Timothy Falcon Crack

Interest Rate Sensitivities of Bond Risk Measures

We present a simple expression for the sensitivity of duration, convexity, and higher-order bond risk measures to changes in term-structure shape parameters. Our analysis enables fixed-income portfolio managers to capture the combined effects of shifts in term-structure level, slope, and curvature on any specific bond risk measure. These results are particularly important in environments characterized by volatile interest rates. We provide simple numerical examples. Building on previous research into the sensitivities of bond risk measures, we present a simple expression for the sensitivity of duration, convexity, and higher-order bond risk measures to nonparallel changes in the shape of the yield curve. Although researchers have analyzed the sensitivity of a bonds duration to changes in the bonds yield, little is known about the interest rate sensitivity of duration, convexity, and so on, to changes in level, slope, and curvature of the term structure. The subject is important because up to 95 percent of returns to portfolios of U.S. Treasury securities are explained by term-structure level, slope, and curvature shifts—and these shifts can be quite extreme in volatile interest rate environments. We captured these parameters of term-structure shape by using a simple polynomial representation of the continuously compounded spot yield curve. Given a noninfinitesimal, nonparallel shift in the yield curve, we were able to derive closed-form expressions for the resulting changes in bond risk measures as a function of changes in the level, slope, and curvature of term structure and as a function of the bond risk measures themselves. Our framework enabled us to answer questions that are relevant to the work of managers who are required to maintain target durations for their bond portfolios and who wish to know how sensitive their bond risk positions are to general interest rate changes: How does the duration of a bond change with respect to a change in the slope of the term structure? How does the convexity of a bond change with respect to a change in the level of the term structure? Do the duration and convexity of a barbell portfolio change more rapidly than those of a bullet portfolio? These questions are relevant to managers of fixed-income portfolios and managers of financial institutions. Shifts in term-structure level, slope, and curvature are not independent. For example, increases in level tend to be associated with decreases in slope. We used such interrelationships to derive a simple but realistic numerical example of the effect of a noninfinitesimal, nonparallel term structure shift on a bullet bond and two barbell bonds. We found that if we ignored the slope and curvature shifts and accounted only for the level shift, we seriously misestimated the effect of the full term-structure shift on bond duration measures for the barbell bonds. The percentage error we made became larger as the cash flow spacing of the barbell became wider. When we added a term (i.e., slope to level) and then two terms (i.e., curvature and slope to level), the magnitude of our estimation errors decreased substantially. Therefore, accounting for the impact of level shifts alone (i.e., parallel shifts) is not sufficient when estimating the effect of changes in term-structure shape on bond risk measures. We also note one simple result: Although the bonds in our numerical example all had the same initial price and duration, the effect of the nonparallel shift in term structure on their prices was quite different. This outcome is a simple reminder that practitioners must look beyond parallel term-structure shifts when analyzing bonds.

Characterising trader manipulation in a limit-order driven market

Use of trading strategies to mislead other market participants, commonly termed trade-based market manipulation, has been identified as a major problem faced by present day stock markets. Although some mathematical models of trade-based market manipulation have been previously developed, this work presents a framework for manipulation in the context of a realistic computational model of a limit-order market. The Maslov limit order market model is extended to introduce manipulators and technical traders. We show that “pump and dump” manipulation is not possible with traditional Maslov (liquidity) traders. The presence of technical traders, however, makes profitable manipulation possible. When exploiting the behaviour of technical traders, manipulators can wait some time after their buying phase before selling, in order to profit. Moreover, if technical traders believe that there is an information asymmetry between buy and sell actions, the manipulator effort required to perform a “pump and dump” is comparatively low, and a manipulator can generate profits even by selling immediately after raising the price.

Inferring Risk-Averse Probability Distributions from Option Prices Using Implied Binomial Trees

We generalize the Rubinstein (1994) risk-neutral implied binomial tree (R-IBT) model to a physical-world risk-averse implied binomial tree (RA-IBT) model. The R-IBT and RA-IBT trees are bound together via a relationship requiring a risk premium (or a risk-adjusted discount rate) on the underlying asset at any node. The RA-IBT provides a powerful numerical platform for many empirical financial option and real option applications; these include probabilistic inference, pricing, and utility theory applications.

A Practical Guide to GMM (with Applications to Option Pricing)

Generalized Method of Moments (GMM) is underutilized in financial economics because it is not adequately explained in the literature. We use a simple example to explain how and why GMM works. We then illustrate practical GMM implementation by estimating and testing the Black-Scholes option pricing model using S&P 500 index options data. We identify problem areas in implementation and we give practical GMM estimation advice, troubleshooting tips, and pseudo code. We pay particular attention to proper choice of moment conditions, exactly-identified versus over-identified estimation, estimation of Newey-West standard errors, and numerical optimization in the presence of multiple extrema.

Central Limit Theorems When Data Are Dependent: Addressing the Pedagogical Gaps

Although dependence in financial data is pervasive, standard doctoral-level econometrics texts do not make clear that the common central limit theorems (CLTs) contained therein fail when applied to dependent data. More advanced books that are clear in their CLT assumptions do not contain any worked examples of CLTs that apply to dependent data. We address these pedagogical gaps by discussing dependence in financial data and dependence assumptions in CLTs and by giving a worked example of the application of a CLT for dependent data to the case of the derivation of the asymptotic distribution of the sample variance of a Gaussian AR(1). We also provide code and the results for a Monte-Carlo simulation used to check the results of the derivation.

The Practical Implications of Modern Portfolio Theory

The practical implications of modern portfolio theory (MPT) are obscured by more than 50 years of academic literature. We shed light on the literature by picking out the few most important implications of MPT. We argue first that what we dub the “Markowitz uncertainty principle” implies that mean-variance efficient portfolios are a practical impossibility, and that attaining “pragmatic diversification” should instead be the goal of investors. We also argue that MPT is silent as to whether any fund manager can or should be able to beat his or her benchmark consistently. We can, however, combine the MPT framework with existing empirical evidence to generate practical advice about active investment for retail investors. For capital budgeting, we argue that the literature implies that the original single-beta capital asset pricing model (CAPM) should typically be implemented using a multi-beta framework.

Evolving trading strategies for a limit-order book generator

A grammatical evolutionary model (GE) is used to evolve trading strategies for a limit-order book model. A modified version of a limit-order book generator, based on the original work of Maslov [1], is used to produce limit-order book tick data. The evolved trading strategies demonstrate profit-making ability even though the Maslov model is fundamentally based on random behaviour.

Inferring Risk-Averse Probability Distributions from Option Prices Using Implied Binomial Trees: Additional Theory and Extensions

Arnold, Crack and Schwartz (ACS) (2010) generalize the Rubinstein (1994) risk-neutral implied binomial tree (R-IBT) model by introducing a risk premium. Their new risk-averse implied binomial tree model (RA-IBT) has both probabilistic and pricing applications. They use the RA-IBT model to estimate the pricing kernel (i.e., marginal rate of substitution) and implied relative risk aversion for a representative agent. They also use the RA-IBT to explore the differences between risk-neutral and risk-averse moments of returns. They also discuss practical applications of the RA-IBT model to Value at Risk and stochastic volatility option pricing models. This paper presents additional theoretical details not contained in ACS. We present a deeper discussion on the assumptions required for the risk-averse trees, we discuss details for extending ACS through the use of general utility functions to generate discount rates in the RA-IBT, and we present further theoretical details on the propagation of risk-averse probabilities through an RA-IBT. We also present an alternate CAPM-driven derivation of the certainty equivalent risk-adjusted discounting formula that is derived using no-arbitrage principles in ACS and an alternate direct estimation routine for the RA-IBT that is similar to Rubinstein’s “one-two-three” technique.

Single-Period Kyle Model Notes (A Clear Summary of the Kyle 1985 Econometrica Paper)

A careful summary of the Kyle (1985) Econometrica model. Also included are some notes that explore properties of the solution. Email the author if you would like the tex code.