Computation and analysis for a constrained entropy optimization problem in finance

In [T. Coleman, C. He, Y. Li, Calibrating volatility function bounds for an uncertain volatility model, Journal of Computational Finance (2006) (submitted for publication)], an entropy minimization formulation has been proposed to calibrate an uncertain volatility option pricing model (UVM) from market bid and ask prices. To avoid potential infeasibility due to numerical error, a quadratic penalty function approach is applied. In this paper, we show that the solution to the quadratic penalty problem can be obtained by minimizing an objective function which can be evaluated via solving a Hamilton–Jacobian–Bellman (HJB) equation. We prove that the implicit finite difference solution of this HJB equation converges to its viscosity solution. In addition, we provide computational examples illustrating accuracy of calibration.     

On the American option‐pricing model with an uncertain volatility

In this paper, we study an American option‐pricing model with an uncertain volatility. Some properties for the option price are derived. Particularly, a global spread for the option price is proved when the volatility depends on the underlying security and time. This result confirms the observed fact from the real financial data in option markets. Copyright © 2015 John Wiley & Sons, Ltd.     

Explicit Density Approximations for Local Volatility Models Using Heat Kernel Expansions

Heat kernel perturbation theory is a tool for constructing explicit approximation formulas for the solutions of linear parabolic equations. We review the crux of this perturbative formalism and then apply it to differential equations which govern the transition densities of several local volatility processes. In particular, we compute all the heat kernel coefficients for the CEV and quadratic local volatility models; in the later case, we are able to use these to construct an exact explicit formula for the processes’ transition density. We then derive low order approximation formulas for the cubic local volatility model, an affine-affine short rate model, and a generalized mean reverting CEV model. We finally demonstrate that the approximation formulas are accurate in certain model parameter regimes via comparison to Monte Carlo simulations.     

不确定微分方程的数值解法及其应用研究

不确定微分方程广泛应用于不确定财政、不确定控制、不确定微分博弈等领域。由于一些不确定微分方程解析解难以实现,本文首先研究了不确定微分方程的Euler方法和Runge-Kutta方法两种数值解法,并进行误差分析。通过比较随机领域Black-Scholes模型和不确定领域Liu模型的欧式期权定价公式,验证不确定微分方程描述证券市场的合理性和实用性。     

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.     

American options under uncertain volatility

The uncertain volatility approach to financial derivatives is extended to American options (which allow early exercise before expiry). The requirement to model at the portfolio level made necessary by the non-linearity of the approach is found to lead to a recursive structure to the exercise possibilities across options. Other novel features include: the optimality sometimes of partial exercise; an interesting resolution to the issues surrounding short options whose exercise is controlled by a buyer counterparty; and the occurrence of a simple game structure for portfolios containing both long and short options. It is demonstrated that the exercise strategies resulting can significantly alter measured uncertain volatility risk. Contrary to the set of attributes for sensible risk measures put forward by Artzner, Delbaen, Eber and Heath, this risk need not be homogenous in portfolio size- forming a convincing argument for weakening this particular requirement.     

随机波动率模型下技术分析指标的一些性质(英文)

一些流行的技术指标(例如布林带,RSI,ROC等)被股市交易者广为使用.交易者将每日(小时,周,……)的实际股价作为计算某个技术指标的样本,通过观察相关频率来指导投资.技术指标的有效性已在广泛的应用中得到了验证.我们已经证明在Black-Scholes模型下,某些技术指标有许多有用的统计性质.作为更一般的情况,随机波动率模型在金融数学中得到了广泛的讨论.本文基于随机波动率模型对技术指标的统计性质进行了研究.研究结果表明,如果股票价格服从随机波动率模型,则技术指标的合理性可以得到有力的证明,从这个角度我们为技术分析奠定理论基础.     

基于期货、期权与现货组合的采购决策和风险控制研究

针对企业大宗原材料采购容易受到上游价格和下游需求的不确定性影响这一问题,本文以制造商(或分销商)多阶段采购决策为研究对象,建立了包含期货、期权和现货三种采购方式的多阶段组合采购决策模型,以此来应对采购中的价格风险和库存风险.结合某钢构厂原材料采购问题,采用蒙特卡罗仿真方法进行求解.通过对比组合方式与现货方式的求解结果,证明了组合采购方式能够在各种采购环境下为采购商带来更高更稳定的利润,所提出的组合采购决策模型能够有效的规避风险.     

Strategic real options with stochastic volatility in a duopoly model

The investment-timing problem has been considered by many authors under the assumption that the instantaneous volatility of the demand shock is constant. Recently, Ting et al. (2013) [12] carried out an asymptotic approach in a monopoly model by letting the volatility parameter follow a stochastic process. In this paper, we consider a strategic game in which two firms compete for a new market under an uncertain demand, and extend the analysis of Ting et al. to duopoly models under different strategic game structures. In particular, we investigate how the additional uncertainty in the volatility affects the investment thresholds and payoffs of players. Several numerical examples and comparison of the results are provided to confirm our analysis.     

Calibrating the Black-Derman-Toy model: some theoretical results

The Black-Derman-Toy (BDT) model is a popular one-factor interest rate model that is widely used by practitioners. One of its advantages is that the model can be calibrated to both the current market term structure of interest rate and the current term structure of volatilities. The input term structure of volatility can be either the short term volatility or the yield volatility. Sandmann and Sondermann derived conditions for the calibration to be feasible when the conditional short rate volatility is used. In this paper conditions are investigated under which calibration to the yield volatility is feasible. Mathematical conditions for this to happen are derived. The restrictions in this case are more complicated than when the short rate volatilities are used since the calibration at each time step now involves the solution of two non-linear equations. The theoretical results are illustrated by showing numerically that in certain situations the calibration based on the yield volatility breaks down for apparently plausible inputs. In implementing the calibration from period n to period n + 1, the corresponding yield volatility has to lie within certain bounds. Under certain circumstances these bounds become very tight. For yield volatilities that violate these bounds, the computed short rates for the period (n, n + 1) either become negative or else explode and this feature corresponds to the economic intuition behind the breakdown.