cổ tức. Điều này làm cho các tùy chọn cuộc gọi rẻ hơn, bởi vì các danh mục đầu tư tái tạo có thể sử dụng cổ tức đến để làm tăng phần cổ phiếu của danh mục đầu tư tái tạo. Điều này được mô tả trong Chương 7, với công thức được đưa ra trong phương trình 7,36 cho một lựa chọn cổ phiếu với Thánh giá tại t, | PTP Option Valuation 245 dividends. This makes call options cheaper because the replicating portfolio can use the incoming dividends to increase the stock part of the replicating portfolio. This is described in Chapter 7 with the formula given in equation for an option on the stock with price St at t where dividends are paid continuously on the stock at a rate of d per year. Using the standard Black-Scholes call option formula with allowance for dividend income the cost of the option at the inception of the contract using PTP indexation assuming index linking to a regular index without reinvested dividends and where dividends are received continuously at a rate of d per year is H0 SP ị S0 e h D d1 - K2C r d2 j P h 2 afc d1 - G - 1 - a e dp a P e dn b d1 - G - P 1 - a e d2 where _ log S0 Kptp r - d a2 2 d1 - ơự _ log q P G - P 1 - a r - d Ơ2 2 1 ơ and d2 d1 ơ n We can use the formula to value the option cost of equity indexation for a standard PTP EIA contract. The details are as follows Seven-year contract with PTP indexation. Sixty percent participation rate. A single premium of 100. Three percent per year minimum return guarantee applied to 95 percent of the premium. In addition we need parameters for the Black-Scholes call option value say a risk-free force of interest of r 6 percent per year a dividend rate of d 2 percent and a volatility of r 20 percent. The cost of the call option for the contract is . Using a 2 percent interest spread the 246 EQUITY-INDEXED ANNUITIES TABLE Break-even participation rates for seven-year PTP EIA. Interest Spread Available Volatility Ơ 1 2 3 Net of non-option expenses. funds available to fund the option are which is substantially on the profitable side of the break-even point. Most authors treat the participation rate as the variable controlling the cost of the embedded option. The guaranteed minimum benefit is generally .
