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FINM 3003/7003 - Continuous Time Finance
创建日期：2024-05-08 15:31:45
所属分类：金融Finance
标签： FINM 3003
Continuous Time Finance

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FINM 3003/7003 - Continuous Time Finance

Assignment 2
Total Marks: 50 marks.
This assignment is worth 10% of the final marks for this course, and covers
materials in Weeks 1-9 of the course.
Due: Friday 13 May 3pm
No late assignments will be accepted, i.e. a grade of 0 will be given.
Please show all workings for full marks.
Notes:
1. For Numerical questions, you can use either R or Excel. You will need to
submit either the R code or the Excel spreadsheet.
2. In your calculations, do NOT round answers until the final answer (using
Excel/R is a good option). For all presentation purposes (i.e. writing values
in your submission) use 4 decimal places.
2. In the assignment the word ‘payoff’ of a derivative implies outcomes at
maturity, and the word ‘value’ implies outcomes at time t < T .
4. Assignments that are not accompanied by a COMPLETED Assignment
cover sheet will get a grade of 0.
1
1. Consider a derivative in the Black-Scholes framework with payoff P = [max(K − ST , 0)]2,
where K is the strike price and T is the exercise date, i.e. the payoff is the square of
the put option. For a given time t(0 ≤ t ≤ T ), we have
ST = Ste
[(
r−σ2
2
)
(T−t)+σ(WT−Wt)
]
where (Wt)(t>0) is a standard Brownian motion under the risk-neutral measure.
(a) (7)Find the expected payoff of the option at maturity T . [7]
(b) Hence or otherwise, find the value of the option at time t = 0. [1]
(c) Compute the expected payoff (at time T ) and the value of the option (at time
0) when S0 = 15, K = 12, r = 5%, σ = 30% and T = 0.5. [2]
This question is worth 10 marks.
(Note: Please show all workings (all derivations) for full marks.)
2
2. Consider a call option with the following parameters: S0 = 100, K = 100, r = 5.5%,
σ = 20%, q = 4% and T = 3 years.
(a) What is the theoretical value (at time 0) of a European call option with the
above parameters. [1]
(b) Construct a Binomial model for a European call option for time steps of 3, 6,
12, 36, 156 and 500, 1000. Estimate the value (at time 0) of the European call
option for each time step.
In your answer sheet, create a table with three columns: Time steps, the BS
value and the Binomial value. The rows represent time steps. Populate this
table with your calculated answers.
Comment on your observations from the table. [7]
(c) Repeat part (b), for an American option (i.e. with early exercise). Add another
column to the table in (b) for the American version of the option.
Comment on your observations from the table. [5]
(d) Repeat part (a) and (c) for q = 3% and q = 5%. Create a table with columns:
Time steps, and the three dividend yields for American style exercise (no need
to do European style exercise).
Comment on your observations from the table. [7]
(Note: For parts (b), (c) and (d), you can add more time steps. The numbers pro-
vided are a minimum, you can choose different ones if you like as well. Support
any statements you make with relevant justifications using either your output or the
theory.)
This question is worth 20 marks.
3
3. The butterfly spread is an option trading strategy where we
• long 1 call option with a strike price of K1,
• short 2 call options with a strike price of K2,
• long 1 call option with a strike price of K3, and
• K3 −K2 = K2 −K1 = h > 0.
Here, the optional all have the same maturity date (T ), the same underlying stock
(which has a value of S), and European style exercise features (can only be exercised
at maturity). Also, we have K1 ≤ S ≤ K3, i.e., the first call with strike price K1 is
“in the money” and the third call with strike price K3 is “out of the money”.
The payoff from the butterfly at maturity is a function f(ST ) of the stock price ST .
Assume the stock pays no dividend.
(a) Write down an expression for the payoff and show that the payoff is sometimes
positive and never negative. [2]
(b) Let
ST = Ste
[(
r−σ2
2
)
(T−t)+σ(WT−Wt)
]
where (Wt)(t>0) is a standard Brownian motion under the risk-neutral measure.
Show that the arbitrage free value at time 0 of the payoff (made at time T), is
strictly positive. [2]
(c) Hence or otherwise, prove that C(K3, S, t) + C(K1, S, t) ≥ 2C(K2, S, t), where
C(K,S, t) is the value at time t < T of a European call option over S with
exercise price of K and maturity T . [3]
(Note: Proving this by contradiction is not acceptable, i.e, you will get a grade
of 0.)
(d) Discuss the result in part (c) when the Black-Scholes option pricing assumptions
are invalid. [2]
(e) Assume S = K3, r, σ, T > 0. Find an expression for the vega of the butterfly
spread. Then find the condition under which this vega is positive or negative.[9]
Hint: consider concavity/convexity.
4
(f) Assume K1 < S < K3. Explain in words whether you think that the impact of
an increase in volatility should be an increase or a decrease in the value of a
butterfly spread and give a reason for your answer. [2]
This question is worth 20 marks.
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