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程序代写案例-APRIL 2021
时间：2022-04-02
LANCASTER UNIVERSITY
APRIL 2021 EXAMINATIONS
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. . .)
6
Question B–4:
On 24 August 2020, HSBC buys a 3-year default protection on NatWest Group
credit-risky bond from Societe Generale, on a notional amount of GBP 10
million, paying a credit-default swap (CDS) premium of 358 basis points per
annum (annually in arrears).
The specifications of NatWest Group credit-risky coupon bond are as follows:
Face Value : GBP 100 (par)
Coupon Rate : 4.222% per annum
Maturity : 3 years
Coupon Frequency : Annual
Credit Rating : BBB+
The 3-year GBP interest-rate-swap (credit rating: AA) is trading at 0.642% per
annum.
Required:
(a) Show clearly how Societe Generale, the seller of the CDS on NatWest
Group credit-risky bond, would hedge the bank’s exposure of the CDS
position.
Your answer should illustrate clearly the asset-swap decomposition of
NatWest Group credit-risky coupon bond into its key components, in
which the CDS cashflows are the residuals.
The cashflows generated by the decomposition should be based on liquid
and tradeable securities. State clearly any assumptions made.
(13 marks)
(b) Comment on the implications of using the asset-swap credit spread as
a measure of a “fair price” for a credit-default swap (CDS), under the
following two scenarios (assume par bond price at GBP 100):
(i) the bond is trading at a discount at GBP 95 at inception, and
(ii) the bond is trading at a premium at GBP 110 at inception.
Assume, in both scenarios, immediately after inception the bond defaults
and that the post-default price (recovery value) of the bond is GBP 40.
Compare both asset swap results to the loss implied in a CDS contract.
(7 marks)
(Total: 20 MARKS)
7
APPENDIX
AcF605 Derivatives Pricing: FORMULA SHEET
Generalized Wiener Process:
∆x = a∆t + b∆z, with ∆z = ε
√
∆t.
Geometric Brownian Motion:
dSt = µStdt + σStdz.
Ito’s Lemma:
dx = a(x, t)dt+ b(x, t)dz,
G = G(x, t),
dG =
(
∂G
∂x
a+ ∂G
∂t
+ 1
2
∂2G
∂x2
b2
)
dt + ∂G
∂x
bdz.
Black–Scholes PDE:
∂f
∂t
+ rSt ∂f
∂St
+ 1
2
∂2f
∂S2t
σ 2S2t = rf .
Θ+ rSt∆+ 1
2
Γσ 2S2t = rΠt.
Black–Scholes without Dividend:
Ct = St ·N(d1)− Ke−rT ·N(d2),
Pt = Ke−rT ·N(−d2)− St ·N(−d1),
where, d1 = ln (St/K)+ [r + (σ
2/2)]T
σ
√
T
,
and, d2 = d1 − σ
√
T .
Black–Scholes with Dividend, q:
Ct = Ste−qT ·N(d1)−Ke−rT ·N(d2),
Pt = Ke−rT ·N(−d2)− Ste−qT ·N(−d1),
where, d1 = ln (St/K)+ [r − q + (σ
2/2)]T
σ
√
T
,
and, d2 = d1 − σ
√
T.
8
Black–Scholes for Currency (FX) Options:
Ct = Ste−rfT ·N(d1)− Ke−rT ·N(d2),
Pt = Ke−rT ·N(−d2)− Ste−rfT ·N(−d1),
where, d1 = ln (St/K)+ [r − rf + (σ
2/2)]T
σ
√
T
,
and, d2 = d1 −σ
√
T .
Black’s Model of a Bond Futures:
Ct = B(0, T)[FBN(d1)− KN(d2)],
Pt = B(0, T)[KN(−d2)− FBN(−d1)],
where, d1 = ln(FB/K)+ (σ
2
BT/2)
σB
√
T
,
and, d2 = d1 −σB
√
T.
Black-Scholes “Greek” Letters:
Greek Letter Call Option Put Option
Delta, ∆ e−qTN(d1) e−qT (N(d1)− 1)
Gamma, Γ (N ′(d1)e−qT)/(Stσ
√
T) (N ′(d1)e−qT)/(Stσ
√
T)
Theta, Θ −StN ′(d1)σe−qT/(2
√
T) −StN ′(d1)σe−qT/(2
√
T)
+qStN(d1)e−qT − rKe−rTN(d2) +qStN(−d1)e−qT + rKe−rTN(−d2)
Vega, V St
√
TN ′(d1)e−qT St
√
TN ′(d1)e−qT
Rho, ρ KTe−rTN(d2) −KTe−rTN(−d2)
Standard Deviation of a Portfolio:
σX+Y =
√
σ 2X + σ 2Y + 2ρσXσY .
9
Put-Call Parity:
Non-Dividend Paying Stock: Ct +K/(1+ r)T = Pt + St (Discrete Time)
Non-Dividend Paying Stock: Ct +Ke−rT = Pt + St (Continuous Time)
Indices: Ct +Ke−rT = Pt + Ste−qT
Foreign Exchange: Ct +Ke−rT = Pt + Ste−rfT
Futures: Ct +Ke−rT = Pt + Fte−rT
Cox-Ross-Rubinstein (CRR) One-Period Binomial Option Pricing Model:
(Non-Dividend Paying Underlying Asset)
Ct = ∆St − B∗t ,
where, ∆ = Cu −Cd
St(u−d) =
Cu − Cd
Su − Sd ,
and, B∗t =
dCu −uCd
(1+ r)(u−d) ;
with, CT =max(0, ST − K), for a Call Option;
or, CT = PT =max(K − ST ,0), for a Put Option.
Cox-Ross-Rubinstein (CRR) One-Period Binomial Option Pricing Model
Martingale (Risk-Neutral) Probability, q:
(Non-Dividend Paying Underlying Asset)
q = (1+ r)−d
u −d ;
u = exp(+σ√T), d = exp(−σ√T); σ =
[
1
2
√
T
]
ln
(
u
d
)
.
Stochastic Floating Cashflow at accrual (settlement) date t:
PVt(L˜t+1) = R˜t,t+1αt,t+1
(1+ R˜t,t+1)
.
Spot-Forward Parity:
Ft,t+1 = Rt,t+1 =
[
B0,t
B0,t+1
− 1
]
× 1
αt,t+1
.
Principal (FRN, Synthetic Bond) Method:
PV0(L˜t+1) = +AB0,t −AB0,t+1.
10
Table for N(x) When X ~ 0
This table shows values of N(x) for x ~ 0. The table should be used with interpolation. For example,
N(0.6278) = N(0.62) + 0.78[N(0.63)- N(0.62)]
= 0.7324 + 0.78 X (0.7357- 0.7324)
= 0.7350
X .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9986 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
11
Table for N(x) When X ~ 0
This table shows values of N(x) for x ~ 0. The table should be used with interpolation. For example,
N(-0.1234) = N(-0.12)- 0.34[N(-0.12)- N(-0.13))
= 0.4522- 0.34 X (0.4522 - 0.4483)
= 0.4509
X .00 .01 .02 .03 .04 .05 .06 .07 .08 . 09
-0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 O.D708 0.0694 0.0681
-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-3.0 0.0014 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
-3.6 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-4.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
12
APRIL 2021 EXAMINATIONS
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. . .)
6
Question B–4:
On 24 August 2020, HSBC buys a 3-year default protection on NatWest Group
credit-risky bond from Societe Generale, on a notional amount of GBP 10
million, paying a credit-default swap (CDS) premium of 358 basis points per
annum (annually in arrears).
The specifications of NatWest Group credit-risky coupon bond are as follows:
Face Value : GBP 100 (par)
Coupon Rate : 4.222% per annum
Maturity : 3 years
Coupon Frequency : Annual
Credit Rating : BBB+
The 3-year GBP interest-rate-swap (credit rating: AA) is trading at 0.642% per
annum.
Required:
(a) Show clearly how Societe Generale, the seller of the CDS on NatWest
Group credit-risky bond, would hedge the bank’s exposure of the CDS
position.
Your answer should illustrate clearly the asset-swap decomposition of
NatWest Group credit-risky coupon bond into its key components, in
which the CDS cashflows are the residuals.
The cashflows generated by the decomposition should be based on liquid
and tradeable securities. State clearly any assumptions made.
(13 marks)
(b) Comment on the implications of using the asset-swap credit spread as
a measure of a “fair price” for a credit-default swap (CDS), under the
following two scenarios (assume par bond price at GBP 100):
(i) the bond is trading at a discount at GBP 95 at inception, and
(ii) the bond is trading at a premium at GBP 110 at inception.
Assume, in both scenarios, immediately after inception the bond defaults
and that the post-default price (recovery value) of the bond is GBP 40.
Compare both asset swap results to the loss implied in a CDS contract.
(7 marks)
(Total: 20 MARKS)
7
APPENDIX
AcF605 Derivatives Pricing: FORMULA SHEET
Generalized Wiener Process:
∆x = a∆t + b∆z, with ∆z = ε
√
∆t.
Geometric Brownian Motion:
dSt = µStdt + σStdz.
Ito’s Lemma:
dx = a(x, t)dt+ b(x, t)dz,
G = G(x, t),
dG =
(
∂G
∂x
a+ ∂G
∂t
+ 1
2
∂2G
∂x2
b2
)
dt + ∂G
∂x
bdz.
Black–Scholes PDE:
∂f
∂t
+ rSt ∂f
∂St
+ 1
2
∂2f
∂S2t
σ 2S2t = rf .
Θ+ rSt∆+ 1
2
Γσ 2S2t = rΠt.
Black–Scholes without Dividend:
Ct = St ·N(d1)− Ke−rT ·N(d2),
Pt = Ke−rT ·N(−d2)− St ·N(−d1),
where, d1 = ln (St/K)+ [r + (σ
2/2)]T
σ
√
T
,
and, d2 = d1 − σ
√
T .
Black–Scholes with Dividend, q:
Ct = Ste−qT ·N(d1)−Ke−rT ·N(d2),
Pt = Ke−rT ·N(−d2)− Ste−qT ·N(−d1),
where, d1 = ln (St/K)+ [r − q + (σ
2/2)]T
σ
√
T
,
and, d2 = d1 − σ
√
T.
8
Black–Scholes for Currency (FX) Options:
Ct = Ste−rfT ·N(d1)− Ke−rT ·N(d2),
Pt = Ke−rT ·N(−d2)− Ste−rfT ·N(−d1),
where, d1 = ln (St/K)+ [r − rf + (σ
2/2)]T
σ
√
T
,
and, d2 = d1 −σ
√
T .
Black’s Model of a Bond Futures:
Ct = B(0, T)[FBN(d1)− KN(d2)],
Pt = B(0, T)[KN(−d2)− FBN(−d1)],
where, d1 = ln(FB/K)+ (σ
2
BT/2)
σB
√
T
,
and, d2 = d1 −σB
√
T.
Black-Scholes “Greek” Letters:
Greek Letter Call Option Put Option
Delta, ∆ e−qTN(d1) e−qT (N(d1)− 1)
Gamma, Γ (N ′(d1)e−qT)/(Stσ
√
T) (N ′(d1)e−qT)/(Stσ
√
T)
Theta, Θ −StN ′(d1)σe−qT/(2
√
T) −StN ′(d1)σe−qT/(2
√
T)
+qStN(d1)e−qT − rKe−rTN(d2) +qStN(−d1)e−qT + rKe−rTN(−d2)
Vega, V St
√
TN ′(d1)e−qT St
√
TN ′(d1)e−qT
Rho, ρ KTe−rTN(d2) −KTe−rTN(−d2)
Standard Deviation of a Portfolio:
σX+Y =
√
σ 2X + σ 2Y + 2ρσXσY .
9
Put-Call Parity:
Non-Dividend Paying Stock: Ct +K/(1+ r)T = Pt + St (Discrete Time)
Non-Dividend Paying Stock: Ct +Ke−rT = Pt + St (Continuous Time)
Indices: Ct +Ke−rT = Pt + Ste−qT
Foreign Exchange: Ct +Ke−rT = Pt + Ste−rfT
Futures: Ct +Ke−rT = Pt + Fte−rT
Cox-Ross-Rubinstein (CRR) One-Period Binomial Option Pricing Model:
(Non-Dividend Paying Underlying Asset)
Ct = ∆St − B∗t ,
where, ∆ = Cu −Cd
St(u−d) =
Cu − Cd
Su − Sd ,
and, B∗t =
dCu −uCd
(1+ r)(u−d) ;
with, CT =max(0, ST − K), for a Call Option;
or, CT = PT =max(K − ST ,0), for a Put Option.
Cox-Ross-Rubinstein (CRR) One-Period Binomial Option Pricing Model
Martingale (Risk-Neutral) Probability, q:
(Non-Dividend Paying Underlying Asset)
q = (1+ r)−d
u −d ;
u = exp(+σ√T), d = exp(−σ√T); σ =
[
1
2
√
T
]
ln
(
u
d
)
.
Stochastic Floating Cashflow at accrual (settlement) date t:
PVt(L˜t+1) = R˜t,t+1αt,t+1
(1+ R˜t,t+1)
.
Spot-Forward Parity:
Ft,t+1 = Rt,t+1 =
[
B0,t
B0,t+1
− 1
]
× 1
αt,t+1
.
Principal (FRN, Synthetic Bond) Method:
PV0(L˜t+1) = +AB0,t −AB0,t+1.
10
Table for N(x) When X ~ 0
This table shows values of N(x) for x ~ 0. The table should be used with interpolation. For example,
N(0.6278) = N(0.62) + 0.78[N(0.63)- N(0.62)]
= 0.7324 + 0.78 X (0.7357- 0.7324)
= 0.7350
X .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9986 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
11
Table for N(x) When X ~ 0
This table shows values of N(x) for x ~ 0. The table should be used with interpolation. For example,
N(-0.1234) = N(-0.12)- 0.34[N(-0.12)- N(-0.13))
= 0.4522- 0.34 X (0.4522 - 0.4483)
= 0.4509
X .00 .01 .02 .03 .04 .05 .06 .07 .08 . 09
-0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 O.D708 0.0694 0.0681
-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-3.0 0.0014 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
-3.6 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-3.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
-4.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
12
