Numerical Methods and Volatility Models for Valuing Cliquet Options

Abstract

Several numerical issues for valuing cliquet options using PDE methods are investigated. The use of a running sum of returns formulation is compared to an average return formulation. Methods for grid construction, interpolation of jump conditions, and application of boundary conditions are compared. The effect of various volatility modelling assumptions on the value of cliquet options is also studied. Numerical results are reported for jump diffusion models, calibrated volatility surface models, and uncertain volatility models.

Pricing cliquet options by tree methods

“…For σ = 0.2 we will compare 1. the singular point method with maximal level of error in each observation period h = 10 −6 (SP); 2. the pure binomial method (BIN); For the first two methods we choose different time steps m = 200, 500, 1,000, 2,000 in every observation period. In the case of FD method we report the values obtained in Windcliff et al (2006) with different numbers of nodes and time steps. Moreover, in order to test the efficiency of the approximation of the continuous value, we also provide the price obtained by using Monte Carlo method with 1.000.000 of trials (hence with very long computation time).…”

“…We use, in both cases, the standard cliquet contract studied in Wilmott (2002) and Windcliff et al (2006):…”

“…We consider two different volatilities σ = 0.2, corresponding to the standard case chosen in Wilmott (2002) and Windcliff et al (2006) and σ = 0.02 in order to test the efficiency of the methods for low volatility cases. For σ = 0.2 we will compare 1. the singular point method with maximal level of error in each observation period h = 10 −6 (SP); 2. the pure binomial method (BIN); For the first two methods we choose different time steps m = 200, 500, 1,000, 2,000 in every observation period.…”

“…Later, Windcliff et al (2006) explored a variety of modelling alternatives including jump diffusion models, local volatility and UVM models, again using finite differences methods.…”

Infinite reload options: Pricing and analysis

“…for a given K j which results in the so-called repeated grids discussed in [13] and [14]. In this case, no interpolation is required to estimate V * .…”