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Richard J. Martin

Publications and source records attributed to Richard J. Martin.

14 recordsLinked to original sources
arXiv2024

The credit spread curve. I: Fundamental concepts, fitting, par-adjusted spread, and expected return

The notion of a credit spread curve is fundamental in fixed income investing, but in practice it is not `given' and needs to be constructed from bond prices either for a particular issuer, or for a sector rating-by-rating. Rather than attempting to fit spreads -- and as we discuss here, the Z-spread is unsuitable -- we fit parametrised survival curves. By deriving a valuation formula for a risky bond, we explain and avoid the problem that bonds with a high dollar price trade at a higher yield or spread than those with low dollar price (at the same maturity point), even though they do not necessarily offer better value. In fact, a concise treatment of this effect is elusive, and much of the academic literature on risky bond pricing, including a well-known paper by Duffie and Singleton (1997), is fundamentally incorrect. We then proceed to show how to calculate carry, rolldown and relative value for bonds/CDS. Also, once curve construction has been programmed and automated we can run it historically and assess the way a curve has moved over time. This provides the necessary grounding for econometric and arbitrage-free models of curve dynamics, which will be pursued in later work, as well as assessing how the perceived relative value of a particular instrument varies over time.

q-fin.PR↗
arXiv2023

Root-finding: from Newton to Halley and beyond

We give a new improvement over Newton's method for root-finding, when the function in question is doubly differentiable. It generally exhibits faster and more reliable convergence. It can be also be thought of as a correction to Halley's method, as this can exhibit undesirable behaviour.

math.OC↗
arXiv2023

Black-Scholes without stochastics or PDEs

We show how to derive the Black-Scholes model and its generalisation to the `exchange-option' (to exchange one asset for another) via the continuum limit of the Binomial tree. No knowledge of stochastic calculus or partial differential equations is assumed, as we do not use them.

q-fin.PR↗
arXiv2023

Design and analysis of momentum trading strategies

We give a complete description of the third-moment (skewness) characteristics of both linear and nonlinear momentum trading strategies, the latter being understood as transformations of a normalised moving-average filter (EMA). We explain in detail why the skewness is generally positive and has a term structure. This paper is a synthesis of two papers published by the author in RISK in 2012, with some updates and comments.

q-fin.GN↗
arXiv2021

Statistics of the first passage area functional for an Ornstein-Uhlenbeck process

We consider the area functional defined by the integral of an Ornstein-Uhlenbeck process which starts from a given value and ends at the time it first reaches zero (its equilibrium level). Exact results are presented for the mean, variance, skewness and kurtosis of the underlying area probability distribution, together with the covariance and correlation between the area and the first passage time. Amongst other things, the analysis demonstrates that the area distribution is asymptotically normal in the weak noise limit, which stands in contrast to the first passage time distribution. Various applications are indicated.

cond-mat.stat-mech↗
arXiv2021

A note on an absorption problem for a Brownian particle moving in a harmonic potential

An analysis is presented of a Brownian particle moving on the half-line, subject to a restoring force proportional to its displacement and an absorbing boundary at the origin. When the initial displacement is large, the central moments of the time to be absorbed tend to finite constants, as do the position moments when evaluated at the most probable absorption time. These quantities are derived explicitly.

cond-mat.stat-mech↗
arXiv2021

Credit migration: Generating generators

Markovian credit migration models are a reasonably standard tool nowadays, but there are fundamental difficulties with calibrating them. We show how these are resolved using a simplified form of matrix generator and explain why risk-neutral calibration cannot be done without volatility information. We also show how to use elementary ideas from differential geometry to make general inferences about calibration stability. This the longer version of an article published by RISK (Feb 2021).

q-fin.RM↗
arXiv2020

Black to Negative: Embedded optionalities in commodities markets

We address the modelling of commodities that are supposed to have positive price but, on account of a possible failure in the physical delivery mechanism, may turn out not to. This is done by explicitly incorporating a `delivery liability' option into the contract. As such it is a simple generalisation of the established Black model.

q-fin.PR↗
arXiv2020

Fixed income portfolio optimisation: Interest rates, credit, and the efficient frontier

Fixed income has received far less attention than equity portfolio optimisation since Markowitz' original work of 1952, partly as a result of the need to model rates and credit risk. We argue that the shape of the efficient frontier is mainly controlled by linear constraints, with the standard deviation relatively unimportant, and propose a two-factor model for its time evolution.

q-fin.MF↗
arXiv2019

On a stochastic version of Lanchester's model of combat

Lanchester's model of combat has certain deficiencies in its standard form arising from the neglect of the influence of random fluctuations. Several approaches to rectify this have been proposed and various results are scattered throughout the literature. Here, a discrete-time stochastic version, which is amenable to exact solution, is revisited with the aim of deriving key results within one setting. The exposition simplifies and provides refinements to earlier derivations and analysis.

physics.soc-ph↗
arXiv2018

Time since maximum of Brownian motion and asymmetric Levy processes

Motivated by recent studies of record statistics in relation to strongly correlated time series, we consider explicitly the drawdown time of a Levy process, which is defined as the time since it last achieved its running maximum when observed over a fixed time period [0,T]. We show that the density function of this drawdown time, in the case of a completely asymmetric jump process, may be factored as a function of $t$ multiplied by a function of T-t. This extends a known result for the case of pure Brownian motion. We state the factors explicitly for the cases of exponential down-jumps with drift, and for the downward Inverse Gaussian Levy process with drift.

math.PR↗
arXiv2012

Optimal multifactor trading under proportional transaction costs

Proportional transaction costs present difficult theoretical problems in trading algorithm design, on account of their lack of analytical tractability. The author derives a solution of DT-NT-DT form for an arbitrary model in which the the traded asset has diffusive dynamics described by one or more stochastic risk factors. The width of the NT zone is found to be, as expected, proportional to the cube root of the transaction cost. It is also proportional to the 2/3 power of the volatility of the target position, thereby causing a faster trading strategy to be buffered more than a slower one. The displacement of the middle of the buffer from the costfree position is found to be proportional to the square of the width, and hence to the 2/3 power of the transaction cost; the proportionality constant depends on the expected short-term change in position.

q-fin.TR↗
arXiv2011

An exactly solvable self-convolutive recurrence

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform we convert this expression into an explicit, non-recursive solution in which the $n$th coefficient is expressed as the $(n-1)$th moment of a measure, and also as the trace of the $(n-1)$th iterate of a linear operator. Applications of these sequences, and hence of the explicit solution provided, are found in quantum field theory as the number of Feynman diagrams of a certain type and order, in Brownian motion theory, and in combinatorics.

math.CO↗