Question Paper
Financial Risk Management – I (231) : April 2005 Section A : Basic Concepts (30 Marks) • This section consists of questions with serial number 1 - 30. • Answer all questions. • Each question carries one mark. • Maximum time for answering Section A is 30 Minutes. 1.
Raising funds through floating interest rate bearing instruments reduces the losses due to interest rate risk. This is an example of managing the risk by which of the following approaches? (a) Avoidance (d) Loss control
2.
(b) d2
(c) N(d1)
6.
(b) Rs.5
(b) Writing a naked put (d) Buying two calls (e) Buying asset.
(c) Rs.7
(d) Rs.8
< Answer >
(e) Rs.10. < Answer >
(b) An intramarket spread (d) A reverse spread < Answer >
(b) Forward contract (e) Swap deal.
(c) Option forward contract < Answer >
A trader who is long on Treasury bond futures expects (a) (b) (c) (d) (e)
9.
< Answer >
Which of the following foreign exchange transaction does not involve credit risk? (a) Spot transaction (d) Futures contract
8.
(b) Infinite risk (c) Necessity to post margin (e) More regulatory hassles.
If somebody buys a soybean futures and sells a soybean meal futures, this is (a) An intermarket spread (c) A calendar spread (e) An intercommodity spread.
7.
< Answer >
If stock is purchased at Rs.50 and a Rs.55 call is written for a premium of Rs.2, the maximum possible gain per share is (a) Rs.2
< Answer >
(e) N(– d1).
Selling stock short and simultaneously buying a call is similar to (a) Writing a naked call (c) Buying a put
5.
(d) N(d2)
The biggest single disadvantage of writing covered calls is (a) Added commissions (d) Opportunity losses if exercise occurs
4.
(c) Transfer
In the Black and Scholes model for call option valuation, the term used to compute option probability of exercise is (a) d1
3.
(b) Separation (e) Combination.
< Answer >
The yield curve to shift parallely upwards The yield curve to become more upward sloping Long-term interest rates to rise Long-term interest rates to decline Both (b) and (c) above.
The two-year spot fixed rate is 5.50%. A swap dealer quotes a price of 23 bp bid, 26 bp ask. Someone who wants to pay the fixed rate would pay (a) 5.24%
(b) 5.27%
(c) 5.50%
(d) 5.73%
(e) 5.76%.
< Answer >
10.
The daily limit of a commodity futures contract is the maximum (a) (b) (c) (d) (e)
11.
< Answer >
Amount by which the maintenance margin can change per day Percentage by which the futures price can increase from the previous day Price increase or decrease relative to the settlement price the previous day Number of contracts allowed to be traded that day Open interest permitted on any trading day.
Which of the following are characteristics of the intrinsic value of a call option?
< Answer >
I.
The intrinsic value is obtained by subtracting the per-share exercise price of an option from the market price of a stock. II. The intrinsic value is obtained by subtracting the market price of a stock from the per-share exercise price of an option. III. The intrinsic value is the maximum price that an option will command when a stock’s market price is below the exercise price. IV. The market price of an option will approach its intrinsic value at expiration. (a) Both (I) and (III) only (c) Both (II) and (III) only (e) (I), (III) and (IV) only. 12.
Which of the following statements is true regarding an in-the-money call option, everything else being equal? (a) (b) (c) (d) (e)
13.
A knock-in barrier option is harder to hedge when it is
< Answer >
Selling put with higher strike price and buying put with lower strike price Buying and selling put of near and long term respectively Writing a call and buying a call with different strike prices but identical maturity Writing a put option at current spot rate Writing a call option and buying a put option of identical strike price.
Which of the following positions has the same exposure to interest rates as the receiver of the floating rate on a standard interest rate swap? (a) (b) (c) (d) (e)
< Answer >
(b) Out-of-the-money (d) Deep out-of-the-money
A Bullish put spread consists of (a) (b) (c) (d) (e)
16.
< Answer >
Require no cash outflow when purchased Permit the adjustment of portfolio risk/return exposure Have no margin requirement Have larger premiums than other options Have more liquidity than index options.
(a) In-the-money (c) At the barrier and near maturity (e) At the inception of trade. 15.
< Answer >
As time passes its delta will approach one As time passes its value will approach zero As time passes its theta will remain constant As time passes its intrinsic value will decline As time passes its vega will increase.
A primary advantage of futures options is that they (a) (b) (c) (d) (e)
14.
(b) Both (I) and (IV) only (d) Both (II) and (IV) only
Long a floating rate note with same maturity Long a fixed rate note with same maturity Short a floating rate note with same maturity Short a fixed rate note with same maturity Both (a) and (d) above.
< Answer >
17.
(a) (b) (c) (d) (e) 18.
Short cap + Long floor = Fixed rate bond Long cap + Short floor = Fixed swap Long cap + Short floor = Floating rate bond Short cap + Short floor = Interest rate collar Long cap + long floor = Caplet.
Which of the following type of option experiences accelerating time decay as expiration approaches in a market where other things remain the same? (a) Deep-in-the-money options (c) Deep-out-of-the-money options (e) At-the-money-options.
19.
21.
A person expects the market to decline sharply in near future. He will want options with
23.
< Answer >
(b) Positive delta, positive rho (d) Positive delta, negative
< Answer >
Which of the following equations is true? (a) (b) (c) (d) (e)
< Answer >
At most $3 million in 1 out of next 100 days At least $3 million in 95 out of next 100 days At least $3 million in 1 out of next 100 days At most $6 million in 2 out of next 100 days Maximum $3 million in a day.
(a) Positive gamma, positive theta (c) Positive theta, negative vega gamma (e) Negative delta, positive gamma. 22.
< Answer >
(b) Monte Carlo simulation (d) Delta/gamma method
What is the correct interpretation of a $3 million overnight VaR figure with 99% confidence level? The institution is expected to lose (a) (b) (c) (d) (e)
< Answer >
(b) In-the-money options (d) Out-of-the-money options
Which VaR methodology is least effective for measuring option risk? (a) Historical simulation (c) Variance/Covariance approach (e) Hybrid method.
20.
< Answer >
The cap-floor parity can be stated as
Long underlying asset + Short call = Long put Short underlying asset + Long put = Short call Long underlying asset + Long call = Short put Short underlying asset + Long put = Long call Short underlying asset + long call = Long put.
A company expecting hot days in summer should
< Answer >
I. Buy call option of CDD in summer. II. Buy call option of HDD in winter. III. Sell CDD indices in summer. (a) Only (I) above (d) Both (I) and (II) above 24.
(b) Only (II) above (c) Only (III) above (e) Both (II) and (III) above.
If two options are bought at two extreme prices, and two options are sold at two intermediate prices, it is known as (a) Bullish spread (d) Condor spread
(b) Ratio spread (e) Gamma spread.
(c) Box spread
< Answer >
25.
< Answer >
Which of the following statements does not indicate the hedge effectiveness? (a) If the principal amount and the notional amount of the swap match (b) If the fair value of the swap is zero in the beginning of the transaction (c) If the net settlements under the swap are computed on each settlement date in the same way as they are calculated on an interest-bearing instrument (d) If there is prepayment facility in the financial instrument (e) If there is no cap or floor on the variable interest rate of the swap.
26.
(a) (b) (c) (d) (e) 27.
VaR measures does not indicate how large the losses can be Stress-testing provides a minimum loss level VaR measures are correct only at 95% confidence interval Stress-testing scenarios incorporate reasonably probable events It performs effective backtesting too.
As per FAS-133 which of the following embedded derivatives is not accounted for separately? (a) (b) (c) (d) (e)
28.
< Answer >
If a day’s temperature is 55º F, then heating degree days is (b) –5
(c) 0
(d) 5
(e) 10.
The US T-bill futures index is 96.2. The yield on holding the bill till maturity of 90 days will be (a) 3.72%
30.
(b) 3.80%
(c) 3.84%
(d) 3.96%
Which of the following losses is excluded from fire insurance policy? (a) (b) (c) (d) (e)
< Answer >
Term extending options having no reset of interest rates Non-leveraged inflation indexed payments Equity indexed interest payments Commodity indexed interest payments Convertible debt qualifying as derivative instrument.
(a) –10 29.
< Answer >
VaR measures should be supplemented by portfolio stress-testing because
Loss due to lightning Loss due to fire resulting from explosion Loss due to fire fighting activities Loss due to explosion of boiler used for domestic purposes Loss by theft during the fires. END OF SECTION A
< Answer >
(e) 4.02%. < Answer >
Section B : Problems (50 Marks) • This section consists of questions with serial number 1 – 6.
• Answer all questions. • Marks are indicated against each question. • Detailed workings should form part of your answer. • Do not spend more than 110 - 120 minutes on Section B. 1.
In January 2005, the T-bill futures on the IMM are trading at the following prices: March futures :
98.25
June futures
97.95
:
A speculator is expecting that the yield curve is about to become steeper. The speculator has no particular views about the level of interest rates, however, he wishes to profit from this view. You are required to show how the speculator can profit from this view? Under what circumstances will he make loss? (8 marks) < Answer > 2.
The stock of a company is currently quoted in the market at Rs.150. The price of the stock is expected to go up or down by 10% in next one year and by 15% in the second year. The risk-free interest rate in the economy is 6%. Required: Using two-step Binomial Model, find out the price of a 2-year American put option on the company’s stock with strike price of Rs.175. (8 marks) < Answer >
3.
A firm in Denmark exports dairy products. On June 15 2004, an order worth $ 5 million to a US super store chain was shipped. The payment was due after 3 months from the day of shipment. The spot DKr/$ was 6.1569 and the 3 month forward rate was 6.1625 at that time. The firm considered hedging the exposure through futures contract. Since futures contract for Danish Kroner was not available, it considered either futures on Swiss Franc or Swedish Kroner on IMM as both the currencies are closely related to Danish Kroner. The spot SFr/$ rate was 1.2743 and September SFr futures were trading at $0.7875. The spot SKr/$ rate was 7.5833 and September SKr futures were trading at $0.13126 at that time. On September 15, 2004, dollar was priced in the spot market as at SFr 1.2678, SKr 7.6166 and DKr 6.1602. In the futures market September SFr future was priced $ 0.7891 and September SKr futures was priced at $ 0.13133. You are required to find out which hedging strategy would have been better for the Danish firm. (Standard size of SFr and SKr futures are 125,000 each). (8 marks) < Answer >
4.
The current ¥/$ spot rate is 112.00. A speculator believes that in the next three months yen will fluctuate significantly against dollar, but he is not sure of the direction of the movement. The following 3-month European put options on yen are traded in the market:
Strike Price Premium $0.0085 $0.00006 $0.0089 $0.00020 $0.0093 $0.00050 You are required to suggest the speculator a spread strategy using all the above options so that the speculator is exposed to a limited loss. Also, prepare the payoff profile of the strategy showing maximum possible profit, maximum possible loss and break-even point(s), if spot rate after 3 months ranges between $0.0082 – $0.0096. (7 marks) < Answer >
5.
A corporation enters into a $10 million notional principal interest rate swap. The swap calls for a corporation to pay fixed rate and receive floating rate on LIBOR. The payment will be made every 90 days for one year and will be based on the adjustment factor 90/360. The term structure of LIBOR when the swap is initiated is as follows:
Days 90 180 270 360 Rate (%) 7.00 7.25 7.45 7.55 such a rate that the value of the swap is zero.
Note that at the initiation of the swap, the fixed rate is set at
You are required to: a. Determine the fixed rate on the swap. b. Calculate the first net payment on the swap. (8 + 1 = 9 marks) < Answer > 6.
The following table lists the deltas, gammas and vegas for long position on three derivative instruments for a notional principal of $1 million for each.
Instrument (underlying 90-day LIBOR) 3-year call option with exercise rate of 4% 3-year swap with fixed rate of 3.25% 2-year FRA with fixed rate of 3%
Delta $40 $152 $72
Gamma $1,343 -$678 -$360
Vega $5.02 $0 $0
Long position on a swap
or FRA means to pay fixed rate and to receive floating rate. An investor is holding $12 million notional principal long position in the 3-year call option, an $8 million notional principal short position in the 3-year swap and an $11 million notional principal long position in the FRA. You are required to: a. b.
Determine the current portfolio delta, gamma and vega. Explain the risk properties of the portfolio. Assume that the investor has to maintain the current position in the call option but is free to change his positions in the swap and FRA. The investor has also identified a 1-year call option with delta of $62, gamma of $2,680 and vega of $2.41 to combine in his portfolio. Find the overall position of notional principals that would make the overall position be delta, gamma and vega-hedged. (2 + 8 = 10 marks) < Answer >
END OF SECTION B
Section C : Applied Theory (20 Marks) • This section consists of questions with serial number 7 - 8.
• Answer all questions. • Marks are indicated against each question. • Do not spend more than 25 -30 minutes on section C.
7.
a. b.
What will happen in the options market if the price of an American call is less than the value Max (0, S – E)? Will your answer differ if the options are European? Explain. Call prices are directly related to the stock’s volatility, yet higher volatility means stock prices can go lower. How will you resolve this apparent paradox? (5 + 5 = 10 marks) < Answer >
8.
‘The list of pure risks suffices to say that doing anything in life involves risk’. Explain the various types of pure
risks. (10 marks) < Answer > END OF SECTION C END OF QUESTION PAPER
Suggested Answers
Financial Risk Management – I (231) : April 2005 Section A : Basic Concepts 1.
Answer : (d) Reason : Loss control measures are used in respect of risks which cannot be avoided. These risks might have been assumed either voluntarily or because they could be avoided. The objective of these measures is either to prevent a loss or to reduce the probability of loss. Insurance, for example, is a loss control measure. Introduction to systems and procedures, internal or external audit help in controlling the losses. Raising funds through floating rate interest bearing instruments reduces the losses due to interest rate risk.
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2.
Answer : (d) Reason : In the Black and Scholes model for call option valuation the term used to compute option probability of exercise is N(d2).
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3.
Answer : (d) Reason : The biggest single disadvantage of writing covered calls is opportunity losses if exercise occurs since the call will be exercised when spot price is more than the strike price. Commissions are not required for writing call, risk is controlled, margin requirement is not there since the call is covered and also there is no regulatory hassle for this.
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4.
Answer : (c) Reason : Selling stock short and simultaneously buying a call is similar to buying a put option.
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5.
Answer : (c) Reason : The maximum possible gain per share when spot price exceeds strike price of Rs.55 is (55 – 50 + 2) = Rs.7.
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6.
Answer : (e) Reason : Buying and selling futures for two different but related commodities is called intercommodity spread.
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7.
Answer : (d) Reason : In futures contract there is no credit risk as the futures clearing house bears the credit risk.
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8.
Answer : (d) Reason : A trader going long Treasury bond futures expects long-term interest rates to decline since the index will be higher then due to fall in interest rate and the trader can sell at a higher price.
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9.
Answer : (e) Reason : The swap quote indicates the bank will receive floating and pay fixed at (5.50 + 0.23) = 5.73%, and will pay floating and receive fixed at (5.50 + 0.26) = 5.76%.
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10.
Answer : (c) Reason : The daily limit of a commodity futures contract is the maximum of price increase or decrease relative
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to the settlement price the previous day. 11.
Answer : (b) Reason : The intrinsic value of a call option is zero until the market price of the underlying stock reaches the strike price; after that point, the intrinsic value is computed by subtracting the strike price from the market price of the stock. At expiration, the option can no longer have a time premium, so its value is equal to its intrinsic value.
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12.
Answer : (a) Reason : For an in-the-money call option, as time passes its delta will approach one. All other alternatives are not correct.
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13.
Answer : (b) Reason : The primary advantage of futures options is that they permit the adjustment of portfolio risk/return exposure.
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14.
Answer : (c) Reason : Knock-in or knock-out options involve discontinuities and are harder to hedge when the spot price is close to the barrier.
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15.
Answer : (a) Reason : Bullish put spread is created by selling put with higher strike price and buying put with lower strike price.
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16.
Answer : (d) Reason : Paying a fixed rate on the swap is same as the being short a fixed rate note.
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17.
Answer : (a) Reason : With the same strike price, a short cap and long floor loses money if rates increase, which is equivalent to a long position in a fixed rate bond.
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18.
Answer : (e) Reason : Time decay describes the loss of option value, which is greatest for at-the-money option with short maturities.
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19.
Answer : (c) Reason : The variance/covariance approach does not take into account second-order curvature effects which is important for measuring option risk.
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20.
Answer : (c) Reason : There will be a loss worse than VaR in on average , n = 1% x 100 = 1 day out of 100.
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21.
Answer : (e) Reason : A person expects the market to decline sharply in the near future. He will want options with negative delta, positive gamma since when stock price decreases the value of a put which is having negative delta will increase and for puts, positive gamma means that their delta will become more negative and move toward –1.00 when the stock price falls.
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22.
Answer : (e) Reason : If we are short in a asset, we cover it by buying call, the payoff is similar to long put.
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23.
Answer : (e) Reason : A company which is expecting hot days in summer would either buy call option of HDD in winter or sell CDD indices in summer.
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24.
Answer : (d) Reason : In a condor spread, two options are bought at two extreme prices, and two options are sold at two intermediate prices.
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25.
Answer : (d) Reason : Though specific conditions apply to the hedge type (fair value/cash flow) we can assume that a
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26.
hedging relationship between an interest bearing financial instrument and an interest rate swap is effective if 1. The principal amount and the notional amount of the swap match. 2. The fair value of the swap is zero in the beginning of the transaction. 3. The net settlements under the swap are computed on each settlement date in the same way as they are calculated on an interest-bearing instrument. 4. There is no prepayment facility in the financial instrument. 5. The terms are typical for both the instruments and they should not invalidate the assumption. 6. The maturity date of the instrument and the expiration date of the swap match. There is no ceiling or floor on the variable interest rate of the swap. The time period between re-pricing is frequent enough to assume that the variable rate is a market rate. Answer : (a) Reason : The goal of stress testing is to identify losses that can go beyond the normal losses measured by VaR.
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27.
Answer : (b) Reason : Non-leveraged inflation indexed payments are not accounted separately under FASB - 133.
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28.
Answer : (e)
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Reason : HDD = 0 or,
65° F− 55° F,
whichever is higher
= 10. 29.
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Answer : (c)
Reason : Price of T-bill = $1,000,000 Yield = 30.
90 1 - 0.038 x 360
1, 000, 000 990, 500 360 990, 500 90
= $990,500
= 3.84%.
Answer : (e) Reason : Loss by theft during the fire is not included in the fire insurance policy.
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Section B : Problems 1.
Rates of T-bill futures in January : March 98.25 June 97.95 If the dealer expects that the yield curve will become steeper it means that spread between the near and far end contract will widen, i.e. Longer term interest rates will rise more than the shorter term interest rates. Hence the speculator will buy a near end contract and sell a far end contract i.e. buying a spread. He buys March contract and sells June contract. Assume in February the new rates are as under: Scenario
I
II
March June
98.35 98.00
98.10 97.75
If
the
speculator
closes
his
position I.
Gain on March contract (98.35 – 98.25) × 100 × 25 =
$ 250
(98.00 – 97.95) × 100 × 25 =
$ 125
Loss on June contract
Net gain
------$ 125
------$ 375 $ 500 ------Net gain $ 125 ------The speculator gains from his expectations that the yield curve becomes steeper under both the scenarios of increasing interest rates and decreasing interest rates. The speculator can make loss with this strategy if yield curve become downward sloping i.e. if the fall in long term interest rate is more than the fall in short term interest rate. II.
Loss on March contract (98.25 – 98.10) x 100 x 25 = Gain on June contract (97.95 – 97.95) x 100 x 25 =
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2.
Current stock price
= Rs.150
Price after first year
=
Price after second year
= ± 15%
±
10%
Risk-free rate = 6% The situation can be represented in the following way:
The value of American put option at node D, E, F and G will be equal to the value of European put option on these nodes. Value at node D : as put is out-of-money, so value is zero Value at node E: 175 – 140.25 = 34.75 Value at node F : 175 – 155.25 = 19.75 Value at node G : 175 – 114.75 = 60.25 1.06 − 0.85 1.15 − 0.85
Probability of price increase in second year, P2 = Probability of price decrease = 1 – P2 = 0.3 Using single-period model, the value of put at node B is P
=
= 0.70
Pu p 2 + Pd (1 − p 2 ) R 0 ×0.70 + 34.75 ×0.30 1.06
= = 9.83 At node B, pay-off from early exercise is Rs.10, which is more than the value calculated as per single-period model. So value of put at node B is 10. The value of put at node C is 19.75 ×0.70 +60.25 ×0.30 1.06
P = = 30.09. Pay-off from early exercise is 40, whereas single-period model gives a value of 30.09 which is lower, so value of put will be 40. Probability of price increase in first year, p1 = Probability of price decrease = 1 – p1 = 0.20. The value of put at mode A,
1.06 − 0.90 1.10 − 0.90
= 0.80.
10 ×0.80 + 40 ×0.20 1.06
P = = 15.09. Whereas the value due to early exercise is Rs.25 which is more than the value given by single period model. Hence, the value of two year American put option is Rs.25. < TOP >
3.
Hedging through SFr futures As the customer had a receivable in $, he would go long in SFr futures as it amounts to go short in USD i.e. buy SFr futures Standard size of SFr future is 125,000. 5, 000, 000 125, 000 × 0.7875
=
The number of SFr futures contracts to be bought = 50.79365079 Gain from SFr futures is = (0.7891 - 0.7875) x 51 x 125,000 = $10,200.00 Gain from SFr futures in DKr = 10,200 x 6.1602 = 62,834.04 Inflow in the spot market = 5,000,000 x 6.1602 = DKr 30,801,000 Total inflow = DKr 30,863,834.04 Hedging through SKr futures Here also as the customer had a receivable in $, he would bought SKr futures. Standard size of SKr future is 125,000. 5, 000, 000 125, 000 × 0.13126
= 51.
=
The number of SKr futures contracts to be bought = 304.7386866 = 305. Gain from SKr futures is = (0.13133 - 0.13126) x 305 x 125,000 = $2,668.75 Gain from SKr futures in DKr =2,668.75 x 6.1602 = DKr 16,440.03 Inflow in the spot market = 5,000,000 x 6.1602 = DKr 30,801,000 Total inflow = DKr 30,817,440.03 So hedging through SFr futures would have given better result since inflow is more there. < TOP >
4.
Appropriate strategy is short butterfly spread. Here, the investor should sell puts at 0.0085 and 0.0093 and buy 2 puts at 0.0089. Initial inflow = 0.00006 + 0.00050 - 2 x 0.00020 = 0.00016.
Spot
Short
Long
Short
Initial
Net Inflow/
P = 0.0085
P = 0.0089
P = 0.0093
Inflow
Outflow
0.0082
-0.0003
0.0014
-0.0011
0.00016
0.00016
0.0083
-0.0002
0.0012
-0.001
0.00016
0.00016
0.0084
-0.0001
0.0010
-0.0009
0.00016
0.00016
0.0085
0
0.0008
-0.0008
0.00016
0.00016
0.0086
0
0.0006
-0.0007
0.00016
0.00006
0.0087
0
0.0004
-0.0006
0.00016
-0.00004
0.0088
0
0.0002
-0.0005
0.00016
-0.00014
0.0089
0
0
-0.0004
0.00016
-0.00024
0.0090
0
0
-0.0003
0.00016
-0.00014
0.0091
0
0
-0.0002
0.00016
-0.00004
0.0092
0
0
-0.0001
0.00016
0.00006
0.0093
0
0
0
0.00016
0.00016
0.0094
0
0
0
0.00016
0.00016
0.0095
0
0
0
0.00016
0.00016
0.0096
0
0
0
0.00016
0.00016
Max. Profit
=
$ 0.00016 Max. Loss = $ 0.00024 Break - even points are $ 0.00866 and $ 0.00914. < TOP >
5.
a.
Let the fixed rate to be received by the bank be ‘R’, and the notional principal be ‘P’. At the first payment date, the fixed payment is = P ´ R ´ (90/360) The present value of the fixed leg we can get by multiplying (P ´ R) by the discounting factor we can get from the LIBOR term structure. Term
Rate
Discounting factor
90 days
7.00%
1/(1 + .07(90/360)) = 0.9828
180 days
7.25%
1/(1 + .0725(180/360)) = 0.9650
270 days
7.45%
1/(1 + .0745(270/360)) = 0.9471
360 days
7.55%
1/(1 + .0755(360/360)) = 0.9298
The present value of fixed leg = P ´ R ´ 0.25 (0.9828 + 0.9650 + 0.9471 + 0.9298) = P ´ R ´ 0.9562 We know that on the date when interest rate is reset, the bond sells at par value. Hence, at time 0, the present value of floating rate payments is the notional principal, P. But, given that there is no principal payment the present value of principal repayment to be subtracted. So, present value of floating payments is = P - P ´ 0.9298 = P ´ 0.0702 Now value of the swap at inception should be zero, hence we will equate present value of fixed payments and present value of floating payments. P ´ R ´ 0.9562 = P ´ 0.0702
or, R = 7.34%
b. The first net payment is based on a fixed rate of 7.34 percent and a floating rate of 7 percent: Fixed payment: $10,000,000(.0734)(90/360) = $183,500
Floating payment: $10,000,000(.07)(90/360) = $175,000 The net is that the party paying fixed makes a payment of $8,500. < TOP >
6.
a.
b.
The delta, gamma and vega of the portfolio are as follows: Delta: 12(40) - 8(152) + 11(72) = 56 Gamma: 12(1,343) - 8(-678) + 11(-360) = 17,580 Vega: 12(5.02) - 8(0) + 11(0) = 60.24 Thus, for each basis point increase in the 90-day LIBOR, the portfolio will increase in value by $56 and the delta will increase by 17,580. For each unit increase in the volatility, the portfolio will increase in value by $60.24. We have to set up the following three equations to make the portfolio delta-, gamma- and vega-hedged using the new 1-year call option: 12(40)
- x1 (152) + x2 (72) + x3 (62)
12(1,343) - x1 (-678) + x2 (-360) + x3 (2,680)
=0
……………. (i) =0
……………. (ii)
12(5.02) - x1 (0) + x2 (0) + x3 (2.41) =0 ……………. (iii) The first equation contains the deltas of the combination of $12 million in the 3-year call option, x 1 million in the 3-year swap, x2 million in the 2-year FRA and x3 million in the 1-year call, which are combined to equal zero so that the position is delta-hedged. The second equation does the same with the gammas. The third equation does the same with the vegas. From (iii) x3 = – 12(5.02)/2.41 = – 25 From (ii) – 678 x1 – 360 x1 = 50,884 From (i) 152 x1 + 72 x2 = 1070 ………………….Multiply with 5 760 x1 + 360 x2 = 5,350 -678 x1 - 360 x1 = 50,884 82 x1 = 56,234 or, x1 = 685.78 From (i) x2 = {1070 – 152(685.78)}/72 = – 1432.90 By solving the equations we get the solutions as x1 = – 685.78, x2 = – 1432.90 and x3 = – 25.00. Thus, a short position in $685.78 million of the 3-year swap, a short position in $1432.90 million of the 2-year FRA and a short position in $25 million of the 1-year call option will combine with the $12 million long position in the 3-year call to set the delta, gamma and vega to zero. < TOP >
Section C: Applied Theory 7.
a.
b.
This would create an arbitrage opportunity. The call would be purchased and immediately exercised. For example, suppose S0 = 44, X = 40, and the call price is Rs.3. Then an investor would buy the call and immediately exercise it. This would cost Rs.3 for the call and Rs.40 for the stock. Then the stock would be immediately sold for Rs.44, netting a risk-free profit of Rs.1. In other words, the investor could obtain an Rs.44 stock for Rs.43. Since everyone would do this, it would drive the price of the call up to at least Rs.4. If the call were European, however, immediate exercise would not be possible (unless, of course, it was the expiration day), so the European call could technically sell for less than the intrinsic value of the American call. We saw, though, that the European call has a lower bound of the stock price minus the present value of the exercise price (assuming no dividends). Since this is greater than the intrinsic value, the European call would sell for more than the intrinsic value. Then at expiration, it would sell for the intrinsic value. The paradox is resolved by recalling that if the option expires out-of-the-money, it does not matter how far out-of-the-money it is. The loss to the option holder is limited to the premium paid. For example, suppose the stock price is Rs.24, the exercise price is Rs.20, and the call price is Rs.6. Higher volatility increases the chance of greater gains to the holder of the call. It also increases the chance of a larger stock price decrease. If, however, the stock price does end up below Rs.20, the investor's loss is the same regardless of whether the
stock price at expiration is Rs.19 or Rs.1. If the stock were purchased instead of the call, the loss would obviously be greater if the stock price went to Rs.1 than if it went to Rs.19. For this reason, holders of stocks dislike volatility, while holders of calls like volatility. A similar argument applies to puts. < TOP >
8.
The main pure risk can be described as under: i. Property Exposure ii. Liability Exposure iii. Life and Health Exposure iv. Financial Exposure Property Exposure Any business or individual that uses any kind of property whether owned, leased, rented or other wise is exposed to the risk of loss, theft and damage that may be caused by man-made reasons or natural reasons. Depending on the extent of exposure and damage, the business may be affected. Liability Exposure Around the world, liability to any business due to litigation, damages, claims, etc. has become a major issue of concern. Millions of dollars are lost by companies over legal suits and settlements. Such risks are there to an individual also. Life and Health Exposure Human beings have a certain death, although the extent of life quality cannot be determined. An individual may die while still young or may be bed-ridden for most of his life. Some people are healthy while others have to spend a major part of their earnings on health related matters. This exposure leads to loss of earnings for the individual, as well as loss of man-hours to the business to which he is associated. Financial Exposure The three exposures mentioned above involve pure risks. Financial exposure can be because of speculative nature also, and should not always be considered as a pure risk, but it still has same problems associated with pure risks. Although the techniques associated with these risks may be different from those uses to manage the other risks mentioned above, it remains critical that these risks be identified as assessed in order for the firm to achieve its business goals. < TOP >
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