Mathematical Trading and Finance

In this coursework, you will be given the opportunity to gain hand-on experience in
using Excel VBA to implement some numerical methods which are useful in finance,
in particular, option valuation. It will also give you the opportunity to develop re-
port writing skill. In this couse work, you are required to write Excel VBA codes
for implementing numerical methods and write a short report, in Word or Adobe
pdf format. The report should not be longer than FOUR pages with double line
spacing. One short report is required for this coursework. You are required to
mention clearly which questions and parts of questions you are answering in differ-
ent parts of the report. The report must be uploaded to Moodle, together with the
Excel VBA files containing the analyses. Both sets of files must be uploaded
to Moodle to obtain the marks. This is an individual coursework. You are
required to work on ALL of the following questions and answer all of them by your-
self. This coursework will constitute 50% of the total marks of the module. Some
marks (25%) will be awarded for the presentation of the report, Excel VBA codes
and spreadsheets, but the bulk of the marks will be based on the appropriateness
of the analyses and the correctness of the results.
Attempt all of the following FOUR questions.
Question 1: (20 Marks)
Consider the following polynomial:
p(x) :137-1-3135-2132-1-6
We wish to evaluate numerically the following definite integral using quadratures with the number
of sub-intervals n : 12:
10
p(x)d.r
Write Excel VBA codes and display the results on spreadsheets for answering the following parts
of the question whenever appropriate.
(i) Compute the definite integral using the trapezoidal rule.
(ii) Compute the definite integral using the general Simpson’s rule and the general Simpson’s
three-eighth rule.
(iii) Compute the definite integral using the general Boole’s rule.
(iv) Comment on the accuracy and computational efficiency of the numerical integration methods
used in Parts (i)-(iii).

Question 2 (20 Marks)
Consider the following complex-valued function:
634-1205
f – 7
We wish to evaluate the following definite integral:
wxndx
-1
where is the real part of
Write Excel VBA codes and display the results on spreadsheets for answering the following parts
of the question whenever appropriate.
(i) Compute the definite integral using the trapezoidal rule with 10 subintervals.
(ii) Compute the definite integral using the 10-point Gauss-Legendre quadrature.
(iii) Compute the definite integral using the general Gauss-Legendre quadrature with six points.
(iv) Comment on the accuracy and computational efficiency of the numerical integral methods
used in Parts – (iii).

Question 3 (30 Marks)
In the Black-Scholes option pricing model, the price process {St} of an underlying security is
governed by a geometric Brownian motion (GBM) as follows:
dSt ∶ −⊢ O’Stth
where ⇂≴ and a are the appreciation rate and the volatility of the underlying security, respectively;
{Wt} is a standard Brownian motion under a real-world probability 73.
For valuing options, instead of using the real-world probability P, the risk-neutral probability Q
is used. Under Q, the price process {St} of the underlying security becomes:
(13): ∶ TStdt −⊢ O’StthQ
where 1″ is the risk-free continuously compounded interest rate and {W9} is a standard Brownian
motion under Q
We consider a European put option and an American put option with common strike price
K : 80 and common maturity T : 0.6 years. Suppose that the current underlying security’s
price So : 100. The risk-free rate 7“ : 2.5% and the volatility a : 25%.
Write Excel VBA codes and display the results on spreadsheets for answering the following parts
of the question whenever appropriate.
(i) Compute the current European and American put prices using the CRR binomial tree with
the number of steps n ranging from 20 to 500 with an increment of 20.
(ii) Compute the current European and American put prices using the trinomial tree with the
number of steps 71 ranging from 20 to 500 with an increment of 20.
(iii) Comment on the accuracy, convergence and computational efficiency of the methods in Parts
(i) and (ii).
Instead of valuing the European and American put options, we now consider a European cash-
or-nothing put option and an American cash-or-nothing put option with common payoff P ∶ 10,
common strike price K : 80 and common maturity T : 0.6 years. Again, the current underlying
security’s price SO : 100. The risk-free rate 7’ : 2.5% and the volatility a : 25%.
(iv) Compute the current European and American cash-or-nothing put prices using the CRR
binomial tree with the number of steps 71 ranging from 20 to 1200 with an increment of 20.
(V) Compute the current European and American cash-or-nothing put prices using the trinomial
tree with the number of steps 71 ranging from 20 to 1200 with an increment of 20.
(vi) Comment on the accuracy, convergence and computational efficiency of the methods in Parts

Question 4 (30 Marks)
Consider an American put option with strike price K : 30 and maturity T : 2.5 years. Suppose
that the current underlying security’s price SO : 50. The risk-free rate r : 2% and the volatility
a ∶ 30%. We wish to use the Tian flexible binomial tree and the CRR binomial tree to price the
American put.
Write Excel VBA codes and display the results on spreadsheets for answering the following parts
of the question whenever appropriate.
(1) Compute the first-period means and standard deviations of the underlying security’s price
under the risk-neutral probability using the CRR binomial tree with the number of steps 71
ranging from 20 to 1000 with an increment of 20.
(ii) Compute the first-period means and standard deviations of the underlying security’s price
under the risk-neutral probability using the Tian flexible binomial tree with the tilting
parameters A : -0.57 A : 0.5 and the number of steps 71 ranging from 20 to 1000 with an
increment of 20.
(iii) Compute the first-period means and standard deviations of the underlying security’s price
under the risk-neutral probability using the Tian flexible binomial tree with the tilting
parameter given by the formula in Tian (1999) and the number of steps 71 ranging from 20
to 1000 with an increment of 20. Comment on the first-period means and standard deviations
obtained in Parts – (iii).
(iv) Compute the current prices of the American put using the CRR binomial tree with the
number of steps 71 ranging from 20 to 1000 with an increment of 20.
(v) Compute the current prices of the American put using the T ian flexible binomial tree with
the number of steps 71 ranging from 20 to 1000 with an increment of 20, the tilting parameter
A : 0.5 and the tilting parameter given by the formula in Tian (1999).
(vi) Comment on the accuracy, convergence and computational efliciency of the American put
prices obtained in Parts (iv)-(v).

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