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.     

On option pricing models in the presence of heavy tails

We propose a modification of the option pricing framework derived by Borland which removes the possibilities for arbitrage within this framework. It turns out that such arbitrage possibilities arise due to an incorrect derivation of the martingale transformation in the non-Gaussian option models which are used in that paper. We show how a similar model can be built for the asset price processes which excludes arbitrage. However, the correction causes the pricing formulas to be less explicit than the ones in the original formulation, since the stock price itself is no longer a Markov process. Practical option pricing algorithms will therefore have to resort to Monte Carlo methods or partial differential equations and we show how these can be implemented. An extra parameter, which needs to be specified before the model can be used, will give market makers some extra freedom when fitting their model to market data.     

Risk measures for derivatives with Markov-modulated pure jump processes

We consider a regime-switching HJB approach to evaluate risk measures for derivative securities when the price process of the underlying risky asset is governed by the exponential of a pure jump process with drift and a Markov switching compensator. The pure jump process is flexible enough to incorporate both the infinite, (small), jump activity and the finite, (large), jump activity. The drift and the compensator of the pure jump process switch over time according to the state of a continuous-time hidden Markov chain representing the state of an economy. The market described by our model is incomplete. Hence, there is more than one pricing kernel and there is no perfect hedging strategy for a derivative security. We derive the regime-switching HJB equations for coherent risk measures for the unhedged position of derivative securities, including standard European options and barrier options. For measuring risk inherent in the unhedged option position, we first need to mark the position into the market by valuing the option. We employ a well-known tool in actuarial science, namely, the Esscher transform to select a pricing kernel for valuation of an option and to generate a family of real-world probabilities for risk measurement. We also derive the regime-switching HJB-variational inequalities for coherent risk measures for American-style options.     

Coincident and leading indicators of the stock market

In this paper we have two goals: first, we want to represent monthly stock market fluctuations by constructing a non-linear coincident financial indicator. The indicator is constructed as an unobservable factor whose first moment and conditional volatility are driven by a two-state Markov variable. It can be interpreted as the investors' real-time belief about the state of financial conditions. Second, we want to explore an approach in which investors may use their perceptions of the state of the economy to form forecasts of financial market conditions and possibly of excess returns. To investigate this, we build leading indicators as forecasts of the estimated coincident financial index. The leading indicators yield better within and out-of-sample performance in forecasting, not only the state of the stock market but also of excess stock returns, as compared with the performance obtained using linear methods that have been proposed in the existing literature.     

Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment

By utilizing information about prices and trading volumes, we discuss the pricing of European contingent claims in a continuous-time hidden regime-switching environment. Hidden market sentiments described by the states of a continuous-time, finite-state, hidden Markov chain represent a common factor for an asset’s drift and volatility, as well as its trading volumes. Using observations about trading volumes, we present a filtered estimate of the hidden common factor. The asset pricing problem is then considered in a filtered market, where the hidden drift and volatility are replaced by their filtered estimates. We adopt the Esscher transform to select an equivalent martingale measure for pricing and derive a partial-differential integral equation for the option price.     

Sources of the stock price fluctuations in Chinese equity market

This paper proposes a latent factor approach based on a state–space framework in order to identify which factor, if any, dominates price fluctuations in the Chinese stock markets. We also illustrate the connection of such stock price decomposition with several general equilibrium asset pricing models and show that the decomposition results can potentially offer useful insights with regard to the empirical relevance of asset pricing models. We use quarterly data of the Chinese A-Share equity market over the period 1995Q3–2011Q1 and find that the estimates of the state–space model suggest that the expected return is the primary driving force behind price fluctuations in the Chinese stock market. We show that the time-varying expected returns appear to be counter-cyclical and this result seems to be consistent with the habit formation model of Campbell and Cochrane [1999. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107, no. 2: 205–51.]. However, we also note that there is a great deal of uncertainty with respect to this variance decomposition due to the resulting small signal-to-noise ratio in the estimated state–space model.     

Information asymmetry,contract design and process of negotiation: The stock options awarding case

Stock option plans are used to increase managerial incentives, and business practices usually set the exercise price equal to the stock market price. The purpose of this paper is to underline the importance of a process of negotiation leading to a possible equilibrium contract satisfying both managers and shareholders. The two key variables of the model are the percentage of equity capital offered by the shareholders to the managers and the exercise price of the options that may be at a discount. We explicitly introduce risk aversion and information asymmetries in the form of (i) an economic uncertainty in the gain of cash flow, (ii) possibly biased information between the two parties and (iii) a noise in the valuation price of the stock in the market. The existence of a process of negotiation between shareholders and managers leading to a possible disclosure of private information is highlighted. As a conclusion, we show that “efficient” stock option plans should be granted in a context of trade-off between the percentage of capital awarded to managers and the discount in stock price.     

Analysis of default data using hidden Markov models

The occurrence of defaults within a bond portfolio is modelled as a simple hidden Markov process. The hidden variable represents the risk state, which is assumed to be common to all bonds within one particular sector and region. After describing the model and recalling the basic properties of hidden Markov chains, we show how to apply the model to a simulated sequence of default events. Then, we consider a real scenario, with default events taken from a large database provided by Standard & Poor's. We are able to obtain estimates for the model parameters and also to reconstruct the most likely sequence of the risk state. Finally, we address the issue of global versus industry-specific risk factors. By extending our model to include independent hidden risk sequences, we can disentangle the risk associated with the business cycle from that specific to the individual sector.     

Stock and option market divergence in the presence of noisy information

We examine market behavior of the stock and option markets upon the arrival of noisy information in the form of CNBC’s Mad Money recommendations. If stock and option markets are not equally efficient, they should respond differently to noisy information, with the less efficient market more susceptible to noise. We find that the stock market is less efficient than the option market. The abnormal difference between option-implied and actual stock returns is negative and significant upon exposure to noisy information. This difference may yield an economically significant monthly trading profit of up to 5%. We conclude that the stock market is more susceptible to noisy information than the option market and is therefore less efficient.     

Does Greater Market Transparency Reduce Information Asymmetry?

This research aims to determine whether the degree of asymmetric information decreases with greater pre-trade transparency in the Taiwan stock market. We used the probability of informed trading based on the Markov regime-switching model in an order-driven auction market to investigate this topic. Information asymmetry showed no conspicuous variations with greater transparency. However, after further grouping, the empirical results revealed that increased transparency facilitated a decrease in information asymmetry in the sub-samples, which originally exhibited greater information asymmetry. In addition, the intraday patterns of probability of informed trading revealed that greater transparency facilitates decreased market information asymmetry after opening.     

信息不对称、过度自信与股价变动

信息传导的非同步和投资者情绪变化是股票市场的两个典型特征,前者会引发投资者之间出现信息不对称问题,后者主要体现为投资者过度自信,两者共同作用影响股票价格变动。本文将信息不对称和投资者过度自信情绪置于同一个分析框架,建立两阶段动态序贯定价理论模型研究现实市场上信息传导过程中股价变动的内在机制。结果表明:(1)面临新信息的进入,投资者对股票收益预期的调整与均衡价格之间具有正相关关系;(2)面临有利消息时,过度自信投资者比例越大,股票的均衡价格越高,投资收益将越低;面临不利消息时则相反;(3)随着过度自信投资者比例以及过度自信程度升高,市场风险溢价将下降;(4)投资者群体在信息传导过程中出现分化,对股价变动形成异质信念,未获取信息和获取信息但未出现过度自信的投资者认为股价被高估,获取信息且出现过度自信的投资者认为价格被低估,促使更多的交易,引发市场成交量和股价变动;(5)过度自信投资者比例与过度自信程度提高均会对市场效率产生正向影响,而对市场深度具有负向效应。最后,基于理论结果对非对称性和持续性等典型的市场波动性特征进行解释。     

Forecasting stock-return variance: toward an understanding of stochastic implied volatilities

We examine the behavior of measured variances from the optionsmarket and the underlying stock market. Under the joint hypothesesthat markets are informationally efficient and that option pricesare explained by a particular asset pricing model, forecastsfrom time-series models of the stock return process should nothave predictive content given the market forecast as embodiedin option prices. Both in-sample and out-of-sample tests suggestthat this hypothesis can be rejected. Using simulations, weshow that biases inherent in the procedure we use to imply variancescannot explain this result. Thus, we provide evidence inconsistentwith the orthogonality restrictions of option pricing modelsthat assume that variance risk is unpriced. These results alsohave implications for optimum variance forecast rules.     

A duscrete-time model of high-frequency stock returns

A model of intraday financial time series is developed. The model is a dynamic factor model consisting of two equations. First, a rate of return of a ‘stock’ in a single day is assumed to be generated by serveral common factors plus some additive erros (‘intraday equation’). Secondly, the joint distribution of those common factors is assumed to depend on the hidden state of the day, which fluctuates according to a Markov chain (‘day-by-day equation’). Together the equations compose a hidden Markov model.

We investigate properties of the model. Among them is a central limit theorem for cumulative returns, which agrees with the well-known empirical phenomenon in the stock markets that the distributions of longer-horizon returns are closer to the normal. We propose a two-step procedure consisting of the method of principal components and the EM algorithm to estimate the model parameters as well as the unboservable states. In addition, we propose a procedure for predicting intraday returns. Finally, the model is fitted to empirical data, the Standard&Poors 500 Index 5 min return data, to see if the model is capable of describing intraday movements of the index.     

Volume,volatility and information linkages in the stock and option markets

This paper examines the relationship between option trading activity and stock market volatility. Although the option market is uniquely suited for trading on volatility information, there is little analysis on how trading activity in this market is linked to stock price volatility. The bulk of the discussion tends to focus on whether trading activity in the stock market is informative about stock volatility. To analyze the information in option trading activity for stock market volatility, a sample of 15 stocks with the highest option trading volume is selected. For each stock, it is noted that the trading activities in the put and call option markets have significant explanatory power for stock market volatility. In addition, the results indicate that the call option trading activity has a stronger impact on stock volatility compared with that of the put options. Our results demonstrate that information and sentiment in the option market is useful for the estimation of stock market volatility. Also, the significance of the effects of option trading activity on stock price volatility is observed to be comparable to that of stock market trading activity. Furthermore, the persistence and asymmetric effects in the volatility of some stocks tend to disappear once option trading activity is taken into account.     

Measuring the impact of option market activity on the stock market: Bivariate point process models of stock and option transactions

I apply the bivariate Autoregressive Conditional Duration model of Engle and Lunde [2003. Trade and quotes: a bivariate point process. Journal of Financial Econometrics 1, 159–188] to stock and option market transactions. The first model uses option trades and stock trades. Shocks to option trade/option trade durations have a significant impact on option trade/stock trade durations. Higher implied volatility, larger stock and option market order imbalances, larger stock trades, larger spreads, smaller depths in the stock market and faster trading in the stock and option markets are all associated with faster trading in both markets. In the second model, option trade/option trade timing leads option trade/stock quote timing and several information-related stock and option market covariates impact the expected inter-market event durations.     

A stochastic differential game for optimal investment of an insurer with regime switching

We introduce a model to discuss an optimal investment problem of an insurance company using a game theoretic approach. The model is general enough to include economic risk, financial risk, insurance risk, and model risk. The insurance company invests its surplus in a bond and a stock index. The interest rate of the bond is stochastic and depends on the state of an economy described by a continuous-time, finite-state, Markov chain. The stock index dynamics are governed by a Markov, regime-switching, geometric Brownian motion modulated by the chain. The company receives premiums and pays aggregate claims. Here the aggregate insurance claims process is modeled by either a Markov, regime-switching, random measure or a Markov, regime-switching, diffusion process modulated by the chain. We adopt a robust approach to model risk, or uncertainty, and generate a family of probability measures using a general approach for a measure change to incorporate model risk. In particular, we adopt a Girsanov transform for the regime-switching Markov chain to incorporate model risk in modeling economic risk by the Markov chain. The goal of the insurance company is to select an optimal investment strategy so as to maximize either the expected exponential utility of terminal wealth or the survival probability of the company in the ‘worst-case’ scenario. We formulate the optimal investment problems as two-player, zero-sum, stochastic differential games between the insurance company and the market. Verification theorems for the HJB solutions to the optimal investment problems are provided and explicit solutions for optimal strategies are obtained in some particular cases.     

Volatility dependences of stock markets with structural breaks

We develop a Vector Heterogeneous Autoregression model with Continuous Volatility and Jumps (VHARCJ) where residuals follow a flexible dynamic heterogeneous covariance structure. We employ the Bayesian data augmentation approach to match the realised volatility series based on high-frequency data from six stock markets. The structural breaks in the covariance are captured by an exogenous stochastic component that follows a three-state Markov regime-switching process. We find that the stock markets have higher volatility dependence during turmoil periods and that breakdowns in volatility dependence can be attributed to the increase in market volatilities. We also find positive correlations between the Asian stock markets, the European stock market, and the UK stock market. The US stock market has positive correlations with all other markets for most of the sample periods, indicating the leading position of US stock market in the global stock markets. In addition, the proposed three-state VHARCJ model with Dynamic Conditional Correlation (DCC) and break structure under student-t distribution has a superior density forecast performance as compared to the competing models. The forecast models with structural breaks outperform those without structural breaks based on the log predicted likelihood, the log Bayesian factor, and the root mean square loss function.     

Volatility and correlation in international stock markets and the role of exchange rate fluctuations

This paper develops a direct, explicit model for the role of exchange rate fluctuations in international stock markets and examines how and to what extent volatility and correlations in equity markets are influenced by exchange rate fluctuations. Evidence presented in this paper indicates that a higher foreign exchange rate variability mostly increases local stock market volatility but decreases volatility for the US stock market. The extent to which stock market volatility is influenced by foreign exchange variability is greater for local markets than for the US market, due to the fact that exchange rate changes are more strongly correlated with local equity market returns than the US market returns. We find that a higher exchange rate fluctuation marginally decreases the US/local equity market correlation. While exchange rate fluctuations held a relatively large fraction of the variation in local stock market returns, there was no significant influence on the US/local equity market correlation.     

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.     

ANALYSIS OF AN OPTION MARKET DYNAMICS BASED ON A HETEROGENEOUS AGENT MODEL

We propose a two‐market model in which an option market and its underlying market interact. Many artificial markets representing stock markets have been developed, and these models have been actively used to investigate the effects of market rules. However, no artificial market model for derivatives has been intensively studied, even though derivative markets are increasingly important. We tested stylized facts that can be observed in an option market and our model can replicate fat‐tailed distributions, positive skew of the return and positive autocorrelation of the square of return of implied volatility. We found that the speed of volatility mean reversion for fundamentalists and the existence of chartists are important factors for replicating the positive skew of an option market. The value of fat‐tailed distributions and positive skewness of the return get closer to the real value by coupling an option market and an underlying market. Copyright © 2014 John Wiley & Sons, Ltd.