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Hi all I have a problem when I have to calculate swaps/swaptions.

n=10-period binomial model for the short-rate, ri,j. The lattice parameters are: r0,0=5%, u=1.1, d=0.9 and q=1−q=1/2.

1.Compute the initial value of a forward-starting swap that begins at t=1, with maturity t=10 and a fixed rate of 4.5%. (The first payment then takes place at t=2 and the final payment takes place at t=11 as we are assuming, as usual, that payments take place in arrears.) You should assume a swap notional of 1 million and assume that you receive floating and pay fixed.)

2.Compute the initial price of a swaption that matures at time t=5 and has a strike of 0. The underlying swap is the same swap as described in the previous question with a notional of 1 million. To be clear, you should assume that if the swaption is exercised at t=5 then the owner of the swaption will receive all cash-flows from the underlying swap from times t=6 to t=11 inclusive. (The swaption strike of 0 should also not be confused with the fixed rate of 4.5% on the underlying swap.)

I have tried calcultating the first one with Forward Equations in the periods 4,3 but I cannot resolve it.

I have tried it once again but the answers are not correct.

Here is what I have done:

excel swaps

Thanks in advance

Edit with Lattice Model:

enter image description here

Katherine99
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2 Answers2

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Start at time $i =10$.

For each node $(10,j)$ with j = $0,1,\ldots,10$, the forward price of the swap (ex payments received at time 10) is a discounted, expected value:

$$S_{10,j} = (1 + r_{10,j})^{-1}\left(\frac1{2}S_{11,j} + \frac1{2}S_{11,j+1}\right),$$

where, for a receive fixed / pay float swap,

$$S_{10,j} = 1,000,000(0.045 - f_{10,11,j}) = 1,000,000(0.045 - r_{10,j}).$$

Note that the forward rate $f_{10,11,j}$ equals $r_{10,j}$ on a tree where the time spacing between the nodes matches the period for floating-rate resets and fixed rate payments.

Now find the forward swap price at each node $(9,j)$ with j = $0,1,\ldots,9$:

$$S_{9,j} = (1 + r_{9,j})^{-1}\left(\frac1{2}S_{10,j} + \frac1{2}S_{10,j+1}+ Q_{10,j}\right),$$

where the net payment received at time $i = 10$ is

$$Q_{10,j} = 1,000,000(0.045 - f_{9,10,j}) = 1,000,000(0.045 - r_{9,j}).$$

Work your way back on the tree until you find the current swap price $S_{0,0}$. Since this is a forward starting swap beginning at time $i=1$, do not include any net payments $Q_{1,j}$.

To price the swaption, set the terminal values at expiry $i = 5$ and $j = 0,1,\ldots,5$ to

$$C_{5,j} = \max(S_{5,j} ,0).$$

Then work backwards from $i = 5$, calculating discounted expected values at each node until you arrive at the current price $C_{0,0}$.

RRL
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  • Thank you very much for the given explanations – Katherine99 Nov 24 '14 at 19:20
  • I have edited the question with the steps I have done – Katherine99 Nov 24 '14 at 20:22
  • @Katherine99: You're welcome. Ill take a look. – RRL Nov 24 '14 at 21:08
  • A few mistakes. (1) You have float minus fixed payments, but that just changes the sign of the swap. (2) You should have $d = 1/u$ exactly so the tree recombines. (3) You have not buolt the short rate tree correctly. It should be a triangle with 0.05 at the vertex. Nodes in the upper half have rates $r_{ij} = ur_{i-1,j-1}$ and nodes in the lower half have rates $r_{ij} = dr_{i-1,j}$. The value of the swap should be close to 0 if you make the fixed rate 0.05. It should be negative with fixed rate 0.045. – RRL Nov 24 '14 at 23:05
  • hi RRL, thank you very much for your help and patience again. Sorry for not having linked the lattice model before, but in my model d=1,1 and u=0.9.I checked also that chaning the fixed rate to 0.05 the swap price is close to 0. However the result is not the correct one. As said, thank you again for your help – Katherine99 Nov 24 '14 at 23:56
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Compute the initial value of a forward-starting swap that begins at t=1, with maturity t=10 and a fixed rate of 4.5%. (The first payment then takes place at t=2 and the final payment takes place at t=11 as we are assuming, as usual, that payments take place in arrears.) You should assume a swap notional of 1 million and assume that you receive floating and pay fixed.)

Submission Guideline: Give your answer rounded to the nearest integer. For example, if you compute the answer to be -220,432.23, submit -220432.

Julio Caesars
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