Risk-adjusted option-implied moments

This paper provides a new way of converting risk-neutral moments into the corresponding physical moments, which are required for many applications. The main theoretical result is a new analytical representation of the expected payoffs of put and call options under the physical measure in terms of current option prices and a representative investor’s preferences. This representation is then used to derive analytical expressions for a variety of ex-ante physical return moments, showing explicitly how moment premiums depend on current option prices and preferences. As an empirical application of our theoretical results, we provide option-implied estimates of the representative stock market investor’s disappointment aversion using S&P 500 index option prices. We find that disappointment aversion has a procyclical pattern. It is high in times of high index levels and declines when the index falls. We confirm the view that investors with high risk aversion and disappointment aversion leave the stock market during times of turbulence and reenter it after a period of high returns.     

Retrieving risk neutral moments and expected quadratic variation from option prices

This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any order from generic European-style option prices. It also provides an exact formula for retrieving the expected quadratic variation of the stock market implied by European option prices, which nowadays is used as an estimate of the implied volatility, and a formula approximating the jump component of this measure of variation. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure and provides upper bounds of its approximation errors. The performance of this procedure is evaluated through a simulation and an empirical exercise. Both of these exercises clearly indicate that the suggested numerical procedure can provide accurate estimates of the risk neutral moments, over different horizons ahead. These can be in turn employed to obtain accurate estimates of risk neutral densities and calculate option prices, efficiently, in a model-free manner. The paper also shows that, in contrast to the prevailing view, ignoring the jump component of the underlying asset can lead to seriously biased estimates of the new volatility index suggested by the Chicago Board Options Exchange.     

Speculative dynamics and price behavior in the Shanghai Stock Exchange

This article examines the extent to which the trading behavior of heterogeneous investors manifests in stock price changes of asset portfolios which constitute the Shanghai Stock Exchange. There are three major findings that materialize. Firstly, reliable statistical evidence of a negative relation between the conditional first and second moments of the return distributions of stock prices lends support to the volatility feedback effect. Secondly, ‘feedback’, or momentum-type investors, are not present in this market as is often detected from the daily price changes of other industrialized markets. Finally, trade volume as a proxy for ‘information-driven’ trading suggests that such investors play a statistically significant role in stock price movements. Parameter estimates from this latter group of investors imply that a rise in stock prices from a high volume trading day is more likely than a rise resulting from a low volume trading day.     

Real Options Premia Implied from Recent Transactions in the Greek Real Estate Market

This research is the first to examine the empirical predictions of a real option-pricing model on market values from the realty market of a Euro area country, namely Greece. Using a manually collected sample of land and property transaction prices, we demonstrate that, a model which incorporates the option to wait to develop land has explanatory power on observed prices over and above the intrinsic value from a simple discounted cash flow (DCF) approach. Recent land transactions in our sample seem to reflect a premium for the option to wait (‘real option premium’) that can be as high as 26.66%–52.38%, especially in the west and north suburbs of Athens. Estimates of annual volatility for specific properties, as implied by transaction prices, are found to range from 15% to 21%.     

Commodity Spread Option with Cointegration

We derive the valuation formula of a European call option on the spread of two cointegrated commodity futures prices, based on the Gibson–Schwartz with cointegration (GSC) model. We also analyze the American commodity spread option including the early exercise premium representation and an analytical approximation valuation formulae with cointegration. In the numerical analysis, we compare the spread option values calculated by the GSC model and the Gibson–Schwartz (GS) model that ignores cointegration. Consistent with the intuition that the cointegration prevents the prices from diverging, the GSC model prices the commodity spread option lower than the GS model which have longer maturity of more than 6 years. In other words, the GS model may overprice the commodity spread options for those with longer maturity without taking account of cointegration. Thus, incorporating cointegration is important for valuation and hedging of long-term commodity spread options such as large scale oil refining plant developments.     

Partial moment volatility indices

Forward‐looking partial moment volatility indices are developed using state‐pricing, called the bear index (BEX) and bull index (BUX). Using S&P 500 index (SPX) option prices, we find that BEX and BUX provide superior forecasts for the lower and upper partial moments of future market realised volatility, respectively. We examine the relation between SPX returns and changes in BEX and BUX at the daily level. Results are consistent with the volatility feedback hypothesis. Further, we show that BEX may be more suitable as the ‘investor fear gauge’ than VIX.     

Credit Derivatives and the Default Risk of Large Complex Financial Institutions

This paper proposes and implements a multivariate model of the coevolution of the first and second moments of two broad credit default swap indices and the equity prices of sixteen large complex financial institutions. We use this empirical model to build a bank default risk model, in the vein of the classic Merton-type, which utilises a multi-equation framework to model forward-looking measures of market and credit risk using the credit default swap (CDS) index market as a measure of the conditions of the global credit environment. In the first step, we estimate the dynamic correlations and volatilities describing the evolution of the CDS indices and the banks’ equity prices and then impute the implied assets and their volatilities conditional on the evolution and volatility of equity. In the second step, we show that there is a substantial ‘asset shortfall’ and that substantial capital injections and/or asset insurance are required to restore the stability of our sample institutions to an acceptable level following large shocks to the aggregate level of credit risk in financial markets.     

Monitoring Financial Distress in a High-Stress Financial World: The Role of Option Prices as Bank Risk Metrics

The current financial crisis offers a unique opportunity to investigate the leading properties of market indicators in a stressed environment and their usefulness from a banking supervision perspective. One pool of relevant information that has been little explored in the empirical literature is the market for bank’s exchange-traded option contracts. In this paper, we first extract implied volatility indicators from the prices of option contracts on financial firms’ equity. We then examine empirically their ability to predict financial distress by applying survival analysis techniques to a sample of large US financial firms. We find that market indicators extracted from option prices significantly explain the survival time of troubled financial firms and do a better job in predicting financial distress than other time-varying covariates typically included in bank failure models. Overall, both accounting information and option prices contain useful information of subsequent financial problems and, more importantly, their combination produces good forecasts in a high-stress financial world.     

The square-root process and Asian options

Although the square-root process has long been used as an alternative to the Black–Scholes geometric Brownian motion model for option valuation, the pricing of Asian options on this diffusion model has never been studied analytically. However, the additivity property of the square-root process makes it a very suitable model for the analysis of Asian options. In this paper, we develop explicit prices for digital and regular Asian options. We also obtain distributional results concerning the square-root process and its average over time, including analytic formulae for their joint density and moments. We also show that the distribution is actually determined by those moments.     

Dividend Forecast Biases in Index Option Valuation

Since the early days of option pricing theory,the assumption that the dividends on the underlying stock or index over the life of the contract are known has not been challenged. We examine the sensitivity of index option prices to the assumption of dividend uncertainty. We consider a number of issues related to the forecasting of dividends and build a dividend forecasting model that passes several rigorous tests for unbiasedness. We then generate option prices using contemporary market levels and interest rates. We find that prices generated with the actual dividends are unbiased with respect to those generated using the forecasted dividends. The magnitudes of the forecast errors, however, are sufficiently large to suggest a concern, but the percentage errors are consistently small, typically amounting to less than two percent of the option price. We conclude that the convenient assumption that the stream of future dividendsis known is probably innocuous. This revised version was published online in November 2006 with corrections to the Cover Date.     

Pricing options on scenario trees

We examine valuation procedures that can be applied to incorporate options in scenario-based portfolio optimization models. Stochastic programming models use discrete scenarios to represent the stochastic evolution of asset prices. At issue is the adoption of suitable procedures to price options on the basis of the postulated discrete distributions of asset prices so as to ensure internally consistent portfolio optimization models. We adapt and implement two methods to price European options in accordance with discrete distributions represented by scenario trees and assess their performance with numerical tests. We consider features of option prices that are observed in practice. We find that asymmetries and/or leptokurtic features in the distribution of the underlying materially affect option prices; we quantify the impact of higher moments (skewness and excess kurtosis) on option prices. We demonstrate through empirical tests using market prices of the S&P500 stock index and options on the index that the proposed procedures consistently approximate the observed prices of options under different market regimes, especially for deep out-of-the-money options.     

Bivariate normal mixture spread option valuation

We discuss the pricing and hedging of European spread options on correlated assets when the marginal distribution of each asset return is assumed to be a mixture of normal distributions. Being a straightforward two-dimensional generalization of a normal mixture diffusion model, the prices and hedge ratios have a firm behavioural and theoretical foundation. In this ‘bivariate normal mixture’ (BNM) model no-arbitrage option values are just weighted sums of different ‘2GBM’ option values that are based on the assumption of two correlated lognormal diffusions, and likewise for their sensitivities. The main advantage of this approach is that BNM option values are consistent with both volatility smiles and with the implied correlation ‘frown’. No other ‘frown consistent’ spread option valuation model has such straightforward implementation. We apply analytic approximations to compare BNM valuations of European spread options with those based on the 2GBM assumption and explain the differences between the two as a weighted sum of six second-order 2GBM sensitivities. We also examine BNM option sensitivities, finding that these, like the option values, can sometimes differ substantially from those obtained under the 2GBM model. Finally, we show how the correlation frown that is implied by the BNM model is affected as we change (a) the correlation structure and (b) the tail probabilities in the joint density of the asset returns.     

A Remark on a Singular Perturbation Method for Option Pricing Under a Stochastic Volatility Model

This paper studies the approximation accuracy of a singular perturbation method for option pricing up to the second order under a stochastic volatility model. First, numerical experiments confirm that the first order approximation provides sufficiently accurate option prices in a fast mean-reversion volatility case. On the other hand, it creates relatively large errors in a non-fast mean-reversion volatility environment. Then, the second order approximation formula is derived and the improvement of the approximation is investigated.     

Index Option Pricing Models with Stochastic Volatility and Stochastic Interest Rates

This paper specifies a multivariate stochasticvolatility (SV) model for the S & P500 index and spot interest rateprocesses. We first estimate the multivariate SV model via theefficient method of moments (EMM) technique based on observations ofunderlying state variables, and then investigate the respective effects of stochastic interest rates, stochastic volatility, and asymmetric S & P500 index returns on option prices. We compute option prices using both reprojected underlying historical volatilities and the implied risk premiumof stochastic volatility to gauge each model's performance through direct comparison with observed market option prices on the index. Our major empirical findings are summarized as follows. First, while allowing for stochastic volatility can reduce the pricing errors and allowing for asymmetric volatility or leverage effect does help to explain the skewness of the volatility smile, allowing for stochastic interest rates has minimal impact on option prices in our case. Second, similar to Melino and Turnbull (1990), our empirical findings strongly suggest the existence of a non-zero risk premium for stochastic volatility of asset returns. Based on the implied volatility risk premium, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information from the options market in pricing options. Finally, both the model diagnostics and option pricing errors in our study suggest that the Gaussian SV model is not sufficientin modeling short-term kurtosis of asset returns, an SV model withfatter-tailed noise or jump component may have better explanatory power.     

The Impact of Irreversibility,Uncertainty and Timing Options on Deprival Valuations and the Detection of Monopoly Profits

This paper has two purposes. The first is to derive rules identifying the deprival value of an asset (i) which is irreversible to one extent or another; (ii) the benefit stream of which is subject to continuing uncertainty; and (iii) for which an option to wait exists as to when to reacquire should the owner be deprived of it. The second is to consider whether accounting rates of return employing these deprival value rules can be developed to help in the detection of monopoly profits in circumstances where investment decision-making takes place in the presence of irreversibility, uncertainty and the existence of timing options. The ‘new’ deprival value rules for an asset differ from the ‘conventional’ ones in that present value less the value of the option to reinvest in the asset appears in the ‘new’ rules wherever present value appears in the ‘conventional’ rules. Examples are provided which suggest that ‘new’ and ‘conventional’ deprival value rules can differ materially. A further result is that accounting rates of return can be developed using the ‘new’ deprival value rules that are, in principle, useful in the detection of monopoly profits. Nonetheless, in practice such use requires a level of information that renders the result superfluous in the sense that the provision of replacement cost balance sheet data, combined with the level of information needed, is sufficient to reveal the presence of monopoly profits directly.     

Leverage effect breakdowns and flight from risky assets

Though part of ‘market lore’, in 1976 Black first reported the inverse relationship between price and volatility, calling it the ‘leverage effect’. Without providing evidence, in 1988 Black claimed that in the months leading up to the October 1987 crash the relationship changed: price and volatility both rose. Using daily data for the Old VIX, derived from S&P 100 Index option market prices, to estimate intra-quarterly regressions of implied volatility against price from Q2 1986 to Q1 2012, the author verifies Black’s claim for the October 1987 crash, and interestingly, for subsequent periods of crisis. He then analyses several constant-elasticity-of-variance optimal portfolio rules, which include the leverage effect, to show the elasticity sign switch implies that investors reduce their risky asset holdings to zero.     

Momentum and reversion in risk neutral martingale probabilities

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 skewness 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.     

Empirical Testing of Real Option-Pricing Models

This research is the first to examine the empirical predictions of a real option-pricing model using a large sample of market prices. We find empirical support for a model that incorporates the option to wait to develop land. The option model has explanatory power for predicting transactions prices over and above the intrinsic value. Market prices reflect a premium for the option to wait to invest that has a mean value of 6% in our sample. We also estimate implied standard deviations for individual commercial property prices ranging from 18 to 28% per year.     

Ex Ante Skewness and Expected Stock Returns

We use option prices to estimate ex ante higher moments of the underlying individual securities’ risk‐neutral returns distribution. We find that individual securities’ risk‐neutral volatility, skewness, and kurtosis are strongly related to future returns. Specifically, we find a negative (positive) relation between ex ante volatility (kurtosis) and subsequent returns in the cross‐section, and more ex ante negatively (positively) skewed returns yield subsequent higher (lower) returns. We analyze the extent to which these returns relations represent compensation for risk and find evidence that, even after controlling for differences in co‐moments, individual securities’ skewness matters.     

Option Prices and the Underlying Asset's Return Distribution

This work examines the relation between option prices and the true, as opposed to risk-neutral, distribution of the underlying asset. If the underlying asset follows a diffusion with an instantaneous expected return at least as large as the instantaneous risk-free rate, observed option prices can be used to place bounds on the moments of the true distribution. An illustration of the paper's results is provided by the analysis of the information concerning the mean and standard deviation of market returns contained in the prices of S&P 100 Index Options.