Differential Swap

[FrankieY18]

on p. 704 in Hull textbook, there is an example on differential swap. I am not understanding what rates are we receiving or paying. Also, I don't understand why year 1 has a net zero cash flow. Can someone please explain? Thanks.

[Car'a'carn]

The example exchanges US LIBOR for UK LIBOR on 10M pounds notional amount. Since initially both rates are 5%, the initial exchanged amount is 0. To calculate the value of the swap we calculate the future forward US rates using the volatility of US rates, dollar to pound exchange rate and correlation between them.

[FrankieY18]

It sounds like you are talking about time 0 has a zero net cashflow. But in the example, it shows 0 in time 1. I do not know how to get the cash flows for years 2 and 3.

[Laurelinda]

It sounds like you are talking about time 0 has a zero net cashflow. But in the example, it shows 0 in time 1. I do not know how to get the cash flows for years 2 and 3.

End-of-period cash flows are determined by the rates observed at the beginning of the period. So since there is no spread between US LIBOR and British LIBOR at time 0, the net cash flow at time 1 will be zero.

For the purpose of valuation, we have to determine expected spreads between US and British LIBOR at all the reset points. Say we're valuing the swap from the perspective of the person receiving US LIBOR. We project the spread of US LIBOR over British LIBOR 1 year from now as the beginning rate (5%) x the volatility of 1-year US forward rates (20%) x the volatility of the forward exchange rate between US Dollars and Pounds Sterling (12%) x the correlation between the two (0.4). So for the time 1 rate we have

5% x 20% x 12% x 0.4 = 0.00048 over British LIBOR, which gives a net positive cash flow of (10,000,000 x 0.00048) = 4,800 at time 2.

For the expected spread at time 2, which will determine the expected cash flow at time 3, we multiply 0.00048 by 2 and get a net cash flow of 9,600 at time 3. If the contract extended another year, we would multiply by 3 for the expected spread at time 3 to project the cash flow at time 4.

Does that help?

[FrankieY18]

I got it now. Thanks.