Blog

Research School of Finance, Actuarial Studies and Statistics
Test 2
Semester 2, 2022
FINM2002 DERIVATIVES / FINM7041 APPLIED DERIVATIVES
Writing Time:
90 minutes
Exam Conditions:
Wattle-based
Permitted Materials:
Any
Instructions to Students:
1. It is not permitted to disclose, disseminate, reproduce, or publish any portion of this test in any manner.
2. This exam consists of a total of THREE questions with subparts. Questions are of unequal value, with marks
indicated for each question. Please answer all questions.
3. All answers must be typed and provided within the Wattle exam interface. Only answers submitted and
typed within the answer space will be read and graded.
4. Incorrect choices for multiple-choice questions will be penalized.
5. Please show all working for marks. A number without intermediate steps and explanations will be awarded
a mark of zero.
6. Emailed answers will not be accepted.
7. Unless stated otherwise in a question, please keep all decimal places for interim numerical answers, final
answers should be rounded up to 4 decimal places. Failure to do so will result in penalties.
8. Please remember to save your answers often by clicking either “next page” or “previous page” as per the
discussion on how to prepare for Test 1 in Lecture 4, including how to work with Wattle quiz interface.
Total Marks = 55
This test is redeemable and counts towards 20% of your final grade for the course.
QUESTION 1. THIS QUESTION HAS 4 SUBPARTS, PLEASE ANSWER ALL SUBPARTS. (Total 8 marks)
I. Which of the following statements is/are correct Select all that apply. (2 marks)
a. Everyone in the class is encouraged to come to the in-person guest lecture with Dr. Adrian Dudek
during our usual workshop time next Monday (Oct. 10, week 10) 9-10am in Kambri T2.
b. Everyone in the class is encouraged to join the live ZOOM post-lecture chat with Dr. Adrian
Dudek during our usual workshop time the following Monday (Oct. 17, week 11) 9-10am via Ding
Ding’s usual ZOOM consult link.
c. Contents from the guest lecture + guest chat are examinable because they are important ways
for us to learn about derivatives trading in real life.
d. We have discussed in class that good assessment-taking strategies include: move on to the next
question if we have a mental block, move on to the next question if we find ourselves spending 10
minutes on a 2-mark question, use Excel for calculations and type out working in dot points to save
time, and prepare for the test as if it’s a closed-book test.
Solution:
Freebie question. Guest lecture + chat information available on Wattle since Day 1 of semester
Test taking strategies discussed in L4 (Test 1 prep) and W8 (Test 2 prep)
II. Which of the following is(are) correct Select all that apply. (2 mark)
a. When we have a position in an option, we can construct a risk-less portfolio with delta-hedge that
involves constructing a hedged portfolio with our original option position and also shares of the
underlying.
b. Risk-neutral probabilities are real-world probabilities, where investors are indifferent between returns
from the stock vs. the risk-free rate.
c. We expect cash dividend payments will decrease the value of both European and American call options,
but the drop on the European option is expected to be bigger because it cannot be exercised early
before the ex-dividend date.
d. Derivatives’ values change because of changes in the underlying stock price. Dividend distribution affects
stock price, which then changes option value. Once we take care of changes in the stock price as a result
of dividends, adjustments to option valuation follow naturally because payoffs are determined solely
from comparing the stock price vs. the strike.
Solution:
Delta-hedged proof of the Binomial tree approach, L5 recording (8-13)
Risk-neutral probabilities. L5 recording (20-23)
Impact of dividend on call value. L4 s14 + W5 recording (12) + T6 Q1e + T6 Q2e
Dividend adjustment in option valuation: verbatim response to [Workshop 5] Put & receiving cash
dividend
III. Which of the following statements is/are correct Select all that apply. (2 marks)
a. In option pricing, stock prices are typically assumed to be normally distributed, and it follows that
their log returns are lognormally distributed.
b. The baseline BSM model relies on five parameters: the underlying asset price, the strike, the risk_x005f free rate, time to maturity, and the volatility, and all parameters can be directly observed.
c. When the underlying asset price is very small, ceteris paribus, we expect the call to be valued
closed to 0, because N(d1) and N(d2) are close to zero, where N(d1) and N(d2) are cumulative
probability functions that capture the probability that the call will be exercised at maturity.
d. Harry exercised an October call option on Copper futures with a strike of $3.40 when the
underlying futures was trading at $3.50. Each contract is for the delivery of 25,000 pounds. As a
result, he received a long position in the underlying October Copper futures contract and also an
amount of $2500 in his margin account.
Solution:
St and return distributions, L6 recording (7-12)
N(d1), N(d2), L6 recording (25)
Testing the BSM properties at extreme values. L6 recording (28-29)
Definition of futures options, L8 (6-8) + T8 Q2
IV. Which of the following is(are) correct Select all that apply. (2 mark)
a. A plot of strike prices against implied volatilities of call options that are otherwise identical except for
their strikes is showing a U-shape. One of the reasons for this shape, instead of the expected flat line, is
that market participants are betting on extreme volatility events with strangles.
b. Chenyang manages an equity portfolio and has taken positions in 3-month SPX put options to insure
against short-term market downturn. He can save insurance costs and still insure the value of his
portfolio by selling 3-month OTM SPX put options.
c. Azwi expected company XYZ’s stock price to increase and sold put options on XYZ. However, market
conditions changed and XYZ’s price decreased. Instead of closing the put at a loss, Azwi could instead
short XYZ so his overall positions synthesized a short call.
d. Elise is bearish on the FINM stock. She can capitalize on her market view by: (1) buying a call on FINM
and also selling a call on FINM with a higher strike (but otherwise identical); or (2) buying a put on FINM
and also selling a put on FINM with a higher strike (but otherwise identical).
Solution:
Implied volatility smile, L7 recording (23-25) + Forum post [Lecture 7] Why is implied volatility so
“happy”
Portfolio insurance with collar, L7 recording (36) + T7 Q6
Synthetic short call, W4 recording (8)
Bull call spread + bull put spread, W4 (10-11)
QUESTION 2. COMPLETE THE QUESTION WITH ALL ITS SUBPARTS. (Total 25 marks)
2002 VERSION
A stock is currently trading at $35 and the stock’s volatility is 10% p.a. The risk-free interest rate is 3% p.a.,
continuously compounded. The stock is also expected to provide a dividend yield of 2% p.a., continuously
compounded.
a) What is the value of a 5-month European call option on this stock with a strike of $30 using the binomial
tree Calculate the corresponding stock price and option value at each node on the tree below, complete
calculation process must be shown for full marks. (15 marks)
b) Without any calculations, if the strike price is instead $40, explain whether you expect the option price to
be greater or lower than (a) and why (3 marks)
c) What is the value of the option in (a) if we used the BSM model Compare this price with what you have
obtained in (a), which price would be more accurate and why (7 marks)
a) L5, T5, T6,L7
MARKS
u 1.04670124 1 u=exp^(0.1*√(2.5/12))=1.04670124408749
d 0.95538245 d=1/u=0.955382450960775
P 0.51142873 1 p=(EXP((r-q) t)-d)/(u-d)=0.511428725897274
1-P 0.48857127 1-p=0.488571274102726
D 38.3454223 1 D=S0*u*u=38.3454223031005
V(D) 8.35 1 V(D)=MAX(D-X,0)=8.34542230310055
E 35 1 E=S0*u*d=35
V(E） 5.00 1 V(E)=MAX(E-X,0)=5
F 31.946447 1 F=S0*d*d=31.9464469661336
V(F) 1.95 1 V(F)=MAX(F-X,0)=1.94644696613365
B 36.6345435 1 B=S0*u=36.6345435430622
V(B) 6.66913246 1.5 V(B)=exp(-r t)(pV(D)+(1-p)V(E))=6.66913245990305
C 33.4383858 1 C=S0*d=33.4383857836271
V(C) 3.48626432 1.5 V(C)=exp(-r t)(pV(E)+(1-p)V(F))=3.48626431853448
A 35 1 A=S0=35 is given in the question
V(A) 5.08221123 1 call price=V(A)=exp(-r t)(pV(B)+(1-p)V(C))=5.08221122754421
b) Original call is ITM now and expected to be ITM in all future states based on the binomial tree, highly
valuable. New call here is OTM at t=0 and will definitely be less valuable in comparison (L4,s14). (3 marks)
(also discussed moniness and option value in W6 s24)
c)
BSM MARKS
d1 2 d1=(ln(S0/X)+(r-q+σ^2/2)T)/(σ*sqrt(T))=2.48491664676528
d2 d2=d1-σ*sqrt(T)=2.42036692432849
call 2 c=S0exp(-qT)N(d1)-Xexp(-rT)N(d2)=5.08700874133163
Both BSM and Binomial tree approaches follow almost identical assumptions, the binomial distribution
converges to the normal distribution as the number of steps in the tree approaches infinity, and the Binomial
tree converges to the BSM model (L6, s39) (1 mark). We also know that the more steps, the more accurate
the price (W5, s16-17) (1 mark). The BSM price, when the situation permits (because there are cases when
the BSM is not suitable), would be more accurate (1 mark).
7041 VERSION
A stock is currently trading at $35 and the stock’s volatility is 15% p.a. The risk-free interest rate is 3% p.a.,
continuously compounded. The stock is also expected to pay a cash dividend of $2 in 2 months.
a) What is the value of a 6-month European call option on this stock with a strike of $30 using the binomial
tree Calculate the corresponding stock price and option value at each node on the total tree below,
complete calculation process must be shown for full marks. (15 marks)
b) Without any calculations, if the strike price is instead $40, explain whether you expect the option price to
be greater or lower than (a) and why (3 marks)
c) What is the value of the option in (a) if we used the BSM model Compare this price with what you have
obtained in (a), which price would be more accurate and why (7 marks)
a) L5, T5, T6,L7
MARKS
u 1.0778841509 1 u=exp^(0.15*√(3/12))=1.07788415088463
d 0.9277434863 d=1/u=0.927743486328553
P 0.5313997334 1 p=(EXP(r t)-d)/(u-d)=0.531399733389223
1-P 0.4686002666 1-p=0.468600266610777
PV(Div.) 1.9900249584 1 PV(Div.)=exp(-
r t)*2=1.99002495838536
S0-PV(Div.) 33.0099750416 S0-PV(Div.)=33.0099750416146
D 38.3521193550 1 D=(S0-PV(Div.))*u*u=38.3521193549539
V(D) 8.3521193550 1 V(D)=MAX(D-X,0)=8.35211935495386
E 33.0099750416 1 E=(S0-PV(Div.))*u*d=33.0099750416146
V(E） 3.0099750416 1 V(E)=MAX(E-X,0)=3.00997504161464
F 28.4119488199 1 F=(S0-PV(Div.))*d*d=28.4119488199098
V(F) 0.0000000000 1 V(F)=MAX(F-X,0)=0
B 35.5809289185 1 B=(S0-PV(Div.))*u=35.5809289184537
V(B) 5.8050872739 1 V(B)=exp(-r t)(pV(D)+(1-p)V(E))=5.80508727387952
C 30.6247893287 1 C=(S0-PV(Div.))*d=30.6247893287261
V(C) 1.5875485588 1 V(C)=exp(-r t)(pV(E)+(1-p)V(F))=1.58754855879395
A 35.0000000000 1 A=S0-PV(Div.)+PV(Div.)=35 adding back the PV(Div.)
V(A) 3.8001393161 1 call price=V(A)=exp(-r t)(pV(B)+(1-p)V(C))=3.80013931606138
b) Original call is ITM now and expected to be ITM at nodes D and E based on the binomial tree. New call
here is OTM at t=0 and will definitely be less valuable in comparison (L4,s14). (3 marks)
(also discussed moneyness and option value in W6 s24)
c)
BSM MARKS
d1 2 d1=(ln((S0-
PV(Div.))/X)+(r+σ^2/2)T)/(σ*sqrt(T))=1.09589679342837
d2 d2=d1-σ*sqrt(T)=0.98983077625039
call 2 c=(S0-PV(Div.))N(d1)-Xexp(-rT)N(d2)=3.7105954902184
Both BSM and Binomial tree approaches follow almost identical assumptions, the binomial distribution
converges to the normal distribution as the number of steps in the tree approaches infinity, and the Binomial
tree converges to the BSM model (L6, s39) (1 mark). We also know that the more steps, the more accurate
the price (W5, s16-17) (1 mark). The BSM price, when the situation permits (because there are cases when
the BSM is not suitable), would be more accurate (1 mark).
QUESTION 3. THIS QUESTION HAS 2 SUBPARTS, PLEASE ANSWER ALL SUBPARTS. (Total 22 marks)
I. It is August, and the S&P/ASX 200 index is trading at 4,000. The dividend yield on the index is 4% p.a., and
the volatility is 15% p.a. A portfolio manager managed a fund valued at $100M with a beta of 1.3 with the
same dividend yield. The manager is worried about wild market movements, and wants to insure against the
value of their portfolio dropping below $90M (excluding dividends) in the next 3 months. The risk-free rate is
3% p.a. All rates and yields are discretely compounded. Both call and put options on the S&P/ASX 200 index
are available. Answer the following
a) What type of contracts and position should the manager use for the desired insurance (2 marks)
b) How many contracts are required for the insurance (3 marks)
c) What is the expected portfolio value at T Write out the complete expression as a function of S_T. (3
marks)
Simplified & identical version of Tutorial 8 Q7, copy & paste from tutorial solution with modifications.
a) Long (1 mark) put (1 mark)
b) The index multiplier for a XJO option is $10. (1 mark)
Optimal # of contracts = β(P/A) = 1.3*100000000/(4000*10) = 3250 (2 marks)
c)
Vp = P_0x(1+E(rp)-q)
Vp = P_0x(1+(rf+ β x((S_T-S_0)/S_0+q-rf))-q)
Vp =100000000x(1+(0.0075+1.3x((S_T-4000)/4000+0.01-0.0075))-0.01) (3 marks)
II. An eight-month European call option on a futures contract with a strike price of $40 is trading for $6. An
eight-month European put option on the same futures contract with the same strike price is also trading for
$6. The futures contract underlying these option contracts is trading at $42. The risk-free interest rate is 5%
per annum continuously compounded. Identify any arbitrage opportunities. If any exist, construct a strategy
that will yield the arbitrageur an immediate profit. (14 marks)
Simplified & identical version of Tutorial 8 Q3, copy & paste from tutorial solution with modifications.
Lower bound check
c_lowerbound = (F0-X)exp(-rT) = (42-40)exp(-0.05×8/12) = 1.934432201, c_mkt > c_lowerbound
p_lowerbound = (X-F0)exp(-rT) = (40-42)exp(-0.05×8/12) < 0, non-binding because option prices must be nonnegative, and in any case p_mkt > p_lowerbound.
The lower bounds hold. (2 marks)
Check put-call parity (PCP): c+Xexp(-rT) = p+F0exp(-rT)
LHS = 6+40exp(-0.05×8/12) = 44.68864402
RHS = 6+42exp(-0.05×8/12) = 46.62307622
LHS < RHS with a difference of 1.9344322, PCP violated, call is undervalued relative to put, so we should buy low (i.e., LONG LHS) and sell high (i.e., SHORT RHS). (2 marks) LONG c+Xexp(-rT) involves: 1. Long one European call futures @ $6 2. Invest an amount equal to Xexp(-rT) SHORT p+F0exp(-rT) involves: 3. Short one European put 4. Short a one-year futures contract @ F0=42 5. Borrow an amount equal to F0exp(-rT) (3 marks) t=0: CF on the LONG portfolio positions are: 1. -6 premium paid for long call 2. -40exp(-0.05x8/12) for cash investment CF on the SHORT portfolio positions are: 3. +6 premium received from selling put 4. 0 value on futures position (futures is always priced so it has 0 value at t=0) 5. +42exp(-0.05x8/12) from cash loan NET CF = (-6+6)+(42-40)exp(-0.05*8/12) = 1.934432201 (note: this is the difference from the put-call parity inequality during our check) (3.5 marks) t=T: FT≥40 [OR] FT<40 CF on the LONG portfolio positions are: 1. Exercise call for FT-40 [OR] 0 2. Cash investment matures to +40 [OR] +40 CF on the SHORT portfolio positions are: 3. Put expires unexercised, 0 to position [OR] put option exercised against the short position, loss of –(40-FT) 4. Futures positions closed for 42-FT [OR] 42-FT 5. Repay cash loan -42 [OR] -42 NET CF: (FT-40+40+0+42-FT-42)=0 [OR] (0+40 – (40-FT) +42- FT- 42)=0 (3.5 marks)