Replicating market makers

Abstract

We present a method for constructing constant function market makers (CFMMs) whose portfolio value functions match a desired payoff. More specifically, we show that the space of concave, nonnegative, nondecreasing, 1-homogeneous payoff functions and the space of convex CFMMs are equivalent; in other words, every CFMM has a concave, nonnegative, nondecreasing, 1-homogeneous payoff function, and every payoff function with these properties has a corresponding convex CFMM. We demonstrate a simple method for recovering a CFMM trading function that produces this desired payoff. This method uses only basic tools from convex analysis and is intimately related to Fenchel conjugacy. We demonstrate our result by constructing trading functions corresponding to basic payoffs, as well as standard financial derivatives, such as options and swaps.

Appendix: Delta-hedging

As in Sect. 4, we have that $c_1 \in {{\textbf {R}}}_+$, $R_1 \in {{\textbf {R}}}_+$, and $R_2 \in {{\textbf {R}}}_+$ are all scalar quantities, where $c_1$ is the price of the asset in question. We seek to construct a trading function such that $R_1=R_1(c_1)$ for all $c_1$. We define

$$\begin{aligned} p(R_1)=R_1^{-1}(c_1), \end{aligned}$$

which is the marginal price of the traded coin. Recalling that $R_2$ can be thought of as an implicit function of $R_1$, we have

$$\begin{aligned} \frac{\text{d}R_2}{\text{d}R_1}=-p(R_1). \end{aligned}$$

Therefore

$$\begin{aligned} \int p(R_1)\text{d}R_1 = -R_2 \end{aligned}$$

will give a family of trading functions with the desired property.

Hedging a covered call Extending the example in Sect. 4, we now look at delta-hedging a covered call. In this case, we hold $\frac{\text{d}}{\text{d}c} V(c)$ units of the risky asset, i.e., $R_1(c)=1-\Phi (d_1)$. We therefore have

$$\begin{aligned} \int Kh(R_1)\,\text{d}R_1 = -R_2, \end{aligned}$$

where h is defined as before where h is defined as $h(R) = \text{e}^{\sigma \sqrt{\tau }\Phi ^{-1}(1-R_1)-\tau \sigma ^2}$. In this case, we have the trading function

$$\begin{aligned} \psi (R_1,R_2) = k-\frac{1}{2}K+K\Phi (\Phi ^{-1}(1-R_1)-\sigma \sqrt{\tau })-R_2=0, \end{aligned}$$

where k is an arbitrary constant. As before, k will control where the hedge will fail as the CFMM runs out of reserves. An appropriate choice would be to select k, such that the initial value of reserves match the initial price of the covered call. In this case, we have

$$\begin{aligned} c_1R_1+R_2=c_1(1-\Phi (d_1))+K\Phi (d_2). \end{aligned}$$

Substituting the values for $R_1(c)=1-\Phi (d_1)$ and $R_2=k-\frac{1}{2}K+K\Phi (\Phi ^{-1}(1-R_1)-\sigma \sqrt{\tau })$ and solving for k, one obtains $k=\frac{K}{2}$. Substituting this value recovers the trading function derived in Sect. 4.

Hedging a log contract We consider the case of delta-hedging a short position in a log contract. As noted in Neuberger (1994), delta-hedging a contract paying the natural logarithm of the futures price will replicate a variance swap. In this case, we seek a trading function for which $R_1(c)=\frac{1}{c_1}$. In other words, to achieve the desired hedge, we seek a CFMM that holds one unit of asset 2’s worth in asset 1. Noting that

$$\begin{aligned} p(R_1)=\frac{1}{R_1}; \end{aligned}$$

we get

$$\begin{aligned} \int p(R_1)\text{d}R_1 =k-\ln {R_1}=R_2, \end{aligned}$$

where k is an arbitrary constant. Any trading function of the form

$$\begin{aligned} \psi (R_1,R_2) = k-\ln {R_1}-R_2=0 \end{aligned}$$

(A1)

will achieve the desired hedge insofar as $k \ge \ln {R_1}$ or $c_1 \ge \text{e}^{-k}$, as the CFMM will otherwise run out of the reserves required to continue hedging (Fig. 3).

Fig. 3

Delta-hedging CFMM for the log contract with different parameter choices for the constant k

Path dependence and arbitrage loss Suppose the price of the asset at time t is $c_1(t)$ and consider delta-hedging a short position in the log contract by holding $\frac{1}{c_1(t)}$ units of the asset under zero transaction costs. The profit-and-loss (PNL) of this strategy over a discrete period $[t,t+1]$ is $\frac{1}{c_1(t)}(c_1(t+1)-c_1(t))$. When continuously rebalancing over [0, T], we have PNL $\int _0^{\textrm{T}} \frac{1}{c_1(t)} \text{d}c_1(s)$. For simplicity of illustration, suppose $c_1(t)$ follows a geometric Brownian motion with stochastic differential

$$\begin{aligned} {\textrm{d}}c_1(t)=\sigma {\textrm{d}}W(t), \end{aligned}$$

where W(t) is a standard Brownian motion. One can check that the expected PNL of the delta-hedging strategy is zero. Now, we contrast this with delta-hedging with the CFMM we recovered in (A1). One can check that this CFMM has payoff

$$\begin{aligned} V(c_1)=1+k+\ln {c_1(t)}. \end{aligned}$$

The expected PNL of this strategy is therefore

$$\begin{aligned} \mathop {{\textbf {E}}{}}[V(c_1(T))-V(c_1(0))]=-\frac{\sigma ^2}{2}T. \end{aligned}$$

In other words, implementing the delta-hedge through a no-fee CFMM instead of continual rebalancing under no transaction costs will result in a supermartingale. This observation is analogous to the result in Angeris and Chitra (2020) and Evans et al. (2020) that the portfolio value of a G3M or constant-mean market maker is a supermartingale under the risk-neutral measure due to arbitrage losses.

More generally, a no-fee CFMM has payoff $V(c)=R_1c + R_2$, which is always path-independent. In contrast, the equivalent delta-hedging strategies continually rebalanced at no cost are path-dependent. When delta-hedging a convex strategy, one’s portfolio will be short gamma and long theta (Shreve, 2004). The equivalent CFMM does not benefit from positive theta due to arbitrage, resulting in the supermartingale behavior. In other words, delta hedging a convex claim with a CFMM with the appropriate concave payoff will underperform the equivalent delta-hedge rebalanced under no transaction costs. We conjecture that a result similar to that of Evans et al. (2021) may allow one to arbitrarily approximate unconstrained delta-hedging strategies with CFMMs by taking the directional limit as the fee approaches zero, but do not pursue this direction further in this paper.