Financial Derivatives
Problem Set 7
Spring 2014
Portfolio Insurance & Exotic Options
1.
Today, a market index is at 1224.36. You manage a portfolio that duplicates the
index and is worth 21,000 times the index (you can think of this as the number of
shares held—that is, 21,000 “shares” of the index). You wish to insure the portfolio
at a particular value over the period from today until 90 days into the future. To
provide portfolio insurance, you plan to use put options on the index future, which
are selling for a premium of $32.70 and have an exercise price of 1210. You plan to
sell some of the stock in order to fund the purchase of puts. How many “shares” will
you have in the revised portfolio, and how many puts? What is the value of the
revised stock portfolio combined with this insurance?
2.
Consider the portfolio you constructed in problem 1. Suppose the index at expiration
is 1201.25. What is the outcome?
3.
Consider the portfolio you constructed in problem 1. Suppose the index at expiration
is 1234.36. Compute the upside capture and the cost of the insurance.
Problems 4 through 7 ask you to use the downloadable option calculator available from
the course website. Please note that this tool will provide answers that are more
precise than you would get if you rounded the intermediate results and then just ran
the rounded numbers through a handheld calculator.
4.
Using the information provided in the problems above, construct the insured
portfolio using equities and treasury bills. In order to do this, you plan to sell some
of the stock and use the money to buy bonds, with the proportions designed to create
a synthetic put in place of the put contract used in problem 1. The volatility
(standard deviation) is 17.5% (0.175). The continuously compounded risk-free rate
is 5%. The dividend yield on the index is 3% annually with continuous
compounding. Use the Portfolio Insurance (initial) worksheet on the downloadable
option calculator (file “Option.xls” on the course website). Please note that when
you use the option calculator you will get answers that are more precise than you
would get if you just ran the rounded numbers used in the first three problems
through a handheld calculator.
5.
Use the information in problems 1&4 to set up a dynamic hedge using stock index
futures. Assume a multiplier of 250. The futures price is 1,230.41. For this
problem, use the option calculator to obtain the delta for a call option with the same
terms as the put in problem 1. Again please not that when you use the options
calculator you will get answers that are more precise than you would get if you just
ran the rounded numbers from each intermediate step through a handheld calculator.
6.
Use the information in problems 1-5 to obtain the delta for the insured portfolio
consisting of stocks and puts. Find the number of futures contracts needed to
construct the hedge using stocks and futures (see the Appendix on portfolio
insurance for details about constructing the hedge).
Prof. Kensinger
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Financial Derivatives
7.
Problem Set 7
Spring 2014
Refer to the portfolio you constructed in problem 6. Let the index rise by $1 to
1225.36 after a moment, and show that the change in the portfolio value is almost
the same as it would have been using the put hedge you constructed in problem 1
(the elapsed time is so short, you don’t need to change t). First, assume that you can
buy fractional contracts, so you can get full precision. How much difference does
the unavailability of fractional contracts make on the outcome?
Problems 8 through 10 refer to the following situation:
A portfolio manager is interested in purchasing an instrument with a payoff that is
like a call option, but does not want to pay money up front. The manager learns
from a banker that one can do this by entering into a break forward contract. The
manager wants to learn if the banker is quoting a fair price. The stock price is 43.75.
The contract expires in 270 days. The volatility is 18% (or 0.18) and the
continuously compounded risk-free rate is 3.75% per year. The exercise price for the
contract will be set at the forward price of the stock. (Use the options calculator for
this problem
8.
Determine the exercise price, assuming the stock does not pay dividends.
9.
For the loan that is implicit in the break forward contract, determine the face value
that would be fair.
10. Using the face value that you calculated in problem 9 and the exercise price you
calculated in problem 8, determine the value of the position if the stock price ends
up at 46.50 at expiration. Repeat for a stock price of 42.50.
Problems 11 through 15 refer to the following situation, which utilize relationships given
in Slide 38 of Topic 8 (slide title is “Decomposition of a European Call”):
These problems refer to digital options on DataWhack, Inc. common stock. Suppose
the stock price is $65, strike price is $65, and the present value of the exercise price
is $64.47 (interest rate is 5% and time remaining is 60 days). Volatility of the stock
is 20%. Delta is 0.5564. Also, the change in value of the call relative to a small
change in the present value of the exercise price is 0.5242. The stock pays no
dividends. Based on this information, estimate the premium for the following digital
options:
11. Calculate the value of an Asset-or-Nothing Option on DataWhack stock. Also, be
prepared to explain what the potential outcomes could be for a holder of such an
option.
12. Calculate the value of a Cash-or-Nothing Option on DataWhack stock. Also, be
prepared to explain what the potential outcomes could be for a holder of such an
option.
13. Combine a portfolio of these digital options to reconstitute a “plain-vanilla”
European call option on DataWhack stock.
Prof. Kensinger
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Financial Derivatives
Problem Set 7
Spring 2014
14. Find the value of a chooser option on DataWhack stock with specifications the same
as the options in problems 11-13, and with the choice of whether to make it a put or
a call to be made on day 30. Also, be prepared to explain what the potential
outcomes could be for a holder of such an option. How is the payoff different from
a “plain-vanilla” straddle?
15. Find the value of a contingent-pay option on DataWhack stock with specifications
the same as the options in problems 11-13. Also, be prepared to explain what the
potential outcomes could be for a holder of such an option.
16. Be prepared to describe how an Asian Option differs from a European Option.
17. Be prepared to describe the payoff from a Lookback Call (lookback options are also
called no-regrets options).
18. Be prepared to describe the payoff from a Lookback Put.
19. Be prepared to describe the payoff from a Fixed-Strike Lookback Call.
20. Be prepared to describe the payoff from a Fixed-Strike Lookback Put.
21. Be prepared to describe the payoff from the following varieties of Barrier Options:
a)
b)
c)
d)
e)
Knock-out option
Knock-in option
Up-option
Down-option
Be prepared to explain why these barrier options are normally
cheaper than “plain-vanilla” options
22. Be prepared to
options:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
describe the payoff from the following varieties of other exotic
Prof. Kensinger
Compound option
Installment option
Multi-asset option
Exchange option
Min-max options (rainbow options)
Alternative options
Outperformance options
Shout options
Cliquet options
Lock-in options
Deferred strike options
Forward-start options
Tandem options
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Financial Derivatives
Solutions: Set 7
Spring 201
1. How many “shares” will you have in the revised portfolio, and how many puts?
NS + NP = 21,000(1224.36)
N(1224.36) + N(32.70) = 21000(1224.36)
N(1224.36+ 32.70) = 21000(1224.36)
N(1257.06) = 25,711,560
N = 20,453.7254
What is the value of the revised stock portfolio combined with this insurance?
NS + NP = N(1224.36) + N(32.70) = 1257.06 * N
= $25,711,560
2. Value of the insured portfolio = (1201.25 * N) + [(1210 – 1201.25) * N]
Value of the insured portfolio = 1210 * N = $24,749,008 (rounded to nearest dollar).
This is the minimum value of the insured portfolio, no matter how far the value of
equity might drop.
Without portfolio insurance, the value of the uninsured portfolio would have been
$1201.25 * 21000 = $25,226,250. Only if the equity fell below the breakeven would the
insured portfolio outperform the uninsured portfolio. The breakeven point would be
24,749,008/21000 = 1178.52
3. Puts would expire worthless, so value of the insured portfolio = 1234.36 * N =
$25,247,260 (rounded to nearest dollar)
Without portfolio insurance, the value of the uninsured portfolio would have been
$1234.36 * 21000 = $25,921,560. The cost of insurance is the difference between the
outcomes; in this case, $25,921,560 – $25,247,260 = $674,300 (or about 2.6% of the
original portfolio value. Upside capture is 100% – 2.6% = 97.4%
4. Nnew = Nold S / ∂1S + (1– ∂2)B(X,t)
Nnew = (21000 * 1224.36)/ [ (.5933 * 1224.36) + (1– .5592)*1204.05]
Nnew = 20,453.71
Number of equity shares = ∂1 * Nnew
Number of equity shares = .5933 * 20453.71
Number of equity shares = 12,134.58
Number of bonds = (1–∂2) * Nnew
Number of bonds = (1 – .5592) * 20453.71
Number of bonds = 9014.98
Page 1 of 3
Financial Derivatives
Solutions: Set 7
Spring 2014
Value of insured portfolio = (12134.58 * $1224.36) + (9014.98 * $(1204.05)
Value of insured portfolio = $25,711,560 (again, recall that the result given by the
options calculator is more precise than you would get by just running the rounded
intermediate values through a handheld calculator).
5. ∂C/ ∂S S/(S+P) = .5933 * 1224.36/(1224.36+32.70) = .5778 (again, recall that the result
given by the options calculator is more precise than you would get by just running the
rounded intermediate values through a handheld calculator).
6. Nf = Nold {[∂C/ ∂S S/(S+P)]–1} e–(r–d)t
Nf = 21000 * [ (0.5778–1) e–(.05–.03)(90/365) ]
Nf = –8822.58
Number of contracts = –8822.58 /250 = –35.29
The futures price in the contract would be $1224.36 e(.05–.03)t =$1230.41
7. For the portfolio in problem 1, the value of the put would become $32.30 and the new
value of the insured portfolio would be ($32.30 + $1225.36) * N = $25,723,732
(rounded to nearest dollar). This is a gain of about $12,172.
For the portfolio in problem 6, the change in futures price would be ($1225.36–
$1224.36) e.05–.03)t = $1.0049 per share of the futures contract. So, you would lose
$8866.20 on the futures contracts and make $21,000 on the stock, netting a gain of
$12,133.80.
Thus the outcomes from the two strategies are close, assuming you can buy fractional
futures contracts. Without fractional contracts, you might have chosen to sell 35
contracts (with the multiplier of 250, this represents 8750 shares). Then the futures
contracts would lose $8793 on the futures contracts and make $21,000 on the stock,
netting a gain of $12,207. So, the lack of fractional shares makes a relatively small
difference in this case (the difference between fractional contracts and whole numbers is
about 0.6% in this case).
8. Forward price = $43.75 e0.0375*(270/365)) = $44.98
9. The puts and calls are each worth $2.70 per share, so at the risk-free rate the face value
of the “loan” would be $2.70 e0.0375*(270/365)) = $2.7759 per share (call it $2.78)
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Financial Derivatives
Solutions: Set 7
Spring 2014
10. Payoff per share = $46.50 – $44.98 – $2.78 = –$1.26
If the stock price at expiration is $42.50, the option expires out of the money. Then the
payoff per share = – $2.78
Breakeven point is $44.98 + $2.78 = $47.76
11. Oaon = S0 N(d1) = .5564 * 65 = $36.17. If the stock price is above $65 at expiration, the
holder of the option would receive the underlying stock. If the stock price is $65 or
lower at expiration, the option-holder would receive nothing.
12. Ocon = e–rt N(d2) = (64.47/65) * .5242 = 52¢. If the stock price is above $65 at
expiration, the holder of the option would receive one dollar. If the stock price is $65 or
lower at expiration, the option-holder would receive nothing.
13. C(S,X,t) = Oaon – XOcon = $36.17 – ($65 * .52) = $2.37
14. Chooser = Call(S,65,60/365) + P(S,65e–.05(30/365) , 30/365) = $2.37 + $1.23 = $3.60
15. Ccp = $2.37/0.52 = $4.56
16-22: For class discussion
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