Showing papers in "Quantitative Finance in 2006"
TL;DR: In this article, an exact and explicit solution of the well-known Black-Scholes equation for the valuation of American put options is presented for the first time, which is based on the homotopy-analysis method.
Abstract: In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, i...
257 citations
TL;DR: In this article, a Levy multivariate model for financial assets is discussed, which incorporates jumps, skewness, kurtosis, and stochastic volatility, and the main feature of the model is the fact that its risk-neutral dependence can be calibrated from univariate derivative prices.
Abstract: We discuss a Levy multivariate model for financial assets which incorporates jumps, skewness, kurtosis and stochastic volatility. We use it to describe the behaviour of a series of stocks or indexes and to study a multi-firm, value-based default model. Starting from an independent Brownian world, we introduce jumps and other deviations from normality, including non-Gaussian dependence. We use a stochastic time-change technique and provide the details for a Gamma change. The main feature of the model is the fact that—opposite to other, non-jointly Gaussian settings—its risk-neutral dependence can be calibrated from univariate derivative prices, providing a surprisingly good fit.
184 citations
TL;DR: In this paper, numerical integration methods for stochastic volatility models in financial markets are discussed, where the volatility is either directly given by a mean-reverting CEV process or as a transformed Ornstein-Uhlenbeck process.
Abstract: Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either directly given by a mean-reverting CEV process or as a transformed Ornstein–Uhlenbeck process. For the latter, we introduce a new model based on a simple hyperbolic transformation. Various numerical methods for integrating mean-reverting CEV processes are analysed and compared with respect to positivity preservation and efficiency. Moreover, we develop a simple and robust integration scheme for the two-dimensional system using the strong convergence behaviour as an indicator for the approximation quality. This method, which we refer to as the IJK (137) scheme, is applicable to all types of stochastic volatility models and can be employed as a drop-in replacement for the standard log-Euler procedure.
181 citations
Abstract: Stock prices are observed to be random walks in time despite a strong, long-term memory in the signs of trades (buys or sells). Lillo and Farmer have recently suggested that these correlations are compensated by opposite long-ranged fluctuations in liquidity, with an otherwise permanent market impact, challenging the scenario proposed in Quantitative Finance, 2004, 4, 176, where the impact is instead transient, with a power-law decay in time. The exponent of this decay is precisely tuned to a critical value, ensuring simultaneously that prices are diffusive on long time scales and that the impact function is nearly lag independent. We provide new analysis of empirical data that confirm and make more precise our previous claims. We show that the power-law decay of the bare impact function comes both from an excess flow of limit order opposite to the market order flow, and to a systematic anti-correlation of the bid–ask motion between trades, two effects that create a ‘liquidity molasses’ which dampens mark...
153 citations
TL;DR: In this article, the authors analyse large stock price changes of more than five standard deviations for (i) TAQ data for the year 1997 and (ii) order book data from the Island ECN, and find that a low density of limit orders in the order book, i.e. a small liquidity, is a necessary prerequisite for the occurrence of extreme price fluctuations.
Abstract: We analyse large stock price changes of more than five standard deviations for (i) TAQ data for the year 1997 and (ii) order book data from the Island ECN for the year 2002. We argue that a large trading volume alone is not a sufficient explanation for large price changes. Instead, we find that a low density of limit orders in the order book, i.e. a small liquidity, is a necessary prerequisite for the occurrence of extreme price fluctuations. Taking into account both order flow and liquidity, large stock price fluctuations can be explained quantitatively.
124 citations
TL;DR: In this paper, a survey and comparison of the Esscher martingale transform for linear and exponential processes, and the minimal entropy measure for exponential Levy models, is presented in order to give a complete characterization of those classes of measures.
Abstract: In this paper we offer a systematic survey and comparison of the Esscher martingale transform for linear processes, the Esscher martingale transform for exponential processes, and the minimal entropy martingale measure for exponential Levy models, and present some new results in order to give a complete characterization of those classes of measures. We illustrate the results with several concrete examples in detail.
119 citations
TL;DR: In this paper, the optimal portfolio problem of an insider with logarithmic utility function is studied in a financial market driven by a Levy process with filtration, where the optimal strategy is adapted to a bigger filter.
Abstract: We consider a financial market driven by a Levy process with filtration . An insider in this market is an agent who has access to more information than an honest trader. Mathematically, this is modelled by allowing a strategy of an insider to be adapted to a bigger filtration . The corresponding anticipating stochastic differential equation of the wealth is interpreted in the sense of forward integrals. In this framework, we study the optimal portfolio problem of an insider with logarithmic utility function. Explicit results are given in the case where the jumps are generated by a Poisson process. §Dedicated to the memory of Axel Grorud.
87 citations
TL;DR: In this paper, the authors focused on the market efficiency and the long-memory of supply and demand and showed that there are waves of buyer-initiated transactions that are highly foreseeable with the use of simple linear algorithm.
Abstract: The article focuses on the market efficiency and the long-memory of supply and demand. The long-memory of supply and demand implies that there are waves of buyer-initiated transactions that are highly foreseeable with the use of simple linear algorithm. The authors stressed that the total price impact can be summed up with bare propagators associated with each transaction.
85 citations
TL;DR: In this article, it was shown that fluctuations in transaction volume, as reflected in the number of transactions and to a lesser extent their size, are not the main cause of clustered volatility.
Abstract: It is widely believed that fluctuations in transaction volume, as reflected in the number of transactions and to a lesser extent their size, are the main cause of clustered volatility. Under this view bursts of rapid or slow price diffusion reflect bursts of frequent or less frequent trading, which cause both clustered volatility and heavy tails in price returns. We investigate this hypothesis using tick by tick data from the New York and London Stock Exchanges and show that only a small fraction of volatility fluctuations are explained in this manner. Clustered volatility is still very strong even if price changes are recorded on intervals in which the total transaction volume or number of transactions is held constant. In addition the distribution of price returns conditioned on volume or transaction frequency being held constant is similar to that in real time, making it clear that neither of these are the principal cause of heavy tails in price returns. We analyse recent results of Ane and Geman (2000...
71 citations
TL;DR: In this paper, the authors introduce the notion of symmetry in a Levy market, which is a particular case of a general known relation between prices of put and call options, of both the European and the American type, and that we call put-call duality.
Abstract: The aim of this paper is to introduce the notion of symmetry in a Levy market. This notion appears as a particular case of a general known relation between prices of put and call options, of both the European and the American type, which is also reviewed in the paper, and that we call put–call duality. Symmetric Levy markets have the distinctive feature of producing symmetric smile curves, in the log of strike/futures prices. Put–call duality is obtained as a consequence of a change of the risk neutral probability measure through Girsanov's theorem, when considering the discounted and reinvested stock price as the numeraire. Symmetry is defined when a certain law before and after the change of measure through Girsanov's theorem coincides. A parameter characterizing the departure from symmetry is introduced, and a necessary and sufficient condition for symmetry to hold is obtained, in terms of the jump measure of the Levy process, answering a question raised by Carr and Chesney (American put call symmetry,...
62 citations
TL;DR: In this paper, the exponential Ornstein-Uhlenbeck stochastic volatility model was studied and the model showed a multiscale behavior in the volatility autocorrelation.
Abstract: We study the exponential Ornstein–Uhlenbeck stochastic volatility model and observe that the model shows a multiscale behaviour in the volatility autocorrelation. It also exhibits a leverage correlation and a probability profile for the stationary volatility which are consistent with market observations. All these features make the model quite appealing since it appears to be more complete than other stochastic volatility models also based on a two-dimensional diffusion. We finally present an approximate solution for the return probability density designed to capture the kurtosis and skewness effects.
TL;DR: In this article, a convergent option pricing scheme in a Black-Merton-Scholes world (Black andScholes 1973, Merton 1973) on a discrete-time lattice is presented.
Abstract: 1. IntroductionThe pricing of American options on stocks wasrendered computable by the work of Cox et al. (1979)(CRR) and the ensuing work of Jarrow and Rudd(1983). They developed a convergent option pricingscheme in a Black–Merton–Scholes world (Black andScholes 1973, Merton 1973) on a discrete-time lattice.In this setting stock prices ½SðtÞ are assumed to followa geometric Brownian motion, i.e.dSðtÞ¼ ðtÞSðtÞdtþ ðtÞSðtÞdZðtÞ , Sð0Þ S
TL;DR: In this article, the authors examine the dynamics of the price evolution of liquid stocks after experiencing a large intra-day price change, using data from the NYSE and the NASDAQ.
Abstract: In our empirical study we examine the dynamics of the price evolution of liquid stocks after experiencing a large intra-day price change, using data from the NYSE and the NASDAQ. We find a significant reversal for both intra-day price decreases and increases. Volatility, volume and, in the case of the NYSE, the bid–ask spread, which increase sharply at the event, stay significantly high days afterwards. The decay of the volatility follows a power law in accordance with the `Omori law'. While on the NYSE the large widening of the bid–ask spread eliminates most of the profits that can be achieved by an outside investor, on the NASDAQ the bid–ask spread stays almost constant, yielding significant short-term profits. The results thus give an insight into the size and speed of the realization of an excess return for providing liquidity in a turbulent market.
TL;DR: In this article, the probability that a drawdown of a units precedes a rally of b units in the geometric Brownian motion model was determined. But the results were not applicable to the stock market.
Abstract: We study drawdowns and rallies of Brownian motion. A rally is defined as the difference of the present value of the Brownian motion and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. This paper determines the probability that a drawdown of a units precedes a rally of b units. We apply this result to examine stock market crashes and rallies in the geometric Brownian motion model.
TL;DR: In this paper, a class of one-factor local volatility function models for stock indices under a benchmark approach is studied without requiring the existence of an equivalent risk-neutral probability measure.
Abstract: 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 in...
TL;DR: In this article, the authors use a reflection result to give simple proofs of valuation formulas and static hedge portfolio constructions for zero-rebate single-barrier options in the Black-Scholes model.
Abstract: We use a reflection result to give simple proofs of (well-known) valuation formulas and static hedge portfolio constructions for zero-rebate single-barrier options in the Black–Scholes model. We then illustrate how to extend the ideas to other model types giving (at least) easy-to-program numerical methods and other option types such as options with rebates, and double-barrier and lookback options.
TL;DR: In this article, a general formula to price European path-dependent options on multidimensional assets is obtained and implemented by means of various flexible and efficient algorithms. And the numerical results are compared with those obtained with other procedures used in quantitative finance and found to be in good agreement.
Abstract: In the framework of the Black–Scholes–Merton model of financial derivatives, a path integral approach to option pricing is presented. A general formula to price European path-dependent options on multidimensional assets is obtained and implemented by means of various flexible and efficient algorithms. As an example, we detail the cases of Asian, barrier knock out, reverse cliquet and basket call options, evaluating prices and Greeks. The numerical results are compared with those obtained with other procedures used in quantitative finance and found to be in good agreement. In particular, when pricing at-the-money and out-of-the-money options, the path integral approach exhibits competitive performances.
TL;DR: In this article, the authors present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk, and use the Cox et al. term structure model to preclude the possibility of negative probabilities of default.
Abstract: In this paper we present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk. We maintain the cause and effect or ‘structural’ definition of default and assume that default is triggered when a state variable reaches a default boundary. However, in our model, the state variable is not interpreted as the assets of the firm, but as a latent variable signalling the credit quality of the firm. Default in our model can also occur according to a doubly stochastic hazard rate. The hazard rate is a linear function of the state variable and the interest rate. We use the Cox et al. (A theory of the term structure of interest rates. Econometrica, 1985, 53(2), 385–407) term structure model to preclude the possibility of negative probabilities of default. We also horse race the proposed valuation model against structural and reduced-form default risky bond pricing models and find that term structures of credit spreads generated using the middle-way appr...
TL;DR: Intelligent finance as discussed by the authors is a new direction recently emerging from the confluence of several distinct disciplines in financial market analysis, investing and trading, removing any historical or artificial barrier between them.
Abstract: Intelligent finance represents a new direction recently emerging from the confluence of several distinct disciplines in financial market analysis, investing and trading, removing any historical or artificial barrier between them. It is conceived as the science, technology and art of the comprehensive, predictive, dynamic and strategic analysis of global financial markets, towards a unification and integration of academic finance and professional finance. As a comprehensive approach, it is a quest for absolute positive and non-trivial returns in investing and trading by exploiting complete information about financial markets from all general perspectives, drawing ideas, theories, models and techniques from many related academic disciplines, such as macroeconomics, microeconomics, academic finance, financial mathematics, econophysics, behavioural finance and computational finance, and from professional schools of thought, such as macrowave investing, trend following, fundamental analysis, technical analysis...
TL;DR: In this article, the Levy Libor market model was extended to a multi-currency setting and closed-form pricing formulas for cross-currency derivatives were derived for foreign caps and floors and crosscurrency swaps.
Abstract: The Levy Libor or market model which was introduced in Eberlein and Ozkan (The Levy Libor model. Financ. Stochast., 2005, 9, 327–348) is extended to a multi-currency setting. As an application we derive closed form pricing formulas for cross-currency derivatives. Foreign caps and floors and cross-currency swaps are studied in detail. Numerically efficient pricing algorithms based on bilateral Laplace transforms are derived. A calibration example is given for a two-currency setting (EUR, USD).
TL;DR: In this article, the authors developed explicit prices for digital and regular Asian options and obtained distributional results concerning the square root process and its average over time, including analytic formulae for their joint density and moments.
Abstract: 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.
TL;DR: In this article, the application of gradient methods to calibrate mean reverting stochastic volatility models is discussed and an extension of the ideas to apply Malliavin calculus methods in the computation of Greek's is presented.
Abstract: We discuss the application of gradient methods to calibrate mean reverting stochastic volatility models. For this we use formulas based on Girsanov transformations as well as a modification of the Bismut–Elworthy formula to compute the derivatives of certain option prices with respect to the parameters of the model by applying Monte Carlo methods. The article presents an extension of the ideas to apply Malliavin calculus methods in the computation of Greek's.
TL;DR: In this paper, the equilibrium asset pricing implications for an economy with single period return exposures to explicit non-Gaussian systematic factors, that may be both skewed and long-tailed, and Gaussian idiosyncratic components are analyzed.
Abstract: We analyse the equilibrium asset pricing implications for an economy with single period return exposures to explicit non-Gaussian systematic factors, that may be both skewed and long-tailed, and Gaussian idiosyncratic components. Investors maximize expected exponential utility and equilibrium factor prices are shown to reflect exponentially tilted prices for non-Gaussian factor risk exposures. It is shown that these prices may be directly estimated from the univariate probability law of the factor exposure, given an estimate of average risk aversion in the economy. In addition, a residual form of the capital asset pricing model continues to hold and prices the idiosyncratic or Gaussian risks. The theory is illustrated on data for the US economy using independent components analysis to identify the factors and the variance gamma model to describe the probability law of the non-Gaussian factors. It is shown that the residual CAPM accounts for no more than 1% of the pricing of risky assets, while the exponen...
TL;DR: In this paper, a simple yet efficient analytic approximation of the utility-based option hedging solution is presented, and compared with the asymptotic and some other well-known strategies and find that their strategy outperforms all the others.
Abstract: One of the most successful approaches to option hedging with transaction costs is the utility-based approach, pioneered by Hodges and Neuberger [Rev. Futures Markets, 1989, 8, 222–239]. Judging against the best possible trade-off between the risk and the costs of a hedging strategy, this approach seems to achieve excellent empirical performance. However, this approach has one major drawback that prevents the broad application of this approach in practice: the lack of a closed-form solution. We overcome this drawback by presenting a simple yet efficient analytic approximation of the solution. We provide an empirical testing of our approximation strategy against the asymptotic and some other well-known strategies and find that our strategy outperforms all the others.
TL;DR: In this paper, the authors analyzed the relationship between diversification and several distributional characteristics that have risk implications for stock returns and developed a flexible three-parameter distribution to model the stock returns.
Abstract: This paper analyzes the relationship between diversification and several distributional characteristics that have risk implications for stock returns. We develop a flexible three-parameter distribution to model the stock returns. Using data on the current 30 DJIA stocks, we show that an investor's strategy on diversification depends on the measures of risk for particular concerns. For example, investors who desire to increase positive skewness would hold a less diversified portfolio, while those who care more about extreme losses would hold a more diversified portfolio. Experimenting with a more general pool of stocks yields the same conclusions.
TL;DR: In this paper, the authors investigated the statistical properties of drawdowns and drawups in interest rates using over 10 years' worth of daily data and provided a coherent explanation for a complex set of empirical observations.
Abstract: We investigate the statistical properties of drawdowns and drawups in interest rates (US$) using over 10 years' worth of daily data. We analyse the nature of the drawdowns in terms of length of runs, magnitude of the individual price moves and coincidence of their occurrence across the maturity spectrum. We document significant positive autocorrelation for several holding periods, pronounced term structure effects and an unexpectedly low degree of coincidence in the occurrence of drawdowns across the maturity spectrum (despite high correlation in daily moves). By drawing on previous work by Rebonato et al. (2005) we try to provide a coherent explanation for a complex set of empirical observations. An essential ingredient of this explanation appears to be the existence of at least two distinct types (normal and excited) of price dynamics, with different serial correlation properties. We concur with the results by Sornette and Johansen (Significance of log-periodic precursors to financial crashes. Quant. Fi...
TL;DR: Kim and Schmidt as discussed by the authors examined the properties of modified Dickey-Fuller unit root tests in the presence of generalized autoregressive conditional heteroskedasticity (GARCH) using Monte Carlo simulation, and found that the modified tests are substantially oversized when the GARCH process exhibits a high degree of volatility.
Abstract: The research of Kim and Schmidt (J. Economet., 1993, 59, 287–300) is extended to examine the properties of modified Dickey–Fuller unit root tests in the presence of generalized autoregressive conditional heteroskedasticity (GARCH). Using Monte Carlo simulation, the properties of the tests are examined for a range of GARCH processes over alternative sample sizes. Oversizing is observed for all tests, with the extent of size distortion driven by the volatility, rather than the persistence, of the underlying GARCH process. While the original Dickey–Fuller test is found to exhibit greater size distortion than the modified tests, the modified tests are found to be substantially oversized when the GARCH process exhibits a high degree of volatility, even for large samples.
TL;DR: In this paper, the authors analyzed the optimal exercise strategies for corporate warrants issued by levered firms and distinguish between two exercise variants, namely the traditional block exercise and competitive exercise in equilibrium.
Abstract: In this paper, we analyse the optimal exercise strategies for corporate warrants issued by levered firms. For the analysis, we distinguish between two exercise variants, namely the traditional block exercise and competitive exercise in equilibrium. We find that the optimal exercise date under the block condition can be before or after an optimal exercise in equilibrium. Surprisingly, optimal block exercise can occur even without any dividend payments in contrast to the competitive exercise. As a consequence, the asset values and the stock volatility under block exercise fundamentally deviate from those under the competitive exercise variant. Moreover, the value of a warrant in the block case and its exercise strategy do not coincide with those of a corresponding call option which contrasts with the assumption of ‘option-like’ warrant valuation.
TL;DR: In this article, the authors consider portfolio optimization problems in the presence of both the censored mean and the conditional value at risk lower bound and show that the latter turn out to imply much tighter bounds for the admissible portfolio set and indeed for the logistic, an upper bound for the portfolio variance that yields a simple portfolio choice rule.
Abstract: Generalized value at risk (GVaR) adds a conditional value at risk or censored mean lower bound to the standard value at risk and considers portfolio optimization problems in the presence of both constraints. For normal distributions the censored mean is synonymous with the statistical hazard function, but this is not true for fat-tailed distributions. The latter turn out to imply much tighter bounds for the admissible portfolio set and indeed for the logistic, an upper bound for the portfolio variance that yields a simple portfolio choice rule. The choice theory in GVaR is in general not consistent with classic Von Neumann–Morgenstern utility functions for money. A re-specification is suggested to make it so that gives a clearer picture of the economic role of the respective constraints. This can be used analytically to explore the choice of portfolio hedges.
TL;DR: In this paper, it is shown that it is possible to have estimators of Expected Shortfall that do not satisfy the sub-additivity condition, which is the most important property a risk measure ought to satisfy.
Abstract: Value at Risk has lost the battle against Expected Shortfall on theoretical grounds, the latter satisfying all coherence properties while the former may, on carefully constructed cases, lack the sub-additivity property that is in a sense, the most important property a risk measure ought to satisfy. While the superiority of Expected Shortfall is evident as a theoretical tool, little has been researched on the properties of estimators proposed in the literature. Since those estimators are the real tools for calculating bank capital reserves in practice, the natural question that one may ask is whether a given estimator of Expected Shortfall also satisfies the coherence properties. In this paper, we show that it is possible to have estimators of Expected Shortfall that do not satisfy the sub-additivity condition. This finding should motivate risk managers and quantitative asset managers to investigate further the properties of the estimators of the risk measures they are currently utilizing.