Testing the Forecasting Performance of Ibex 35 Option-implied Risk-neutral Densities

Published also as: Documento de Trabajo Banco de España 0504/2005.

Estimating Relative Risk Aversion, Risk-Neutral and Real-World Densities using Brazilian Real Currency Options

Building Risk-Neutral Densities (RND) from options data can provide market-implied expectations about the future behavior of a financial variable. And market expectations on financial variables may influence macroeconomic policy decisions. It can be useful also for corporate and financial institutions decision making. This paper uses the Liu et all (2007) approach to estimate the option-implied Risk-neutral densities from the Brazilian Real/US Dollar exchange rate distribution. We then compare the RND with actual exchange rates, on a monthly basis, in order to estimate the relative risk-aversion of investors and also obtain a Real-world density for the exchange rate. We are the first to calculate relative risk-aversion and the option-implied Real World Density for an emerging market currency. Our empirical application uses a sample of Brazilian Real/US Dollar options traded at BM&F-Bovespa from 1999 to 2011. The RND is estimated using a Mixture of Two Log-Normals distribution and then the real-world density is obtained by means of the Liu et al. (2007) parametric risktransformations. The relative risk aversion is calculated for the full sample. Our estimated value of the relative risk aversion parameter is around 2.7, which is in line with other articles that have estimated this parameter for the Brazilian Economy, such as Araújo (2005) and Issler and Piqueira (2000). Our out-of-sample evaluation results showed that the RND has some ability to forecast the Brazilian Real exchange rate. Abe et all (2007) found also mixed results in the out-of-sample analysis of the RND forecast ability for exchange rate options. However, when we incorporate the risk aversion into RND in order to obtain a Real-world density, the out-of-sample performance improves substantially, with satisfactory results in both Kolmogorov and Berkowitz tests. Therefore, we would suggest not using the “pure” RND, but rather taking into account risk aversion in order to forecast the Brazilian Real exchange rate.

Option-Implied Preferences Adjustments and Risk-Neutral Density Forecasts

The main objective of this paper is to analyse the value of information contained in prices of options on the IBEX 35 index at the Spanish Stock Exchange Market. The forward looking information is extracted using implied risk-neutral density functions estimated by a mixture of two-lognormals and three alternative risk-adjustments: the classic power and exponential utility functions and a habit-based specification that allows for a counter-cyclical variation of risk aversion. Our results show that at four-week horizon we can reject the hypothesis that between October 1996 and March 2000 the risk-neutral densities provide accurate predictions of the distributions of future realisations of the IBEX 35 index at a four-week horizon. When forecasting through risk-adjusted densities the performance of this period is statistically improved and we no longer reject that hypothesis. All risk-adjusted densities generate similar forecasting statistics. Then, at least for a horizon of four-weeks, the actual risk adjustment does not seem to be the issue. By contrast, at the one-week horizon risk-adjusted densities do not improve the forecasting ability of the risk-neutral counterparts.

Density forecasts of crude-oil prices using option-implied and ARCH-type models

The predictive accuracy of competing crude-oil price forecast densities is investigated for the 1994–2006 period. Moving beyond standard ARCH type models that rely exclusively on past returns, we examine the benefits of utilizing the forward-looking information that is embedded in the prices of derivative contracts. Risk-neutral densities, obtained from panels of crude-oil option prices, are adjusted to reflect real-world risks using either a parametric or a non-parametric calibration approach. The relative performance of the models is evaluated for the entire support of the density, as well as for regions and intervals that are of special interest for the economic agent. We find that non-parametric adjustments of risk-neutral density forecasts perform significantly better than their parametric counterparts. Goodness-of-fit tests and out-of-sample likelihood comparisons favor forecast densities obtained by option prices and non-parametric calibration methods over those constructed using historical returns and simulated ARCH processes. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:727–754, 2011

Faster parameter estimation using risk-sensitive filters

In this paper, we propose a risk-sensitive approach to parameter estimation for hidden Markov models (HMMs). The parameter estimation approach considered exploits estimation of various functions of the state, based on model estimates. We propose certain practical suboptimal risk-sensitive filters to estimate the various functions of the state during transients, rather than optimal risk-neutral filters as in earlier studies. The estimates are asymptotically optimal, if asymptotically risk neutral, and can give significantly improved transient performance, which is a very desirable objective for certain engineering applications. To demonstrate the improvement in estimation simulation studies are presented that compare parameter estimation based on risk-sensitive filters with estimation based on risk-neutral filters.

Risk-sensitivity and the mean-variance trade-off: decision making in sensorimotor control.

Numerous psychophysical studies suggest that the sensorimotor system chooses actions that optimize the average cost associated with a movement. Recently, however, violations of this hypothesis have been reported in line with economic theories of decision-making that not only consider the mean payoff, but are also sensitive to risk, that is the variability of the payoff. Here, we examine the hypothesis that risk-sensitivity in sensorimotor control arises as a mean-variance trade-off in movement costs. We designed a motor task in which participants could choose between a sure motor action that resulted in a fixed amount of effort and a risky motor action that resulted in a variable amount of effort that could be either lower or higher than the fixed effort. By changing the mean effort of the risky action while experimentally fixing its variance, we determined indifference points at which participants chose equiprobably between the sure, fixed amount of effort option and the risky, variable effort option. Depending on whether participants accepted a variable effort with a mean that was higher, lower or equal to the fixed effort, they could be classified as risk-seeking, risk-averse or risk-neutral. Most subjects were risk-sensitive in our task consistent with a mean-variance trade-off in effort, thereby, underlining the importance of risk-sensitivity in computational models of sensorimotor control.

Risk-sensitive optimal feedback control accounts for sensorimotor behavior under uncertainty.

Many aspects of human motor behavior can be understood using optimality principles such as optimal feedback control. However, these proposed optimal control models are risk-neutral; that is, they are indifferent to the variability of the movement cost. Here, we propose the use of a risk-sensitive optimal controller that incorporates movement cost variance either as an added cost (risk-averse controller) or as an added value (risk-seeking controller) to model human motor behavior in the face of uncertainty. We use a sensorimotor task to test the hypothesis that subjects are risk-sensitive. Subjects controlled a virtual ball undergoing Brownian motion towards a target. Subjects were required to minimize an explicit cost, in points, that was a combination of the final positional error of the ball and the integrated control cost. By testing subjects on different levels of Brownian motion noise and relative weighting of the position and control cost, we could distinguish between risk-sensitive and risk-neutral control. We show that subjects change their movement strategy pessimistically in the face of increased uncertainty in accord with the predictions of a risk-averse optimal controller. Our results suggest that risk-sensitivity is a fundamental attribute that needs to be incorporated into optimal feedback control models.

Risk sensitivity in a motor task with speed-accuracy trade-off.

When a racing driver steers a car around a sharp bend, there is a trade-off between speed and accuracy, in that high speed can lead to a skid whereas a low speed increases lap time, both of which can adversely affect the driver's payoff function. While speed-accuracy trade-offs have been studied extensively, their susceptibility to risk sensitivity is much less understood, since most theories of motor control are risk neutral with respect to payoff, i.e., they only consider mean payoffs and ignore payoff variability. Here we investigate how individual risk attitudes impact a motor task that involves such a speed-accuracy trade-off. We designed an experiment where a target had to be hit and the reward (given in points) increased as a function of both subjects' endpoint accuracy and endpoint velocity. As faster movements lead to poorer endpoint accuracy, the variance of the reward increased for higher velocities. We tested subjects on two reward conditions that had the same mean reward but differed in the variance of the reward. A risk-neutral account predicts that subjects should only maximize the mean reward and hence perform identically in the two conditions. In contrast, we found that some (risk-averse) subjects chose to move with lower velocities and other (risk-seeking) subjects with higher velocities in the condition with higher reward variance (risk). This behavior is suboptimal with regard to maximizing the mean number of points but is in accordance with a risk-sensitive account of movement selection. Our study suggests that individual risk sensitivity is an important factor in motor tasks with speed-accuracy trade-offs.

Risk-sensitive optimal feedback control accounts for sensorimotor behavior under uncertainty

Many aspects of human motor behavior can be understood using optimality principles such as optimal feedback control. However, these proposed optimal control models are risk-neutral; that is, they are indifferent to the variability of the movement cost. Here, we propose the use of a risk-sensitive optimal controller that incorporates movement cost variance either as an added cost (risk-averse controller) or as an added value (risk-seeking controller) to model human motor behavior in the face of uncertainty. We use a sensorimotor task to test the hypothesis that subjects are risk-sensitive. Subjects controlled a virtual ball undergoing Brownian motion towards a target. Subjects were required to minimize an explicit cost, in points, that was a combination of the final positional error of the ball and the integrated control cost. By testing subjects on different levels of Brownian motion noise and relative weighting of the position and control cost, we could distinguish between risk-sensitive and risk-neutral control. We show that subjects change their movement strategy pessimistically in the face of increased uncertainty in accord with the predictions of a risk-averse optimal controller. Our results suggest that risk-sensitivity is a fundamental attribute that needs to be incorporated into optimal feedback control models. © 2010 Nagengast et al.

The Effect of Risk Sensitivity on Collaborative New Product Development in a Supply Chain

The increasing risks and costs of new product development require firms to collaborate with their supply chain partners in product management. In this paper, a supply chain model is proposed with one risk-neutral supplier and one risk-averse manufacturer. The manufacturer has an opportunity to enhance demand by developing a new product, but both the actual demand for new product and the supplier’s wholesale price are uncertain. The supplier has an incentive to share risks of new product development via an advance commitment to wholesale price for its own profit maximization. The effects of the manufacturer’s risk sensitivity on the players’ optimal strategies are analyzed and the trade-off between innovation incentives and pricing flexibility is investigated from the perspective of the supplier. The results highlight the significant role of risk sensitivity in collaborative new product development, and it is found that the manufacturer’s innovation level and retail price are always decreasing in the risk sensitivity, and the supplier prefers commitment to wholesale price only when the risk sensitivity is below a certain threshold.

Risk Aversion in Inventory Management

Traditional inventory models focus on risk-neutral decision makers, i.e., characterizing replenishment strategies that maximize expected total profit, or equivalently, minimize expected total cost over a planning horizon. In this paper, we propose a framework for incorporating risk aversion in multi-period inventory models as well as multi-period models that coordinate inventory and pricing strategies. In each case, we characterize the optimal policy for various measures of risk that have been commonly used in the finance literature. In particular, we show that the structure of the optimal policy for a decision maker with exponential utility functions is almost identical to the structure of the optimal risk-neutral inventory (and pricing) policies. Computational results demonstrate the importance of this approach not only to risk-averse decision makers, but also to risk-neutral decision makers with limited information on the demand distribution.

The lottery-panel task for bi-dimensional parameter-free elicitation of risk attitudes

We propose first, a simple task for the eliciting attitudes toward risky choice, the SGG lottery-panel task, which consists in a series of lotteries constructed to compensate riskier options with higher risk-return trade-offs. Using Principal Component Analysis technique, we show that the SGG lottery-panel task is capable of capturing two dimensions of individual risky decision making i.e. subjects’ average risk taking and their sensitivity towards variations in risk-return. From the results of a large experimental dataset, we confirm that the task systematically captures a number of regularities such as: A tendency to risk averse behavior (only around 10% of choices are compatible with risk neutrality); An attraction to certain payoffs compared to low risk lotteries, compatible with over-(under-) weighting of small (large) probabilities predicted in PT and; Gender differences, i.e. males being consistently less risk averse than females but both genders being similarly responsive to the increases in risk-premium. Another interesting result is that in hypothetical choices most individuals increase their risk taking responding to the increase in return to risk, as predicted by PT, while across panels with real rewards we see even more changes, but opposite to the expected pattern of riskier choices for higher risk-returns. Therefore, we conclude from our data that an “economic anomaly” emerges in the real reward choices opposite to the hypothetical choices. These findings are in line with Camerer's (1995) view that although in many domains, paid subjects probably do exert extra mental effort which improves their performance, choice over money gambles is not likely to be a domain in which effort will improve adherence to rational axioms (p. 635). Finally, we demonstrate that both dimensions of risk attitudes, average risk taking and sensitivity towards variations in the return to risk, are desirable not only to describe behavior under risk but also to explain behavior in other contexts, as illustrated by an example. In the second study, we propose three additional treatments intended to elicit risk attitudes under high stakes and mixed outcome (gains and losses) lotteries. Using a dataset obtained from a hypothetical implementation of the tasks we show that the new treatments are able to capture both dimensions of risk attitudes. This new dataset allows us to describe several regularities, both at the aggregate and within-subjects level. We find that in every treatment over 70% of choices show some degree of risk aversion and only between 0.6% and 15.3% of individuals are consistently risk neutral within the same treatment. We also confirm the existence of gender differences in the degree of risk taking, that is, in all treatments females prefer safer lotteries compared to males. Regarding our second dimension of risk attitudes we observe, in all treatments, an increase in risk taking in response to risk premium increases. Treatment comparisons reveal other regularities, such as a lower degree of risk taking in large stake treatments compared to low stake treatments and a lower degree of risk taking when losses are incorporated into the large stake lotteries. Results that are compatible with previous findings in the literature, for stake size effects (e.g., Binswanger, 1980; Antoni Bosch-Domènech & Silvestre, 1999; Hogarth & Einhorn, 1990; Holt & Laury, 2002; Kachelmeier & Shehata, 1992; Kühberger et al., 1999; B. J. Weber & Chapman, 2005; Wik et al., 2007) and domain effect (e.g., Brooks and Zank, 2005, Schoemaker, 1990, Wik et al., 2007). Whereas for small stake treatments, we find that the effect of incorporating losses into the outcomes is not so clear. At the aggregate level an increase in risk taking is observed, but also more dispersion in the choices, whilst at the within-subjects level the effect weakens. Finally, regarding responses to risk premium, we find that compared to only gains treatments sensitivity is lower in the mixed lotteries treatments (SL and LL). In general sensitivity to risk-return is more affected by the domain than the stake size. After having described the properties of risk attitudes as captured by the SGG risk elicitation task and its three new versions, it is important to recall that the danger of using unidimensional descriptions of risk attitudes goes beyond the incompatibility with modern economic theories like PT, CPT etc., all of which call for tests with multiple degrees of freedom. Being faithful to this recommendation, the contribution of this essay is an empirically and endogenously determined bi-dimensional specification of risk attitudes, useful to describe behavior under uncertainty and to explain behavior in other contexts. Hopefully, this will contribute to create large datasets containing a multidimensional description of individual risk attitudes, while at the same time allowing for a robust context, compatible with present and even future more complex descriptions of human attitudes towards risk.

Nonparametric tail risk, macroeconomics and stock returns: predictability and risk premia

This paper proposes a new novel to calculate tail risks incorporating risk-neutral information without dependence on options data. Proceeding via a non parametric approach we derive a stochastic discount factor that correctly price a chosen panel of stocks returns. With the assumption that states probabilities are homogeneous we back out the risk neutral distribution and calculate five primitive tail risk measures, all extracted from this risk neutral probability. The final measure is than set as the first principal component of the preliminary measures. Using six Fama-French size and book to market portfolios to calculate our tail risk, we find that it has significant predictive power when forecasting market returns one month ahead, aggregate U.S. consumption and GDP one quarter ahead and also macroeconomic activity indexes. Conditional Fama-Macbeth two-pass cross-sectional regressions reveal that our factor present a positive risk premium when controlling for traditional factors.