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AND ACCOUNTING & FINANCIAL MANAGEMENT
创建日期：2024-04-30 18:12:24
所属分类：会计Accounting
标签： ACCOUNTING
AND ACCOUNTING & FINANCIAL MANAGEMENT

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EXAMINATIONS FOR THE MASTER OF SCIENCE IN FINANCE,

AND ACCOUNTING & FINANCIAL MANAGEMENT,
AND ADVANCED FINANCIAL ANALYSIS,
AND FINANCIAL MATHEMATICS,
AND MONEY, BANKING & FINANCE
AcF 605 Derivatives Pricing
Time: 3 hours
INSTRUCTIONS
• Section A—COMPULSORY. Answer ALL questions—(60 marks).
• Section B—Answer any TWO questions out of four—(40 marks).
• Relevant formulae and statistical tables are provided in the Appendix.
• Begin answers to each question on a fresh page.
• The examination is open book.
1
Section A (Compulsory)—60 marks
Answer ALL questions
A–1. (a) It is said that the prospect theory overcomes some weaknesses of the
expected utility theory. Explain.
(5 marks)
(b) If the implied volatility is a decreasing function of the strike price, should
the left tail of the implied distribution be heavier than that of the
lognormal distribution? Explain.
(5 marks)
A–2. (a) Explain what a straddle strategy is?
When should we use such a trading strategy, and why?
(5 marks)
(b) Explain which of the following two stochastic processes:
(1) dSt = µStdt +σStdWt,
(2) dSt = µdt + σdWt,
where µ and σ are constants, and Wt is a standard Wiener process,
is more appropriate to model stock price movements?
(5 marks)
A–3. (a) Three put options on an underlying stock, all have the same expiration
date, and strike (exercise) prices of $45, $50 and $55, respectively.
The market prices of the put options are $2, $4, and $7, respectively.
Explain how a butterfly spread can be created using these put options.
Construct a table showing the profit from such a trading strategy.
(5 marks)
(b) Suppose that the market prices of the three put options, in question part
(a) above, are changed to $2, $5, and $7, respectively.
Is there an arbitrage opportunity? Explain.
(5 marks)
(Section A continued on next page. . .)
2
A–4. (a) Given a term-structure of recently reset par swap rates, show clearly and
mathematically, how the unknown cashflows of the LIBOR leg of a plain
vanilla interest-rate-swap are determined at spot date, and deduce the
corresponding term-structure of forward rates.
State clearly the critical assumptions necessary in your derivation.
(8 marks)
(b) Show clearly and mathematically how the deduced implied spot-forward
relationship, in question part (a) above, is traded in the form of an
interest-rate derivative product known as a Forward Rate Agreement
(FRA).
(7 marks)
A–5. The economics of a convertible bond is often (erroneously) cited as:
“Cheap, Deferred Financing”.
Critically evaluate this economic characterisation of a convertible bond and
its insights on credit risk.
In particular, the puzzling existence of complex financing vehicles
(a) in the light of the Modigliani–Miller (M&M) irrelevance propositions,
(b) the insights provided by the agency-asymmetric information framework,
and (c) the roles of the embedded optionalities.
(6 marks)
A–6. Using the following input parameters in a one-period binomial option pricing
model, with step-size of 1-year period:
Current underlying stock price, S0 = $100.00,
Stock price movement: Up-factor, u = 1.45,
Stock price movement: Down-factor, d = 0.90;
Put option’s strike (exercise) price, K = $93.00,
Put option’s time-to-maturity, τ = 1-year;
Riskfree (logarithmic) interest rate, r = 2.50% per annum;
and, by means of (i) the changes in the underlying asset hedge-ratios, and
(ii) the cash funding requirements, show the distinctive attributes of the
(a) self-financing, and (b) path-independence
characteristics inherent in the pricing of the European put option.
(9 marks)
(Section A, Total: 60 MARKS)
3
Section B—40 marks
Answer any TWO questions out of four
Question B–1:
(a) The six-month forward price of the underlying S&P 500 index is 1,500,
and the volatility of the S&P 500 index is 20.00% per annum.
Calculate the price of a six-month European put option on the spot value
of the underlying S&P 500 index, with a strike price of 1,550.
The risk-free rate is 5.00% per annum.
(8 marks)
(b) Explain what the theta of an option position is?
If a trader feels that neither the underlying stock price, nor its implied
volatility will change, what type of theta option position is desirable?
(3 marks)
(c) It is said that the implied volatilities using call options on an underlying asset
should be the same as the implied volatilities using put options on the same
underlying asset. Explain.
(3 marks)
(d) Suppose that the price of an asset at the close of trading yesterday was $600,
and its daily volatility was estimated at 1.00% per day at that time.
The price of the asset at the close of trading today is $588.
Update the volatility estimate using:
(i) The EWMA model with parameter: λ = 0.93.
(3 marks)
(ii) The GARCH(1,1) model with parameters:
ω = 0.000002, α = 0.04, and β = 0.93.
(3 marks)
(Total: 20 MARKS)
(Section B continued on next page. . .)
4
Question B–2:
(a) Explain what the gamma of a European call option position on a non-dividend
underlying stock under the Black–Scholes model is.
(3 marks)
(b) Assume that the stock St follows the following Ito process:
dSt = µdt +σStdWt.
What is the process followed by the variable: S2t ?
(6 marks)
(c) Explain the difference between the cost of a butterfly spread created using
European call options, and the cost of a butterfly spread using European put
options.
(5 marks)
(d) An investment bank has the following portfolio of over-the-counter European
options on the US dollar:
Type Position Delta Gamma
of an Option of an Option
Call −1,000 0.50 2.20
Put −2,000 −0.40 1.30
Call −500 0.70 1.80
Now assume that a traded option is available with a delta of 0.60, and
a gamma of 1.50.
What positions in the traded option and in the US dollar, when added to the
over-the-counter portfolio, would make the combined portfolio both gamma-
neutral and delta-neutral?
(6 marks)
(Total: 20 MARKS)
(Section B continued on next page. . .)
5
Question B–3:
(a) In September 2018, Hangzhou All Venture Capital (HAVC), an investment arm
of Alibaba, purchased 218 million shares of One97Communications (One97C)
in a private placement, which represents 25% of the total shares issued at the
time.
The One97C stock price was $0.68 per share. The stock was restricted and
could not be resold publicly for 3 years even if One97C were to go public.
In March 2019, One97C issued shares publicly. Following the shares issues,
One97C’s stock price rose as high as $2.75 per share.
In May 2020, the One97C stock price was $8.30 per share, and HAVC had
approximately $1,662 million in capital gains on the One97C stock, if not for
the regulatory restrictions. If the shares had been sold on the open market,
the tax liability (at a 25% capital gains tax rate) would be approximately $416
million.
Required:
Explain how you would financially engineer a derivative structured-finance
vehicle with embedded optionalities, in such a way for HAVC to hedge
its holdings of One97C shares without triggering immediate taxation of
gains.
Explain clearly the regulatory, tax and accounting arbitrages that can be
derived from what is effectively a tax deferral strategy.
(13 marks)
(b) Financial institutions acquire assets that are difficult to resell individually,
examples are auto loans, credit card receivables, home equity loans, and
mortgages.
Required:
Explain the process of securitisation which provides a mechanism for
off-loading such on-balance-sheet underlying risky loans and debt through
the credit tranching special purpose vehicles (SPVs).
(7 marks)
(Total: 20 MARKS)
(Section B continued on next page. . .)
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