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Summative project
Computational Finance with Python
Table of contents
General information 1
Marking and credit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Academic integrity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Submission instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 Black-Scholes model 3
1.1 Historical volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Binomial tree approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Local volatility model 8
2.1 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Monte Carlo method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Finite difference methods . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Theta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
References 10
General information
Each student has a unique individualised assignment. Download your assignment directly from the submission point on Moodle.
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Marking and credit
This project contributes 90% towards the final mark for the module.
The project will be marked anonymously. Do not include your name, student number or
any identifying details in your work.
In each exercise, marks are allocated for both coding (style, clarity and correctness) and
accuracy of numerical results, as well as explanations and written work (including comments and docstrings in the code). The marks indicated are indicative only.
Partial credit will be given for partial completion of an exercise, for example, if you are
able to do a Monte Carlo estimate with fewer paths or time steps.
It may prove impossible to examine projects containing code that does not run, or does
not allow data to be changed. In such cases a mark of 0 will be recorded. It is therefore
essential that all files are tested thoroughly in Colab before being submitted in Moodle.
Academic integrity
You may use and adapt any code submitted by yourself as part of this module, including
your own solutions to the exercises. You may use and adapt all Python code provided
in Moodle or on GitHub as part of this module, without the need for acknowledgement.
Any code not written by yourself must be acknowledged and referenced.
This is an individual project. You must not collaborate with anybody else or use code
that has not been provided in Moodle without acknowledgment. This includes, for example, discussing the questions with other students, actively using discussion forums or
using generative artificial intelligence (such as ChatGPT). Please consult the University’s
Student guidance on using AI and translation tools and Policy on Acceptable Assistance
with Assessment. Collusion and plagiarism detection software will be used to detect
academic misconduct, and suspected offences will be investigated.
Submission instructions
Submit your work by uploading it in Moodle by 4pm on 31 May. Work should be submitted as follows:
• Code (and numerical results) should be submitted in the form of Colab notebooks.
Code will be tested using unseen test data: use variables to store parameters rather
than values hard-wired into your code.
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• Written text should be submitted in pdf format. Scans of handwritten text are acceptable, provided that they are legible.
• Use a separate set of files (ipynb and pdf, as appropriate) for parts 1 (Black-Scholes
model) and 2 (Local volatility model). “Part1.ipynb” or “part1.pdf” are examples of
sensible filenames.
Late submissions will incur a penalty according to the standard rules for late submission
of assessed work. It is advisable to allow enough time (at least one hour) to upload your
files to avoid possible congestion in Moodle. In the unlikely event of technical problems
in Moodle please email your files in a zip archive to alet.roux@york.ac.uk before the
deadline.
1 Black-Scholes model
Consider a risky asset with stock price S that follows geometric Brownian motion below
under the risk neutral measure Q, in other words,
dS_t = r S_t dt + \sigma S_t dW^Q_t
or, equivalently,
S_t = S_0e^{(r-\sigma ^2/2)t + \sigma W^Q_t},
where r is the risk free rate, \sigma is the volatility and W^Q is a standard Brownian motion
under Q.
The following code accesses data about a company Warner Bros. Discovery, Inc. using
Yahoo Finance.
import yfinance as yf 1
# create Ticker object which will allow data access
data = yf.Ticker('WBD')
# name of company
print("Name of company:", data.info['longName'])
# summary of company business
import textwrap 2
print("\n", textwrap.fill(data.info['longBusinessSummary'], 65))
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1 See Arrousi (2024) for more details and usage instructions. This package is available
in Google Colab, but on local installations (such as Anaconda) it may need to be
installed before use—Arrousi (2024) provides instructions.
2 See The Python Software Foundation (2024) for more details on wrapping text.
Name of company: Warner Bros. Discovery, Inc.
Warner Bros. Discovery, Inc. operates as a media and
entertainment company worldwide. It operates through three
segments: Studios, Network, and DTC. The Studios segment produces
and releases feature films for initial exhibition in theaters;
produces and licenses television programs to its networks and
third parties and direct-to-consumer services; distributes films
and television programs to various third parties and internal
television; and offers streaming services and distribution
through the home entertainment market, themed experience
licensing, and interactive gaming. The Network segment comprises
domestic and international television networks. The DTC segment
offers premium pay-tv and streaming services. In addition, the
company offers portfolio of content, brands, and franchises
across television, film, streaming, and gaming under the Warner
Bros. Motion Picture Group, Warner Bros. Television Group, DC,
HBO, HBO Max, Max, Discovery Channel, discovery+, CNN, HGTV, Food
Network, TNT Sports, TBS, TLC, OWN, Warner Bros. Games, Batman,
Superman, Wonder Woman, Harry Potter, Looney Tunes, HannaBarbera, Game of Thrones, and The Lord of the Rings brands.
Further, it provides content through distribution platforms,
including linear network, free-to-air, and broadcast television;
authenticated GO applications, digital distribution arrangements,
content licensing arrangements, and direct-to-consumer
subscription products. Warner Bros. Discovery, Inc. was
incorporated in 2008 and is headquartered in New York, New York.
The following code downloads 6 months of daily prices and stores it in a NumPy array S.
# Download 6 months of price data (daily closing prices)
hist = data.history(period="6mo")
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# Store closing prices in NumPy array S
import numpy as np
S = np.array(hist["Close"])
# Plot closing prices
import matplotlib.pyplot as plt
fig, ax = plt.subplots(figsize = (8,6))
ax.set(title = f"Closing prices of {data.info['longName']}",
ylabel = "Price", xlabel = "Time step")
ax.xaxis.grid(True)
ax.yaxis.grid(True)
ax.plot(S)
plt.show()
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1.1 Historical volatility
A popular way of calibrating the volatility \sigma is to estimate it from historical data. Suppose
that stock prices S_i (i=0,1,\ldots ,m ) have been recorded at some fixed time intervals of
length \tau >0 measured in years (for example, \tau =1/12 for monthly recorded stock prices).
The log returns on these prices over each time interval of length \tau are
R_i=\ln (S_i/S_{i-1})
for i=1,\ldots ,m . The sample variance of these returns is s^2 , where
s^2=\frac {1}{m-1}\sum _{i=1}^m\left (R_{i}-\overline {R}\right )^{2},
where \protect \overline {R} = \frac {1}{m}\sum _{i=1}^m R_{i}. The sample variance s^2 estimates the variance \sigma ^2\tau of the log return
on the stock price over a time interval of length \tau . It follows that
\hat {\sigma }_{\text {hist}}=\frac {s}{\sqrt {\tau }}
is an estimate for the volatility \sigma . This is called historical volatility.
Exercise 1.1 (Coding task: 10% of project mark). Write a function that calculates the historical volatility of a given set of stock prices. It should have two arguments: a NumPy array
of stock prices, as well as \tau , and return the estimate \protect \hat {\sigma }_{\text {hist}} .
Then use your code to calculate and display the historical volatility of the Warner
Bros. Discovery, Inc. stock prices. You should copy and paste the code given above into
your Colab notebook in order to download the data.
1.2 Binomial tree approximation
The continuous process S can be approximated by a binary tree. One approach is based
on the log price process x defined as
x_t = \ln S_t.
A binomial tree for x on the interval [0,T] can be built by taking \Delta t = T/M , where M is
the number of steps in the tree. The steps are at the equidistant times
t_i = i\Delta t \text { for } i=0,\ldots ,M,
and x^{(M)}_i is the relevant approximation of x_{t_i} .
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Consider the binomial tree approximation of x in which, at any time t_i and over a small
time interval \Delta t, the approximation x^{(M)}_i of x_{t_i} can go up by \Delta x (the space step, to be
determined), or go down by \Delta x, with probabilities q and 1-q , respectively, i.e.
x^{(M)}_{i+1} = \begin {cases} x^{(M)}_i + \Delta x & \text {with probability } q \\ x^{(M)}_i - \Delta x & \text {with probability } 1-q. \end {cases}
The drift and volatility parameters of the continuous time process are now captured by
\Delta x and \Delta t, together with the parameters r and \sigma . We take x^{(M)}_0 = x_0 .
Exercise 1.2 (Binomial tree approximation: 25% of project mark). Design a binary tree
method that approximates the Black-Scholes model and can be used to price European
options. Your work should include the following:
1. Written work:
(a) Compute the conditional first and second moments of x_{t+\Delta t} - x_t in the
continuous-time model and x^{(M)}_{i+1}-x^{(M)}_i in the binomial tree model.
(b) Given the value of \Delta t, derive formulae for \Delta x and q in terms of r and \sigma by
matching the first and second moments of the continuous-time model and the
binomial tree model.
(c) State the algorithm used for pricing a European-style derivative security with
payoff at time T given by f(S_T) .
2. Write a Python function that uses the tree approximation to calculate the price at
time 0 of a European style derivative security with payoff at time T given by f(S_T) .
Your function should take the arguments S_0 , T, r, \sigma , M and f.
3. Use your code to price a derivative with payoff
f(S) = \begin {cases} S^{ 1.9 } & \text {if } S < K, \\ 0 & \text {otherwise}\end {cases}
and the following data:
• T = 0.5 .
• r = 0.03 .
• \sigma = \hat {\sigma }_{\text {hist}} as calculated for the Warner Bros. Discovery, Inc. data (use \sigma =0.3 if
you were not able to calculate \protect \hat {\sigma }_{\text {hist}} ).
• S_0 as the final item in the array of Warner Bros. Discovery, Inc. data.
• M = 180 .
• K = S_0 .
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2 Local volatility model
A model for a stock S is given by the stochastic differential equation
\phantomsection \label {eq-local-vol-model}{dS_t = r S_t dt + \sigma (t, S_t) S_t dW_t^Q,} (1)
where r is the risk-free interest rate and W^Q is a Brownian motion under the risk-neutral
probability Q. The function \sigma is defined as
\sigma (t,S) = 0.25 e^{ -0.03 t} \left (\frac { 105 }{S}\right )^{ 0.4 }.
The aim of this part of the project is to use Monte Carlo and finite difference techniques to
study a European style derivative where the payoff at time T=1 is
X = \begin {cases} (S_T - 110)^+ & \text {if } S_t \ge 100 \text { for all } t\in [0,T], \\ 0 & \text {otherwise.} \end {cases}
Take S_0=Â£105 and r=0.05 .
2.1 Option pricing
The aim in this section is to approximate the price of the put option at time 0 for a range
of stock price values.
2.1.1 Monte Carlo method
Exercise 2.1 (Monte Carlo estimate: 30% of project mark). Implement Monte Carlo simulation to estimate the price of this derivative at time 0 under Q. Use the Euler method, and
antithetic variates to reduce the variance of the estimate.
Your work should include a statement of the Monte Carlo algorithm that you are using.
Report the Monte Carlo estimate of the price, the standard error and an approximate 95%
confidence interval with 10000 paths and 20000 steps per path.
Hint: To reduce the discretization error, apply the Euler method to the process Y defined
as Y_t=\ln S_t instead of to S directly. Use the Itô formula to derive a stochastic differential
equation for Y.
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2.1.2 Finite difference methods
The price of a path-independent option at time t can be expressed as V(t,S_t) , where the
function V satisfies the partial differential equation
\phantomsection \label {eq-local-vol-pde}{\frac {\partial V}{\partial t}(t,S) + r S \frac {\partial V}{\partial S}(t,S) + \tfrac {1}{2}\sigma ^2(t,S)S^2\frac {\partial ^2 V}{\partial S^2}(t,S) - rV(t,S) = 0} (2)
for all t\in [0,T) and S>0 , together with boundary condition
V(t, 100) = 0
and final value
V(T, S) = \begin {cases} (S - 110)^+ & \text {if } S \ge 100, \\ 0 & \text {otherwise}. \end {cases}
Exercise 2.2 (Backward Euler method: 25% of project mark). Design an implicit (backward Euler) finite difference scheme to approximate the solution to the partial differential
equation in Equation 2. Your work should include the following:
1. Convert the final value problem into an initial value problem, and define a suitable
grid that contains the points (0,100) and (0,S_0) .
2. Propose an appropriate boundary condition for S\rightarrow \infty .
3. Derive the iterative scheme in matrix form that would allow you to approximate
the value of V at the grid points. Give the lower diagonal, main diagonal and upper
diagonal elements of the matrix.
4. Write a Python function to implement the iterative scheme. Your function should
have suitable arguments and return values.
5. Compare your results to the Monte Carlo estimate. Is it possible to choose values
for the step sizes that result in the price V(0,S_0) being within the confidence interval
produced by the Monte Carlo method?
6. Report numerical results in an appropriate way, for example, via a 3d surface representing option prices for all t\in [0,T] and a 2d graph representing option prices at
time t=0 , all for a range of S.
2.2 Theta
Exercise 2.3 (Theta: 10% of project mark). Approximate the value of the theta \Theta = \frac {\partial V}{\partial \tau } at
all the grid points used for the finite difference approximation in Exercise 2.2. Your work
should include the following:
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1. Design a suitable algorithm for approximating the theta, paying special attention to
the boundary points. It is not necessary to repeat any information already provided
in Exercise 2.2.
2. Implement the algorithm in Python.
3. Report numerical results appropriately, for example, via a 3d surface representing
the value of theta at all grid points.
References
Arrousi, Ran. 2024. “Download Market Data from Yahoo! Finance’s API.” https://github
.com/ranaroussi/yfinance.
The Python Software Foundation. 2024. “textwrap: Text Wrapping and Filling.” https:
//docs.python.org/3/library/textwrap.html.
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