Description
• For these problems use the Salomon model and take the relative volatility of the short rate to be 25%
annually. We will consider only flat par yield curves, i.e., of the form c(t) = r for all t. I have given you
starter code DandC.cpp and some TeX files for graphical display. (You should probably also refer to the code
SalomonModel.cpp in the Tool Box — specifically to the callable bond valuation function.)
1. Suppose Dollars & Cents Finance wants to sell you a security that works as follows. D&C will make
a payment into an account every 6 months starting immediately. The first payment (made today) is $300.
Subsequent payments decline by $5 every 6 months to 0 at the end of 30 years. The account initially has
nothing in it and it earns no interest on its balance. You have the option to withdraw the entire balance
at any time — but when you withdraw the balance the payments stop and the arrangement is terminated.
Calculate and plot the fair price for this security as a function of the rate r for values of r ranging from 0
to 10% in steps of 1%. For r = 5% graphically display the nodes in the lattice at which it is optimal to
withdraw (as we did in class for when to call a callable bond). Explain the results. [Hint: when r = 5% the
fair value is 3737.87.]
2. Duration is a measure of a fixed-income security’s price sensitivity to changing interest rates. Its units
are years and it is defined by D = −
1
P
dP
dr , where P is the security’s value and r is the rate level. Numerically
estimate the duration of the security in problem 1 if r = 5% as follows. Shift the entire par curve up by an
amount ∆r and re-price the security — call this price P
+. Now shift the entire par curve down by ∆r and
compute P
−. If P is the price with the 5% par curve, we have
D ≈ −
1
P
P
+ − P
−
2∆r
.
Estimate the duration first by taking ∆r to be 100 basis points and then with ∆r = 1 bp. Which estimate
do you think is best?
3. A fixed-income security’s convexity is defined by C =
1
P
d
2P
dr2 and may be similarly numerically approximated:
C ≈
1
P
P
+−P
∆r −
P −P
−
∆r
∆r
=
1
P
P
+ − 2P + P
−
∆r
2
.
Estimate the convexity of the security in problem 1 if r = 5% by taking ∆r to be 100 basis points and also
1 bp. Which estimate do you think is best?
4. Superimpose on your price plot in problem 1 a quadratic fit of the form Pb(r) = a(r−.05)2+b(r−.05)+c.
Take c = P(.05) = 3737.87; b should be your numerical estimate of P
0
(.05) from problem 2; a should be
1
2
of your estimate of P
00(.05) from problem 3. Do this first for estimates coming from the 100 bp shift and
then for the estimates coming from the 1 bp shift. Which produces the better fit near r = .05? Explain.
I challenge you to solve this!:
5. Suppose the security permits you withdraw the money k times before D&C stops making the payments
and terminates the arrangement. Here 1 ≤ k ≤ 60. For example, if k = 3, if you so choose you could withdraw
300 + 295 + 290 = 885 at time 1 year, then 285 at time 1.5 years, and then 280 + 275 = 555 at time 2.5 years
at which point the arrangement would terminate. Show the fair price as a function of k.
• In addition to numerical and graphical data, please carefully write up the algorithm that you use to
solve problem 1 and, hopefully, problem 4. Your write-up should be sufficiently detailed and clear that I
could code it up! Submit your code in a separate file.
