MATH60082 (Computational Finance)
Main Assignment: Simulation Methods
Version 10794803
1 Background
1.1 European style Financial Contracts on a Single Underlying Asset
The trader has calibrated a specialised risk neutral process for some underlying asset price. Given the current
asset is S0, market prices indicate the risk-neutral distribution of the asset price at time T is given by:
ST ∼ N(f(S0, T ), v2(S0, T )T ) (1)
for some calibrated functions f and v2.
Consider a financial contract C(S, t) written on the underlying asset S with payoff on expiry
C(S, T ) = g(S).
Then for any given payoff, the analytic solution may be found by carrying out the numerical integration
C(S0, t = 0) =
e−rT
v
√
2πT
∫ ∞
−∞
g(z) · exp
[
− (z − f)
2
2v2T
]
dz. (2)
To carry out a Monte Carlo valuation of the financial contract, we may use samples from a standard
random normal distribution
ϕ ∼ N(0, 1) (3)
to write the equation
SiT = f(S0, T ) + v(S0, T )
√
Tϕi. (4)
Then we can simply average out the discounted payoff over n to get our approximation to the value of the
financial contract:
C(S0, t = 0) ≈ e−rT 1
n
n∑
i=1
g(SiT ) (5)
where g(S) is the desired payoff of the contract.
1.2 Path Dependent Options
Now assume that the risk neutral stochastic process follows the SDE
dS = f(S, t)dt+ v(S, t)dW.
The path dependent options you will be pricing depends on S(tk) which are the share prices at K+1 equally
spaced sampling times t0, t1,..., tk = k∆t, ..., tK with t0 = 0, tK = T and
∆t =
T
K
.
1
Use an Euler type scheme to write
Si(tk) = S
i(tk−1) + f(Si(tk−1), tk−1)∆t+ v(Si(tk−1), tk−1)
√
∆tϕi,k−1
for k = 1, 2, ...,K to estimate the underlying asset values at each time. Here ϕi,k is a random draw from a
Normal distribution.
For path dependent options, the payoff function can be written g(S(t0), ..., S(tk), ..., S(tK)) and so the
value of the path dependent option can be approximated by
V (S(t0), t = 0) ≈ e−rT 1
n
n∑
i=1
g(Si(t0), ..., S
i(tk), ..., S
i(tK)). (6)
Asian Option
Assume that a discretely sampled Asian option has a payoff depending on the discretely sampled average
given by
A =
1
K
K∑
k=1
S(tk).
Then we can write
g(S(t0), ..., S(tK)) = G(S(tK), A),
where G(S,A) is the payoff function depending the type of option.
There are different classes of Asian option, resulting in different payoff conditions. In this coursework we
look at simple European style call or put options. A fixed strike call option will have the payoff
G(S,A) = max(A−X, 0)
where X is the strike price and a floating strike call option would be
G(S,A) = max(S −A, 0).
where A is sometimes called the average strike price.
A fixed strike put option will have the payoff
G(S,A) = max(X −A, 0)
where X is the strike price and a floating strike put option would be
G(S,A) = max(A− S, 0).
where A is the strike price.
Lookback Option
The discretely sampled Lookback option has a payoff depending on the discretely sampled maximum or
minimum given by
A = max
k
S(tk),
or
A = min
k
S(tk).
Then we can write
g(S(t0), ..., S(tK)) = G(S(tK), A),
where G(S,A) is the payoff function depending the type of option.
2
There are different classes of Lookback option, resulting in different payoff conditions. In this coursework
we look at simple European style call or put options. We can either have a floating strike S or a fixed strike
X. For example a floating strike Lookback call option would give
G(S,A) = max(S −A, 0)
where A must be the minimum, and a floating strike Lookback put option would be
G(S,A) = max(A− S, 0).
where A must be the maximum.
A fixed strike call option will have the payoff
G(S,A) = max(A−X, 0)
where X is the strike price and A must be the maximum. and a fixed strike put option will have the payoff
G(S,A) = max(X −A, 0)
where X is the strike price and A must be the minimum.
Barrier Options
The discretely sampled knock-out barrier option will be knocked out (and return a value of zero) if the a
barrier asset price B is crossed before the maturity date.
The option will be an “up” option if the knock out condition is on S > B, or a “down” option if the
condition is on S < B.
Let the variable A be a binary variable such that
A =
{
1 if the barrier is never crossed,
0 if the barrier is crossed at least once.
Then we can write
g(S(t0), ..., S(tK)) = A ·G(S(tK)),
where G(S) is the payoff function depending the type of option.
So for example an up-and-out knockout barrier call option has the payoff
G(S) = max(S −X, 0)
where
A =
{
0 if S(tk) > B for any k = 1, 2, . . . ,K
1 otherwise
and a down-and-out knockout barrier put option will be
G(S) = max(X − S, 0)
where
A =
{
0 if S(tk) < B for any k = 1, 2, . . . ,K
1 otherwise
3
2 Tasks
2.1 European style Financial Contracts on a Single Underlying Asset
You must value a financial contract written on an underlying asset S where the distribution of the stock
price at time T is described by the functions
f(S0, T ) = S0(2− eαT − eβT ) + θ cos(αT + βT ), v(S0, T ) = σe(2α−β)T 1
4
(3S0 + θ)
γ .
The payoff on expiry for the function is
C(ST , T ) = g(ST ) =
 X1 − ST if ST < X1X2 −X1 if X1 ≤ ST < X2
X2 if ST ≥ X2
The stock price at time t = 0 is S0 = 451.022, and the risk-free interest rate is r = 0.05. According to
the contract, the option matures at T = 1.25 with strike prices of X1 = 450 and X2 = 550. The market
fitted parameters are θ = 451, α = 0.03, β = 0.04, γ = 1.02, and the volatility of the option is σ = 0.35.
• Write a program that uses Monte Carlo simulation (5) to approximate the value of the financial contract
C and also calculate the value using numerical quadrature (2) both with the parameters stated above.
You should state in your report the value of the contract using the analytic solution, and your best
estimate for the value of the financial contract using a Monte Carlo approximation. You need only
include the code in the appendix of your report. (Coding 3 marks)
• Plot a single figure showing your Monte Carlo approximate of the financial contract value C(S0, t = 0)
with increasing n (n = 1000, 2000, . . ., 50000, or more!), alongside the exact value from the analytical
formula. Comment on the appearance and accuracy of your result. (Understanding 6 marks)
• Write a new program (or programs) that use different random number generators, antithetic variables,
moment matching and/or Halton sequences to approximate the financial contract C and investigate
how the accuracy of the approximation changes with different value of n and how long each calculation
takes. You should produce at most 4 plots or tables of your results, and write about them in the
text to justify how the methods rank against each other in terms of their efficiency. (Understanding 4
marks, Originality/Initiative 4 marks)
2.2 Path Dependent Options
Now assume that our underlying asset follows the risk neutral process:
dS = (αθ − βS)dt+ σ(|S|)γdW (7)
where W is a Wiener process and α, β, γ, θ, and σ are all constant parameters determined by the market.
Your task is to price a discrete maximum fixed-strike lookback call option V with the following parameters.
As before, the risk free interest rate is r = 0.05, the market parameters are θ = 451, α = 0.03, β = 0.04,
γ = 1.02 and volatility is σ = 0.35. The stock price is currently S0 = 451.022, the option matures at T = 1.25
with K = 75 equally spaced observations after the initial time. The fixed strike price is X = 450. We do
not have an analytic formula to price this option.
• Write a program to approximate the path dependent option V using (6) with the model and parameters
as stated above. You should state in the report your best estimate for value of the option using a Monte
Carlo approximation. You need only include the code in the appendix of your report. (Coding 2 marks)
4
• Use your program to generate approximations to the path dependent option V with different values of
n. With the outputs from your calculations you should generate atmost 2 plots or tables to investigate
the convergence of your result. Comment on your results, state in the text your best estimate of the
option value. (Understanding 6 marks)
• Finally, estimate the value of the partial derivative ∂V∂σ at S0 and t = 0 using Monte Carlo simulations.
You should be able to generate values of V (S0, t = 0;σ = 0.35) and V (S0, t = 0;σ = 0.35 + ∆σ), so a
simple finite difference formula can approximate the derivative:
∂V
∂σ
≈ V (S0, t = 0;σ = 0.35 + ∆σ)− V (S0, t = 0;σ = 0.35)
∆σ
.
What type of Monte Carlo simulation, what values of N , and what values of ∆σ give the most accurate
estimate of the value of the derivative? Explain any problems you may find using this formula. State
your estimate for the value in the text, you should include up to 2 plots or tables of results to support
your result. (Understanding 4 marks, Originality/Initiative 6 marks)
3 Instructions
The deadline for this assignment is 11am on Monday 18th March, and as part of a 15 credit course unit
you should expect this may take up to 20 hours to complete. Unless you have an agreed extension
on coursework deadlines with DASS reports handed in AFTER 11am Monday 18th March
will be docked 4 marks plus an additional 4 marks each day thereafter until a mark of zero is
reached. Reports handed in after 5pm Friday 29th March will be awarded a mark of zero and
will not be marked.
You must use LATEX to create the document you hand in. Please write your own code and report, you
may work together but do not transfer your codes or report files to another student for them to use, via
email or by sharing them on cloud hosting websites. To do so would be considered plagiarism and it
would carry severe penalties. Please see the university guidance and associated documents on plagiarism:
https://documents.manchester.ac.uk/display.aspx?DocID=2870
In order that your report conforms to the standards for a technical report, you should use the following
structure:
• Typeset using LaTeX, so the only allowed submission filetype is pdf, and it must be submitted without
your name, but with your university ID number online through the TurnItIn system.
• approximately 8 - 10 pages long (excluding appendices)
• be written in continuous prose
• give a brief introduction stating the problem you are solving and the parameters you are using (from
the model or method),
• present your results in the form of figures and tables, using the order of items in the bullet points as a
guide as to the order of your document
• absolutely NO screenshots of running code need to be included,
• do not include overly long tables – a table should never cross over a page,
• present the results for any methods you have implemented, there is no credit for a discussion of a
method that has not been shown to be implemented by you (through results) for your problem
• refer to and discuss each of your results in the text, part of the marks available in each bullet point
are for interpreting the results
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• try to keep to the page limit, removing any unnecessary results from the main text
• number and caption your figures and tables and refer to them by their number (not their position in
the text),
• number any equations to which you refer,
• use consistent internal (and external) referencing.
RUBRIC
This assignment will account for 40% of your final mark in this module. The total number of marks in this
assessment is 40, and they will be awarded as follows:
(i) 5 for working codes;
Grade Description
0-50% Little or no attempt, codes not working
50%-70% One or two bugs in the code are affecting the results
70%-100% The results in 2.1 and 2.2 from the codes appear correct
(ii) 5 for the presentation of your written report;
Grade Description
0-50% Poorly presented work. Significant amount of text unreferenced.
Graphs and tables poorly labelled making it difficult to interpret
them.
50%-70% Good presentation. Text is readable. Graphs are ok, maybe miss-
ing labels and not always referenced correctly. Report is overly
long and unnecessarily repeats the same (or similar) results.
70%-100% Excellent presentation, well written and well referenced. Graphs
are clear, tables used when appropriate. Report keeps within the
page limit.
(iii) 20 for the understanding of the problems involved;
Grade Description
0-50% Results are poorly presented or they are without supporting text.
The methods are described but are not shown to be implemented
through results. The student is unable to demonstrate they can
correctly interpret results.
50%-70% Demonstrates a good understanding of the standard methods. Is
able to generate standard results and discuss them. Results are
well presented.
70%-100% Student is able to correctly interpret standard results and evaluate
the efficiency of the standard methods.
(iv) 10 for originality/initiative.
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Grade Description
0-50% Little or no attempt at implementing any of the new methods.
Those that have been implemented have poorly presented results
or the student is unable to demonstrate they can correctly inter-
pret results.
50%-70% Demonstrates a good understanding of new or alternative meth-
ods. Is able to implement new methods, present results and dis-
cuss them. Results are well presented.
70%-100% Has implemented difficult algorithms not detailed in the course.
Presentation of the results is excellent. Student is able to correctly
interpret results and compare methods in a coherent way.
See individual bullet points in the task section for a break down of the marks.