Showing papers in "Quantitative Finance in 2004"
TL;DR: In this article, the authors provide an empirical analysis of the network structure of the Austrian interbank market based on Austrian Central Bank (OeNB) data and find that the degree distributions of the interbank network follow power laws.
Abstract: We provide an empirical analysis of the network structure of the Austrian interbank market based on Austrian Central Bank (OeNB) data. The interbank market is interpreted as a network where banks are nodes and the claims and liabilities between banks define the links. This allows us to apply methods from general network theory. We find that the degree distributions of the interbank network follow power laws. Given this result we discuss how the network structure affects the stability of the banking system with respect to the elimination of a node in the network, i.e. the default of a single bank. Further, the interbank liability network shows a community structure that exactly mirrors the regional and sectoral organization of the current Austrian banking system. The banking network has the typical structural features found in numerous other complex real-world networks: a low clustering coefficient and a short average path length. These empirical findings are in marked contrast to the network structures th...
836 citations
TL;DR: In this paper, the cause of large fluctuations in prices on the London Stock Exchange is studied at the microscopic level of individual events, where an event is the placement or cancellation of an order to buy or sell, and it is shown that price fluctuations caused by individual market orders are essentially independent of the volume of orders.
Abstract: We study the cause of large fluctuations in prices on the London Stock Exchange. This is done at the microscopic level of individual events, where an event is the placement or cancellation of an order to buy or sell. We show that price fluctuations caused by individual market orders are essentially independent of the volume of orders. Instead, large price fluctuations are driven by liquidity fluctuations, variations in the market's ability to absorb new orders. Even for the most liquid stocks there can be substantial gaps in the order book, corresponding to a block of adjacent price levels containing no quotes. When such a gap exists next to the best price, a new order can remove the best quote, triggering a large midpoint price change. Thus, the distribution of large price changes merely reflects the distribution of gaps in the limit order book. This is a finite size effect, caused by the granularity of order flow: in a market where participants place many small orders uniformly across prices, such large...
352 citations
TL;DR: In this article, the random walk nature of traded prices results from a very delilcated interplay between two opposite tendencies: long-range correlated market orders that lead to super-diffusion (or persistence), and mean revrting limit orders that leads to sub-diffusions (or anti-persistence).
Abstract: Using trades and quotes data from the Paris stock market, we show that the random walk nature of traded prices results from a very delilcated interplay between two opposite tendencies: long-range correlated market orders that lead to super-diffusion (or persistence), and mean revrting limit orders that lead to sub-diffusion (or anti-persistence). We define and study a model where the price, at any instant, is the result of the impact of all past trades, mediated by a non-constant ‘propagator’ in time that describes the response of the market to a single trade. Within this model, the market is shown to be, in a precise sense, at a critical point, where the price is purely diffusive and the average response function almost constant. We find empirically, and discuss theoretically, a fluctuation-response relation. We also discuss the fraction of truly informed market orders, that correctly anticipate short-term moves, and find that it is quite small.
307 citations
TL;DR: In this paper, the authors present a general model that simultaneously takes into account the following features: seasonal patterns, price spikes, mean reversion, price dependent volatilities and long term non-stationarity.
Abstract: In this paper, we analyse the evolution of prices in deregulated electricity markets. We present a general model that simultaneously takes into account the following features: seasonal patterns, price spikes, mean reversion, price dependent volatilities and long term non-stationarity. We estimate the parameters of the model using historical data from the European Energy Exchange. Finally, we demonstrate how it can be used for pricing derivatives via Monte Carlo simulation.
238 citations
TL;DR: Three sampling algorithms for multivariate Archimedean copulas are developed that entail drawing from a one-dimensional distribution and then scaling the result to create random deviates distributed according to the copula.
Abstract: We develop sampling algorithms for multivariate Archimedean copulas. For exchangeable copulas, where there is only one generating function, we first analyse the distribution of the copula itself, deriving a number of integral representations and a generating function representation. One of the integral representations is related, by a form of convolution, to the distribution whose Laplace transform yields the copula generating function. In the infinite-dimensional limit there is a direct connection between the distribution of the copula value and the inverse Laplace transform. Armed with these results, we present three sampling algorithms, all of which entail drawing from a one-dimensional distribution and then scaling the result to create random deviates distributed according to the copula. We implement and compare the various methods. For more general cases, in which an N-dimensional Archimedean copula is given by N−1 nested generating functions, we present algorithms in which each new variate is drawn ...
160 citations
TL;DR: In this article, the origin of power-law tails in price fluctuations is discussed and a discussion of the relationship between power law and price fluctuations and its relationship to price changes is presented.
Abstract: (2004). On the origin of power-law tails in price fluctuations. Quantitative Finance: Vol. 4, No. 1, pp. 7-11.
118 citations
TL;DR: In this article, the authors present a model for risk management in a firm which exercises control of its risk as well as potential profit by choosing different business activities among those available to it.
Abstract: This paper represents a model for risk management in a firm which exercises control of its risk as well as potential profit by choosing different business activities among those available to it. Furthermore, the firm has an option of investing its reserve in a financial market consisting of a risk-free asset (bond) and a risky asset (stock). The example we consider is that of a large corporation such as an insurance company, whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, while drift is associated with potential profit. At each moment of time there is an option to reduce risk exposure, simultaneously reducing the potential profit, like using proportional reinsurance with another carrier for an insurance company. The company invests its reserve in a financial market, which is described by a classical Black–Scholes model. The management of...
97 citations
95 citations
TL;DR: In this article, the authors derived a nonlinear expectation representation for the claim's ask price and a formula for the optimal hedging strategy and generated a perturbation expansion for the price and hedging strategies in powers of � 2 = 1 − ρ 2, where terms in the price expansion are proportional to the central moments of the claim payoff under the minimal martingale measure.
Abstract: performance of optimal strategies for hedging a claim on a non-traded asset is analysed. The claim is valued and hedged in a utility maximization framework, using exponential utility. A traded asset, correlated with that underlying the claim, is used for hedging, with the correlation ρ typically close to 1. Using a distortion method (Zariphopoulou 2001 Finance Stochastics 5 61-82) we derive a nonlinear expectation representation for the claim's ask price and a formula for the optimal hedging strategy. We generate a perturbation expansion for the price and hedging strategy in powers of � 2 = 1 − ρ 2 . The terms in the price expansion are proportional to the central moments of the claim payoff under the minimal martingale measure. The resulting fast computation capability is used to carry out a simulation-based test of the optimal hedging program, computing the terminal hedging error over many asset price paths. These errors are compared with those from a naive strategy which uses the traded asset as a proxy for the non-traded one. The distribution of the hedging error acts as a suitable metric to analyse hedging performance. We find that the optimal policy improves hedging performance, in that the hedging error distribution is more sharply peaked around a non-negative profit. The frequency of profits over losses is increased, and this is measured by the median of the distribution, which is always increased by the optimal strategies. An empirical example illustrates the application of the method to the hedging of a stock basket using index futures.
95 citations
TL;DR: In this paper, the valuation and hedging of volatility swaps within the frame of a GARCH(1,1) stochastic volatility model are discussed, and a closed-form approximate solution for the so-called convexity correction is provided.
Abstract: This article discusses the valuation and hedging of volatility swaps within the frame of a GARCH(1,1) stochastic volatility model. First we use a general and flexible partial differential equation (PDE) approach to determine the first two moments of the realized variance in a continuous or discrete context. Next, and also the main contribution of the paper, is a closed-form approximate solution for the so-called convexity correction, when the risk-neutral process for the instantaneous variance is a continuous time limit of a GARCH (1,1) model. Following this, we provide a numerical example using S&P 500 data.
86 citations
TL;DR: A novel algorithm is developed for the problem of finding a low-rank correlation matrix nearest to a given correlation matrix, which is based on majorization and, therefore, it is globally convergent.
Abstract: A novel algorithm is developed for the problem of finding a low-rank correlation matrix nearest to a given correlation matrix. The algorithm is based on majorization and, therefore, it is globally convergent. The algorithm is computationally efficient, is straightforward to implement, and can handle arbitrary weights on the entries of the correlation matrix. A simulation study suggests that majorization compares favourably with competing approaches in terms of the quality of the solution within a fixed computational time. The problem of rank reduction of correlation matrices occurs when pricing a derivative dependent on a large number of assets, where the asset prices are modelled as correlated log-normal processes. Such an application mainly concerns interest rates.
TL;DR: Two different approximated dynamics for basket-options pricing are presented, based on a moment-matching procedure and on the space of probability densities, namely the Kullback–Leibler information (KLI) and the Hellinger distance.
Abstract: The aim of this paper is to present two different approximated dynamics for basket-options pricing. The approximations are based on a moment-matching procedure. Single names in the basket are modelled according to geometric Brownian motions (GBMs), as in the classical Black and Scholes framework, driven by instantaneously correlated Brownian motions. The basket distribution resulting from such GBMs for the single assets is unknown. By resorting to Monte Carlo simulation, we compare this distribution to the known approximated distribution coming from the moment-matching dynamics. The comparison is carried out through distances on the space of probability densities, namely the Kullback–Leibler information (KLI) and the Hellinger distance (HD). We are interested in measuring the KLI and the HD between the ‘real’ simulated basket distribution at terminal time and the distributions used for the approximation, both in the log-normal and shifted log-normal families. We isolate influences of instantaneou...
TL;DR: In this paper, it was shown that the inherently path dependent problem of pricing Asian options can be transformed into a problem without path dependency in the payo function, and that the price satisfies a simpler integro-dierenti al equation in the case the stock price is driven by a process with independent increments.
Abstract: In this article we study arithmetic Asian options when the underlying stock is driven by special semimartingale processes. We show that the inherently path dependent problem of pricing Asian options can be transformed into a problem without path dependency in the payo function. We also show that the price satisfies a simpler integro-dierenti al equation in the case the stock price is driven by a process with independent increments, Levy process being a special case.
TL;DR: In this paper, the authors define the class of local Levy processes, which are Levy processes time changed by an inhomogeneous local speed function, and show how to reverse engineer the local speed functions from traded option prices of all strikes and maturities.
Abstract: We define the class of local Levy processes. These are Levy processes time changed by an inhomogeneous local speed function. The local speed function is a deterministic function of time and the level of the process itself. We show how to reverse engineer the local speed function from traded option prices of all strikes and maturities. The local Levy processes generalize the class of local volatility models. Closed forms for local speed functions for a variety of cases are also presented. Numerical methods for recovery are also described.
TL;DR: In this article, the authors apply continuous-time random walks (CTRWs) as phenomenological models of the high-frequency price dynamics and show that the waiting-time survival probability for highfrequency data is non-exponential.
Abstract: In high-frequency financial data not only returns, but also waiting times between consecutive trades are random variables. Therefore, it is possible to apply continuous-time random walks (CTRWs) as phenomenological models of the high-frequency price dynamics. An empirical analysis performed on the 30 DJIA stocks shows that the waiting-time survival probability for high-frequency data is non-exponential. This fact imposes constraints on agent-based models of financial markets.
TL;DR: In this paper, the authors extend the non-Gaussian option pricing model of L. Borland (Quantitative Finance, 2, 415-431, 2002) to include volatility-stock correlations consistent with the leverage effect.
Abstract: Closed form option pricing formulae explaining skew and smile are obtained within a parsimonious non-Gaussian framework. We extend the non-Gaussian option pricing model of L. Borland (Quantitative Finance, 2, 415-431, 2002) to include volatility-stock correlations consistent with the leverage effect. A generalized Black-Scholes partial differential equation for this model is obtained, together with closedform approximate solutions for the fair price of a European call option. In certain limits, the standard Black-Scholes model is recovered, as is the Constant Elasticity of Variance (CEV) model of Cox and Ross. Alternative methods of solution to that model are thereby also discussed. The model parameters are partially fit from empirical observations of the distribution of the underlying. The option pricing model then predicts European call prices which fit well to empirical market data over several maturities.
TL;DR: In this article, a new family of processes that include the long memory (LM) (power law) in the volatility correlation is introduced by measuring the historical volatilities on a set of increasing time horizons and by computing the resulting effective volatility by a sum with power law weights.
Abstract: We introduce a new family of processes that include the long memory (LM) (power law) in the volatility correlation. This is achieved by measuring the historical volatilities on a set of increasing time horizons and by computing the resulting effective volatility by a sum with power law weights. The processes have two parameters (linear processes) or four parameters (affine processes). In the limit where only one component is included, the processes are equivalent to GARCH(1, 1) and I-GARCH(1). Volatility forecast is discussed in the context of processes with quadratic equations, in particular as a means to estimate process parameters. Using hourly data, the empirical properties of the new processes are compared to existing processes (GARCH, I-GARCH, FIGARCH,…), in particular log-likelihood estimates and volatility forecast errors. This study covers time horizons ranging from 1 h to 1 month. We also study the variation of the estimated parameters with respect to changing sample by introducing a na...
TL;DR: In this paper, the authors used importance sampling to increase the rate of convergence of Monte Carlo simulations for pricing nth to default credit swaps in the Li model and combined with the likelihood ratio and pathwise methods for computing the sensitivities of these products to changes in the hazard rates of the underlying obligors.
Abstract: New techniques are introduced for pricing nth to default credit swaps in the Li model. We demonstrate the use of importance sampling to greatly increase the rate of convergence of Monte Carlo simulations for pricing. This technique is combined with the likelihood ratio and pathwise methods for computing the sensitivities of these products to changes in the hazard rates of the underlying obligors. In particular the extension of the pathwise method has wider significance in that it is shown that the method can be used even when the pay-off is discontinuous.
TL;DR: In this paper, the authors discuss the problems of defining suitable performance objectives and tracking error that scale properly over the entire management period and implement an optimal investment strategy when full replication of an index is not deemed suitable.
Abstract: With the increased acceptance of capital market efficiency, there has been a significant increase in the money managed on an indexed basis. Several methodologies are available to replicate the target index. In this paper, we discuss the problems of (1) defining suitable performance objectives and tracking error that scale properly over the entire management period and (2) implementing an optimal investment strategy when full replication of an index is not deemed suitable. We then argue that clustering might be a viable methodology for building parsimonious tracking portfolios. With suitably defined distances between the time series of asset prices, clustering ‘discovers’ the correlation and cointegration structure of an index. Sampling the clusters with appropriate heuristics and optimization techniques, an optimal tracking portfolio can be constructed. One advantage of this approach is that it eschews the difficulties and computational burden of density forecasts and full optimization.
TL;DR: In this paper, the authors compare the profit and loss arising from the delta-neutral dynamic hedging of options, using two possible values for the delta of the option: the Black-Scholes implied delta and the local delta.
Abstract: In this article we compare the profit and loss arising from the delta-neutral dynamic hedging of options, using two possible values for the delta of the option. The first is the Black–Scholes implied delta, while the second is the local delta, namely the delta of the option in a generalized Black–Scholes model with a local volatility, recalibrated to the market smile every day. We explain why, in negatively skewed markets, the local delta should provide a better hedge than the implied delta during slow rallies or fast sell-offs, and a worse hedge, although to a smaller extent, during fast rallies or slow sell-offs. Since slow rallies and fast sell-offs are more likely to occur than fast rallies or slow sell-offs in negatively skewed markets (provided we have physical as well as implied negative skewness), we conclude that, on average, the local delta provides a better hedge than the implied delta in negatively skewed markets. We obtain the same conclusion in the case of positively skewed markets. We illus...
TL;DR: In this paper, variance reduction methods for Monte Carlo simulations to evaluate European and Asian options in the context of multiscale stochastic volatility models were presented, where geometric average Asian options (GAOs) were used as control variates.
Abstract: We present variance reduction methods for Monte Carlo simulations to evaluate European and Asian options in the context of multiscale stochastic volatility models European option price approximations, obtained from singular and regular perturbation analysis [Fouque J P, Papanicolaou G, Sircar R and Solna K 2003 Multiscale stochastic volatility asymptotics, SIAM J Multiscale Modeling and Simulation 2], are used in importance sampling techniques, and their efficiencies are compared Then we investigate the problem of pricing arithmetic average Asian options (AAOs) by Monte Carlo simulations A two-step strategy is proposed to reduce the variance where geometric average Asian options (GAOs) are used as control variates Due to the lack of analytical formulas for GAOs under stochastic volatility models, it is then necessary to consider efficient Monte Carlo methods to estimate the unbiased means of GAOs The second step consists in deriving formulas for approximate prices based on perturbation techniques, a
TL;DR: For many years, researchers and practitioners alike have tried to build trading models, but history has not been kind to their efforts as discussed by the authors, and history has also proven that such models are notoriously difficult to predict.
Abstract: Foreign exchange markets are notoriously difficult to predict. For many years academics and practitioners alike have tried to build trading models, but history has not been kind to their efforts. C...
TL;DR: In this article, an appropriate trade-investment model, depending on three adjustable parameters associated with the total wealth of a society, a social differentiation among agents, and economic volatility referred to as investment, can successfully reproduce the distribution of empirical wealth data in the low, medium and high ranges.
Abstract: The distribution of wealth among the members of a society is herein assumed to result from two fundamental mechanisms: trade and investment. An empirical distribution of wealth shows an abrupt change between the low–medium range, that may be fitted by a non-monotonic function with an exponential-like tail such as a gamma distribution, and the high wealth range, that is well fitted by a Pareto or inverse power-law function. We demonstrate that an appropriate trade-investment model, depending on three adjustable parameters associated with the total wealth of a society, a social differentiation among agents, and economic volatility referred to as investment, can successfully reproduce the distribution of empirical wealth data in the low, medium and high ranges. Finally, we provide an economic interpretation of the mechanisms in the model and, in particular, we discuss the difference between classical and neoclassical theories regarding the concepts of value and price. We consider the importance that out-of-e...
TL;DR: In this article, the authors extend their theory by proving the existence and uniqueness of this reliability and provide a formula to estimate both the cost of capital and its reliability for individual firms.
Abstract: Gordon and Shapiro (1956 Management Sci. 10 102–10) first equated the price of a share with the present value of future dividends and derived the well known relationship. Since then, there have been many improvements on the theory. For example, Thompson (1985 Managerial Decis. Economics 6 132–40, 1987 Managerial Decis. Economics 8 321–32) combined the ‘dividend yield plus growth’ method with Box–Jenkins time series analysis of past dividend experience to estimate the cost of capital and its ‘reliability’ for individual firms. Thompson and Wong (1991 Managerial Decis. Economics 12 27–42, 1996 Eng. Economist 41 123–47) proved the existence and uniqueness of the cost of capital and provided a formula to estimate both the cost of capital and its reliability. However, their approaches cannot be used if the ‘reliability’ does not exist or if there are multiple solutions for the ‘reliability’. In this paper, we extend their theory by proving the existence and uniqueness of this reliability. In addition, we propo...
TL;DR: In this paper, a new technique for pricing a class of exotic options that are characterized by two expiry dates is developed. But the method, based on the partial differential equation approach to option pricing, however requires no formal solution of such equations and does not depend on any particular underlying asset price dynamics.
Abstract: This paper develops a new technique for pricing a class of exotic options that are characterized by two expiry dates. Examples of such exotics include compound options, chooser options, extendable options, shout options, partial barrier options and others. The method, based on the partial differential equation approach to option pricing, however requires no formal solution of such equations. Instead, the method exploits the observation that dual expiry options have payoffs that can be perfectly replicated by a particular set of first and second order binary options. Hence, in order to avoid arbitrage, the exotic option prices are obtained by static replication with respect to this family of binaries. The representation of prices in terms of these binaries is also quite general and does not depend on any particular underlying asset price dynamics. Closed form expressions agreeing with published results are given for the case of log-normal asset price dynamics and standard Black–Scholes assumptions.
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TL;DR: In this paper, a Monte Carlo simulation method based on importance sampling is applied to the problem of determining individual risk contributions of the obligors in a credit portfolio, where the risk measure adopted is expected shortfall, a particualr coherent risk measure.
Abstract: A Monte Carlo simulation method based on importance sampling is applied to the problem of determining individual risk contributions of the obligors in a credit portfolio. The effectiveness of the method is benchmarked against standard Monte Carlo techniques and the asymptotic optimality of the method is proved. The risk measure adopted is expected shortfall, a particualr coherent risk measure. The concept of a coherent risk spectrum is discussed on the basis of some numerical examples.
TL;DR: In this paper, a lattice method for the valuation of Bermudan options is proposed, which can be used when the returns process is Levy, based upon an approximation to the transition density function of the Levy process.
Abstract: Evidence from the financial markets suggests that empirical returns distributions, both historical and implied, do not arise from diffusion processes. A growing literature models the returns process as a Levy process, finding a number of explicit formulae for the values of some derivatives in special cases. Practical use of these models has been hindered by a relative paucity of numerical methods which can be used when explicit solutions are not present. In particular, the valuation of Bermudan options is problematical. This paper investigates a lattice method that can be used when the returns process is Levy, based upon an approximation to the transition density function of the Levy process. We find alternative derivations of the lattice, stemming from alternative representations of the Levy process, which may be useful if the transition density function is unknown or intractable. We apply the lattice to models based on the variance-gamma and normal inverse Gaussian processes. We find that the l...
TL;DR: In this paper, the effect of stochastic volatility on geometric Asian options (GAO) in a mean-reverting volatility economy was studied. But the authors focused on the effects of volatility on averaging type options.
Abstract: This paper studies continuously sampled geometric Asian options (GAO) in a stochastic volatility economy. The underlying asset price is assumed to follow a geometric Brownian motion with stochastic volatility driven by a mean-reverting process. Semi-analytical pricing formulae for GAO are derived in a fast mean-reverting stochastic volatility economy by the means of a perturbation method. The effects of stochastic volatility on averaging type options are examined. A unified regression approach is proposed to capture smiles of some geometric Asian options and European options in one shot.
TL;DR: In this article, the early exercise boundary of the American put is separated from the strike by a non-vanishing margin, and as the riskless rate vanishes, the boundary tends to 0 uniformly over the interval [0, T].
Abstract: Pricing and hedging of European, American, barrier options and interest rate derivatives for wide classes of Levy driven models is considered in situations where qualitative and quantitative differences between Gaussian and Levy modelling are most prominent, and the dependence on the choice of a family of Levy processes is analysed. Asymptotics of option prices near the barrier and expiry are calculated; for American options, two fast numerical methods are constructed. It is shown that for many classes of Levy processes, the early exercise boundary of the American put is separated from the strike by a non-vanishing margin, and as the riskless rate vanishes, the early exercise boundary tends to 0 uniformly over the interval [0, T). Implications for fitting of parameters are discussed.
TL;DR: The authors compare the sensitivities of option prices to shifts in skewness and kurtosis using parameter values from Corrado and Su (1996) and Brown and Robinson (2002), and market data from the French options market.
Abstract: Several authors have proposed series expansion methods to price options when the risk-neutral density is asymmetric and leptokurtic. Among these, Corrado and Su (1996) provide an intuitive pricing formula based on a Gram–Charlier Type A series expansion. However, their formula contains a typographic error that can be significant. Brown and Robinson (2002) correct their pricing formula and provide an example of economic significance under plausible market conditions. The purpose of this comment is to slightly modify their pricing formula to provide consistency with a martingale restriction. We also compare the sensitivities of option prices to shifts in skewness and kurtosis using parameter values from Corrado and Su (1996) and Brown and Robinson (2002), and market data from the French options market. We show that differences between the original, corrected and our modified versions of the Corrado and Su (1996) original model are minor on the whole sample, but could be economically significant in specific ...