A Two-Factor Model for Low Interest Rate Regimes

This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

Pricing of index options under a minimal market model with log-normal scaling

Abstract

This paper describes a two-factor model for a diversified market index using the growth optimal portfolio with a stochastic and possibly correlated intrinsic timescale. The index is modelled using a time transformed squared Bessel process with a log-normal scaling factor for the time transformation. A consistent pricing and hedging framework is established by using the benchmark approach. Here the numeraire is taken to be the growth optimal portfolio. Benchmarked traded prices appear as conditional expectations of future benchmarked prices under the real world probability measure. The proposed minimal market model with log-normal scaling produces the type of implied volatility term structures for European call and put options typically observed in real markets. In addition, the prices of binary options and their deviations from corresponding Black–Scholes prices are examined.     

Intraday Empirical Analysis and Modeling of Diversified World Stock Indices

This paper proposes an approach to the intraday analysis of diversified world stock accumulation indices. The growth optimal portfolio (GOP) is used as reference unit or benchmark in a continuous financial market model. Diversified portfolios, covering the world stock market, are constructed and shown to approximate the GOP, providing the basis for a range of financial applications. The normalized GOP is modeled as a time transformed square root process of dimension four. Its dynamics are empirically verified for several world stock indices. Furthermore, the evolution of the transformed time is modeled as the integral over a rapidly evolving mean-reverting market activity process with deterministic volatility. The empirical findings suggest a rather simple and robust model for a world stock index that reflects the historical evolution, by using only a few readily observable parameters. Mathematics Subject Classification: (1991) primary 90A12, secondary 60G30,62P20 JEL Classification: G10, G13     

A Benchmark Approach to Filtering in Finance

The paper proposes the use of the growth optimal portfolio for pricing and hedging in incomplete markets when there are unobserved factors that have to be filtered. The proposed filtering framework is applicable also in cases when there does not exist an equivalent risk neutral martingale measure. The reduction of the variance of derivative prices for increasing degrees of available information is measured. 1991 Mathematics Subject Classification: primary 90A09; secondary 60G99; 62P20 JEL Classification: G10, G13     

A Fair Pricing Approach to Weather Derivatives

This paper proposes a consistent approach to the pricing of weather derivatives. Since weather derivatives are traded in an incomplete market setting, standard hedging based pricing methods cannot be applied. The growth optimal portfolio, which is interpreted as a world stock index, is used as a benchmark or numeraire such that all benchmarked derivative price processes are martingales. No measure transformation is needed for the proposed fair pricing. For weather derivative payoffs that are independent of the value of the growth optimal portfolio, it is shown that the classical actuarial pricing methodology is a particular case of the fair pricing concept. A discrete time model is constructed to approximate historical weather characteristics. The fair prices of some particular weather derivatives are derived using historical and Gaussian residuals. The question of weather risk as diversifiable risk is also discussed. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: C16, G10, G13     

Modelling the implied volatility surface: Does market efficiency matter?: An application to MIB30 index options

We analyze the volatility surface vs. moneyness and time-to-expiration implied by MIBO options written on the MIB30, the most important Italian stock index. We specify and fit a number of models of the implied volatility surface and find that it has a rich and interesting structure that strongly departs from a constant volatility, Black-Scholes benchmark. This result is robust to alternative econometric approaches, including generalized least squares approaches that take into account both the panel structure of the data and the likely presence of heteroskedasticity and serial correlation in the random disturbances. Finally we show that the degree of pricing efficiency of this options market can strongly condition the results of the econometric analysis and therefore our understanding of the pricing mechanism underlying observed MIBO option prices. Applications to value-at-risk and portfolio choice calculations illustrate the importance of using arbitrage-free data only.     

Local volatility function models under a benchmark approach

Without requiring the existence of an equivalent risk-neutral probability measure this paper studies a class of one-factor local volatility function models for stock indices under a benchmark approach. It is assumed that the dynamics for a large diversified index approximates that of the growth optimal portfolio. Fair prices for derivatives when expressed in units of the index are martingales under the real-world probability measure. Different to the classical approach that derives risk-neutral probabilities the paper obtains the transition density for the index with respect to the real-world probability measure. Furthermore, the Dupire formula for the underlying local volatility function is recovered without assuming the existence of an equivalent risk-neutral probability measure. A modification of the constant elasticity of variance model and a version of the minimal market model are discussed as specific examples together with a smoothed local volatility function model that fits a snapshot of S&P500 index options data.     

Stochastic orders in dynamic reinsurance markets

We consider a dynamic reinsurance market, where the traded risk process is driven by a compound Poisson process and where claim amounts are unbounded. These markets are known to be incomplete, and there are typically infinitely many martingale measures. In this case, no-arbitrage pricing theory can typically only provide wide bounds on prices of reinsurance claims. Optimal martingale measures such as the minimal martingale measure and the minimal entropy martingale measure are determined, and some comparison results for prices under different martingale measures are provided. This leads to a simple stochastic ordering result for the optimal martingale measures. Moreover, these optimal martingale measures are compared with other martingale measures that have been suggested in the literature on dynamic reinsurance markets.Received: March 2004, Mathematics Subject Classification (2000): 62P05, 60J75, 60G44JEL Classification: G10     

Hazard rate for credit risk and hedging defaultable contingent claims

We provide a concise exposition of theoretical results that appear in modeling default time as a random time, we study in details the invariance martingale property and we establish a representation theorem which leads, in a complete market setting, to the hedging portfolio of a vulnerable claim. Our main result is that, to hedge a defaultable claim one has to invest the value of this contingent claim in the defaultable zero-coupon.Received: April 2003Mathematics Subject Classification: 91B24, 91B29, 60G46JEL Classification: G10The authors would like to thank D. Becherer and J.N. Hugonnier for interesting discussions and the anonymous referee whose pertinent questions on the first version of this paper help them to clarify the proofs. All remaining errors are ours.     

Mean-reversion properties of implied volatilities

In this paper, we present a new stylized fact for options whose underlying asset is a stock index. Extracting implied volatility time series from call and put options on the Deutscher Aktien index (DAX) and financial times stock exchange index (FTSE), we show that the persistence of these volatilities depends on the moneyness of the options used for its computation. Using a functional autoregressive model, we show that this effect is statistically significant. Surprisingly, we show that the diffusion-based stochastic volatility models are not consistent with this stylized fact. Finally, we argue that adding jumps to a diffusion-based volatility model help recovering this volatility pattern. This suggests that the persistence of implied volatilities can be related to the tails of the underlying volatility process: this corroborates the intuition that the liquidity of the options across moneynesses introduces an additional risk factor to the one usually considered.     

An Intraweek Seasonality in the Implied Volatilities of Individual and Index Options

This paper studies intraweek seasonalities in the implied volatilities of options on stock market indices. Oneway analysis of variance isolates the daily behavior of implied volatilities. The differential between call implied volatility and put implied volatility tends to drop on Friday and rise on Monday. Relying on a synthetic futures contract created from options, an explanatory model is proposed. The model complements previous research on the difference between the intraweek behavior of stock market indices and that of derivative instruments based on the indices.     

Diversified Portfolios with Jumps in a Benchmark Framework

This paper considers diversified portfolios in a sequence of jump diffusion market models. Conditions for the approximation of the growth optimal portfolio (GOP) by diversified portfolios are provided. Under realistic assumptions, it is shown that diversified portfolios approximate the GOP without requiring any major model specifications. This provides a basis for systematic use of diversified stock indices as proxies for the GOP in derivative pricing, risk management and portfolio optimization. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

An investor sentiment barometer — Greek Implied Volatility Index (GRIV)

In this paper, we propose a new measure of Greek equity market volatility based on the prices of FTSE/ATHEX-20 index options. Greek Implied Volatility Index is calculated using the model-free methodology that involves option prices summations and is independent from the Black and Scholes pricing formula. The specific method is applied for the first time in a peripheral and illiquid market as the Athens Exchange.The empirical findings of this paper show that the proposed volatility index includes information about future realized volatility beyond that contained in past volatility. In addition, our analysis indicates that there is a statistically significant negative and asymmetric contemporaneous relationship between the returns of the implied volatility index and the underlying equity index. Finally, the volatility transmission effects on the Greek stock exchange from two leading markets, namely the New York Stock Exchange and the Deutsche Börse, are tested and documented.     

New evidence on the implied-realized volatility relation

We consider the relation between the volatility implied in an option's price and the subsequently realized volatility. Earlier studies on stock index options have found biases and inefficiencies in implied volatility as a forecast of future volatility. More recently, Christensen and Prabhala find that implied volatility in at-the-money one-month OEX call options on the S&P 100 index in fact is an unbiased and efficient forecast of ex-post realized index volatility after the 1987 stock market crash. In this paper, the robustness of the unbiasedness and efficiency result is extended to a more recent period covering April 1993 to February 1997. As a new contribution, implied volatility is constructed as a trade weighted average of implied volatilities from both in-the-money and out-of-the-money options and both puts and calls. We run a horse race between implied call, implied put, and historical return volatility. Several robustness checks, including a new simultaneous equation approach, underscore our conclusion, that implied volatility is an efficient forecast of realized return volatility.     

Ex Ante Stock Market Return Volatility Implied by the OEX Option Premium

Previous empirical evidence suggests that stock return volatility expectations change over time, but the existing models of time-varying variance lack a theoretical structure that is rigorously linked to the efficient markets dividend discount model. This paper develops and tests such a model. The conditional forecast variance of the return on the stock market portfolio is expressed as a linear combination of the adjusted conditional forecast variance of the interest rate and the dividend growth rate. An empirical test using the implied variance of the S&P 100 index option provides evidence that supports the model's predictions.     

Zum fairen Wert der Aktienanlagen eines Lebensversicherungsunternehmens aus ökonomisch-statistischer Perspektive

The present contribution deals with the problem of an adequate determination of the “fair value” of the stock portfolio of a life insurance company taking an economical and statistical point of view. The starting point consists of three standard models from mathematical finance for the development of stock prices, the martingale model, the random walk model and the AR(l)-process, the latter as a basic model capturing mean reversion effects. The best (with respect to mean square error) forecast of the future value of the process given the information about its history then is a natural and model-based concept of a “fair value”. The question remaining now is which of the alternative models gives the best representation of the data. First of all it is explained, that the martingale hypothesis is valid neither from a theoretical from an empirical point of view. However, only in the case of a martingale the market value is the best forecast, i.e. only in this case the market value would correspond with the fair value in an economical-statistical sense. This implies that the common understanding in accounting practice that the market value is a natural candidate for the fair value is not justified in theoretical terms. With respect to the question “random walk or mean reversion?” we perform a statistical-econometrical analysis of a thirty year time series of the monthly price earnings ratios (PER) of the German stock index DAX. As a result, the AR(l)-process reveals to be a better statistical representative of the time series as the random walk. This implies, that the long term equilibrium value of the AR(l)-process is the best forecast of the fair value of the DAX-PER. Given that DAX-PER then — based on different assumptions about the earnings growth rate and the time horizon — corresponding forecasts are calculated for the DAX itself. Finally it is explained that because of the pronounced long term character of life insurance business this model based conception of a long-term fair value is ideally suited for an application to life insurance companies.     

The predictive power of the implied volatility of options traded OTC and on exchanges

This paper investigates the efficiency of stock index options traded over-the-counter (OTC) and on the exchanges in Hong Kong and Japan. Our findings suggest that implied volatility is superior to either historical volatility or a GARCH-type volatility forecast in predicting future volatility in both the OTC and exchange markets. This paper is also one of the first to compare the predictive power of the implied volatility of stock index options traded OTC to that of exchange-traded stock index options. Our evidence suggests that the OTC market is more efficient than the exchanges in Japan, but that the opposite is true in Hong Kong.     

Price discovery in the U.S. stock and stock options markets: A portfolio approach

Option prices vary with not only the underlying asset price, but also volatilities and higher moments. In this paper, we use a portfolio of options to seclude the value change of the portfolio from the impact of volatility and higher moments. We apply this portfolio approach to the price discovery analysis in the U.S. stock and stock options markets. We find that the price discovery on the directional movement of the stock price mainly occurs in the stock market, more so now than before as an increasing proportion of options market makers adopt automated quoting algorithms. Nevertheless, the options market becomes more informative during periods of significant options trading activities. The informativeness of the options quotes increases further when the options trading activity generates net sell or buy pressure on the underlying stock price, even more so when the pressure is consistent with deviations between the stock and the options market quotes. JEL Classification C52, G10, G13, G14     

Pricing volatility options under stochastic skew with application to the VIX index

In recent years, there has been a remarkable growth of volatility options. In particular, VIX options are among the most actively trading contracts at Chicago Board Options Exchange. These options exhibit upward sloping volatility skew and the shape of the skew is largely independent of the volatility level. To take into account these stylized facts, this article introduces a novel two-factor stochastic volatility model with mean reversion that accounts for stochastic skew consistent with empirical evidence. Importantly, the model is analytically tractable. In this sense, I solve the pricing problem corresponding to standard-start, as well as to forward-start European options through the Fast Fourier Transform. To illustrate the practical performance of the model, I calibrate the model parameters to the quoted prices of European options on the VIX index. The calibration results are fairly good indicating the ability of the model to capture the shape of the implied volatility skew associated with VIX options.     

Valuation of American options in the presence of event risk

This paper studies the valuation of American options in the presence of external/non-hedgeable event risk. When the event occurs, the American option is terminated and a rebate is paid instead of the promised pay-off profile. Consequently, the presence of event risk influences the exercise strategy of the option holder. For the financial market in a diffusion setting, the probabilistic structure in terms of equivalent martingale measures is briefly analysed. Then, for a given equivalent martingale measure the optimal stopping problem of the American option is solved. As a main result, no-arbitrage bounds for American option values in the presence of event risk are derived, as well as hedging strategies corresponding to the no-arbitrage bounds.Received: May 2004, Mathematics Subject Classification: 90C47, 60H30, 60G40JEL Classification: G13, D52, D81The author thanks John Gould and Ross Maller for useful discussions. The author is also grateful to a referee for helpful comments. This research was partially supported by University of Western Australia Research Grant RA/1/485.