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Numerical Methods in Finance
Columbia University
MATH GR5030 Spring 2022
Numerical Methods in Finance
Professor Tat Sang Fung
Homework 2: Interpolation

Distribution date Due date
Homework Feb 7, 2022 (Mon) Feb 28, 2022 (Mon) 7:30pm
Note: please upload all homework solution before the due date and time from CourseWorks. If you suspect
there are typos in this homework, or some questions are wrong, please feel free to email the instructor.

PRACTICE
QUESTION 1
Suppose you are given the market data of the volatility smile by delta for a one year1 option:
Make an Excel spreadsheet that calculates
a) the linear interpolated vol
b) the polynomial interpolated vol using Neville’s algorithm2
c) the cubic spline interpolated vol 3
for any given delta that lies between the smallest and largest delta
In particular, calculate the value of the vol that corresponds to the Delta of 45 and save the spreadsheet
showing the value and calculation (it is preferable to show some intermediate calculations in some
cells)
d) Produce a graph of the interpolated smile
When you plot the graph, plot it in such a way that the values of Delta decreases when you look at the x-
axis from left to right. This corresponds to increasing strikes for call options.
The spreadsheet should have a toggle (or some control) to let user choose which interpolation he would like
to use. And the graph shows the entire interpolated smile of his choice. The user should in general be able
to change the values of the input Deltas, and the input vols (then spreadsheet would recalculated the
interpolated vol upon request)
For the purpose of this homework, you can assume that the user always gives you nine points, and will not
change the number of given points.
For the cubic spline, you should provide a choice for users to control the “2 degree of freedom” discussed
in class. If you feel that some combinations of the choice should not be allowed, please remember to rule
them out. Please save the spreadsheet when handing in to default to the natural cubic spline if chosen.

1 Let’s assume one year means 365 calendar days
2 Although there are other methods to find this value, you have to use this method for the purpose of this
home work to show the understanding of this particular method. You should have the “tree like” picture
displayed somewhere in the spreadsheet
3 During the computation, you will need to invert a “tri-diagonal” matrix. Please do so without using
Excel’s matrix inversion routine (you can do it within Excel with row reduction as discussed in class etc)
Delta 86.03543 81.03543 76.03543 61.03543 50 25 10 5 0
Vol 20.14% 18.64% 17.14% 15.64% 14.28% 15.04% 15.74% 16.44% 17.14%

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Numerical Methods in Finance
THEORY
QUESTION 2
The Vandermonde Matrix is one that has the form

a) How is Vandermonde Matrix related to the existence and uniqueness of polynomial interpolation4?
b) Find the polynomial expression for the determinant of this matrix for
c) Show that the determinant of this matrix is not zero if are distinct for
d) Find the polynomial expression for the determinant of this matrix for general . Show that the
determinant of this matrix is not zero if are distinct.
QUESTION 3
The purpose of this question is to illustrate that for linear interpolation, the two points (that determines the
line) can be chosen strategically to give the smallest theoretical upper bound for the interpolation error.
This question is taken from [StBu] page 134 question 3
a) Consider a function that is twice continuously differentiable on the interval .
Interpolate the function by a linear polynomial through that given points and
where . Prove that

is an upper bound for the maximal absolute interpolation error on the interval .
b) Which values minimize ?
i. Imagine answering if a trader asks you the question and is only interested in the answer
ii. Now, imagine answering the above if a risk manager asks you this and is interested to know
the explanation (perhaps some intuition, but not necessarily a proof)
iii. Now, imagine a desk Quant or a mathematician asks you this and is interested in a proof when
you are to answer
c) Create a spreadsheet and confirm visually that your minimize with a numerical
validation, for example, like this:

4 Hint: we discussed that in class…
ú
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ê
ë
é
n
nnn
n
n
xxx
xxx
xxx
2
1
2
11
0
2
00
1
1
1
1
!!!
!!
"
"
2,1=n
nxxx ,,, 10 ! 2,1=n
n
nxxx ,,, 10 !
[ ]1,1I = -
( )( )0 0,x f x
( )( )1 1,x f x 0 1,x x IÎ
( ) ( )( )0 1
1 max max
2 I x I
f x x x x
x
a x
Î Î
¢¢= × - -
I
0 1,x x IÎ a
0 1,x x IÎ a
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Numerical Methods in Finance

Your spreadsheet should allow user to specific the range of the x-axis and y-axis so that it can be zoomed
to see more dights. The above shows both axis starting from -1 to 1. For example if user want to zoom to
see it so that x-axis showings from 0.6 to 0.8 and y-axis showings from -0.6 to -0.8:

This way users can patiently get to a very precise answer for .
You do not need to submit the spreadsheet, please submit screen shots that would show a four decimal
accuracy for that minimize
QUESTION 4
a. Let the continuous compounding zero rate for maturity . Let and be the
forward zero rate for the period . Prove that5 .
b. Let be the instantaneous forward rate6 for maturity . Suppose is once differentiable.
Prove that .

5 See [Hull] book if you need a refresh on this topic
0 1,x x IÎ
0 1,x x IÎ a
( )Tr T 21 TT < ( )21,TTrF
[ ]21,TT ( )
( ) ( )
12
1122
21, TT
TTrTTrTTrF -
-
=
( )Tf T ( )Tr
( ) ( ) ( )
T
TrTTrTf
¶
¶
+=
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Numerical Methods in Finance
c. Consider a set of zero rates are given: and piecewise linear interpolation is
used on . Discuss under what condition the instantaneous forward rate would be defined
everywhere and is continuous between and . Show that if all given are non-negative, then
is also non-negative for all
d. Consider a set of zero rates are given: and piecewise linear interpolation is
used on . Derive an expression7 for . Show that if all given are non-negative, then
is also non-negative for all
QUESTION 5
This question examines the “internal consistency” of a chosen interpolation. Loosely speaking: does
interpolation based on interpolated values gives you the original interpolated values too?
Let !! be the value of corresponding to some time ". For # ≤ % ≤ & and given values !" and !# , denote the
interpolated value at % by '(")*[%, (#, !"), (&, !#)].
Let 0 ≤ 1 ≤ 2 ≤ 3 ≤ 4.
Suppose we consider !$ and !% are known. We can compute a value for !& with 5 ='(")*[1, (0, !$), (4, !%)] , a value for !'with 7 = '(")*[2, (0, !$), (4, !%)] and a value for !(with 8 ='(")*[3, (0, !$), (4, !%)].
Noting that now if we consider !& and !( as known with given values of 5 and 8 respectively, we can also
compute a value for !' with 7∗ = '(")*[2, (1, 5), (3, 8)].
i. If the chosen interpolation above is “piecewise linear” interpolation, do you think 7=7∗ always? If
yes, please prove it mathematically; if no please provide a counterexample
Recall in class we discussed “Linear interpolation on the variance” with formula

ii. If the chosen interpolation above is “Linear interpolation on the variance”, do you think 7=7∗
always? If yes, please prove it mathematically; if no please provide a counterexample
Let & > 0 be a constant. Let :* be the volatility for term ;. Consider a new interpolation method where we
set :* = < *+*!*"+*!=# :*" + <*"+**"+*!=# :*! for ;$ ≤ ; ≤ ;,
iii. If the chosen interpolation is the above interpolation with & = 2, do you think 7=7∗ always? If
yes, please prove it mathematically; if no please provide a counterexample
QUESTION 6
This question relates to the way to determine the cubic spline coefficients. The textbook [StBu] did not
supply a complete proof. You are therefore asked to prove it here.
For notations, please refer to section [StBu]2.4.2.

6 See [Hull] page 85 for a definition. Here we take the limit as .
7 This is sometimes known as the “RT” interpolation among the practitioners…
( )( ) ( )( )nn trttrt ,,,, 00 !
( )tr ( )Tf
0t nt ( )itr
( )tr nttt ££0
( )( ) ( )( )nn trttrt ,,,, 00 !
( ) ttr × ( )tr ( )itr
( )tr nttt ££0
1 0
2 20 1
1 0
1 0 1 0
1
T T T
T T T TT T
T T T T T
s s s
æ ö- -
= +ç ÷- -è ø
+® 12 TT
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Numerical Methods in Finance
let be the target spline function and we let for . These are the
unknowns to be solved.
Prove that:
for lying in where




where the unknowns ’s are the solutions of the linear system

with the quantities being for . with



Show also that for the two degrees of freedom:
• If for the beginning we have the condition on the first derivative, we set and
; if we have the condition on the second derivative, we set and

( )S x ( )j jM S x¢¢= 0,1, ,j n= !
( ) ( ) ( ) ( )2 3j j j j j j jS x x x x x x xa b g d= + - + - + - x 1,j jx x +é ùë û
j jya =
1 1
1
1
2
6
j j j j
j j
j
y y M M
h
h
b + + +
+
- +
= -
2
j
j
M
g =
1
16
j j
j
j
M M
h
d +
+
-
=
1n + jM
0 0 0
1 1 1 1
2
1 1 1
2
2
2
2
2
n n n
n n n
M d
M d
M d
M d
l
µ l
µ
l
µ
- - -
é ù é ù é ù
ê ú ê ú ê ú
ê ú ê ú ê ú
ê ú ê ú ê ú
ê ú ê ú ê ú=ê ú ê ú ê ú
ê ú ê ú ê ú
ê ú ê ú ê ú
ê ú ê ú ê ú
ê ú ê ú ê úë û ë û ë û
! ! " "
! ! !
!
1,2, , 1j n= -!
1
1
j
j
j j
h
h h
l +
+
=
+
1j jµ l= -
1 1
1 1
6 j j j j
j
j j j j
y y y y
d
h h h h
+ -
+ +
æ ö- -
= -ç ÷ç ÷+ è ø
0 1l =
1 0
0 0
1 1
6 y yd y
h h
æ ö- ¢= -ç ÷
è ø
0 0l =
00 2yd ¢¢=
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Numerical Methods in Finance
• If for the end we have the condition on the first derivative, we set and
; if we have the condition on the second derivative, we set
and
REFERENCES
[StBu] J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Springer
[Hull] John Hull, Options, Futures and other derivatives, Prentice Hall, 8th edition (you will find similar
discussion in the latest editions also)

1nµ =
16 n n
n n
n n
y yd y
h h
-æ ö-¢= -ç ÷
è ø
0nµ =
nn yd ¢¢= 2