Downside Risk Neutral Probabilities

Downside Risk Neutral Probabilities

The fundamental theory of asset pricing has been developed under the two main assumptions that markets are frictionless and have no arbitrage opportunities. In this case the market enforces that replicable assets are valued by a linear function of their payoffs, or as the discounted expectation with respect to the so-called risk-neutral probability. Important evidence of the presence of frictions in financial markets has led to study market pricing rules in such a framework. Recently, Cerreia-Vioglio et al. (J Econ Theory 157:730–762, 2015) have extended the Fundamental Theorem of Finance by showing that, with markets frictions, requiring the put–call parity to hold, together with the mild assumption of translation invariance, is equivalent to the market pricing rule being represented as a discounted Choquet expectation with respect to a risk-neutral nonadditive probability. This paper continues this study by characterizing important properties of the (unique) risk-neutral nonadditive probability v_f associated with a Choquet pricing rule f, when it is not assumed to be subadditive. First, we show that the observed violation of the call–put parity, a condition considered by Chateauneuf et al. (Math Financ 6:323–330, 1996) similar to the put–call parity in Cerreia-Vioglio et al. (2015), is consistent with the existence of bid-ask spreads. Second, the balancedness of v_f—or equivalently the non-vacuity of its core—is characterized by an arbitrage-free condition that eliminates all the arbitrage opportunities that can be obtained by splitting payoffs in parts; moreover the (nonempty) core of v_f consists of additive probabilities below v_f whose associated (standard) expectations are all below the Choquet pricing rule f. Third, by strengthening again the previous arbitrage-free condition, we show the existence of a strictly positive risk-neutral probability below v_f, which allows to recover the standard formulation of the Fundamental Theorem of Finance for frictionless markets.

In a viable single-period model with one stock and k ≥ 2 scenarios the completeness of the market is equivalent to the uniqueness of the risk neutral probability; this equivalence allows to price every derivative security with a unique fair price. When the market is incomplete, the set of all possible risk neutral probabilities is not a singleton and for every non attainable derivative security we have a bid-ask interval of possible prices. In literature, different methods have been proposed in order to select a unique risk neutral probability starting with the real world probability p. Contrary to the complete case, in all these models \(\textbf{p}\) is really used for the option pricing and its elicitation is a crucial point for every criterion used to select a risk neutral probability. We propose a method for the valuation problem in incomplete markets which can be used when p is a partial conditional probability assessment as well as when we have different expert opinions expressed through conditional probability assessments. In fact, it is not always possible to elicit a probability distribution p over all the possible states of the world: the information that we have could be partial, conditional or even not coherent. Therefore we will select a risk neutral probability by minimizing a discrepancy measure introduced in [2] and analized in [3] between p and the set of all possible risk neutral probability, where p can be a partial conditional probability assessments or it can be given by the fusion of different expert opinions.KeywordsExpert OpinionCall OptionConditional EventMartingale MeasureStrike PriceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Affine multivariate GARCH models

‘Forward and Futures Prices of Contingent Claims’ applies the multi-period rational expectations model to analyse the pricing of forward and futures contracts and derives an expression for the difference between the prices of the two, which is commonly known as the convexity adjustment. These are illustrated using an example where the asset price, the pricing kernel and the bond prices are joint lognormal variables. The authors then consider the futures and forward prices of contingent claims, such as call options. Futures prices of options can be significantly different from the forward prices of the contingent claims, especially when the options are on interest-rate related assets such as bonds or swaps. Forward prices of long-term bond can be obtained by taking appropriate expected values of future spot pries under the risk-neutral measure.

Credit Risk and Risk Neutral Default Probabilities: Information About Rating Migrations and Defaults

We present a pricing kernel that summarizes well the main features of the dynamics of interest rates and risk in postwar U.S. data and use it to uncover how the pricing kernel has moved with the short rate in this data. Our findings imply that standard monetary models miss an essential link between the central bank instrument and the economic activity that monetary policy is intended to affect, and thus we call for a new approach to monetary policy analysis. We sketch a new approach using an economic model based on our pricing kernel. The model incorporates the key relationships between policy and risk movements in an unconventional way: the central bank's policy changes are viewed as primarily intended to compensate for exogenous business cycle fluctuations in risk that threaten to push inflation off target. This model, while an improvement on standard models, is considered just a starting point for their revision. It leads to critical questions that researchers need to answer as they continue to revise their approach to monetary policy analysis. Copyright © 2009 by the National Bureau of Economic Research.

On Dynamic Measures of Risk

The paper studies connections between risk aversion and martingale measures in a discrete‐time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log‐utility, the optimal dynamic portfolio defines the numéraire portfolio.

This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling, and risk measurement. It is based on the natural assumption that investors prefer more rather than less, in the sense that given two portfolios with the same diffusion coefficient value, the one with the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account. The market portfolio of investable wealth is shown to equal a combination of the growth optimal portfolio (GOP) and the savings account. In this setup the capital asset pricing model follows without the use of expected utility functions, Markovianity, or equilibrium assumptions. The expected increase of the discounted value of the GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk‐neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real‐world probability measure as pricing measure. An equivalent risk‐neutral martingale measure does not exist under the derived minimal market model.

A key question for understanding the cross-section of expected returns of equities is the following: which factors, from a given collection of factors, are risk factors, equivalently, which factors are in the stochastic discount factor (SDF)? Though the SDF is unobserved, assumptions about which factors (from the available set of factors) are in the SDF restricts the joint distribution of factors in specific ways, as a consequence of the economic theory of asset pricing. A different starting collection of factors that go into the SDF leads to a different set of restrictions on the joint distribution of factors. The conditional distribution of equity returns has the same restricted form, regardless of what is assumed about the factors in the SDF, as long as the factors are traded, and hence the distribution of asset returns is irrelevant for isolating the risk-factors. The restricted factors models are distinct (nonnested) and do not arise by omitting or including a variable from a full model, thus precluding analysis by standard statistical variable selection methods, such as those based on the lasso and its variants. Instead, we develop what we call a Bayesian model scan strategy in which each factor is allowed to enter or not enter the SDF and the resulting restricted models (of which there are 114,674 in our empirical study) are simultaneously confronted with the data. We use a Student-t distribution for the factors, and model-specific independent Student-t distribution for the location parameters, a training sample to fix prior locations, and a creative way to arrive at the joint distribution of several other model-specific parameters from a single prior distribution. This allows our method to be essentially a scaleable and tuned-black-box method that can be applied across our large model space with little to no user-intervention. The model marginal likelihoods, and implied posterior model probabilities, are compared with the prior probability of 1/114,674 of each model to find the best-supported model, and thus the factors most likely to be in the SDF. We provide detailed simulation evidence about the high finite-sample accuracy of the method. Our empirical study with 13 leading factors reveals that the highest marginal likelihood model is a Student-t distributed factor model with 5 degrees of freedom and 8 risk factors.

This paper analyzes an interest rate model with self-exciting jumps, in which a jump in the interest rate model increases the intensity of jumps in the same model. This self-exciting property leads to clustering effects in the interest rate model. We obtain a closed-form expression for the conditional moment-generating function when the model coefficients have affine structures. Based on the Girsanov-type measure transformation for general jump-diffusion processes, we derive the evolution of the interest rate under the equivalent martingale measure and an explicit expression of the zero-coupon bond pricing formula. Furthermore, we give a pricing formula for the European call option written on zero-coupon bonds. Finally, we provide an interpretation for the clustering effects in the interest rate model within a simple framework of general equilibrium. Indeed, we construct an interest rate model, the equilibrium state of which coincides with the interest rate model with clustering effects proposed in this paper.

Suppose that discounted asset prices in a financial market are given by a P ‐semimartingale S . Among all probability measures Q that turn S into a local Q ‐martingale, the minimal entropy martingale measure is characterized by the property that it minimizes the relative entropy with respect to P . Via convex duality, it is intimately linked to the problem of maximizing expected exponential utility from terminal wealth. It also appears as a limit of p ‐optimal martingale measures as p decreases to 1. Like for most optimal martingale measures, finding its explicit form is easy if S is an exponential Lévy process and quite difficult otherwise.

This chapter introduces the use of a time-homogenous Markov chain for the valuation of options. In the computational finance literature, three categories of numerical techniques have been explored extensively for the valuation of contingent claims. The first category involves the use of a lattice structure to approximate the price movement of the underlying asset under the risk neutral probability measure and then computes the price of a contingent claim as a discounted expected payoff. The lattice approach essentially discretizes both time and state in a particular way. The second category is the finite difference/element approach. This technique numerically solves the partial differential equation that the value function of a contingent claim must obey under the no-arbitrage condition. The third category is the Monte Carlo method, which simulates the system under the risk-neutral probability measure so that the expectation of a contingent payoff can be approximated by the sample average.

In this article we use a novel clustering approach to study the role of heterogeneity in asset pricing. We present evidence that the equity premium is consistent with a stochastic discount factor (SDF) calculated as the average of the household clusters’ intertemporal marginal rates of substitution in the 1984–2002 period. The result is driven by the skewness of the cluster-based cross-sectional distribution of consumption growth, but cannot be explained by the cross-sectional variance and mean alone. We find that nine clusters are sufficient to explain the equity premium with relative risk aversion coefficient equal to six. The result is robust to various averaging schemes of cluster-based consumption growth used to construct the SDF. Lastly, the analysis reveals that standard approximation schemes of the SDF using individual household data produce unreliable results, implying a negative SDF.