts. At the time of writing, it is estimated that nearly 1 billion USD [1] isbeing utilized by DeFi systems. DeFi applications often employ automated market makers (AMMs) to offerstandard financial services such as trading [16] and lending [19], as well as less conventional products suchas perpetual swaps [4] and flash loans [17].Among AMM designs, geometric mean market makers (G3Ms) are most common to Decentralized Ex-changes (DEXs) such as Uniswap [6] and Balancer [25]. In G3Ms, liquidity providers deposit assets into thereserves of a smart contract. This contract permits third parties to submit trades against supplied reserves,executing a trade only if the weighted geometric mean of reserves after the trade is equal to the one before.In exchange for supplying reserves to the contract, liquidity providers are issued liquidity pool (LP) shares inproportion to their contributions. LP shares may be redeemed for a proportional share of the pool’s reservesat any time. The marginal prices offered by G3Ms are known to closely track prices on more liquid tradingvenues [9]. This occurs because arbitrageurs are incentivized to respond to price fluctuations by submittingtrades that rebalance reserves to target weights [25]. This activity is akin to automated Exchange TradedFund (ETF) rebalancing.
Numerical example.
While a formal definition of G3Ms is provided in § A and 5 units of asset B to a G3M that assigns weights w A = 1 / A and w B = 2 / B . The weighted geometricmean of reserves is then 10 / / = 10. If a trader sends 1 unit of asset A to the smart contract anddemands 5 units of asset B in exchange, the trade will be rejected, as the post-trade weighted mean wouldbe 11 / / (cid:54) = 10. However, a trade that adds 1 unit of asset A in exchange for 0.466 units of asset B willbe accepted, as 11 / . / = 10. Clearly, the price offered by the G3M in this trade is 1 unit of asset ∗ [email protected] a r X i v : . [ q -f i n . M F ] J u l which is added to the LP, for 0.466 units of asset B which is removed from the LP. This price dependsonly on the pre-trade reserves R A = R B = 10 and the weights w A = 1 / w B = 2 /
3. After the trade, eachinvestor’s LP shares are redeemable for half of the reserves, namely 5.5 units of asset A and 4.767 units ofasset B . We refer to the total value of reserves that the LP shares can be redeemed for as their “payoff.”The marginal price offered by the G3M is the amount of asset B a trader receives in exchange for asmall quantity of asset A (and vice versa). When the marginal price offered by the G3M doesn’t reflect thetrue market price, an arbitrage opportunity results to adjust the reserves of the G3M. For example, consideragain the case where the LP consists of 10 units of asset A and 10 units of asset B . If the price of asset B is S B = $2 and the price of asset A is S A = $1, then the LP holds $30 worth of assets, of which 1 / A and 2 / B . This allocation agrees with the respective weight of each asset, w A = 1 / w B = 2 /
3. If the external price of asset B drops to S (cid:48) B = $1, then, to restore the allocation so that1 / A and 2 / B , a trader sends 2 . B to the smartcontract. In exchange, the contract sends 3 . A to the trader, maintaining the geometric meanof (10 − . / (10 + 2 . / = 10. The trader thus makes an arbitrage profit of 3 . − . .
1. After thetrade, the reserves are updated to R A = 6 . A and R B = 12 . B . The total value heldin the LP is S A R A + S (cid:48) B R B = $6 . . .
9, of which 6 . / . / A and 2 / B (again corresponding to the respective weights of the two assets). One can check that sending anyamount of asset B to the G3M other than 2 . / A and 2 / G3Ms in practice.
The most well-studied examples of G3Ms are the Uniswap and Balancer protocols.Uniswap exclusively supports LPs consisting of two assets whose reserves are equally weighted. This simplifiesthe geometric mean to a “constant product rule” that allows traders to perform any trade that preserves theproduct of reserves. The simplicitly and apparent effectiveness of Uniswap has spurred other applications toadopt the constant product rule [3, 4, 5, 15].Balancer generalizes the constant product formula by allowing pools of multiple assets as well as config-urable weights. Balancer also supports dynamic weights that can be updated according to a set of rules [23].For example, this allows the LP to gradually decrease its exposure to an asset over time [26] or to adjustweights to favor assets that exhibit lower volatility [14].As of this writing, Uniswap has nearly 60 million USD in reserves and facilitates 10 million USD in dailytrading volume, while Balancer has approximately 30 million USD in reserves and facilitates nearly 1 millionUSD in daily trading volume [1, 2, 7, 21]. Amid growing interest in G3Ms, DeFi lending platforms havestarted accepting LP shares as collateral for secured loans [22]. As G3Ms are attracting larger amounts ofcapital and their LP shares are being used in increasingly complex financial transactions, there is a risingneed for a unified framework to study the return and price characteristics of LP shares in G3Ms.
Prior work.
AMMs have been widely studied since the the introduction of the popular logarithmic marketscoring rule [18]. The present paper focuses on LP share returns in G3Ms, which are a popular class ofAMMs pioneered by [6, 25]. The most relevant prior work in this context is that of [8, 9, 13]. Specifically, [8]derives returns and prices of LP shares in Uniswap, which consists of two equally-weighted assets, while [9]derives an expression for LP share returns in constant-weight G3Ms consisting of more than two assets. ForUniswap, [13] replicates LP share payoffs with the spanning formula of [12] and demonstrates approximatehedging techniques using portfolios consisting of Uniswap LP shares and positions in futures contracts.
Overview.
This paper studies LP share returns in generalized no-fee G3Ms. The static-weight payoffresults in [8] and [9] are extended to G3Ms with dynamic weights. In a parametric setting, the no-arbitrageprices of LP shares are shown to follow directly from these payoff solutions. The resulting prices can beused to analyze certain properties of LP share returns, such as per-trade losses and value leakage from2olatility. This paper also shows how to use LP shares to replicate target payoffs. We show that settingthe weight of a G3M equal to the elasticity of a given payoff function ensures that the LP shares replicatethe payoff. The elasticity of a derivative’s payoff is defined as the percent change in the derivative’s valueper percent change in the price of the underlying asset it references. For differentiable payoff functions thathave elasticity between zero and one, the resulting hedges are exact and do not depend on the model oneuses for the underlying asset price. Replication is also studied under more general assumptions by utilizingparametric hedges. G3M LPs are therefore shown to recreate the payouts of dynamic trading strategiesthrough passive positions in LP shares. Rather than using dynamic trading to replicate a desired payoff,a user may instead purchase and hold the corresponding LPs, while rebalancing is handled by an externalgroup of arbitrage-seeking traders.
A Geometric Mean Market Maker (G3M) is an Automated Market Maker (AMM) [18] whose feasible tradeset is determined by the weighted geometric mean of its reserves. Specifically, for a set of n assets withcorresponding weight vector w ( t ) = ( w ( t ) , . . . , w n ( t )) and reserve vector R ( t ) = ( R ( t ) , . . . , R n ( t )) with R ( t ) ∈ R n + , a G3M enforces the geometric mean V ( t ) = n (cid:89) i =1 R i ( t ) w i ( t ) (1)for all t ≥
0. By assumption, the weight vector is satisfies n (cid:88) i =1 w i ( t ) = 1 , (2) w i ( t ) ≥ . (3)A feasible trade is one that results in an updated reserve vector R (cid:48) ( t ) = ( R (cid:48) ( t ) , . . . , R (cid:48) n ( t )) for which V ( t ) = n (cid:89) i =1 R (cid:48) i ( t ) w i ( t ) . In this paper, we work with G3Ms with no fees. This allows us to greatly simplify the results, while providinga close approximation for many real-world G3Ms that charge traders a small fee. In this setting, let thefeasible trades for a G3M be defined as the set of vectors of the form ∆( t ) = (∆ ( t ) , . . . , ∆ n ( t )) with∆( t ) ∈ R n + that satisfy V ( t ) = n (cid:89) i =1 R i ( t ) w i ( t ) = n (cid:89) i =1 ( R i ( t ) + ∆ i ( t )) w i ( t ) , with ∆ i ( t ) representing the amount of asset i that a trader will deposit into the pool. (Negative valuesindicate amounts the trader removes from the pool.)For a given weighted geometric mean, V ( t ), the price offered by a G3M depends only on the size of thetrade and the balances of reserves in the LP. Denote the prices of the assets in the reserve by the vector S ( t ) = ( S ( t ) , . . . , S n ( t )) with S ( t ) ∈ R n + . As shown in [25, Eq. 7], no-arbitrage requires that for all i (cid:54) = j , R i ( t ) /w i ( t ) R j ( t ) /w j ( t ) = S j ( t ) S i ( t ) . (4)That is, if the weight-normalized ratio of reserves for two assets in the LP is equal to the ratio of their prices,then no arbitrage opportunity exists. We denote the payoff of the LP at time t by G ( t ). Since LP shares3an be redeemed at any time for their underlying assets, their payoff is equal to the value of the underlyingreserves: G ( t ) = n (cid:88) i =1 R i ( t ) S i ( t ) . (5)From (4) and (5), we have for all j ∈ (1 , . . . , n ) G ( t ) = R j ( t ) S j ( t ) w j ( t ) . (6)Note that (6) is equivalent to R i ( t ) S j ( t ) = w j ( t ) G ( t ). In other words, the no-arbitrage condition ensuresthat the value of the position in asset i represents a proportion w i of the LP’s overall value. As shown in [9]and [25], should asset values in the LP deviate from the target weights, an arbitrage opportunity is createdto restore (6). To preclude arbitrage, the G3M LP is therefore continually rebalanced so that the proportionof value allocated to each asset j matches its target weight, w j ( t ), akin to an ETF. Using (4) and (6), andnoting the restriction (2), one can derive the LP share payoff (total value of assets it can be redeemed for)as a function of the weighted geometric mean V ( t ): G ( t ) = R j ( t ) S j ( t ) w j ( t ) (cid:89) ≤ j
Corollary 1.1 (Pricing Uniswap LP shares) . Define Uniswap as a G3M with n = 2 assets, a and b , and w a = w b = . Then the Uniswap LP has η U = − σ r ab T − t ) . (15) ,where σ r ab = (cid:113) σ a + σ b − σ a σ b ρ ab . (16) In particular, we prove that σ r ab is the volatility of the price ratio S a /S b for the two assets in the LP. olatility Losses. To understand the content of η in (13) and (14), recall the observation in § dN ( t ) = N ( t ) n (cid:88) i =1 w i dS i S i . (17)In Appendix A.4, we show that this portfolio strategy has value˜ E (cid:104) e − r ( T − t ) N ( T ) |F ( t ) (cid:105) = e − η ( T − t ) f ( t, S ( t )) . (18)This shows that e η ( T − t ) represents the expected loss LP shares incur relative to a constant-mix portfolio withequivalent weights. This coincides with the well-documented result of volatility harvesting [30] which statesthat a continuously-rebalanced constant-mix portfolio has a greater growth rate than the weighted averageof its component assets. The constant-mix portfolio in (18) benefits from volatility through the e − η ( T − t ) term, while the no-fee LP in (7) does not. This is the cause of the supermartingale behavior observed in(13). One can therefore replicate the value of a fixed-weight G3M LP share with less initial capital bycontinuously rebalancing to the same target weights in a frictionless market. Informally, this occurs becausethe G3M lags the market during rebalancing. LP rebalancing occurs through arbitrage which results whenLP reserves do not reflect updated market prices. LP shares therefore rebalance at suboptimal prices relativeto conventional constant-mix portfolios.Figure 1 plots η using an example of a two-asset LP share with assets a and b . Note that (15) is minimizedin the Uniswap configuration, where w a = ; this represents the maximum loss relative to the constant-mixportfolio. Meanwhile, η is zero when w a = 0 and when w a = 1; in these cases the LP shares coincide withbuy-and-hold portfolios, and there are no opportunities for trading against the assets of the pool (hence noarbitrage losses). The quantity η is increasing with respect to the correlation coefficient ρ ab . The higher thecorrelation coefficient, the smaller the price deviations are expected to be for the assets in the LP; thus, highvalues of ρ ab limit arbitrage losses. Similarly, higher levels of volatility for one of the two assets in the LPproduce greater volatility losses. In the case two-asset case, when σ a = σ b and ρ ab = 1, η is zero regardless ofthe choice of weight, as there is no expected trading (price moves are expected to have identical magnitudeand direction). LP share gamma.
Taking the first derivative of (12) with respect to the stock price (“delta” in optionsterminology) yields f S i = w i S − i f , which is non-negative. Taking the second derivative (“gamma”) gives f S i S i = w i ( w i − S − i f , which, by the restrictions on w i , is non-positive. The constant-weight LP willtherefore decrease its unit position in asset i as its price increases (and conversely increase its unit positionas price declines). The resulting payoff is concave in S i , an effect Uniswap traders refer to as “impermanentloss.” Specifically, regardless of the direction of a price movement, the LP share will decrease in value relativeto the buy-and-hold portfolio, which has a gamma of zero. Note that the constant-mix portfolio withoutrebalancing costs described above also exhibits a negative gamma, but, unlike the G3M LP share, it benefitsfrom volatility in exchange (this is the content of (18)). The LP share’s gamma is minimized (impermanentloss is highest) when the weight of asset i is w i = , while it is zero when w i = 1 (LP holds only asset i ) andwhen w i = 0 (no exposure to i ). As noted in [24], this comes with a direct trade-off to the slippage offeredto traders in the pool. 6igure 1: The left figure plots η (defined in (14)) for a two-asset LP share ( n = 2) with asset volatilities σ a = 0 . σ b = 0 .
2, given different choices for the weight w a of asset a and for the correlation coefficient ρ ab . The right figure holds ρ ab = 0 and plots η for different choices of w a and volatility levels σ a . In this section, the payoffs for G3M LP shares are derived for the case where the weight vector w ( t ) isan F ( t )-measurable process. From an initial weighted geometric mean V (0), assume the process V ( t ) isgenerated by updating the weight vector at a sequence of re-weighting times 0 = t < t < . . . < t s = T .The weight vector is updated at the left endpoint of each interval [ t k , t k +1 ) and is then held constant untilthe next re-weighting time. This ensures that V ( t ) remains constant on each interval but is allowed to varyacross intervals. Assume the initial weighted geometric mean V (0) = V ( t ) is given by V ( t ) = n (cid:89) i =1 R i (0) w i (0) . By assumption, updating satisfies V ( t k − ) = (cid:81) ni =1 R i ( t k ) w i ( t k − ) and V ( t k ) = (cid:81) ni =1 R i ( t k ) w i ( t k ) . Since theweighted geometric mean is constant within each interval, at each t k we have V ( t k ) = n (cid:89) i =1 R i ( t k ) w i ( t k ) = n (cid:89) i =1 R i ( t k ) w i ( t k − ) R i ( t k ) ∆ w i ( t k ) = V ( t k − ) n (cid:89) i =1 R i ( t k ) ∆ w i ( t k ) , where ∆ w i ( t k ) = w i ( t k ) − w i ( t k − ) and (cid:80) ni =1 ∆ w i ( t ) = 0. Repeating this procedure starting from t s we get V ( t s ) = V (0) s (cid:89) k =1 n (cid:89) i =1 R i ( t k − ) ∆ w i ( t k ) . Solving for R i ( t k − ) in the no-arbitrage condition of (6), we have7 i ( t k − ) = w i ( t k − ) S i ( t k − ) G ( t k − ) . Again using (cid:80) ni =1 ∆ w i ( t ) = 0, n (cid:89) i =1 R i ( t k − ) ∆ w i ( t k ) = n (cid:89) i =1 (cid:18) w i ( t k − ) S i ( t k − ) (cid:19) ∆ w i ( t k ) . This provides the discrete-time formula for the weighted geometric mean at time T : V ( T ) = V ( t ) n (cid:89) i =1 s (cid:89) k =1 (cid:18) w i ( t k − ) S i ( t k − ) (cid:19) ∆ w i ( t k ) . (19)Note that this discrete-time formulation is the most realistic setting for G3Ms deployed on public blockchainssuch as Ethereum that have positive-length time intervals between blocks. In this setting, each weightadjustment will present an arbitrage opportunity that results in some value loss for LP shares. This section studies LP returns in the case where weights are allowed to vary continuously. The key resultof this section is the following.
Proposition 2 (Payoff for dynamic-weight LPs) . Assume each component weight function w i ( s ) , i ∈{ , . . . , n } , is continuous and has bounded variation, and denote the length of the longest interval in (19) by || Π || = max k =0 ,...,s − ( t k +1 − t k ) . Then taking the limit in (19) as || Π || → gives the weighted geometricmean for all T ≥ t ≥ V ( T ) = V ( t ) n (cid:89) i =1 (cid:18) w i ( T ) S i ( T ) (cid:19) w i ( T ) (cid:18) S i ( t ) w i ( t ) (cid:19) w i ( t ) e (cid:82) Tt w i ( t ) d log( S i ( t )) with corresponding payoff function G ( T ) = G ( t ) n (cid:89) i =1 e (cid:82) Tt w i ( t ) d log( S i ( t )) . (20)This is the payoff function we work with in the remaining sections.LP prices computed by taking discounted risk-neutral expectations in (20) will depend on the stochasticprocess chosen for the weight vector w ( t ) = ( w ( t ) , . . . , w n ( t )). However, if the weight vector is a deterministicfunction of time, the solution can be simplified. In this case, LP prices can be computed directly given themodel in § Proposition 3 (Pricing LPs with deterministic time-varying weights) . If each component of w ( t ) = ( w ( t ) , . . . , w n ( t )) is an F ( t ) -measurable deterministic function of t , then the corresponding LP share price is given by the dis-counted expectation under the risk-neutral measure of (20) and is equal to ˜ E (cid:104) e − r ( T − t ) G ( T ) |F ( t ) (cid:105) = G ( t ) e η ( t,T ) , (21) where ( t, T ) = n (cid:88) i =1 σ i (cid:90) Tt [ w i ( t ) − w i ( t )] dt + 12 (cid:88) i (cid:54) = j σ i σ j ρ ij (cid:90) Tt w i ( t ) w j ( t ) dt. (22)These prices are relevant to applications that require G3M weights to be adjusted according to a fixedschedule. Typically, an LP will reduce the weight of one of its assets until some target weight is reached. Thiscreates an arbitrage opportunity to remove units of the asset whose weight is declining in favor of the otherreserve assets. This has been proposed as a mechanism for bootstrapping liquidity in nascent markets [26].Similarly, it may be desirable for an LP to decrease its exposure to assets with fixed maturities, such asoptions and bonds, as these near expiry. This section shows how to select G3M weight functions to ensure that the resulting payoffs of the LP sharesreplicate the payoffs of derivative claims on the price of an asset. We work with a two-asset G3M thatconsists of a risky asset with weight w ( x, t ) and a position in the risk-free asset with weight 1 − w ( x, t ),where x = S α ( t ) is the price of the risky asset. Consider a contract with payoff given by the real-valuedfunction g ( x, t ). Rewriting (20) as G ( t ) = G (0) e (cid:82) t w ( x,s ) d log( x ) , (23)we solve for the weight w ∗ ( x, t ) such that the LP and the derivative contract have the same payoff for all t ≥ G ( t ) = g ( S α ( t ) , t ) for all t ≥ . (24) Proposition 4 (Replicating weight function) . Let g be differentiable with respect to x for x ∈ R + . Thenthe solution for w ( x, t ) in (24) with initial condition G (0) = g ( S α (0) , is given by w ∗ ( x, t ) = d log( g ( x, t ))) d log( x ) = xg x ( x, t ) g ( x, t ) , (25) where g x is the partial derivative of g with respect to x . The payoff g ( x, t ) can be replicated by a G3M LPprovided that w ∗ ( x, t ) is continuous in x and ≤ w ∗ ( x, t ) ≤ for all x, t ∈ R + . (26)Equation (25) is the elasticity of a contingent claim, i.e. the percent change in the value of the derivative givena one-percent change in the price of the risky asset (it is also termed “lambda” or “omega” in derivativesparlance). The condition in (26) is due to the restrictions (2) and (3) on the weights of the G3M. Note thatif short-selling an LP share is possible, one can also replicate claims with − ≤ w ∗ ( x ) ≤
0. The condition(26) states that the G3M cannot be used to gain leverage on its reserve assets. The maximum elasticity of acontingent claim with respect to the risky asset is therefore attained when w ( x, t ) = 1 , when the pool consistsexclusively of the risky asset. For differentiable claims where (26) is satisfied, (25) guarantees that holdingan LP share provides an exact static hedge of the contingent claim regardless of the model one uses for theunderlying asset price. In practice, continuous weight adjustments will not be possible in the discrete-time For example, a forward contract expiring at time T has g ( x, T ) = S α ( T ) − K , and an option expiring at time T has g ( x, T ) = max { S α ( T ) − K, } , where, in both examples, K is the strike price. In practice, enforcing weight updates of this form may require the use of a “price oracle” such as [20] that reports the priceof the asset to the G3M smart contract. g ( x, t ), thoughthe introduction of fees can be used to offset all or part of these relative losses.It will often be possible to relax the assumptions of Proposition 4 by instead replicating the value of thecontract by replacing g ( x, t ) in (25) with its discounted expectation under the risk-neutral measure. Suchpricing formulae will typically require the use of a model such as that of § Example (Protective put). A protective put [27] is a popular risk-management strategy wherein aninvestor buys an asset alongside a put option on the same asset. In exchange for the option premium, thestrategy allows the investor to profit from price appreciation while being protected from losses. Given amodel for the option price, we can show that a G3M LP can be programmed to synthetically replicate aprotective put. For example, using the Black–Scholes formula [11] for the value of a put option, we have P ( x, t ) = Ke − r ( T − t ) Φ( − d ) − x Φ( − d ) , where T > K ≥ · ) is the standard normal CDF, and d = log( x/K ) + ( r + σ α / T − t ) σ α √ T − t ,d = d − σ α √ T − t. where σ α is the volatility of the risky asset. It can be shown that the protective put claim g ( x, t ) = x + P ( x, t )has elasticity w ∗ pp ( x, t ) = x (1 − Φ( − d )) P ( x, t ) + x . (27)Note that the numerator is equal to the price of the asset multiplied by one plus the “put delta,” thefirst derivative of the put with respect to x . This quantity is always non-negative, as 0 ≤ Φ( · ) ≤ x ∈ R + . The denominator is also non=negative, as the value of the option is given by the time- t risk-neutralexpectation of g ( x, T ) = max { S α ( T ) − K, } . Therefore w ∗ pp ( x, t ) ≥
0. Furthermore, w ∗ pp ( x, t ) ≤ w ∗ pp ( x, t ) + Ke − r ( T − t ) Φ( − d ) P ( x, t ) + x = 1 . We conclude that setting the G3M’s weight for the risky asset to (27) replicates a protective put on therisky asset with strike K and expiry T . Using the same procedure, we can show that an LP can replicate acovered call, which consists of a long position in an asset alongside a short position in a call option writtenon the same asset. Figure 2 shows the weight function that replicates a protective put. As the price of theunderlying asset increases, the weight tends to one, where the LP consists entirely of the risky asset. As pricedeclines, the LP increases the weight of the money market (risk-free) asset. The relationship with time tomaturity depends on whether the put option is “in the money” (above the strike price K ). If the put is “atthe money” ( S α = K ), then the G3M weight is 0 . S α > K , then the G3M places a greater weight on the risky asset. If the put is near expiry and S α < K ,then the G3M places a greater weight on the risk-free asset. The replicating weight of the protective put inthe risky asset is therefore increasing with respect to the probability that the put will expire out of the money.A number of interesting derivative contracts, such as pure (“naked”) options, often exhibit elasticityfar greater than one. There are two approaches to replicating such contracts. The first involves taking10igure 2: Replicating weights w ( x, t ) for a protective put given a strike price of K = 100 USD for the putoption and a risky asset with monthly volatility of σ α = 0 .
2. The left figure plots the replicating weightas a function of asset price for different maturities, while the right figure plots the replicating weight as afunction of time to maturity for different values of the asset price.offsetting positions in addition to the LP. For example, holding an LP that replicates a protective put whilealso establishing a short position in the underlying asset will replicate the payoff of the put option. Usingthe approach of the proceeding example, it can be shown that a portfolio consisting of a call option plusa position worth e r ( T − t ) K in the money market satisfies (26). Holding the replicating LP share of thisportfolio in addition to an offseting short position of e r ( T − t ) K in the money market will replicate the purecall option. The offsetting positions in the risky or money market assets can be interpreted as borrowingthe respective assets and placing them in the replicating LP. This could be facilitated by an existing lendingprotocol such as [22] that accepts LP shares as collateral for secured loans. For example, to replicate anaked put option, the investor would place an amount of capital equal to the initial price of the option in aG3M that replicates a protective put. At the same time, the lending protocol would supply one unit of therisky asset to the G3M, while taking the corresponding LP shares as collateral. Even if the option expiresworthless, the lender can be assured that the replicating LP will be at least as valuable as the risky asset thatwas lent, ensuring repayment of the loan. At expiration, after repaying the borrowed asset to the lendingprotocol, the investor’s remaining position will have equal value to that of the pure put option (assumingthe model used in constructing the hedge was correctly calibrated).A second approach to replicating claims with elasticity greater than one involves adding derivatives to aG3M’s reserves. The use of levered assets can expand the range of derivatives that an LP share can be usedto replicate. For example, in place of the risky asset one can include a derivative on the risky asset in theLP’s reserves with time- t price z ( S α ( t )). In this case (23) becomes G ( t ) = G (0) e (cid:82) t w ( z ( x )) d log( z ( x )) , (28)and we have the following solution. For simplicity, we work with the single-variable payoff, g ( x ). Corollary 4.1 (Replication with derivative assets) . Let g and z be differentiable on R + . Then the solutionto G ( t ) = g ( S α ( t )) when G ( t ) is given by (28) and with initial condition G (0) = g ( S α (0)) is ∗ ( z ( x )) = d log( g ( x )) d log( z ( x )) . (29) Replication with a G3M LP requires that ≤ d log( g ( x )) d log( z ( x )) ≤ for all x ∈ R + . (30)G3Ms can therefore replicate any claim whose logarithmic derivative is no larger than that of its reserve assetprice function. The logarithmic derivatives of the payoff g ( x ) and price z ( x ) determine their infinitesimalrelative changes and can informally be thought of as a measure of leverage. When the target claim is nomore levered than the reserve claim, replication will be possible through a static position in the LP. This work studies the returns investors receive for contributing reserves to G3Ms. We derive explicit payoffand pricing functions for LP shares in G3Ms that utilize both static and dynamic weights. We show thatLP share payoffs of G3Ms that do not charge fees are supermartingales under the risk-neutral probabilitymeasure, due to having higher rebalancing costs than constant-mix portfolios. Utilizing dynamic weights, weshow that G3M LP shares can be used to provide exact static hedges for arbitrary financial contracts whosepayoffs have elasticity between zero and one. In a parametric setting, we demonstrate how to use offsettingpositions and external leverage to replicate more general financial contracts, such as standard options.A question left open by this paper concerns fees. In practice, most G3Ms charge fees that introduce pathdependencies in LP share payoffs [9]. As fees may alter both the frequency and the cost of G3M rebalancing,it may be instructive to consider the corresponding constant-mix portfolio under rebalancing restrictions andtransaction costs [28].
Acknowledgement
The author would like to thank Guillermo Angeris, Tarun Chitra, Alexandre Obadia and Assimakis Kattisfor their feedback on this paper. 12
Proofs
A.1 Combining Brownian Motions
We establish a definition that will be useful in the proofs of Propositions 1 and 3. For n ≤ d , and given thatthe components of w ( t ) are square-integrable by the restrictions in (2) and (3), we can define Z P ( t ) = n (cid:88) i =1 (cid:90) t w i ( u ) σ i ( u ) σ P ( u ) dW j ( u ) , with σ P ( t ) = (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) i =1 w i ( t ) σ i ( t ) + (cid:88) i (cid:54) = j w i ( t ) w j ( t ) σ i ( t ) σ j ( t ) ρ ij ( t ) , which we assume is non-zero. (As will be discussed in the proofs of Propositions 1 and 3, σ P represents thevolatility of the weighted geometric mean of the risky asset prices.) We can use these definitions to write σ P ( t ) dZ P ( t ) = n (cid:88) i =1 w i ( t ) σ i ( t ) dW i ( t ) . It is trivial to verify that Z P has quadratic variation (cid:104) Z P ( t ) (cid:105) = t . Being the sum of continuous martingales, Z P ( t ) is therefore a Brownian motion by L´evy’s theorem. A.2 Proof of Proposition 1
The proof of Proposition 1 has two parts: first we prove (13), and then we prove that the quantity η definedin (14) is at most zero. i) The proof of (13) runs as follows: the differential for the weighted geometric mean of the prices will givea geometric Brownian motion, from which (13) follows immediately by taking expectations in (11).Note that S i ( t ) w i is given by S w i i ( t ) = S w i i (0) e w i ( r − σ i / t + w i σ i W i ( t ) . Applying Itˆo’s lemma results in the differential dS w i i ( t ) = S w i i ( t ) (cid:20) ( w i r + σ i w i − w i )) dt + w i σ i dW i ( t ) (cid:21) , which defines a geometric Brownian motion with mean ( w i r + σ ( w i − w i )) and volatility w i σ i . Note furtherthat d ( S w i i ( t ) S w j j ( t )) = S w i i ( t ) dS w j j ( t ) + dS w i i ( t ) S w j j ( t ) + dS w i i ( t ) dS w j j ( t )= S w i i ( t ) S w j j ( t )[( r ( w i + w j ) + σ i w i − w i ) + σ j w j − w j ) + w i w j σ i σ j ρ ij ) dt + w i σ i dW i ( t ) + w j σ j dW j ( t )] . Iterating gives 13 (cid:32) n (cid:89) i =1 S w i i ( t ) (cid:33) = n (cid:89) i =1 S w i i ( t ) r + n (cid:88) i =1 σ i w i − w i ) + 12 (cid:88) i (cid:54) = j w i w j σ i σ j ρ ij dt + n (cid:88) i =1 w i σ i dW i ( t ) . (31)As shown shown in § A.1, we may define σ P = (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) i =1 w i σ i + (cid:88) i (cid:54) = j w i w j σ i σ j ρ ij and Z P ( t ) = n (cid:88) i =1 (cid:90) t w i σ i σ P dW j ( u ) , which is a Brownian motion. We can then rewrite (31) as d (cid:32) n (cid:89) i =1 S w i i ( t ) (cid:33) = n (cid:89) i =1 S w i i ( t ) r + n (cid:88) i =1 σ i w i − w i ) + 12 (cid:88) i (cid:54) = j w i w j σ i σ j ρ ij dt + σ P dZ P ( t ) , (32)which is a geometric Brownian motion with mean r + (cid:80) ni =1 σ i ( w i − w i ) + (cid:80) i (cid:54) = j w i w j σ i σ j ρ ij and volatility σ P . We obtain the result (13) by taking the expectation in (11). The result in (12) follows from noting that V (0) = G (0) (cid:81) ni =1 (cid:16) w i (0) S i (0) (cid:17) w i (0) , which follows from (7). ii) Next, we show that η ≤ η is defined in (14)). Since ( T − t ) ≥
0, this is equivalent to showingthat n (cid:88) i =1 σ i ( w i − w i ) + (cid:88) i (cid:54) = j σ i σ j ρ ij w i w j ≤ . Recall the restrictions (2) and (3) and the assumption that σ , . . . , σ n are positive constants. Since thesecond summand is positive and 0 ≤ ρ ij ( t ) ≤
1, it suffices to show that n (cid:88) i =1 σ i ( w i − w i ) + (cid:88) i (cid:54) = j σ i σ j w i w j ≤ . The left-hand side can be rewritten as n (cid:88) i =1 σ i ( w i − w i ) + (cid:88) i (cid:54) = j σ i σ j w i w j = − n (cid:88) i =1 σ i w i (1 − w i ) + (cid:88) i (cid:54) = j σ i σ j w i w j = − n (cid:88) i =1 σ i w i i − (cid:88) j =1 w j + n (cid:88) j = i +1 w j + 2 (cid:88) ≤ i The volatility σ ab will follow from the expression for the price ratio. The stochastic differential for the ratioof the prices of two assets S r ab ( t ) = S a ( t ) /S b ( t ) is given by S r ab ( t ) = (1 /S b ( t )) dS a ( t ) − ( S a ( t ) /S b ( t )) dS b ( t ) − (1 /S b ( t )) dS a ( t ) dS b ( t ) + ( S a ( t ) /S b ( t ))( dS b ) = S r ab ( σ b ( t ) − σ a ( t ) σ b ( t ) ρ ab ( t )) dt + S r ab σ r ab ( t ) dZ r ( t ) , where σ r ab ( t ) = (cid:113) σ a ( t ) + σ b ( t ) − σ a ( t ) σ b ( t ) ρ ab ( t )and Z r ( t ) = 1 σ r ab (cid:18)(cid:90) t σ a ( u ) dW a ( u ) − (cid:90) t σ b ( u ) W b ( u ) (cid:19) ;note that Z r ( t ) is a Brownian motion. Therefore S r ab is a geometric Brownian motion with drift σ b ( t ) − σ a ( t ) σ b ( t ) ρ ab ( t ) and volatility σ r ab ( t ). Assuming constant volatilities and taking n = 2 and w a = w b = in(13), we have η = (cid:18) − σ a − σ a σ a σ b ρ ab (cid:19) ( T − t ) = σ r ab T − t ) , as desired. A.4 Payoff of Constant-Mix Portfolio From (17) we have dN ( t ) = N ( t )( rdt + n (cid:88) i =1 w i σ i dW i )= N ( t )( rdt + σ P dZ P ) , (33)which gives N ( t ) = N (0) e ( r − σ P ) t + σ P Z P ( t ) . Comparing (33) with (32) shows that the difference between their drift terms is (14), from which the resultin (18) follows by taking expectations. 15 .5 Proof of Proposition 2 Take the limit as the quantity || Π || = max k =0 ,...,s − ( t k +1 − t k ) (the size of the longest time interval in (19))tends to zero: V ( T ) = V ( t ) lim || Π ||→ s (cid:89) k =1 n (cid:89) i =1 w i ( t k − ) S i ( t k − ) ∆ w i ( t k ) . We have log[ V ( T ) /V ( t )] = lim || Π ||→ n (cid:88) i =1 s (cid:88) k =1 log (cid:18) w i ( t k − ) S i ( t k − ) (cid:19) ∆ w i ( t k )= n (cid:88) i =1 (cid:90) Tt log (cid:18) w i ( t ) S i ( t ) (cid:19) dw i ( t )= n (cid:88) i =1 (cid:34) log (cid:18) w i ( T ) w i ( T ) w i (0) w i (0) (cid:19) + w i ( T ) − w i (0) − (cid:90) Tt log( S i ( t )) dw i ( t ) (cid:35) . Note that (cid:80) ni =1 [ w i ( T ) − w i ( t )] = 0, and integrate by parts:log[ V ( T ) /V ( t )] = n (cid:88) i =1 (cid:34) log (cid:18) w i ( T ) w i ( T ) w i ( t ) w i ( t ) (cid:19) − w i ( T ) log( S i ( T )) + w i ( t ) log( S i ( t )) + (cid:90) Tt d log( S i ( t )) w i ( t ) (cid:35) . Setting t = t , V ( T ) = V ( t ) n (cid:89) i =1 (cid:18) w i ( T ) S i ( T ) (cid:19) w i ( T ) (cid:18) S i ( t ) w i ( t ) (cid:19) w i ( t ) e (cid:82) Tt w i ( t ) d log( S i ( t )) . Using the payoff function in (7), G ( T ) = V ( t ) n (cid:89) i =1 (cid:18) S i ( t ) w i ( t ) (cid:19) w i ( t ) e (cid:82) Tt w i ( t ) d log( S i ( t )) . Noting that (7) also implies V ( t ) = G ( t ) n (cid:89) i =1 (cid:18) w i ( t ) S i ( t ) (cid:19) w i ( t ) gives G ( T ) = G ( t ) n (cid:89) i =1 e (cid:82) Tt w i ( t ) d log( S i ( t )) , as desired. 16 .6 Proof of Proposition 3 Expanding in (20), we have G ( T ) = G ( t ) n (cid:89) i =1 e ( r − σ i ) (cid:82) Tt w i ( t ) dt + σ i (cid:82) Tt w i ( t ) dW − i ( t ) = G ( t ) e r ( T − t ) − (cid:80) ni =1 σ i (cid:82) Tt w i ( t ) dt + σ i (cid:82) Tt w i ( t ) dW i ( t ) . Taking expectations, we obtain˜ E (cid:104) e − r ( T − t ) G ( T ) |F ( t ) (cid:105) = ˜ E (cid:20) G ( t ) e − r ( T − t ) e r − (cid:80) ni =1 σ i (cid:82) Tt w i ( t ) dt + σ i (cid:82) Tt w i ( t ) dW i ( t ) (cid:21) = G ( t ) e − (cid:80) ni =1 σ i (cid:82) Tt w i ( t ) dt ˜ E (cid:104) e σ i (cid:82) Tt w i ( t ) dW i ( t ) |F ( t ) (cid:105) . (34)Following the process outlined in § A.1 now define the processes σ P ( t ) = (cid:118)(cid:117)(cid:117)(cid:116) n (cid:88) i =1 w i ( t ) σ i + (cid:88) i (cid:54) = j w i ( t ) w j ( t ) σ i σ j ρ ij and Z P ( t ) = n (cid:88) i =1 (cid:90) t w i ( t ) σ i σ P ( t ) dW j ( u ) . where Z P ( t ) is a Brownian motion. Equation (34) can now be written as˜ E (cid:104) e − r ( T − t ) G ( T ) |F ( t ) (cid:105) = G ( t ) e − (cid:80) ni =1 σ i (cid:82) Tt w i ( t ) dt + (cid:82) Tt σ P ( t )2 dt = G ( t ) e (cid:80) ni =1 σ i (cid:82) Tt [ w i ( t ) − w i ( t )] dt + (cid:80) i (cid:54) = j σ i σ j ρ ij (cid:82) Tt w i ( t ) w j ( t ) dt , as desired. A.7 Proof of Proposition 4 We seek a solution for w ( x, t ) that satisfies G (0) e (cid:82) t w ∗ ( x,s ) d log( x ) = g ( x, t )with initial condition g ( S α (0)) = G (0). This is equivalent to (cid:90) t w ∗ ( x, s ) d log( x ) = log g ( x, t ) G (0) , which is solved by w ∗ ( x, t ) = d log( g ( x, t )) d log( x ) = xg x ( x, t ) g ( x, t ) . .8 Proof of Corollary 4.1 The proof is identical to that of Proposition 4, except that we replace x by z ( x ) and w ( x, t ) by w ( z ( x )). 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