Optimal strategies for asset allocation and consumption under stochastic volatility

Selecting optimal asset allocation and consumption strategies is an important, but difficult, topic in modern finance. The dynamics is governed by a nonlinear partial differential equation. Stochastic volatility adds further complication. Even to obtain a numerical solution is challenging. Here, we develop a closed-form approximate solution. We show that our theoretical predictions for the optimal asset allocation strategy and the optimal consumption strategy are in surprisingly good agreement with the results from full numerical computations.     

Asymptotic expansion for pricing options for a mean-reverting asset with multiscale stochastic volatility

This work investigates the valuation of options when the underlying asset follows a mean-reverting log-normal process with a stochastic volatility that is driven by two stochastic processes with one persistent factor and one fast mean-reverting factor. Semi-analytical pricing formulas for European options are derived by means of multiscale asymptotic techniques. Numerical examples demonstrate the use of the model and the quality of the numerical scheme.     

Uncertain volatility and the risk-free synthesis of derivatives

To price contingent claims in a multidimensional frictionless security market it is sufficient that the volatility of the security process is a known function of price and time. In this note we introduce optimal and risk-free strategies for intermediaries in such markets to meet their obligations when the volatility is unknown, and is only assumed to lie in some convex region depending on the prices of the underlying securities and time. Our approach is underpinned by the theory of totally non-linear parabolic partial differential equations (Krylov and Safanov, 1979; Wang, 1992) and the non-stochastic approach to Itô's formation first introduced by Föllmer (1981a,b).

In these more general conditions of unknown volatility, the optimal risk-free trading strategy will, necessarily, produce an unpredictable surplus over the minimum assets required at any time to meet the liabilities. This surplus, which could be released to the intermediary or to the client, is not required to meet the contingent claim. One sees that the effect of unknown volatility is the creation of a ‘with profits’ policy, where a premium is paid at the beginning, the contingent claim is collected at the terminal time, but that in addition an unpredictable surplus available as well.

The risk-free initial premium required to meet the contingent claim is given by the solution to the Dirichlet problem for a totally non-linear parabolic equation of the Pucci-Bellman type. The existence of a risk-free strategy starting with this minimum sum is dependent upon theorems ensuring the regularity of the solution and upon a non-probabilistic understanding of Itô's change of variable formulae.

To illustrate the ideas we give a very simple example of a one-dimensional barrier option where the maximum Black-Scholes price of the option over different fixed values for the volatility lying in an interval always underestimates the risk-free ‘price’ under the assumption that the volatility can vary within the same interval.

This paper puts together rather standard mathematical ideas. However, the author hopes that the overall result is more than the sum of its parts. The ability to hedge under conditions of uncertain volatility seems to be of considerable practical importance.

In addition it would be interesting if these ideas explained some features in the design of existing contracts.     

Portfolio optimization when asset returns have the Gaussian mixture distribution

In this paper we consider a portfolio optimization problem where the underlying asset returns are distributed as a mixture of two multivariate Gaussians; these two Gaussians may be associated with “distressed” and “tranquil” market regimes. In this context, the Sharpe ratio needs to be replaced by other non-linear objective functions which, in the case of many underlying assets, lead to optimization problems which cannot be easily solved with standard techniques. We obtain a geometric characterization of efficient portfolios, which reduces the complexity of the portfolio optimization problem.     

Statistical modelling of asymmetric risk in asset returns

The purpose of this article is to provide a straightforward model for asset returns which captures the fundamental asymmetry in upward versus downward returns. We model this feature by using scale gamma distributions for the conditional distributions of positive and negative returns. By allowing the parameters for positive returns to differ from parameters for negative returns we can test the hypothesis of symmetry. Some applications of this process to expected utility and semi-variance calculations are considered. Finally we estimate the model using daily UK FT100 index and Futures data.     

Estimating the instantaneous volatility and covariance of risky assets

Using the Mihlstein approximation for solutions to stochastic differential equations and the stochastic calculus an estimate for the volatility is obtained. The estimate is also valid for stochastic, Markov, volatilities. If the process has jumps, these reduce the previous estimate. The instantaneous covariance of two risky assets is also calculated.     

Option pricing with mean reversion and stochastic volatility

Many underlying assets of option contracts, such as currencies, commodities, energy, temperature and even some stocks, exhibit both mean reversion and stochastic volatility. This paper investigates the valuation of options when the underlying asset follows a mean-reverting lognormal process with stochastic volatility. A closed-form solution is derived for European options by means of Fourier transform. The proposed model allows the option pricing formula to capture both the term structure of futures prices and the market implied volatility smile within a unified framework. A bivariate trinomial lattice approach is introduced to value path-dependent options with the proposed model. Numerical examples using European options, American options and barrier options demonstrate the use of the model and the quality of the numerical scheme.     

The implied volatility of Forward-Start options: ATM short-time level,skew and curvature

Using Malliavin Calculus techniques, we derive closed-form expressions for the at-the-money behaviour of the forward implied volatility, its skew and its curvature, in general Markovian stochastic volatility models with continuous paths.     

Exact and approximate solutions for options with time-dependent stochastic volatility

In this paper it is shown how symmetry methods can be used to find exact solutions for European option pricing under a time-dependent 3/2-stochastic volatility model $\mathit{dv}=\mathit{kv}(A(t)-v)\mathit{dt}+{\mathit{bv}}^{\frac{3}{2}}\mathit{dZ}$. This model with $A(t)$ constant has been proven by many authors to outperform the Heston model in its ability to capture the behaviour of volatility and fit option prices. Further, singular perturbation techniques are used to derive a simple analytic approximation suitable for pricing options with short tenor, a common feature of most options traded in the market.     

基于连续红利支付和随机波动率的未定权益定价模型

研究了具有连续红利支付和随机波动率的未定权益定价问题,利用等价鞅测度的方法推导了风险中性下的欧式未定权益定价公式．     

A mathematical analysis of the Gumbel test for jumps in stochastic volatility models

This article gives an exhaustive mathematical analysis of the Gumbel test for additive jump components based on extreme value theory. The Gumbel test was first introduced by Lee and Mykland in 2008 from an economical point of view. They consider a continuous-time stochastic volatility model with a general continuous volatility process and observe it under a high-frequency sampling scheme. The test statistics based on the maximum of increments converges to the Gumbel distribution under the null hypothesis of no additive jump component and to infinity otherwise. Our article presents a moment method based technique that provides some deeper mathematical insights into the convergence and divergence case of the test statistics. In the non-jump case we are able to prove the convergence to the Gumbel distribution under greatly weak assumptions: The volatility process has to be merely pathwise Hölder continuous with an arbitrary random Hölder exponent and we have no restrictions concerning an additional drift term. Therefore, for example, we are allowing for long and short-range dependence. In the case of existing additive jumps, we give divergence results in a general semimartingale setting and investigate the speed of divergence depending on the jump activity. As a by-product of our analysis we also deduce an optimal pathwise estimator for the spot volatility process. Moreover, we provide a detailed simulation study that compares the power of the Gumbel test with the power of the jump test proposed by Barndorff–Nielsen and Shephard in 2006 for Hölder exponents close to zero. Finally, both tests are applied to a real dataset.     

Approximations for Asian options in local volatility models

We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.     

Pricing VXX option with default risk and positive volatility skew

This paper proposes and makes a comparative study of alternative models for VXX option pricing. Factors such as mean-reversion, jumps, default risk and positive volatility skew are taken into consideration. In particular, default risk is characterized by jump-to-default framework and the “positive volatility skew” issue is addressed by stochastic volatility of volatility and jumps. Daily calibration is conducted and comparative study of the models is performed to check whether they properly fit market prices and generate reasonable positive volatility skews and deltas. Overall, jump-to-default extended LRJ model with positive correlated stochastic volatility (called JDLRJSV in the paper) serves as the best model in all the required aspects.     

The Heston stochastic volatility model in Hilbert space

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein–Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modeling of forward curves in energy and commodity markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics.     

An optimal portfolio model with stochastic volatility and stochastic interest rate

We consider a portfolio optimization problem under stochastic volatility as well as stochastic interest rate on an infinite time horizon. It is assumed that risky asset prices follow geometric Brownian motion and both volatility and interest rate vary according to ergodic Markov diffusion processes and are correlated with risky asset price. We use an asymptotic method to obtain an optimal consumption and investment policy and find some characteristics of the policy depending upon the correlation between the underlying risky asset price and the stochastic interest rate.     

Numerical solution of European call option with dividends and variable volatility

In this paper, the effect of strike price, interest rate, dividends and maturities on European call option with dividends is discussed. The volatility for the data of ONGC Ltd. listed in National Stock Exchange, India, during 03-01-2000 to 30-03-2009 is forecasted by GJR-GARCH method. The option price and Greeks are determined by solving modified Black-Scholes partial differential equation by adjusting forecasted volatility at each grid point of finite difference method. It is observed that call option premium decreases as strike price and dividend increases but it increases as rate of interest and time of maturities increases. Hence call option is more profitable for a long maturity, high interest rate and low dividend.     

上海股市波动的周日效应检验

与以往日历异常现象的研究大多集中在股市收益率上不同,本文对上海股市波动的周日效应进行实证研究,无条件波动的修正Levene检验和条件波动的GARCH模型被应用。结果显示上海股市存在显著的星期一高波动现象,利用混合分布模型对此现象进行了解释,周末信息的积累对星期一交易的影响可能是其高波动的原因。