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程序代写案例-7CCMFM06

时间：2021-03-29

7CCMFM06 Coursework 2020/21

John Armstrong

December 14, 2020

General Coursework Instructions

For your coursework you should submit four documents via Keats:

1. A pdf write-up of your coursework. This must be generated using LaTeX.

It should contain a mathematical description of how you answered the

question, together with any plots you have generated and the interpreta-

tion of your results. You should focus on the mathematics used rather

than the details of how it was implemented in Python. For example, you

should explicitly state any difference equations used in your calculations.

You should divide your write-up into sections corresponding to the differ-

ent parts of the question.

2. A Jupyter notebook containing the Python code that calculates any nu-

merical values used in your write-up. This notebook must run on CoCalc

without requiring any additional files or libraries.

3. A pdf version of your Jupyter notebook. This may be generated using the

options on the File menu in the Jupyter notebook.

4. A signed departmental coversheet indicating that this is all your own work.

The pdf file of your coursework should be generated with the default LaTeX

fonts and margins and must not exceed 10 pages. You should not see 10 pages

as a target: if you can express yourself concisely you should do so. You will be

given credit for concise and interesting writing and will lose marks for long and

uninteresting writing and for including irrelevant details.

Be sure to reference all of your sources.

You may discuss the problem with other students but must not share any of

your code or written work with other students.

Specific Question for Shengkai Wang - Corrected

Version

1. You must show how to price a specific knock-out derivative on a stock.

1

The stock price process St for times t ∈ [0, T ] is known to follow the

stochastic differential equation

dSt = St(µdt+ σ(t) dWt)

where Wt is a Brownian motion, S0 = 270.0000, µ = 0.0900 and

σ(t) = 0.16× (1.6667× t)1.5.

A trader may also invest in a risk-free bank account that grows at the

continuously compounded rate r = 0.0425. The units of time for this

question are years and all prices are in dollars.

The contract of this derivative is defined as follows. If the stock ever hits

the barrier level B = 303.0000 on or before the maturity T = 1.0000 of

the option then the payoff of the derivative will be 0. Otherwise the payoff

of the option will be f(ST ) where ST is the stock price at maturity and

the function f is defined by

f(ST ) = max{(0.0033× ST )1.5 × ST − 185, 0}.

You should price this option using the Crank-Nicolson finite difference

method and may assume that the price V (t, S) satisfies the partial differ-

ential equation

∂V

∂t

+

1

2

σ(t)2S2

∂2V

∂S2

+ rS

∂V

∂S

− rV = 0

with boundary condition V (t, B) = 0, plus additional boundary conditions

you should identify. You should use the Crank-Nicolson finite difference

method with a uniform grid containing 1001 time points and 1001 stock

price values. x

Record the price of the derivative at time 0 and the delta of the derivative

at time 0 in the table of results at the end of your essay. [20%]

2. Suppose that a trader attempts to replicate this option using a discrete-

time version of the delta-hedging trading strategy, rehedging at the same

time-points you used when implementing the finite difference method. The

trader uses the value for the price and the value of delta computed by the

finite difference method, using linear interpolation to calculate values at

points which are not in the grid.

Calculate an SDE satisfied by Zt = logSt and simulate Zt using the

Euler scheme for this SDE using the same time points as above. Using

the stock prices arising from this simulation, perform 1000 simulations of

the trader’s strategy and plot a histogram of the error in this replication

strategy. You should compute the 25th and 75th percentiles of the error

and record these values in the table at the end of your essay.[20%]

2

3. Interpret your results. This means that you should make some financially

relevant observations based on your simulation, plotting charts you find

interesting. In order to make interesting observations you might want to

run variations on the simulation, for example varying the barrier. Please

remember to edit your findings and to only comment on the most inter-

esting points. [15%]

4. Describe how you have tested your code is correct, for example explaining

how you have checked that you have priced the derivative correctly. [15%]

5. You must finish your coursework by completing the following table of

results:

Price

Delta

25th percentile

75th percentile

The remaining 10% of your marks will be allocated based on the quality of

your code and your write-up.

3

学霸联盟

John Armstrong

December 14, 2020

General Coursework Instructions

For your coursework you should submit four documents via Keats:

1. A pdf write-up of your coursework. This must be generated using LaTeX.

It should contain a mathematical description of how you answered the

question, together with any plots you have generated and the interpreta-

tion of your results. You should focus on the mathematics used rather

than the details of how it was implemented in Python. For example, you

should explicitly state any difference equations used in your calculations.

You should divide your write-up into sections corresponding to the differ-

ent parts of the question.

2. A Jupyter notebook containing the Python code that calculates any nu-

merical values used in your write-up. This notebook must run on CoCalc

without requiring any additional files or libraries.

3. A pdf version of your Jupyter notebook. This may be generated using the

options on the File menu in the Jupyter notebook.

4. A signed departmental coversheet indicating that this is all your own work.

The pdf file of your coursework should be generated with the default LaTeX

fonts and margins and must not exceed 10 pages. You should not see 10 pages

as a target: if you can express yourself concisely you should do so. You will be

given credit for concise and interesting writing and will lose marks for long and

uninteresting writing and for including irrelevant details.

Be sure to reference all of your sources.

You may discuss the problem with other students but must not share any of

your code or written work with other students.

Specific Question for Shengkai Wang - Corrected

Version

1. You must show how to price a specific knock-out derivative on a stock.

1

The stock price process St for times t ∈ [0, T ] is known to follow the

stochastic differential equation

dSt = St(µdt+ σ(t) dWt)

where Wt is a Brownian motion, S0 = 270.0000, µ = 0.0900 and

σ(t) = 0.16× (1.6667× t)1.5.

A trader may also invest in a risk-free bank account that grows at the

continuously compounded rate r = 0.0425. The units of time for this

question are years and all prices are in dollars.

The contract of this derivative is defined as follows. If the stock ever hits

the barrier level B = 303.0000 on or before the maturity T = 1.0000 of

the option then the payoff of the derivative will be 0. Otherwise the payoff

of the option will be f(ST ) where ST is the stock price at maturity and

the function f is defined by

f(ST ) = max{(0.0033× ST )1.5 × ST − 185, 0}.

You should price this option using the Crank-Nicolson finite difference

method and may assume that the price V (t, S) satisfies the partial differ-

ential equation

∂V

∂t

+

1

2

σ(t)2S2

∂2V

∂S2

+ rS

∂V

∂S

− rV = 0

with boundary condition V (t, B) = 0, plus additional boundary conditions

you should identify. You should use the Crank-Nicolson finite difference

method with a uniform grid containing 1001 time points and 1001 stock

price values. x

Record the price of the derivative at time 0 and the delta of the derivative

at time 0 in the table of results at the end of your essay. [20%]

2. Suppose that a trader attempts to replicate this option using a discrete-

time version of the delta-hedging trading strategy, rehedging at the same

time-points you used when implementing the finite difference method. The

trader uses the value for the price and the value of delta computed by the

finite difference method, using linear interpolation to calculate values at

points which are not in the grid.

Calculate an SDE satisfied by Zt = logSt and simulate Zt using the

Euler scheme for this SDE using the same time points as above. Using

the stock prices arising from this simulation, perform 1000 simulations of

the trader’s strategy and plot a histogram of the error in this replication

strategy. You should compute the 25th and 75th percentiles of the error

and record these values in the table at the end of your essay.[20%]

2

3. Interpret your results. This means that you should make some financially

relevant observations based on your simulation, plotting charts you find

interesting. In order to make interesting observations you might want to

run variations on the simulation, for example varying the barrier. Please

remember to edit your findings and to only comment on the most inter-

esting points. [15%]

4. Describe how you have tested your code is correct, for example explaining

how you have checked that you have priced the derivative correctly. [15%]

5. You must finish your coursework by completing the following table of

results:

Price

Delta

25th percentile

75th percentile

The remaining 10% of your marks will be allocated based on the quality of

your code and your write-up.

3

学霸联盟