Showing papers in "Quantitative Finance in 2002"
TL;DR: In this paper, the authors use time series of option prices on the SP500 and FTSE indices to study the deformation of the implied volatility surface and show that it may be represented as a randomly fluctuating surface driven by a small number of orthogonal random factors.
Abstract: The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. However, the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modelling approaches, giving rise to `Vega' risk in option portfolios. Using time series of option prices on the SP500 and FTSE indices, we study the deformation of this surface and show that it may be represented as a randomly fluctuating surface driven by a small number of orthogonal random factors. We identify and interpret the shape of each of these factors, study their dynamics and their correlation with the underlying index. Our approach is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data. A simple factor model compatible with the empirical observations is proposed. We illustrate how this ...
439 citations
TL;DR: In this article, an analytic formula for the time-dependent probability distribution of stock price changes (returns) was proposed, which is in excellent agreement with the Dow Jones index for time lags from 1 to 250 trading days.
Abstract: study the Heston model, where the stock price dynamics is governed by a geometrical (multiplicative) Brownian motion with stochastic variance. We solve the corresponding Fokker-Planck equation exactly and, after integrating out the variance, find an analytic formula for the time-dependent probability distribution of stock price changes (returns). The formula is in excellent agreement with the Dow-Jones index for time lags from 1 to 250 trading days. For large returns, the distribution is exponential in log-returns with a time-dependent exponent, whereas for small returns it is Gaussian. For time lags longer than the relaxation time of variance, the probability distribution can be expressed in a scaling form using a Bessel function. The Dow-Jones data for 1982-2001 follow the scaling function for seven orders of magnitude.
313 citations
TL;DR: In this article, the authors investigate several statistical properties of the order book of three liquid stocks of the Paris Bourse and find that the statistics of incoming limit order prices follow a power-law around the current price with a diverging mean.
Abstract: We investigate several statistical properties of the order book of three liquid stocks of the Paris Bourse. The results are to a large degree independent of the stock studied. The most interesting features concern (i) the statistics of incoming limit order prices, which follows a power-law around the current price with a diverging mean; and (ii) the shape of the average order book, which can be quantitatively reproduced using a ‘zero intelligence’ numerical model, and qualitatively predicted using a simple approximation. Financial markets offer an amazing source of detailed data on the collective behaviour of interacting agents. It is possible to find many reproducible patterns and even to perform experiments, which bring this atypical subject into the realm of experimental science. The situation is simple and well defined, since many agents, with all the same goal, trade the very same asset. As such, the statistical analysis of financial markets also offers an interesting testing ground not only for economic theories, but also for more ambitious theories of human activities. One may indeed wonder to what extent it is necessary to invoke human intelligence or rationality to explain the various universal statistical laws which have been recently unveiled by the systematic analysis of very large data sets. Many statistical properties of financial markets have already been explored, and have revealed striking similarities between very different markets (different
310 citations
TL;DR: In this article, an order-driven market model with heterogeneous agents trading via a central order matching mechanism is introduced, where traders set bids and asks and post market or limit orders according to exogenously fixed rules.
Abstract: We introduce an order-driven market model with heterogeneous agents trading via a central order matching mechanism. Traders set bids and asks and post market or limit orders according to exogenously fixed rules. We investigate how different trading strategies may affect the dynamics of price, bid-ask spreads, trading volume and volatility. We also analyse how some features of market design, such as tick size and order lifetime, affect market liquidity. The model is able to reproduce many of the complex phenomena observed in real stock markets. *Paper presented at Applications of Physics in Financial Analysis (APFA) 3, 5–7 December 2001, Museum of London, UK.
273 citations
TL;DR: In this paper, the authors derive a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation, and establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in-and out-of-the-money options.
Abstract: We derive a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation. Using this equation we establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in- and out-of-the-money options. This in turn leads us to propose a new formulation near expiry of the calibration problem for the local volatility model, which we show to be well posed.
217 citations
TL;DR: In this paper, the dynamics of temperature can be modelled by means of a stochastic process known as fractional Brownian motion, which is used to price two types of contingent claims: one based on heating and cooling degree days, and another based on cumulative temperature.
Abstract: The dynamics of temperature can be modelled by means of a stochastic process known as fractional Brownian motion. Based on this empirical observation, we characterize temperature dynamics by a fractional Ornstein–Uhlenbeck process. This model is used to price two types of contingent claims: one based on heating and cooling degree days, and one based on cumulative temperature. We derive analytic expressions for the expected discounted payoffs of such derivatives, and discuss the dependence of the results on the fractionality of the temperature dynamics.
213 citations
TL;DR: In this article, the authors demonstrate a striking regularity in the way people place limit orders in financial markets, using a data set consisting of roughly two million orders from the London Stock Exchange and demonstrate that the unconditional cumulative distribution of relative limit prices decays roughly as a power law with exponent approximately 1.5.
Abstract: In this paper we demonstrate a striking regularity in the way people place limit orders in financial markets, using a data set consisting of roughly two million orders from the London Stock Exchange. We define the relative limit price as the difference between the limit price and the best price available. Merging the data from 50 stocks, we demonstrate that for both buy and sell orders, the unconditional cumulative distribution of relative limit prices decays roughly as a power law with exponent approximately –1.5. This behaviour spans more than two decades, ranging from a few ticks to about 2000 ticks. Time series of relative limit prices show interesting temporal structure, characterized by an autocorrelation function that asymptotically decays as C(τ)∼τ−0.4. Furthermore, relative limit price levels are positively correlated with and are led by price volatility. This feedback may potentially contribute to clustered volatility.
148 citations
TL;DR: In this paper, a generalized form of the BlackScholes differential equation is found, and a martingale measure which leads to closed-form solutions for European call options is derived from a non-Gaussian model of stock returns.
Abstract: Option pricing formulae are derived from a non‐Gaussian model of stock returns. Fluctuations are assumed to evolve according to a nonlinear Fokker‐Planck equation which maximizes the Tsallis nonextensive entropy of index q. A generalized form of the Black‐Scholes differential equation is found, and we derive a martingale measure which leads to closed‐form solutions for European call options. The standard Black‐Scholes pricing equations are recovered as a special case (q = 1). The distribution of stock returns is well modelled with q circa 1.5. Using that value of q in the option pricing model we reproduce the volatility smile. The partial derivatives (or Greeks) of the model are also calculated. Empirical results are demonstrated for options on Japanese Yen futures. Using just one value of σ across strikes we closely reproduce market prices, for expiration times ranging from weeks to several months.
146 citations
TL;DR: In this article, the authors advocate the use of semi-parametric models for distributions, where the mean vector μ and covariance Σ are parametric components and the so-called density generator (function) g is the nonparametric component.
Abstract: The benchmark theory of mathematical finance is the Black-Scholes-Merton theory, based on Brownian motion as the driving noise process for asset prices. Here the distributions of returns of the assets in a portfolio are multivariate normal. The two most obvious limitations here concern symmetry and thin tails, neither being consistent with real data. The most common replacements for the multinormal are parametric—stable, generalized hyperbolic, variance gamma. In this paper we advocate the use of semi-parametric models for distributions, where the mean vector μ and covariance Σ are parametric components and the so-called density generator (function) g is the non-parametric component. We work mainly within the family of elliptically contoured distributions, focusing particularly on normal variance mixtures with self-decomposable mixing distributions. We show how the parametric cases can be treated in a unified, systematic way within the non-parametric framework and obtain the density generators fo...
141 citations
TL;DR: This article developed a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime.
Abstract: This paper develops a family of option pricing models when the underlying stock price dynamic is modelled by a regime switching process in which prices remain in one volatility regime for a random amount of time before switching over into a new regime. Our family includes the regime switching models of Hamilton (Hamilton J 1989 Econometrica 57 357–84), in which volatility influences returns. In addition, our models allow for feedback effects from returns to volatilities. Our family also includes GARCH option models as a special limiting case. Our models are more general than GARCH models in that our variance updating schemes do not only depend on levels of volatility and asset innovations, but also allow for a second factor that is orthogonal to asset innovations. The underlying processes in our family capture the asymmetric response of volatility to good and bad news and thus permit negative (or positive) correlation between returns and volatility. We provide the theory for pricing options under...
119 citations
TL;DR: In this article, two performance measures advocated for asymmetric return distributions, the Sortino ratio and the power utility, were examined and compared with the maximum principle in the context of asymmetric distributions.
Abstract: We examine two performance measures advocated for asymmetric return distributions: the Sortino ratio—originally introduced by Sortino and Price (Sortino F and Price L 1994 J. Investing 59–65)—and a measure based on power utility introduced in Leland (Leland H 1999 Financial Analysts J. 27–36). In particular, we investigate the role of the maximum principle in this context, and assess the conditions under which the measures satisfy it. Our results add further motivation for the use of a modified Sortino ratio, by placing it on a sound theoretical foundation. In this light, we discuss its relative merits compared with alternative approaches.
TL;DR: The authors reviewed and put in context some of the recent work on stochastic volatility (SV) modelling for financial economics, focusing on the relationship between subordination and SV, and the OOU based volatility models, exact option pricing, realized power variation and realized variance.
Abstract: This paper reviews and puts in context some of our recent work on stochastic volatility (SV) modelling for financial economics. Here our main focus is on: (i) the relationship between subordination and SV, (ii) OU based volatility models, (iii) exact option pricing, (iv) realized power variation and realized variance, (v) building multivariate models.
TL;DR: The authors predict an overall increasing market until the end of the year 2002 or until the first quarter of 2003; they predict a severe following descent (with maybe one or two severe ups and downs in the middle) which stops during the first semester of 2004.
Abstract: A remarkable similarity in the behaviour of the US SP we predict an overall increasing market until the end of the year 2002 or until the first quarter of 2003; we predict a severe following descent (with maybe one or two severe ups and downs in the middle) which stops during the first semester of 2004. Beyond this, we cannot ...
TL;DR: In this article, the authors identify economic sectors as clusters of assets with a similar economic dynamics and characterize market efficiency by analysing the market's predictability and find that the market is indeed close to being efficient.
Abstract: By analysing a large data set of daily returns with the maximum likelihood data clustering technique, we identify economic sectors as clusters of assets with a similar economic dynamics. The sector size distribution follows Zipf's law. Secondly, we find that patterns of daily market-wide economic activity cluster into classes that can be identified with market states. The distribution of frequencies of market states shows scale-free properties and the memory of the market state process extends to long times (∼50 days). Assets in the same sector behave similarly across states. We characterize market efficiency by analysing the market's predictability and find that the market is indeed close to being efficient. We find evidence of the existence of a dynamic pattern after the market's crashes.
TL;DR: In this article, the authors introduce a simple model to investigate the interplay of contrarian and imitative behaviour in a stock market where agents can take only two states, bullish or bearish, and show that in the limit where the number of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behaviour in the parameter space.
Abstract: Imitative and contrarian behaviours are the two typical opposite attitudes of investors in stock markets. We introduce a simple model to investigate their interplay in a stock market where agents can take only two states, bullish or bearish. Each bullish (bearish) agent polls m ‘friends’ and changes her opinion to bearish (bullish) if (i) at least mρ hb (mρ bh ) among the m agents inspected are bearish (bullish) or (ii) at least mρ hh >mρ hb (mρ bb >mρ bh ) among the m agents inspected are bullish (bearish). The condition (i) ((ii)) corresponds to imitative (antagonistic) behaviour. In the limit where the number N of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behaviour in a significant domain of the parameter space {ρ hb ,ρ bh ,ρ hh ,ρ bb ,m}. A typical chaotic trajectory is characterized by intermittent phases of chaos, quasi-periodic behaviour and super-exponentially growing bubbles followed by crashes. A typical bubble starts initia...
TL;DR: In this paper, the authors generalize the construction of the multifractal random walk (MRW) due to Bacry et al. to take into account the asymmetric character of financial returns.
Abstract: We generalize the construction of the multifractal random walk (MRW) due to Bacry et al (Bacry E, Delour J and Muzy J-F 2001 Modelling financial time series using multifractal random walks Physica A 299 84) to take into account the asymmetric character of financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. We compute the leading behaviour of q-moments of the process, which behave as power laws of the time lag with an exponent ζ q =p−2p(p−1)λ2 for even q=2p, as in the symmetric MRW, and as ζ q =p + 1−2p 2λ2−α (q=2p + 1), where λ and α are parameters. We show that this extended model reproduces the ‘HARCH’ effect or ‘causal cascade’ reported by some authors. We illustrate the usefulness of this ‘skewed’ MRW by computing the resulting shape of the volatility smiles generated by such a process, which we compare with approximate cumulant...
TL;DR: In this article, a two-factor Vasicek model is proposed to capture the behaviour of monetary authorities who normally set a reference rate which changes from time to time. But the model is not suitable for the case of fixed rates, since the mean reversion level changes according to a continuous time finite state Markov chain.
Abstract: A two‐factor Vasicek model, where the mean reversion level changes according to a continuous time finite state Markov chain, is considered. This model could capture the behaviour of monetary authorities who normally set a reference rate which changes from time to time. We derive the term structure via the analytic expression of the bond price that involves a fundamental matrix. The validity of the bond price closed form solution is verified via the forward rate dynamics.
TL;DR: A new term structure calibration methodology based on maximization of entropy is introduced, and some new families of interest rate models arising naturally in this context are presented.
Abstract: with every positive interest term structure there is a probability density function over the positive half-line. This fact can be used to turn the problem of term structure analysis into a problem in the comparison of probability distributions, an area well developed in statistics, known as information geometry. The information-theoretic and geometric aspects of term structures thus arising are here illustrated. In particular, we introduce a new term structure calibration methodology based on maximization of entropy, and also present some new families of interest rate models arising naturally in this context.
TL;DR: In this article, the authors characterize two deterministic implied volatility models, defined by assuming that either the per-delta or per-strike implied volatility surface has a deterministic evolution.
Abstract: In this paper, we characterize two deterministic implied volatility models, defined by assuming that either the per-delta or the per-strike implied volatility surface has a deterministic evolution. Practitioners have recently proposed these two models to describe two regimes of implied volatility (see Derman (1999 Risk 4 55–9)). In an arbitrage-free sticky-delta model, we show that the underlying asset price is the exponential of a process with independent increments under the unique risk neutral measure and that any square-integrable claim can be replicated up to a vanishing risk by trading portfolios of vanilla options. This latter result is similar in nature to the quasi-completeness result obtained by Bjork et al (1997 Finance Stochastics 1 141–74) for interest rate models driven by Levy processes. Finally, we show that the only arbitrage-free sticky-strike model is the standard Black-Scholes model.
TL;DR: In this article, the authors explore the consequences of two simple hypotheses about risk: common sense invariance principle and the perception of time, which leads directly to the well known Sharpe ratio and the classic risk-return relationship of arbitrage pricing theory and the capital asset pricing model.
Abstract: What return should you expect when you take on a given amount of risk? How should that return depend upon other people's behaviour? What principles can you use to answer these questions? In this paper, I approach these topics by exploring the consequences of two simple hypotheses about risk. The first is a common-sense invariance principle: assets with the same perceived risk must have the same expected return. It leads directly to the well known Sharpe ratio and the classic risk-return relationships of arbitrage pricing theory and the capital asset pricing model. The second hypothesis concerns the perception of time. I conjecture that in times of speculative excitement, short-term investors may instinctively imagine stock prices to be evolving in a time measure different from that of calendar time. They may perceive and experience the risk and return of a stock in intrinsic time, a dimensionless time scale that counts the number of trading opportunities that occur, but pays no attention to the c...
TL;DR: In this paper, the optimal structure of derivatives written on an illiquid asset, such as a catastrophic or a weather event, is determined based on a utility maximization point of view.
Abstract: The aim of this paper is to determine the optimal structure of derivatives written on an illiquid asset, such as a catastrophic or a weather event. This transaction involves two agents: a bank which wants to hedge its initial exposure towards this illiquid asset and an investor which may buy the contract. Both agents also have the opportunity to invest their residual wealth on a financial market. Based on a utility maximization point of view, we determine an optimal profile (and its value) such that it maximizes the bank's utility given that the investor decides to make the deal only if it increases its utility. In the case of exponential utility, we show that the pricing rule is a non-linear function of the structure and that the bank always transfers the same proportion of its initial exposure. In the general case, an additional term appears, depending only on the relative log-likelihood of the two agents' views of the distribution of the illiquid asset. * Presented at the special session on ‘M...
TL;DR: In this paper, the authors derived exact analytical pricing formulae in terms of the factors in the Wiener-Hopf factorization (WFH) factorization for a wide class of processes and payoffs.
Abstract: We consider perpetual Bermudan options and more general perpetual American options in discrete time. For wide classes of processes and pay‐offs, we obtain exact analytical pricing formulae in terms of the factors in the Wiener‐Hopf factorization formulae. Under additional conditions on the process, we derive simpler approximate formulae.
TL;DR: In this article, a new and powerful variance reduction technique is presented, based directly on the Ito calculus and is used to find unbiased variance-reduced estimators for the expectation of functionals of Ito diffusion processes.
Abstract: Standard Monte Carlo methods can often be significantly improved with the addition of appropriate variance reduction techniques. In this paper a new and powerful variance reduction technique is presented. The method is based directly on the Ito calculus and is used to find unbiased variance-reduced estimators for the expectation of functionals of Ito diffusion processes. The approach considered has wide applicability: for instance, it can be used as a means of approximating solutions of parabolic partial differential equations or applied to valuation problems that arise in mathematical finance. We illustrate how the method can be applied by considering the pricing of European-style derivative securities for a class of stochastic volatility models, including the Heston model.
TL;DR: In this paper, the authors study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes, and they conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in perturbation of volatility.
Abstract: We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in perturbation of volatility. We obtain an integral equation for this functional and we show that under some natural conditions it can be inverted for volatility. We demonstrate the stability of the linearized problem, and we propose a numerical algorithm which is accurate for volatility functions with different properties.
TL;DR: In this article, the authors examined the asymmetry of several individual stock returns at different investment horizons: daily, weekly, and monthly, and found that while some asymmetries are observed in daily returns, they disappear almost completely in weekly and monthly returns.
Abstract: This paper examines the (a)symmetry of several individual stock returns at different investment horizons: daily, weekly and monthly. While some asymmetries are observed in daily returns, they disappear almost completely in weekly and monthly returns. The explanation for this fact lies in the convergence to normality that takes place when the investment horizon increases. These features allow one to question several financial models; in particular, they question the preference for positive skewness as a factor for investments in stock markets.
TL;DR: In this article, a new method for finding upper bounds for Bermudan swaptions in a swap-rate market model was developed, and the bounds were well within bid-offer spread.
Abstract: We develop a new method for finding upper bounds for Bermudan swaptions in a swap-rate market model. By comparing with lower bounds found by exercise boundary parametrization, we find that the bounds are well within bid-offer spread. As an application, we study the dependence of Bermudan swaption prices on the number of instantaneous factors used in the model. We also establish an equivalence with LIBOR market models and show that virtually identical lower bounds for Bermudan swaptions are obtained.
TL;DR: In this paper, the authors show that the diffusion equation which gives the value of a European option on S can be represented, upon expanding in Laguerre polynomials, by a tridiagonal infinite matrix.
Abstract: We consider a risky asset following a mean-reverting stochastic process of the form We show that the (singular) diffusion equation which gives the value of a European option on S can be represented, upon expanding in Laguerre polynomials, by a tridiagonal infinite matrix. We analyse this matrix to show that the diffusion equation does indeed have a solution and truncate the matrix to give a simple, highly efficient method for the numerical calculation of the solution.
TL;DR: The authors proposed a variance reduction method for Monte Carlo computation of option prices in the context of stochastic volatility, which is based on importance sampling using an approximation of the option price obtained by a fast mean-reversion expansion introduced in Fouque et al (2000 Derivatives in Financial Markets with Stochastic Volatility).
Abstract: We propose a variance reduction method for Monte Carlo computation of option prices in the context of stochastic volatility. This method is based on importance sampling using an approximation of the option price obtained by a fast mean-reversion expansion introduced in Fouque et al (2000 Derivatives in Financial Markets with Stochastic Volatility (Cambridge: Cambridge University Press)). We compare this with the small noise expansion method proposed in Fournie et al (1997 Asymptotic Anal. 14 361–76) and demonstrate numerically the efficiency of our method, in particular in the presence of a skew.
TL;DR: In this article, a modification of the well known constant elasticity of variance model where it is used to model the growth optimal portfolio (GOP) is considered. But, for this application, there is no equivalent risk neutral pricing measure and therefore the classical risk neutral methodology fails.
Abstract: This paper considers a modification of the well known constant elasticity of variance model where it is used to model the growth optimal portfolio (GOP). It is shown that, for this application, there is no equivalent risk neutral pricing measure and therefore the classical risk neutral pricing methodology fails. However, a consistent pricing and hedging framework can be established by application of the benchmark approach. Perfect hedging strategies can be constructed for European style contingent claims, where the underlying risky asset is the GOP. In this framework, fair prices for contingent claims are the minimal prices that permit perfect replication of the claims. Numerical examples show that these prices may differ significantly from the corresponding ‘risk neutral’ prices.
TL;DR: In this article, the authors use Malliavin calculus to smoothen the payoff to estimate and apply this method to the Heston model and show the efficiency of this method and discuss when it is appropriate or not to use it.
Abstract: Current Monte Carlo pricing engines may face a computational challenge for the Greeks, not only because of their time consumption but also their poor convergence when using a finite difference estimate with a brute force perturbation. The same story may apply to conditional expectation. In this short paper, following Fournie et al (Fournie E, Lasry J M, Lebuchoux J, Lions P L and Touzi N 1999 Finance Stochastics 3 391-412), we explain how to tackle this issue using Malliavin calculus to smoothen the payoff to estimate. We discuss the relationship with the likelihood ratio method of Broadie and Glasserman (Broadie M and Glasserman P 1996 Manag. Sci. 42 269-85). We show by numerical results the efficiency of this method and discuss when it is appropriate or not to use it. We see how to apply this method to the Heston model. 1The views herein are the authors' own and do not necessarily reflect those of Goldman Sachs.