Fixed Income

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Part III. Pricing (15 marks) Assume that these rates are annual bond versus 1 year LIBOR annual bond.

Maturity Swap Rate

1 Year 6.00%

2 Year 6.25%

3 Year 6.50%

4 Year 6.75%

5 Year 7.00%

1. What is the 5 year swap rate? Please build up the term structure of the forward rates and corresponding zero coupon bond prices.

2. What is the 2x5 swap rate? Here 2x5 swap refers to the 3-year swap, two years forward.

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Appendix Bootstrap Method Following Wilmott (2007), the bootstrap method for deriving the yield curve from market swap rates is based upon rearranging the formula for the swap rate to solve for the longest discount factor. The time 𝑡 swap rate is

𝑆 = 1 − 𝛿(𝑡 )

∑ 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 )    ,

Where 𝛼 (𝑡 ,𝑡 )is the day count fraction between times 𝑡 and 𝑡 , and 𝛿(𝑡 ) is the discount factor for cash flows occurring at time 𝑡 . Rearranging the above equation,

𝑆 = 1 − 𝛿(𝑡 )

𝛼 (𝑡 ,𝑡 )𝛿(𝑡 ) + ∑ 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 )

⇒ 𝑆 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 ) + 𝑆 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 ) = 1 − 𝛿(𝑡 )    ,

so

𝛿(𝑡 ) = 1 − 𝑆 ∑ 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 )

1 + 𝑆 𝛼 (𝑡 ,𝑡 )    .

If we have available the swap rates for many equally-spaced maturities, we may set 𝛼 (𝑡 ,𝑡 ) = 𝛼 and it simplifies to

𝛿(𝑡 ) = 1 − 𝑆 𝛼 ∑ 𝛿(𝑡 )

1 + 𝑆 𝛼

Then, starting with the one-period swap rate, 𝑆 and noting that 𝛿(𝑡 ) = 𝛿(0) = 1, we have

𝛿(𝑡 ) = 1

1 + 𝑆 𝛼  . The remaining discount factors  𝛿(𝑡 ), 𝑗 = 2,…,𝑛 can then be found sequentially by applying

𝛿 𝑡 = 1 − 𝑆 𝛼 ∑ 𝛿(𝑡 )

1 + 𝑆 𝛼    .

Note that in practice the swap rates we observe in the market may not be equally spaced, nor may they have identical payment frequencies and/or day count bases. Almost certainly the swap contracts will not actually start on the pricing date 𝑡 = 𝑡 . Furthermore, we may wish to introduce elaborate schemes for interpolating between the discrete maturity dates available to us. Such considerations lead to much more complicated bootstrapping algorithms but which are all founded on the principles presented here.

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Reference D. Brigo, F. Mercurio. Interest Rate Models: Theory and Practice. Springer, 2nd edition, 2006. A. Cairns. Interest Rate Models. Princeton University Press, 2004. F. A. Longstaff and E.S. Schwartz. Interest rate volatility and the term structure: A two-factor general equilibrium model. The Journal of Finance, 47:1259–1282, 1992. F. A. Longstaff and E.S. Schwartz. Implementation of the longstaff-schwartz interest rate model. Journal of Fixed Income, September: 7–14, 1993.11 Riccardo Rebonato. Interest-Rate Option Models. John Wiley and Sons, 2nd edition, 1996. P. Wilmott. Paul Wilmott introduces Quantitative Finance. John Wiley and Sons, 2nd edition, 2007.