Madan D. Social Science Research Network Working Paper Series [Internet]. 2014. Publisher's VersionAbstract
Minimal discounted distorted expectations across a range of stress levels are employed to model risk acceptability in markets. Interactions between discounting and stress levels used in measure changes are accommodated by lowering discount rates for the higher stress levels. Acceptability parameters represent a maximal and minimal discount rate, a maximal stress level and the speed of rate reduction in response to stress. An explicit model relating credit default swap (CDS) prices to default probabilities is formulated with a view to making the default risk market acceptable. Data on CDS prices and default probabilities for the six major US banks obtained from the Risk Management Institute of the National University of Singapore is employed to estimate parameters defining acceptability and the movements in market implied recovery rates. We observe that the financial crisis saw an increase in the maximal discount rate and its spread over the minimal rate along with an increase in the maximal stress level being demanded for acceptability and a stable pattern for the speed of rate adjustment through the period. The maximal rate, rate spread and stress levels have come down but with periods in the interim where they have peaked as they did in the crisis. Recovery rates have oscillated and they did fall substantially but have recovered towards 40 percent near the end of the period.
Madan D. Social Science Research Network Working Paper Series [Internet]. 2014. Publisher's VersionAbstract
We show that nonlinearly discounted nonlinear martingales are related to no arbitrage in two price economies as linearly discounted martingales were related to no arbitrage in economies satisfying the law of one price. Furthermore, assuming risk acceptability requires a positive physical expectation, we demonstrate that expected rates of return on ask prices should be dominated by expected rates of return on bid prices. A preliminary investigation conducted here, supports this hypothesis. In general we observe that asset pricing theory in two price economies leads to asset pricing inequalities. A model incorporating both nonlinear discounting and nonlinear martingales is developed for the valuation of contingent claims in two price economies. Examples illustrate the interactions present between the severity of measure changes and their associated discount rates. As a consequence arbitrage free two price economies can involve unique discount curves and measure changes that are however specific to both the product being priced and the trade direction.
Madan D, Pistorius M. Social Science Research Network Working Paper Series [Internet]. 2014. Publisher's VersionAbstract
Eberlein, Kallsen and Kristen (2003) argued that the VIX index is a good way to devolatize SPX returns. Adopting this approach we construct a risk neutral model for SPX returns as a variance gamma process scaled by the VIX. We model the risk neutral evolution of the squared VIX as a mean reverting finite state continuous time Markov chain which we calibrate to VIX options and the forward variance swap curve. We derive closed forms for the characteristic function of the logarithm of the SPX as a VIX scaled variance gamma process with up jumps in the VIX directly impacting the SPX downwards. The SPX parameters are calibrated to SPX options. Simulated sample paths are used to comment on the gap between the forward VIX in the model and the expected level of the volatility swap. The simulated path space is also used to price and statically hedge equity and volatility linked notes. The hedge is further enhanced by delta strategies extracted by an application of the SPSA technique with a view to lowering the ask price based on distorted expectations. A thrust of the paper is the demonstration of the desirability of zero cost hedging with a view to lowering the ask price required to make the unhedged risk acceptable, moving beyond hedging for replication which is viewed as too limited a perspective for realistic applications.
Eberlein E, Madan D, Pistorius M, Yor M. Social Science Research Network Working Paper Series [Internet]. 2013. Publisher's VersionAbstract
Probability distortions for constructing nonlinear G-expectations for the bid and ask or lower and upper prices in continuous time are here extended to the direct use of measure distortions. Fairly generally measure distortions can be constructed as probability distortions applied to an exponential distribution function on the half line. The valuation methodologies are extended beyond contract valuation to the valuation of potentially infinitely lived economic activities. Explicit computations illustrate the procedures for stock indices and insurance loss processes.
Carr P, Madan D. Social Science Research Network Working Paper Series [Internet]. 2013. Publisher's VersionAbstract
A double gamma model is proposed for the VIX. The VIX is modeled as gamma distributed with a mean and variance that respond to a gamma distributed realized variance over the preceeding month. Conditionally on VIX and the realized variance, the logarithm of the stock is variance gamma distributed with affine conditional drift and quadratic variation. The joint density for the triple, realized variance, VIX and the SPX is in closed form. Calibrating the model jointly to SPX and VIX options a risk management application illustrates a hedge for realized volatility options.
Eberlein E, Madan D, Schoutens W. Social Science Research Network Working Paper Series [Internet]. 2013. Publisher's VersionAbstract
The use of internal Bank models for meeting capital requirements has been approved for some time. Regulators then face issues of model approval necessitating some public domain analysis of model performance. This paper presents a new approach to risk model evaluation using forward looking risk neutral probabilities. In addition to VaR and CVaR we analyse a new measure termed here RWAVaR that was proposed in Carr, Madan and Vicente Alvarez (2011). The new measure is a formalization of the popular concept of risk weighted assets used in the Basel approach to capital requirements. The formalization allows for a possible leveraging of information contained in bid and ask prices and the study reports on the potential of this approach. Capital measures using RWAVaR are observed to be sensitive to volatility, the volatility of volatility, the skewness of return distributions and the volatility spread across maturities. Movements in bid ask spreads also strongly influence capital requirements. Additionally there is a potential for some procyclicality to be built into the requirements, particularly when one adapts requirements to movements in liquidity spreads.
Madan D. Social Science Research Network Working Paper Series [Internet]. 2013. Publisher's VersionAbstract
Statistical theory has been relatively absent in the exercise of estimating parameters of an option pricing model from cross-sectional data at a fixed point of calendar time. The cross-sectional data typically consists of prices for options at various strikes and maturities at market close. The problem has been the formulation of an error model consistent with no arbitrage conditions satisfied by models and possibly also market data. The paper presents such requisite error specifications consistent with no arbitrage conditions. The properties of such estimators are then analyzed on simulated data to evaluate biases in parameter estimates and their volatilities. The use of such error specifications coupled with maximum likelihood estimation makes available standard statistical tests for testing hypotheses on model parameter values. The methods proposed are finally illustrated on four popular models from the literature on data for options on the S&P 500 index at market close on April 30, 2013. It is shown that one can then conclude the risk neutral statistical significance of skewness, excess kurtosis, the presence of jump components, the negative correlation between volatility and the stock price and the presence of volatility of volatility.
Madan D, Yor M. Social Science Research Network Working Paper Series [Internet]. 2013. Publisher's VersionAbstract
For longer horizons, assuming no dividend distributions, equilibrium models for discounted stock prices are formulated as conditional expectations of nontrivial terminal random variables defined at infinity. Observing that extant models fail to have this property, new models are proposed. The new equilibrium concept proposed here permits a distinction between unduly optimistic or pessimistic disequilibria. A tractable example is provided by the discounted variance gamma model. Calibrations to market data provide empirical support. Additionally, procedures are presented for the valuation of path dependent stochastic perpetuities. For these new discounted models, implied volatility curves do not flatten out at the larger maturities. Evidence is provided for long dated claims, paying coupon for the time spent by the stock price above a lower barrier, being underpriced by extant models relative to the new discounted ones. Given that such claims are now issued quite regularly, the resulting mispricing could possibly take some corrections.
Madan D, Schoutens W. Social Science Research Network Working Paper Series [Internet]. 2013. Publisher's VersionAbstract
Postulating additivity of bid and ask prices for claims comonotone with a long or short stock position, two pricing processes are identified from data on bid and ask prices for options. It is observed that there are two separate put call parity relations in place, with the ask price for call less bid prices for put delivering an ask price for the forward stock. Likewise, the ask for puts less the bid for calls identifies the bid for the forward stock. Two processes are introduced to determine bid and ask prices for claims comonotone with a long or short position in the stock. For a claim comonotone with a long position, one uses the so-called increasing process for the ask price and the so-called decreasing process for the bid price, and vice versa for a claim comonotone with a short position. As candidates for the two processes, one may employ any of the traditional one dimensional Markov processes. We illustrate the theory by using a Sato process, a model known to produce a smile conforming fit over strike and maturity. The two processes are observed to have marginals related by first order stochastic dominance. The increasing process dominates the decreasing process in this sense. These two processes are also used to construct upper and lower bounds for bid and ask prices for claims not comonotone with a long or short stock position. The two processes and their properties are illustrated with data on bid and ask prices for options on the exchange traded fund, SPY.
Madan D. Social Science Research Network Working Paper Series [Internet]. 2013. Publisher's VersionAbstract
A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewenss and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.