An optimal investment model with Markov-driven volatilities

We consider a multi-stock market model. The processes of stock prices are governed by stochastic differential equations with stock return rates and volatilities driven by a finite-state Markov process. Each volatility is also disturbed by a Brownian motion; more exactly, it follows a Markov-driven Ornstein–Uhlenbeck process. Investors can observe the stock prices only. Both the underlying Brownian motion and the Markov process are unobservable. We study a discretized version, which is a discrete-time hidden Markov process. The objective is to control trading at each time step to maximize an expected utility function of terminal wealth. Exploiting dynamic programming techniques, we derive an approximate optimal trading strategy that results in an expected utility function close to the optimal value function. Necessary filtering and forecasting techniques are developed to compute the near-optimal trading strategy.     

Pure jump Lévy processes for asset price modelling

The goal of the paper is to show that some types of Lévy processes such as the hyperbolic motion and the CGMY are particularly suitable for asset price modelling and option pricing. We wish to review some fundamental mathematic properties of Lévy distributions, such as the one of infinite divisibility, and how they translate observed features of asset price returns. We explain how these processes are related to Brownian motion, the central process in finance, through stochastic time changes which can in turn be interpreted as a measure of the economic activity. Lastly, we focus on two particular classes of pure jump Lévy processes, the generalized hyperbolic model and the CGMY models, and report on the goodness of fit obtained both on stock prices and option prices.     

The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S & P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.     

The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gammaprocess, that generalizes Brownian motion is developed as amodel for the dynamics of log stock prices. Theprocess is obtainedby evaluating Brownian motion with drift at a random time givenby a gamma process. The two additional parameters are the driftof the Brownian motion and the volatility of the time change.These additional parameters provide control over the skewnessand kurtosis of the return distribution. Closed forms are obtainedfor the return density and the prices of European options.Thestatistical and risk neutral densities are estimated for dataon the S&P500 Index and the prices of options on this Index.It is observed that the statistical density is symmetric withsome kurtosis, while the risk neutral density is negativelyskewed with a larger kurtosis. The additional parameters alsocorrect for pricing biases of the Black Scholes model that isa parametric special case of the option pricing model developedhere.     

On modifications of the Bachelier model

Annals of Finance - Mathematically, stock prices described by a classical Bachelier model are sums of a Brownian motion and an absolute continuous drift. Hence, stock prices can take negative...     

Pricing and hedging basket options to prespecified levels of acceptability

The concept of stress levels embedded in S&P500 options is defined and illustrated with explicit constructions. The particular example of a stress function used is MINMAXVAR. Seven joint laws for the top 50 stocks in the index are considered. The first time changes a Gaussian one factor copula. The remaining six employ correlated Brownian motion independently time changed in each coordinate. Four models use daily returns, either run as Lévy processes or scaled to the option maturity. The last two employ risk-neutral marginals from the VGSSD and CGMYSSD Sato processes. The smallest stress function uses CGMYSSD risk-neutral marginals and Lévy correlation. Running the Lévy process yields a lower stress surface than scaling to the option maturity. Static hedging of basket options to a particular level of acceptability is shown to substantially lower the price at which the basket option may be offered.     

Probability distribution of returns in the Heston model with stochastic volatility*

Abstract

We 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.     

Estimation and pricing under long-memory stochastic volatility

We treat the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. We model the price process as a Geometric Brownian Motion with volatility evolving as a fractional Ornstein–Uhlenbeck process. We assume that the model has long-memory, thus the memory parameter H in the volatility is greater than 0.5. Although the price process evolves in continuous time, the reality is that observations can only be collected in discrete time. Using historical stock price information we adapt an interacting particle stochastic filtering algorithm to estimate the stochastic volatility empirical distribution. In order to deal with the pricing problem we construct a multinomial recombining tree using sampled values of the volatility from the stochastic volatility empirical measure. Moreover, we describe how to estimate the parameters of our model, including the long-memory parameter of the fractional Brownian motion that drives the volatility process using an implied method. Finally, we compute option prices on the S&P 500 index and we compare our estimated prices with the market option prices.     

Lévy processes driven by stochastic volatility

In this paper we extend option pricing under Lévy dynamics, by assuming that the volatility of the Lévy process is stochastic. We, therefore, develop the analog of the standard stochastic volatility models, when the underlying process is not a standard (unit variance) Brownian motion, but rather a standardized Lévy process. We present a methodology that allows one to compute option prices, under virtually any set of diffusive dynamics for the parameters of the volatility process. First, we use ‘local consistency’ arguments to approximate the volatility process with a finite, but sufficiently dense Markov chain; we then use this regime switching approximation to efficiently compute option prices using Fourier inversion. A detailed example, based on a generalization of the popular stochastic volatility model of Heston (Rev Financial Stud 6 (1993) 327), is used to illustrate the implementation of the algorithms. Computer code is available at www.theponytail.net/     

The pricing of supershares

The new ‘supershare’ securities proposed by Hakansson (1977, 1976) are subject to the same sort of rickless-hedge combinations as are other forms of secondary securities such as stock options. In consequence, the prices of supershares must, even in the absence of distributional assumptions, obey certain pricing relationships with each other and with the underlying primary security. When the primary security is assumed in addition to follow a geometric Brownian motion process, exact supershare valuation formulae of the Black-Scholes (1973) type are obtained. The ‘hedge portfolio algebra’ of Garman (1976) is employed to make the analysis concise.     

Skew Brownian Motion and Pricing European Options

Abstract

The volatility smile and systematic mispricing of the Black–Scholes option pricing model are the typical motivation for examining stochastic processes other than geometric Brownian motion to describe the underlying stock price. In this paper a new stochastic process is presented, which is a special case of the skew-Brownian motion of Itô and McKean. The process in question is the sum of a standard Brownian motion and an independent reflecting Brownian motion that is similar in construction to the stochastic representation of a skew-normal random variable. This stochastic process is taken in its exponential form to price European options. The derived option price nests the Black–Scholes equation as a special case and is flexible enough to accommodate stochastic volatility as well as stochastic skewness.     

When are Options Overpriced? The Black--Scholes Model and Alternative Characterisations of the Pricing Kernel

An important determinant of option prices is the elasticityof the pricing kernel used to price all claims in the economy.In this paper, we first show that for a given forward priceof the underlying asset, option prices are higher when the elasticityof the pricing kernel is declining than when it is constant.We then investigate the implications of the elasticity of thepricing kernel for the stochastic process followed by the underlyingasset. Given that the underlying information process followsa geometric Brownian motion, we demonstrate that constant elasticityof the pricing kernel is equivalent to a Brownian motion forthe forward price of the underlying asset, so that the Black–Scholesformula correctly prices options on the asset. In contrast,declining elasticity implies that the forward price processis no longer a Brownian motion: it has higher volatility andexhibits autocorrelation. In this case, the Black–Scholesformula underprices all options.     

A one-factor volatility smile model with closed-form solutions for European options

The common practice of using different volatilities for options of different strikes in the Black-Scholes (1973) model imposes inconsistent assumptions on underlying securities. The phenomenon is referred to as the volatility smile. This paper addresses this problem by replacing the Brownian motion or, alternatively, the Geometric Brownian motion in the Black-Scholes model with a two-piece quadratic or linear function of the Brownian motion. By selecting appropriate parameters of this function we obtain a wide range of shapes of implied volatility curves with respect to option strikes. The model has closed-form solutions for European options, which enables fast calibration of the model to market option prices. The model can also be efficiently implemented in discrete time for pricing complex options.
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On Pricing Exponential Square Root Barrier Knockout European Options

A barrier option is one of the most popular exotic options which is designedto give a protection against unexpected wild fluctuation of stock prices.Protection is given to both the writer and holder of such an option.Kunitomo and Ikeda (1992) analytically obtained a pricing formula forexponential double barrier knockout options. Since the logarithm of theirproposed barriers for the stock price process S(t), whichisassumed to be geometric Brownian motion, are nothing but straight lineboundaries, the protection provided by them is not uniform over time. Toremedy this problem, we propose square root curved boundaries±btfor the underlying Brownian motion process W(t). Since thestandarddeviation of Brownian motion is proportional to t, theseboundaries(after transformation) can be made to provide more uniform protectionthroughout the life time of the option. We will apply asymptoticexpansions of certain conditional probabilities obtained by Morimoto (1999)to approximate pricing formulae for exponential square root double barrierknockout European call options. These formulae allow us to computenumerical values in a very short time (t < 10^–6sec), whereas it takesmuch longer to perform Monte Carlo simulations to determine optionpremiums.     

基于MF-DFA的中国股指期货价格的多重分形实证研究

本文通过应用多重分形谱分析法和多重分形消除趋势波动分析（MF-DFA）法,研究了新产生的中国股指期货市场的多重分形性。通过对2942个股指期货最后十分钟结算价格的分析,我们发现中国股指期货的收益率具有长程相关性和多重分形性,期货价格波动并不能用单一的标度指数进行充分描述。进一步通过将原始序列和转换后的收益序列进行比较,转换过程包括重排以及相位随机化,我们发现导致中国股指期货市场多重分形性的两种不同成因。研究结果表明,虽然厚尾分布是造成多重分形性的一个方面,但长程相关性才是引起中国股指期货市场多重分形的主要原因。     

On The Decomposition Of The Ruin Probability For A Jump-Diffusion Surplus Process Compounded By A Geometric Brownian Motion

Abstract

If one assumes that the surplus of an insurer follows a jump-diffusion process and the insurer would invest its surplus in a risky asset, whose prices are modeled by a geometric Brownian motion, the resulting surplus for the insurer is called a jump-diffusion surplus process compounded by a geometric Brownian motion. In this resulting surplus process, ruin may be caused by a claim or oscillation. We decompose the ruin probability in the resulting surplus process into the sum of two ruin probabilities: the probability that ruin is caused by a claim, and the probability that ruin is caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When claim sizes are exponentially distributed, asymptotical formulas of the ruin probabilities are derived from the integro-differential equations, and it is shown that all three ruin probabilities are asymptotical power functions with the same orders and that the orders of the power functions are determined by the drift and volatility parameters of the geometric Brownian motion. It is known that the ruin probability for a jump-diffusion surplus process is an asymptotical exponential function when claim sizes are exponentially distributed. The results of this paper further confirm that risky investments for an insurer are dangerous in the sense that either ruin is certain or the ruin probabilities are asymptotical power functions, not asymptotical exponential functions, when claim sizes are exponentially distributed.     

Leveraged Lévy processes as models for stock prices

Adopting a constant elasticity of variance formulation in the context of a general Lévy process as the driving uncertainty we show that the presence of the leverage effect? ?One explanation of the documented negative relation between market volatilities and the level of asset prices (the ‘smile’ or ‘skew’), we term the ‘leverage effect’, argues that this negative relation reflects greater risk taking by the management, induced by a fall in the asset price, with a view of maximizing the option value of equity shareholders. in this form has the implication that asset price processes satisfy a scaling hypothesis. We develop forward partial integro-differential equations under a general Markovian setup, and show in two examples (both continuous and pure-jump Lévy) how to use them for option pricing when stock prices follow our leveraged Lévy processes. Using calibrated models we then show an example of simulation-based pricing and report on the adequacy of using leveraged Lévy models to value equity structured products.