Non-recombining trinomial tree pricing model and calibration for the volatility smile

Abstract

In this paper, we consider the non-recombining trinomial tree pricing model under the volatility, which is a function of time, establish the option pricing model and give the convergence rates of the non-recombining trinomial tree method. In addition, we research the calibration problem of volatility and adopt an exterior penalty method to transform this problem into a nonlinear unconstrained optimization problem. For the optimization problem, we use the quasi-Newton algorithm. Finally, we test our model by numerical examples and options data on the S&P 500 index. The results show the effectiveness of the non-recombining trinomial tree pricing model.

An alternative tree method for calibration of the local volatility

“…Lok and Lyuu [20] find a potentially fundamental reason why the negative probabilities linger and construct a waterline tree for separable local volatility which combines two kinds of binomial trees by matching the moments of the underlying asset price and logarithmic asset price. Gong and Xu [17] propose an implied non-recombining trinomial tree and calibrate the volatility smile. Table 1 shows some tree methods for calibration of the local volatility.…”