arXiv subjects

Roger Lee

Publications and source records attributed to Roger Lee.

5 recordsLinked to original sources

arXiv2026

Portfolio Optimization under Fast and Slow Latent Mean-Reverting and Momentum Drift

We consider a class of partial-information portfolio optimization problems in which the drift of a risky asset is driven by two latent stochastic factors evolving at distinct time scales. We show that the filtered estimate of the latent mean-reversion level is driven by the difference between fast and slow exponential moving average (EMA)-type processes of the trailing price history, yielding a Moving Average Convergence Divergence (MACD)-type signal, along with a deterministic Volterra correction. Under logarithmic, power, and exponential utility, we derive candidate optimal strategies in explicit feedback form and establish admissibility and verification results. In particular, the results provide a mathematical foundation for the endogenous emergence of MACD-type trading signals as estimators of latent drift information contained in observed price paths.

q-fin.MF↗

arXiv2023

All AMMs are CFMMs. All DeFi markets have invariants. A DeFi market is arbitrage-free if and only if it has an increasing invariant

In a universal framework that expresses any market system in terms of state transition rules, we prove that every DeFi market system has an invariant function and is thus by definition a CFMM; indeed, all automated market makers (AMMs) are CFMMs. Invariants connect directly to arbitrage and to completeness, according to two fundamental equivalences. First, a DeFi market system is, we prove, arbitrage-free if and only if it has a strictly increasing invariant, where arbitrage-free means that no state can be transformed into a dominated state by any sequence of transactions. Second, the invariant is, we prove, unique if and only if the market system is complete, meaning that it allows transitions between all pairs of states in the state space, in at least one direction. Thus a necessary and sufficient condition for no-arbitrage (respectively, for completeness) is the existence of the increasing (respectively, the uniqueness of the) invariant, which, therefore, fulfills in nonlinear DeFi theory the foundational role parallel to the existence (respectively, uniqueness) of the pricing measure in the Fundamental Theorem of Asset Pricing for linear markets. Moreover, a market system is recoverable by its invariant if and only if it is complete; and in all cases, complete or incomplete, every market system has, and is recoverable by, a multi-invariant. A market system is arbitrage-free if and only if its multi-invariant is increasing. Our examples illustrate (non)existence of various specific types of arbitrage in the context of various specific types of market systems -- with or without fees, with or without liquidity operations, and with or without coordination among multiple pools -- but the fundamental theorems have full generality, applicable to any DeFi market system and to any notion of arbitrage expressible as a strict partial order.

q-fin.TR↗

arXiv2021

Robust Replication of Volatility and Hybrid Derivatives on Jump Diffusions

We price and replicate a variety of claims written on the log price $X$ and quadratic variation $[X]$ of a risky asset, modeled as a positive semimartingale, subject to stochastic volatility and jumps. The pricing and hedging formulas do not depend on the dynamics of volatility process, aside from integrability and independence assumptions; in particular, the volatility process may be non-Markovian and exhibit jumps of unknown distribution. The jump risk may be driven by any finite activity Poisson random measure with bounded jump sizes. As hedging instruments, we use the underlying risky asset, a zero-coupon bond, and European calls and puts with the same maturity as the claim to be hedged. Examples of contracts that we price include variance swaps, volatility swaps, a claim that pays the realized Sharpe ratio, and a call on a leveraged exchange traded fund.

q-fin.MF↗

arXiv2017

Pricing Variance Swaps on Time-Changed Markov Processes

We prove that the variance swap rate (fair strike) equals the price of a co-terminal European-style contract when the underlying is an exponential Markov process, time-changed by an arbitrary continuous stochastic clock, which has arbitrary correlation with the driving Markov process, provided that the payoff function $G$ of the European contract satisfies an ordinary integro-differential equation, which depends only on the dynamics of the Markov process, not on the clock. We present examples of Markov processes where the function $G$ that prices the variance swap can be computed explicitly. In general, the solutions $G$ are not contained in the logarithmic family previously obtained in the special case where the Markov process is a L\'evy process.

q-fin.MF↗

arXiv2015

Robust replication of barrier-style claims on price and volatility

We show how to price and replicate a variety of barrier-style claims written on the $\log$ price $X$ and quadratic variation $\langle X \rangle$ of a risky asset. Our framework assumes no arbitrage, frictionless markets and zero interest rates. We model the risky asset as a strictly positive continuous semimartingale with an independent volatility process. The volatility process may exhibit jumps and may be non-Markovian. As hedging instruments, we use only the underlying risky asset, zero-coupon bonds, and European calls and puts with the same maturity as the barrier-style claim. We consider knock-in, knock-out and rebate claims in single and double barrier varieties.

q-fin.MF↗