arXiv subjects

Matthew Lorig

Publications and source records attributed to Matthew Lorig.

At least 19 recordsLinked to original sources

arXiv2026

An Introduction to Financial Derivatives: Open Lecture Notes

These lecture notes give a broad introduction to financial derivatives at the level of a Masters program in quantitative finance. Topics include forward and futures contracts, no-arbitrage bounds and put-call parity, the binomial model, the fundamental theorems of asset pricing, Brownian motion and Itô calculus, the Black-Scholes model, the Greeks, variance swaps, local and stochastic volatility models including Heston, and static hedging of barrier options. Each chapter ends with exercises, and complete solutions, including Python code for the computational exercises, are provided in an appendix. The notes are open source: the LaTeX source is available at https://github.com/mattlorig/intro-financial-derivatives, and readers are invited to submit corrections and additions.

q-fin.MF↗

arXiv2026

Optimal Control of the Ethena Yield-Bearing Stablecoin

We formulate and solve stochastic control problems that model the core yield-generating strategy of the Ethena protocol, a decentralized finance (DeFi) stablecoin that earns yield by combining a long position in staked Ethereum (stETH) with an equal-sized short position in ETH perpetual futures. The combined position is delta-neutral with respect to the ETH spot price, yet earns carry from two sources: staking rewards on the stETH leg, and funding-rate payments received from long perpetual holders when the perpetual trades at a premium to spot. A key feature of our model is that the control -- the rate of simultaneously buying stETH and shorting the perpetual -- exerts two distinct types of price impact. \textit{Permanent} impact shifts the mid-market prices of both legs, compressing the basis and permanently eroding future funding income. \textit{Temporary} impact reflects execution slippage on each leg. We study both an infinite-horizon discounted problem and a finite-horizon problem in which the protocol maximizes total wealth up to a fixed date $T$, subject to a terminal cost for liquidating any remaining position. In both cases the optimal control is obtained explicitly.

q-fin.MF↗

arXiv2026

Publishing Without Journals: An Open, Forkable Archive with Attributed Review

The journal is a seventeenth-century technology asked to do four modern jobs at once: disseminate results, certify their quality, allocate scholarly attention, and confer career credit. It does none of them well. Pre-publication peer review is slow, only weakly reliable, demonstrably biased toward established authors and institutions, and expensive, while the reviewing effort it consumes is spent largely on work that will never matter. We argue that these are not defects to be patched but consequences of bundling dissemination and certification into a single gated act, and we propose unbundling them. Under the proposal, authors deposit papers in an open archive; certification happens \emph{after} deposit, continuously, through attributed and up- or down-voted public commentary to which authors may reply; and papers are version-controlled objects that any qualified reader may \emph{fork}, so that the lineage of an idea -- and hence the credit for it -- is recorded automatically. None of the individual components is speculative: each already exists somewhere in the scholarly ecosystem. The contribution here is to argue that assembling them into a single venue that \emph{replaces} rather than supplements the journal is both feasible and preferable, and to confront the objections -- sparse participation, the chilling effect of real names, the loss of the certification signal, and the non-meritocratic distribution of attention -- that any honest version of the argument must answer.

cs.DL↗

arXiv2026

Short-Rate-Dependent Volatility Models

We price European options in a class of models in which the volatility of the underlying risky asset depends on the short rate of interest. Our study results in an explicit pricing formula that is expressed in terms of a characteristic function. We provide examples of models in which the characteristic function can be computed analytically and, thus, the value of European options is explicit. Numerical implementation to produce the implied volatility is also presented.

q-fin.MF↗

arXiv2026

How to Cook a Soft-Boiled Egg Optimally: A Laplace-Transform Solution of a Two-Domain Heat Equation

We study the problem of cooking the yolk and albumen of a hen's egg to their respective optimal temperatures of $T_Y^* = 65^\circ$C and $T_W^* = 85^\circ$C, subject to the requirement that neither temperature ever exceed its target at any time during cooking, since temporary overshoot still overcooks the egg even if the final reading is correct. We model the egg as a two-domain sphere with distinct thermal diffusivities, and take the Laplace transform of the heat equation in each domain, reducing the problem to a $3 \times 3$ linear system in the transform variable $s$ with hyperbolic-trigonometric solutions. The resulting transform is inverted numerically via Talbot's method and validated against a finite-difference solver. A single boiling phase cannot satisfy the no-overshoot requirement: the thin outer albumen heats far faster than the insulated yolk and necessarily overshoots $T_W^*$ before the yolk approaches $T_Y^*$. We show that a three-phase protocol resolves this: a sous-vide pre-soak at exactly $65^\circ$C (which cannot overshoot since the bath temperature equals the target), a short boil to bring the albumen toward $T_W^*$, and an ice-water bath that arrests the albumen's residual overshoot while residual heat continues raising the yolk to its target. Optimizing the phase durations gives $17.26$ minutes of sous-vide, $66$ seconds of boiling, and an ice bath, achieving both targets at $T^* \approx 20.67$ minutes with neither constraint violated at any time. This compares favorably with the periodic protocol of Di Lorenzo et al. (2025), which requires 32 minutes and misses both targets substantially.

math.AP↗

arXiv2026

A Calculus of Variations Approach to Stochastic Control

We use classical tools from calculus of variations to formally derive necessary conditions for a Markov control to be optimal in a standard finite time horizon stochastic control problem. As an example, we solve the well-known Merton portfolio optimization problem.

math.OC↗

arXiv2026

Optimal Liquidation of Perpetual Contracts

An agent holds a position in a perpetual contract with payoff function $ψ$ and attempts to liquidate the position while managing transaction costs, inventory risk, and funding rate payments. By solving the agent's stochastic control problem we obtain a closed-form expression for the optimal trading strategy when the payoff function is given by $ψ(s) = s$. When the payoff function is non-linear we provide approximations to the optimal strategy which apply when the funding rate parameter is small or when the length of the trading interval is small. We further prove that when $ψ$ is non-linear, the short time approximation can be written in terms of the closed-form trading strategy corresponding to the case of the identity payoff function.

q-fin.MF↗

arXiv2025

Short-Rate Derivatives in a Higher-for-Longer Environment

We introduce a class of short-rate models that exhibit a ``higher for longer'' phenomenon. Specifically, the short-rate is modeled as a general time-homogeneous one-factor Markov diffusion on a finite interval. The lower endpoint is assumed to be regular, exit or natural according to boundary classification while the upper endpoint is assumed to be regular with absorbing behavior. In this setting, we give an explicit expression for price of a zero-coupon bond (as well as more general interest rate derivatives) in terms of the transition density of the short-rate under a new probability measure, and the solution of a non-linear ordinary differential equation (ODE). We then narrow our focus to a class of models for which the transition density and ODE can be solved explicitly. For models within this class, we provide conditions under which the lower endpoint is regular, exit and natural. Finally, we study two specific models -- one in which the lower endpoint is exit and another in which the lower endpoint is natural. In these two models, we give an explicit solution of transition density of the short-rate as a (generalized) eigenfunction expansion. We provide plots of the transition density, (generalized) eigenfunctions, bond prices and the associated yield curve.

q-fin.MF↗

arXiv2024

Interest rate derivatives in a CTMC setting: pricing, replication and Ross recovery

We consider a financial market in which the short rate is modeled by a continuous time Markov chain (CTMC) with a finite state space. In this setting, we show how to price any financial derivative whose payoff is a function of the state of the underlying CTMC at the maturity date. We also show how to replicate such claims by trading only a money market account and zero-coupon bonds. Finally, using an extension of Ross' Recovery Theorem due to Qin and Linetsky, we deduce the real-world dynamics of the CTMC.

q-fin.MF↗

arXiv2024

Rooftop and Community Solar Adoption with Income Heterogeneity

Each household in a population characterized by income heterogeneity faces random demand for electricity and decides if and when it should adopt a solar product, rooftop solar or community solar. A central planner, aiming to meet an adoption level target within a set time, offers net metering and subsidy on solar products and minimizes its total cost. Our focus is on analyzing the interactions of three new features we add to the literature: income diversity, availability of community solar, and consideration of adoption timing. {Methodology and results:} We develop a bilevel optimization formulation to derive the optimal subsidy policy. The upper level (planner's) problem is a constrained non-linear optimization model in which the planner aims to minimize the average subsidy cost. The lower level (household's) problem is an optimal stopping formulation, which captures the adoption decisions of the households. We derive a closed-form expression for the distribution of optimal adoption time of households for a given subsidy policy. We show that the planner's problem is convex in the case of homogeneous subsidy for the two products. {Managerial implications:} Our results underscore the importance for planners to consider three factors - adoption level target, time target, and subsidy budget - simultaneously as they work in tandem to influence the adoption outcome. The planners must also consider the inclusion of community solar in their plans because, as we show, community and rooftop solar attract households from different sides of the income spectrum. In the presence of income inequality, the availability of community makes it easier to meet solar adoption targets.

econ.GN↗

arXiv2024

Optimal positioning in derivative securities in incomplete markets

This paper analyzes a problem of optimal static hedging using derivatives in incomplete markets. The investor is assumed to have a risk exposure to two underlying assets. The hedging instruments are vanilla options written on a single underlying asset. The hedging problem is formulated as a utility maximization problem whereby the form of the optimal static hedge is determined. Among our results, a semi-analytical solution for the optimizer is found through variational methods for exponential, power/logarithmic, and quadratic utility. When vanilla options are available for each underlying asset, the optimal solution is related to the fixed points of a Lipschitz map. In the case of exponential utility, there is only one such fixed point, and subsequent iterations of the map converge to it.

q-fin.MF↗

arXiv2022

Explicit Caplet Implied Volatilities for Quadratic Term-Structure Models

We derive an explicit asymptotic approximation for implied volatilities of caplets under the assumption that the short-rate is described by a generic quadratic term-structure model. In addition to providing an asymptotic accuracy result, we perform experiments in order to gauge the numerical accuracy of our approximation.

q-fin.MF↗

arXiv2022

A primer on perpetuals

We consider a continuous-time financial market with no arbitrage and no transactions costs. In this setting, we introduce two types of perpetual contracts, one in which the payoff to the long side is a fixed function of the underlyers and the long side pays a funding rate to the short side, the other in which the payoff to the long side is a fixed function of the underlyers times a discount factor that changes over time but no funding payments are required. Assuming asset prices are continuous and strictly positive, we derive model-free expressions for the funding rate and discount rate of these perpetual contracts as well as replication strategies for the short side. When asset prices can jump, we derive expressions for the funding and discount rates, which are semi-robust in the sense that they do not depend on the dynamics of the volatility process of the underlying risky assets, but do depend on the intensity of jumps under the market's pricing measure. When asset prices can jump and the volatility process is independent of the underlying risky assets, we derive an explicit replication strategy for the short side of a perpetual contract. Throughout the paper, we illustrate through examples how specific perpetual contracts relate to traditional financial instruments such as variance swaps and leveraged exchange traded funds.

q-fin.MF↗

arXiv2022

Optimal times to buy and sell a home

We consider a financial market in which the risk-free rate of interest is modeled as a Markov diffusion. We suppose that home prices are set by a representative home-buyer, who can afford to pay only a fixed cash-flow per unit time for housing. The cash-flow is a fraction of the representative home-buyer's salary, which grows at a rate that is proportional to the risk-free rate of interest. As a result, in the long-run, higher interest rates lead to faster growth of home prices. The representative home-buyer finances the purchase of a home by taking out a mortgage. The mortgage rate paid by the home-buyer is fixed at the time of purchase and equal to the risk-free rate of interest plus a positive constant. As the home-buyer can only afford to pay a fixed cash-flow per unit time, a higher mortgage rate limits the size of the loan the home-buyer can take out. As a result, the short-term effect of higher interest rates is to lower the value of homes. In this setting, we consider an investor who wishes to buy and then sell a home in order to maximize his discounted expected profit. This leads to a nested optimal stopping problem. We use a nonnegative concave majorant approach to derive the investor's optimal buying and selling strategies. Additionally, we provide a detailed analytic and numerical study of the case in which the risk-free rate of interest is modeled by a Cox-Ingersoll-Ross (CIR) process. We also examine, in the case of CIR interest rates, the expected time that the investor waits before buying and then selling a home when following the optimal strategies.

q-fin.MF↗

arXiv2022

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↗

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↗

arXiv2021

Options on Bonds: Implied Volatilities from Affine Short-Rate Dynamics

We derive an explicit asymptotic approximation for the implied volatilities of Call options written on bonds assuming the short-rate is described by an affine short-rate model. For specific affine short-rate models, we perform numerical experiments in order to gauge the accuracy of our approximation.

q-fin.MF↗

arXiv2021

Optimal Bookmaking

We introduce a general framework for continuous-time betting markets, in which a bookmaker can dynamically control the prices of bets on outcomes of random events. In turn, the prices set by the bookmaker affect the rate or intensity of bets placed by gamblers. The bookmaker seeks a price process that maximizes his expected (utility of) terminal wealth. We obtain explicit solutions or characterizations to the bookmaker's optimal bookmaking problem in various interesting models.

q-fin.MF↗