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:
Thanks in advance
Edit with Lattice Model:
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 $C_{0,0}$.