Analysing the bias in the primal-dual upper bound method for early exercisable derivatives: bounds,estimation and removal

We analyse the primal-dual upper bound method for Bermudan options and prove that its bias is inversely proportional to the number of paths in sub-simulations for a large class of cases. We develop a methodology for estimating and reducing the bias. We present numerical results showing that the new technique is indeed effective.     

The representation of American options prices under stochastic volatility and jump-diffusion dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square-root process as used by Heston [Rev. Financial Stud., 1993, 6, 327–343], and by a Poisson jump process as introduced by Merton [J. Financial Econ., 1976, 3, 125–144]. Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalizes in an intuitive way the structure of the solution to the corresponding European option pricing problem obtained by Scott [Math. Finance, 1997, 7(4), 413–426], but here in the case of a call option and constant interest rates.     

Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy

In this paper, we use a Markov-modulated regime switching approach to model various states of the economy, and study the pricing of vulnerable European options when the dynamics of the underlying asset value and the asset value of the counterparty follow two correlated jump-diffusion processes under regime switching. The correlation is modelled by both the diffusion parts and the pure jump parts which describe the uncertainty of the value of the risky assets. We develop a method to determine an equivalent martingale measure and a parsimonious representation of the risk-neutral density is provided. Based on this, we derive an analytical pricing formula for vulnerable options via two-dimensional Laplace transforms, and implement the formula through numerical Laplace inversion.     

Smoothing the payoff for efficient computation of Basket option prices

We consider the problem of pricing basket options in a multivariate Black–Scholes or Variance-Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi Monte Carlo methods in dimensions up to 35.     

A systematic and efficient simulation scheme for the Greeks of financial derivatives

Greeks are the price sensitivities of financial derivatives and are essential for pricing, speculation, risk management, and model calibration. Although the pathwise method has been popular for calculating them, its applicability is problematic when the integrand is discontinuous. To tackle this problem, this paper defines and derives the parameter derivative of a discontinuous integrand of certain functional forms with respect to the parameter of interest. The parameter derivative is such that its integration equals the differentiation of the integration of the aforesaid discontinuous integrand with respect to that parameter. As a result, unbiased Greek formulas for a very broad class of payoff functions and models can be systematically derived. This new method is applied to the Greeks of (1) Asian options under two popular Lévy processes, i.e. Merton's jump-diffusion model and the variance-gamma process, and (2) collateralized debt obligations under the Gaussian copula model. Our Greeks outperform the finite-difference and likelihood ratio methods in terms of accuracy, variance, and computation time.     

Moments of renewal shot-noise processes and their applications

In this paper, we study the family of renewal shot-noise processes. The Feynmann–Kac formula is obtained based on the piecewise deterministic Markov process theory and the martingale methodology. We then derive the Laplace transforms of the conditional moments and asymptotic moments of the processes. In general, by inverting the Laplace transforms, the asymptotic moments and the first conditional moments can be derived explicitly; however, other conditional moments may need to be estimated numerically. As an example, we develop a very efficient and general algorithm of Monte Carlo exact simulation for estimating the second conditional moments. The results can be then easily transformed to the counterparts of discounted aggregate claims for insurance applications, and we apply the first two conditional moments for the actuarial net premium calculation. Similarly, they can also be applied to credit risk and reliability modelling. Numerical examples with four distribution choices for interarrival times are provided to illustrate how the models can be implemented.     

互联网企业价值评估模型比较研究

基于互联网企业轻资产、高估值、迭代快以及风险大等特点,比较传统价值评估模型与Schwar-tz-Moon等实物期权价值评估模型,分别运用于评估案例企业泛微网络价值.结果发现,相较于传统现金流贴现模型,实物期权价值评估模型评估结果更接近于公司实际价值.三种实物期权模型敏感性分析表明:Schwartz-Moon模型评估误差最小,且模型稳健性最强,适用于不确定性高的互联网企业估值.