In this paper we propose an estimation procedure which uses joint data on the underlying asset and option prices to extract market prices of return and volatility risks in the context of the G.A.R.C.H. diffusion model. The procedure is flexible and simple to implement. Firstly, a quasi-closed form pricing formula for European options in the G.A.R.C.H. diffusion model is derived. This result greatly eases the computational burden for computing option prices, and well suited for our model estimation. Then, based upon the joint data, we develop an efficient importance sampling-based maximum likelihood (E.I.S.-M.L.) estimation method for the objective and risk-neutral parameters of the G.A.R.C.H. diffusion model and a particle filter algorithm for latent state variable. Hence, this allows us to infer the market prices of risks that link the objective measure and the risk-neutral measure. Finally, we illustrate our approach using actual data on the Hang Seng Index (H.S.I.) and index warrant prices. The results show that both the return and volatility risks are priced by the market. Moreover, an option pricing study demonstrates that the market price of the volatility risk plays an important role in fitting option prices.

In this paper we introduce a triple-threshold leverage stochastic volatility (TTLSV) model for financial return time series. The main feature of the model is to allow asymmetries in the leverage effect as well as mean and volatility. In the model the asymmetric effect is modeled by a threshold nonlinear structure that the two regimes are determined by the sign of the past returns. The model parameters are estimated using maximum likelihood (ML) method based on the efficient importance sampling (EIS) technique. Monte Carlo simulations are presented to examine the accuracy and finite sample properties of the proposed methodology. The EIS-based ML (EIS-ML) method shows good performance according to the Monte Carlo results. The proposed model and methodology are applied to two stock market indices for China. Strong evidence of the mean and volatility asymmetries is detected in Chinese stock market. Moreover, asymmetries in the volatility persistence and leverage effect are also discovered. The log-likelihood and Akaike information criterion (AIC) suggest evidence in favor of the proposed model. In addition, model diagnostics suggest that the proposed model performs relatively well in capturing the key features of the data. Finally, we compare models in a Value at Risk (VaR) study. The results show that the proposed model can yield more accurate VaR estimates than the alternatives.

The changes of numeraire can be used as a very powerful mean in pricing contingent claims in the context of a complete market. We apply the method of nurmeraire changes to evaluate convertible bonds when the instantaneous growth and variance of the value of issuer and those of zero-coupon bonds follow a general adapted stochastic process in this paper. A closed-form solution is derived when the instantaneous growth and variance of the value of issuer and those of zero-coupon bonds are deterministic function of time. We also consider a special case when the asset price follows GBM (Geometric Brownian Motion) and interest rate follows Vasicek's model.

Valuation of American options is a difficult and challenging problem encountered in financial engineering. Longstaff and Schwartz [Longstaff, FA and ES Schwartz (2001). Valuing American Options by Simulation: A Simple Least-squares Approach, Review of Financial Studies, 14(1), 113-147.] Proposed the least-squares Monte Carlo (LSM) method for valuing American options. As this approach is intuitive and easy to apply, it has received much attention in the finance literature. However, a drawback of the LSM method is the low efficiency. In order to overcome this problem, we propose the least-squares randomized quasi-Monte Carlo (LSRQM) methods which can be viewed as a use low-discrepancy sequences as a variance reduction technique in the LSM method for valuing American options in this paper. Numerical results demonstrate that our proposed LSRQM methods are more efficient than the LSM method in terms of the valuation accuracy, the computation time and the convergence rate.