The numéraire portfolio in semimartingale financial models

We study the existence of the numéraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numéraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numéraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free-lunch-type notion. In particular, the full strength of the “No Free Lunch with Vanishing Risk” (NFLVR) condition is not needed, only the weaker “No Unbounded Profit with Bounded Risk” (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks.      

Relative asset price bubbles

In models of financial bubbles, the price of a stock is typically unbounded, and this plays a fundamental role in the analysis of finite horizon local martingale bubbles. It would seem that price bubbles do not apply to a priori bounded risky asset prices, such as bond prices. To avoid this limitation, to characterize, and to identify bond price mispricings consistent with an absence of arbitrage, we develop the concept of a relative asset price bubble. This notion uses a risky asset’s price as the numéraire instead of the money market account’s value. This change of numéraire generates some interesting mathematical complexities because many important numéraires, including risky bonds, can vanish with positive probability over the model’s horizon.     

No arbitrage of the first kind and local martingale numéraires

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind (\(\mbox{NA}_{1}\)) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under \(\mbox{NA}_{1}\), a local martingale numéraire may fail to exist. In this work, we establish that under \(\mbox{NA}_{1}\), a supermartingale numéraire under the original probability \(P\) becomes a local martingale numéraire for equivalent probabilities arbitrarily close to \(P\) in the total variation distance.     

Recovering the real-world density and liquidity premia from option data

In this paper, we develop a methodology for simultaneous recovery of the real-world probability density and liquidity premia from observed S&P 500 index option prices. Assuming the existence of a numéraire portfolio for the US equity market, fair prices of derivatives under the benchmark approach can be obtained directly under the real-world measure. Under this modelling framework, there exists a direct link between observed call option prices on the index and the real-world density for the underlying index. We use a novel method for the estimation of option-implied volatility surfaces of high quality, which enables the subsequent analysis. We show that the real-world density that we recover is consistent with the observed realized dynamics of the underlying index. This admits the identification of liquidity premia embedded in option price data. We identify and estimate two separate liquidity premia embedded in S&P 500 index options that are consistent with previous findings in the literature.     

In which financial markets do mutual fund theorems hold true?

The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are:

On the implied market price of risk under the stochastic numéraire

This papers addresses the stock option pricing problem in a continuous time market model where there are two stochastic tradable assets, and one of them is selected as a numéraire. An equivalent martingale measure is not unique for this market, and there are non-replicable claims. Some rational choices of the equivalent martingale measures are suggested and discussed, including implied measures calculated from bond prices constructed as a risk-free investment with deterministic payoff at the terminal time. This leads to possibility to infer a implied market price of risk process from observed historical bond prices.     

On the hedging of options on exploding exchange rates

We study a novel pricing operator for complete, local martingale models. The new pricing operator guarantees put-call parity to hold for model prices and the value of a forward contract to match the buy-and-hold strategy, even if the underlying follows strict local martingale dynamics. More precisely, we discuss a change of numéraire (change of currency) technique when the underlying is only a local martingale, modelling for example an exchange rate. The new pricing operator assigns prices to contingent claims according to the minimal cost for superreplication strategies that succeed with probability one for both currencies as numéraire. Within this context, we interpret the lack of the martingale property of an exchange rate as a reflection of the possibility that the numéraire currency may devalue completely against the asset currency (hyperinflation).     

Slowing investment rates in developing economies: Evidence from a Bayesian hierarchical model

Using a large unbalanced panel of 11,812 publicly listed firms covering 11 major developing economies between 1997–2017, we detail a slowdown in investment rates post-2008: from 2013 for Chinese incorporated firms, and 2008 for others. We test competing explanations for slowing investment rates using a Bayesian ‘mixed effects’ model consisting of time-varying and country-varying coefficients. Firms’ estimated underlying mean impetus to invest (their ‘animal spirits’) falls more sharply than raw investment rates from 2008 to record lows by 2017. One-third of the variation in falling ‘animal spirits’ over time is statistically explained by the corporate sector’s changing median leverage, which declines by 40% since 2008. Firms’ investment rates have increasingly been sustained through external financing constraints loosening (as cash flow coefficients decline), and firms becoming more responsive to investment opportunities — reflected by time-varying Q regression coefficients increasing. At the country-level, we find that loosening external financing constraints is associated with greater responsiveness of firms to investment opportunities.     

Risk-sensitive benchmarked asset management

This paper extends the risk-sensitive asset management theory developed by Bielecki and Pliska and by Kuroda and Nagai to the case where the investor's objective is to outperform an investment benchmark. The main result is a mutual fund theorem. Every investor following the same benchmark will take positions, in proportions dependent on his/her risk sensitivity coefficient, in two funds: the log-optimal portfolio and a second fund which adjusts for the correlation between the traded assets, the benchmark and the underlying valuation factors.     

The Role of Real Options in Capital Budgeting: Theory and Practice1

Most discussions of capital budgeting take for granted that discounted cash flow (DCF) and real options valuation (ROV) are very different methods that are meant to be applied in different circumstances. Such discussions also typically assume that DCF is “easy” and ROV is “hard”—or at least dauntingly unfamiliar—and that, mainly for this reason, managers often use DCF and rarely ROV. This paper argues that all three assumptions are wrong or at least seriously misleading. DCF and ROV both assign a present value to risky future cash flows. DCF entails discounting expected future cash flows at the expected return on an asset of comparable risk. ROV uses “risk‐neutral” valuation, which means computing expected cash flows based on “risk‐neutral” probabilities and discounting these flows at the risk‐free rate. Using a series of single‐period examples, the author demonstrates that both methods, when done correctly, should provide the same answer. Moreover, in most ROV applications—those where there is no forward price or “replicating portfolio” of traded assets—a “preliminary” DCF valuation is required to perform the risk‐neutral valuation. So why use ROV at all? In cases where project risk and the discount rates are expected to change over time, the risk‐neutral ROV approach will be easier to implement than DCF (since adjusting cash flow probabilities is more straightforward than adjusting discount rates). The author uses multi‐period examples to illustrate further both the simplicity of ROV and the strong assumptions required for a typical DCF valuation. But the simplicity that results from discounting with risk‐free rates is not the only benefit of using ROV instead of—or together with—traditional DCF. The use of formal ROV techniques may also encourage managers to think more broadly about the flexibility that is (or can be) built into future business decisions, and thus to choose from a different set of possible investments. To the extent that managers who use ROV have effectively adopted a different business model, there is a real and important difference between the two valuation techniques. Consistent with this possibility, much of the evidence from both surveys and academic studies of managerial behavior and market pricing suggests that managers and investors implicitly take account of real options when making investment decisions.     

Implementing Fischer Black's Simple Discounting Rule

Corporate managers typically estimate the value of capital projects by discounting the project's expected future net cash flows at the cost of capital. The capital asset pricing model (CAPM) is generally used to estimate that cost. But, as anyone who has worked on the finance or business development staff of a public company can attest, there are major challenges in applying the CAPM, including largely unresolved questions about what constitutes the “market portfolio,” how to estimate market risk premiums, and how to estimate the betas of projects. In a short article published in Financial Management in 1988, Fischer Black proposed a valuation “discounting rule” that avoids all these problems—one that involves discounting a relatively certain (as opposed to an expected or average) level of operating cash flows at the risk-free rate. But Black's article does not address the question of how to calculate these “certainty equivalent” or “conditional” cash flows. In this article, the authors propose a way of implementing Black's rule that involves estimating the “conditional” cash flows in a three-step procedure:

• • Find a benchmark security that correlates with the project's cash flows;
• • Estimate the percentiles of the distribution in which the benchmark return equals the risk-free rate over different investment horizons;
• • Use information from corporate managers to assess the cash flows that define the same percentiles in the cash flow distributions.

As the authors point out, the virtue of Black's rule is that it shifts the focus of the analyst away from the assessment of discount factors and puts it squarely on the more challenging, and arguably more relevant, problem of estimating the project's cash flows.     

Extreme Value Dependence in Financial Markets: Diagnostics, Models, and Financial Implications

This article presents a general framework for identifying andmodeling the joint-tail distribution based on multivariate extremevalue theories. We argue that the multivariate approach is themost efficient and effective way to study extreme events suchas systemic risk and crisis. We show, using returns on fivemajor stock indices, that the use of traditional dependencemeasures could lead to inaccurate portfolio risk assessment.We explain how the framework proposed here could be exploitedin a number of finance applications such as portfolio selection,risk management, Sharpe ratio targeting, hedging, option valuation,and credit risk analysis.     

Risk-adjusted value allocation for (non-traded) assets with performance ratios

We propose a new valuation principle for possibly non-traded assets based on an implicit definition of a benchmark. The valuation principle allows taking (default and shortfall) risk constraints explicitly into account. The resulting risk-adjusted value functional is monotonic, positively homogeneous, partially concave and allows for an additive allocation of risk-adjusted values of non-traded assets in a portfolio. The valuation principle is applied to the problem of hedging and pricing in incomplete markets. Furthermore, accounting for non-traded assets is considered and we derive a risk-adjusted balance sheet for non-deterministic cash streams.     

Political risk and international valuation

Measuring the impact of political risk on investment projects is one of the most vexing issues in international business. One popular approach is to assume that the sovereign yield spread captures political risk and to augment the project discount rate by this spread. We show that this approach is flawed. While the sovereign spread is influenced by political risk, it also reflects other risks that are likely included in the valuation analysis — leading to the double counting of risks. We propose to use “political risk spreads” to undo the double counting in the evaluation of international investment projects.     

Investment timing and optimal capital structure under liquidity risk

Deterioration in debt market liquidity reduces debt values and affects firms' decisions. Considering such risk, we develop an investment timing model and obtain analytic solutions. We carry out a comprehensive analysis in optimal financing, default, and investment strategies, and stockholder–bondholder conflicts. Our model explains stylized facts and replicates empirical findings in credit spreads. We obtain six new insights for decision makers. We propose a ‘new trade-off theory’ for optimal capital structure, a new tax effect, and new explanations of ‘debt conservatism puzzle’ and ‘zero-leverage puzzle’. Failure in recognizing liquidity risk results in substantially over-leveraging, early bankruptcy or investment, overpriced options, and undervalued coupons and credit spreads. In addition, agency costs are surprisingly small for a high liquidity risk or a low project risk. Interestingly, the risk shifting incentive and debt overhang problem decrease with liquidity risk under moderate tax rates while they increase under high tax rates.     

The Three‐Factor Model: A Practitioner's Guide

The three‐factor model (3FM) has slowly but steadily become a popular alternative to the CAPM for measuring risk from the perspective of both corporate finance and portfolio management. The evidence clearly shows a negative relationship between market capitalization and returns, and a positive relationship between the book‐to‐market ratio and returns. Under the assumption that size and value are risk factors, the 3FM incorporates a market risk premium, a size premium, and a value premium into a model that aims to assess risk in a more comprehensive way, and ultimately to provide a more reliable estimation of required return. The required return produced by the 3FM has corporate finance applications (such as cost of capital estimation, project evaluation, and firm valuation) as well as portfolio management applications (such as performance evaluation). This article discusses the foundations and intuition behind the 3FM, as well as its application to the estimation of the cost of equity and excess returns.     

Consistent pricing and hedging for a modified constant elasticity of variance model

Abstract

This paper considers a modification of the well known constant elasticity of variance model where it is used to model the growth optimal portfolio (GOP). It is shown that, for this application, there is no equivalent risk neutral pricing measure and therefore the classical risk neutral pricing methodology fails. However, a consistent pricing and hedging framework can be established by application of the benchmark approach.

Perfect hedging strategies can be constructed for European style contingent claims, where the underlying risky asset is the GOP. In this framework, fair prices for contingent claims are the minimal prices that permit perfect replication of the claims. Numerical examples show that these prices may differ significantly from the corresponding ‘risk neutral’ prices.