Pricing the equity-linked and principal-protected securities with cap and path dependence

It is difficult for the capped and path dependent models to find its closed-form solution. This paper extends the work by Brennan and Schwartz [M.J. Brennan, E.S. Schwartz, The pricing of equity-linked insurance policies with an asset value guarantee, Journal of Financial Economics 3 (1976) 195–213] to develop a model for the capped, equity-linked and principal-protected securities with path dependence, and proposes a closed-form approximation using the 2nd-order Taylor approximation and the method of Vorst [T. Vorst, Prices and hedge ratios of average exchange rate option, International Review of Financial Analysis 1(3) (1992) 179–194] for the valuation of the securities in order to overcome problems of the sum of lognormal distribution. The numerical results indicate that based on Quasi Monte Carlo simulation criterion, in spite of various maturities or various volatility levels, this analytic approximation solution has higher accuracy than the binomial tree model.