plain vanilla swap,

问题如下：

Consider a swap involving two currencies, the US dollar ($) and the Swiss franc (SF). The exchange rate is now 0.80 $/SF. As we do in a plain vanilla swap, we can get the fixed rate that will make the fixed $ payments equal to $1, and the fixed rate that will make the fixed SF payments equal to SF1.

For example, if the US term structure is:

l L₀(90)=3.45%

l L₀(180)= 3.58%

l L₀(270)= 3.7%

l L₀(360)= 3.75%

The present value factors are obtained as follows:

l B₀(90)=1/[1+0.0345(90/360)]

l B₀(180)=1/[1+0.0358(180/360)]

l B0(270)=1/[1+0.0370(270/360)]

l B0(360)=1/[1+0.0375(360/360)]

we can get the fixed rate of 3.68% for US dollar.

At day 60, we face a new term structure of Libors.

l L60(30) = 0.0425

l L60(120) = 0.0432

l L60(210) = 0.0437

l L60(300) = 0.0444

The new set of discount factors is

l B0(30)=1/[1+0.0425(30/360)]=0.9965

l B0(120)=1/[1+0.0432(150/360)]=0.9858

l B0(210)=1/[1+0.0437(240/360)]=0.9751

l B0(300)=1/[1+0.0444(330/360)]=0.9643

If the SF term structure is:

l R(90-day)=5.2%

l R(180-day)=5.4%

l R(270-day)=5.55%

l R(360-day)=5.7%

We would have a fixed rate of 5.56 percent in Swiss francs.

The notional principal would be $1.0 and 1/$0.80 = SF1.25. Summarizing, we have the following terms for the four swaps:

Swap 1: Pay dollars fixed at 3.68 percent, receive SF fixed at 5.56 percent.

Swap 2: Pay dollars fixed at 3.68 percent, receive SF floating.

Swap 3: Pay dollars floating, receive SF fixed at 5.56 percent.

Swap 4: Pay dollars floating, receive SF floating.

选项：

解释：

In the swap 4 (floating for floating), there is no pricing problem because there is no fixed rate.

In each case, the notional principal is $1 and SF1.25, or more generally, SF1.25 for every dollar of notional principal.

As we did with interest rate swaps, we move 60 days forward in time. We have a new US term structure, given in the interest rate swap problem, and a new Swiss franc term structure, which is given below:

l LSF60(30)=0.0600

l LSF60(120)=0.0615

l LSF60(210)=0.0635

l LSF60(300)=0.0653

The new set of discount factors is

l BSF60(30)=1/[1+0.0600(30/360)]=0.9950

l BSF60(120)=1/[1+0.0615(120/360)]=0.9799

l BSF60(210)=1/[1+0.0635(270/360)]=0.9643

l BSF60(300)=1/[1+0.0653(360/360)]=0.9484

The new exchange rate is $0.82. Now let us value each swap in turn, taking advantage of what we already know about the values of the US dollar interest rate swaps calculated in the previous section. Recall we found that

l Present value of dollar fixed payments = 1.0004

l Present value of dollar floating payments = 1.0051

Let us find the comparable numbers for the Swiss franc payments. In other words, we position ourselves as a Swiss resident or institution and obtain the values of the fixed and floating streams of Swiss franc payments per SF1 notional principal. The present value of the remaining Swiss fixed payments is

0.0139(0.9950 + 0.9799 + 0.9643 + 0.9484) + 1.0(0.9484) = 1.0024

Recall that in finding the present value of the floating payments, we simply recognize that on the next payment date, we shall receive a floating payment of 0.052(90/360) = 0.013, and the market value of the remaining payments will be 1.0. Thus, we can discount 1.0130 back 30 days to obtain 1.0130 (0.9950) = 1.0079.

These two figures are based on SF1 notional principal. We convert them to the actual notional principal in Swiss francs by multiplying by SF1.25. Thus,

l Present value of SF fixed payments = 1.0024(1.25) = SF1.2530

l Present value of SF floating payments = 1.0079(1.25) = SF1.2599

Now we need to convert these figures to dollars by multiplying by the current exchange rate of $0.82. Thus,

l Present value of SF fixed payments in dollars = 1.2530($0.82) = $1.0275

l Present value of SF floating payments in dollars = 1.2599($0.82) = $1.0331

Now we can value the four currency swaps:

l Value of swap to receive SF fixed, pay $ fixed = +$1.0275 – $1.0004 = +$0.0271

l Value of swap to receive SF floating, pay $ fixed = +$1.0331 – $1.0004 = +$0.0327

l Value of swap to receive SF fixed, pay $ floating = +$1.0275 – $1.0051 = +$0.0224

l Value of swap to receive SF floating, pay $ floating = +$1.0331 – $1.0051 = +$0.0280