gặp khó khăn và không có ảnh hưởng đến giá các sản phẩm được. Một lần nữa, tùy chọn cao cấp sẽ phải được điều chỉnh nếu điều này không phải là trường hợp. Con đường ngẫu nhiên liên tục. Black-Scholes giả định rằng giá của các ngành nghề cơ bản liên tục và di chuyển qua tất cả các cấp mà không cần nhảy đột ngột. Tài sản không thanh khoản làm | Appendix A 209 Table Present value of floating rate cash flows Year Notional m Floating cash flow m Discount factor Present value m 1 100 2 100 3 100 Sum Table Fixed rate cash flows on swap Notional Fixed cash flow Present value Year m m Discount factor m 1 100 2 100 3 100 Sum years time which we assume is established by the forward rate F2x3 which we calculated as . The next step is to discount these cash flows at the zero-coupon or spot rates for each time period - or to make the calculation easier to multiply each cash flow by the discount factor for that time period. The results and the sum of the present values is shown in Table . A par swap is one in which the present value of the floating and fixed legs sum to zero. If a swap is entered into at exactly par the expected payout to both sides is zero and neither side pays a premium to the other. The fixed rate on a par swap is the single rate such that if the fixed cash flows are calculated at that rate the present value of the fixed cash flows completely offsets the present value of the floating rate cash flows. As a result the net present value of the swap is zero. In our example assuming the swap is agreed at par we need to find a fixed rate such that the present value of the fixed cash flows on the swap equals minus million. In our example the fixed cash flows are negative because we are paying fixed on the swap. At that rate the net present value - . the sum of the PVs of the fixed and floating cash flows - is zero. A direct way to calculate the rate is shown below but it can also be found by trial-and-error. Either way as Table shows the answer is . The fixed cash flows are minus million each year for three years. The present values are established by multiplying each cash flow by the appropriate discount factor. The .
