股市波动性预测模型改进研究

本文从参数估计准则和收益率波动性的定量表达这两方面来探讨股市收益的波动性预测改进方法,并由此提出了以收益率偏差绝对值为代替偏差平方并利用非线性最小二乘法来估计参数的(A)GARCH-NLS模型。最后,我们以国泰君安指数、上证综指以及深证成指1998 1 5-2002 9 25的每日收盘价格为样本来考察各种合理替代所产生的模型对股市波动性的预测性能影响。结果发现,(A)GARCH-NLS模型对股市波动性的预测精确度有了较显著的提高。     

Towards a self-consistent theory of volatility

In this paper, we propose a new theory for the formation of volatility which takes into account the influence of option hedging on the assets price dynamics. By analogy with statistical mechanics, we build a self-consistent equation for the volatility, we show it is well-posed and we explain how it can be solved.     

Salient features of dependence in daily US stock market indices

This paper deals with the analysis of long range dependence in the US stock market. We focus first on the log-values of the Dow Jones Industrial Average, Standard and Poors 500 and Nasdaq indices, daily from February, 1971 to February, 2007. The volatility processes are examined based on the squared and the absolute values of the returns series, and the stability of the parameters across time is also investigated in both the level and the volatility processes. A method that permits us to estimate fractional differencing parameters in the context of structural breaks is conducted in this paper. Finally, the “day of the week” effect is examined by looking at the order of integration for each day of the week, providing also a new modeling approach to describe the dependence in this context.     

Investment volatility: A critique of standard beta estimation and a simple way forward

Beta is a widely used quantity in investment analysis. We review the common interpretations that are applied to beta in finance and show that the standard method of estimation – least squares regression – is inconsistent with these interpretations.     

Estimation of stochastic volatility in the Hull-White model

Estimation of the stochastic volatility in the Hull-White framework is considered. Stock price is taken as the observation and the estimation problem is posed for the stochastic volatility. It is first shown that it is not possible to formulate this as the usual filtering problem, and an alternative formulation is proposed. A robust filtering equation is then derived suitable for real observation data.     

Passport options with stochastic volatility

A passport option is a call option on the profits of a trading account. In this article, the robustness of passport option pricing is investigated by incorporating stochastic volatility. The key feature of a passport option is the holders' optimal strategy. It is known that in the case of exponential Brownian motion the strategy is to be long if the trading account is below zero and short if the account is above zero. Here this result is extended to models with stochastic volatility where the volatility is defined via an autonomous SDE. It is shown that if the Brownian motions driving the underlying asset and the volatility are independent then the form of the optimal strategy remains unchanged. This means that the strategy is robust to misspecification of the underlying model. A second aim of this article is to investigate some of the biases which become apparent in a stochastic volatility regime. Using an analytic approximation, comparisons are obtained for passport option prices using the exponential Brownian motion model and some well-known stochastic volatility models. This is illustrated with numerical examples. One conclusion is that if volatility and price are uncorrelated, then prices are sometimes lower in a model with stochastic volatility than in a model with constant volatility.     

Trading volume in models of financial derivatives

This paper develops a subordinated stochastic process model for an asset price, where the directing process is identified as information. Motivated by recent empirical and theoretical work, the paper makes use of the under-used market statistic of transaction count as a suitable proxy for the information flow. An option pricing formula is derived, and comparisons with stochastic volatility models are drawn. Both the asset price and the number of trades are used in parameter estimation. The underlying process is found to be fast mean reverting, and this is exploited to perform an asymptotic expansion. The implied volatility skew is then used to calibrate the model.     

The European options hedge perfectly in a Poisson-Gaussian stock market model

It is shown that n + 1 European call options written on a stock S with different strike prices (or the stock and n calls) are non-redundant assets in a model for the stock driven by a Brownian motion and n independent Poisson processes. That extends the result obtained for n = 1 by Pham and implies that the proposed model can price and perfectly hedge any integrable derivative on S.     

Equivalent Black volatilities

We consider European calls and puts on an asset whose forward price F(t) obeys dF(t)=α(t)A(F)dW(t,) under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic formulas for the implied volatility σ _B in terms of today's forward price F ₀ ≡ F(0), the strike K of the option, and the time to expiry t_ex . The price of any call or put can then be calculated simply by substituting this implied volatility into Black's formula. For example, for a power law (constant elasticity of variance) model dF(t)=aF^β dW(t) we obtain σ _B = a/f _aυ ^1? β {1 + (1?β)(2+β)/24 (F ₀ ? K/f _aυ)² + (1 ? β)²/24 a ² t_ex /f _aυ ^2?2β +…} where f _aυ = ½(F ₀ + K). Our formula for the implied volatility is not exact. However, we show that the error is insignificant, rarely approaching 1/1000 of the time value of the option. We also present more accurate (albeit more complicated) formulas which can be used for the implied volatility.