Topic 8
Portfolio Insurance
& Other Advanced Derivatives Strategies
Several Routes to Insured
Portfolios
•
•
•
•
Put options on the index
Synthetic put using dynamic hedging
Index futures using dynamic hedging
Break forward contract
1
Put options on the index (Prob 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
•  1210 puts on the index future, expiring in
90 days, have premium of $32.70
•  You plan to sell some of the stock in
order to fund the purchase of puts
Put options on the index (Prob 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 = 20,453.73
•  What is the value of the revised stock portfolio
combined with this insurance?
NS + NP = N(1224.36) + N(32.70) = 1257.06 * 20,453.73
= $25,711,560
2
What is the result of this strategy?
•  If puts expire worthless, you will have less stock
and no more protection
•  If index stays above the breakeven point, you
would still have reduced portfolio value with no
more protection
Breakeven: 1210 – 32.70 = 1177.30
•  If index falls below the breakeven, you would
be thankful for the protection
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Further Practice (problem 2)
•  Suppose the index falls to 1201.25 at expiration
Ø Value of the insured portfolio = (1201.25 *
20,453.73) + [(1210 – 1201.25) * 20,453.73]
Ø Value of the insured portfolio = 1210 * 20,453.73 =
$24,749,000
Ø This is the minimum value of the insured portfolio,
no matter how far the value of equity might drop
Ø Without portfolio insurance, value of uninsured
portfolio would have been $1201.25 * 21000 =
$25,226,250
Ø Breakeven be 24,749,000/21000 = 1177.52
Further Practice (problem 3)
•  Suppose the index rises to 1234.36 at expiration
Ø Put expires worthless. Value of the insured portfolio
= 1234.36 * 20,453.73 = $25,247,266
Ø Without portfolio insurance, value of uninsured
portfolio would have been $ 1234.36 * 21000 =
$25,921,560
Ø Cost of insurance: $25,921,560 – $25,247,266 =
$674,294 (about 2.6% of the original portfolio value)
Ø Upside capture is 100% – 2.6% = 97.4%
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Basic Premise of Synthetic
Options
•  We know that
C(S,X,t) = S – B(X,t) + P(S,X,t)
•  We assume that for a very short time
C(S,X,t) = ∂1S – ∂2B(X,t)
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Here’s a Picture of ∂1
Call
Call
∆C/∆S
C2
C1
Stock
B
B(X,t)
(X,t)
S1
Stock
S2
Here’s a Picture of ∂2
Call
Call
∆C/∆X
C1
C2
BB(X,t)
(X1,t)
S
B (X2,t) Stock
Stock
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Important Relationships
•  ∂1 and ∂2 are always less than 1
•  ∂2 is always less than ∂1
•  Exact values can be estimated using the
Black-Scholes OPM
•  Proportions must be continuously
adjusted
Example
•  Suppose
∂1 = 0.7
∂2 = 0.6
S = $50
B(X,t) = $46
•  Then C(S,X,t) = (.
7*50) – (.6*46) =
$7.40
•  Then, sell 100-option
contract (receive $740)
Ø  Sell 60 bonds (receive
another $2760, making
total $3500)
Ø  Buy 70 shares stock (pay
$3500)
Ø  Zero net investment
•  After a moment
Ø  S = $50.125
Ø  B(X,t) = $46.10
Ø  C(S,X,t)=7.4275
•  Close position, net
zero
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Example
•  Now, suppose calls selling on CBOE for
$7.75 per share
Ø Sell 100-share call contract for $775
Ø Create synthetic for $740
Ø Pocket profit of $35
•  How will adjustment process work?
Basic Premise for Portfolio
Insurance
•  From put-call parity, we know that
C(S,X,t) + B(X,t) = S + P(S,X,t)
•  Then let’s make a simple rearrangement
S + P(S,X,t) = C(S,X,t) + B(X,t)
•  We assume that for a very short time
C(S,X,t) = ∂1S – ∂2B(X,t)
•  Then for a short time
S + P(S,X,t) = B(X,t) + ∂C(S,X,t)
1S – ∂2B(X,t)
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
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How does insurance work?
Call
Call
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
∂ close to 1
∂ declines
Less in stock
More in bonds
B (X,t)
B(X,t)
Stock
S2
Stock
S1
How does insurance work?
Call
Call
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
∂ closer to 1
Sell bonds
Buy more stock
∂ less than 1
B (X,t)
B(X,t)
Stock
S1
Stock
S2
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How might this process lead to
runup?
Call
Call
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
∂ closer to 1
Sell bonds
Buy more stock
∂ less than 1
B (X,t)
B(X,t)
Stock
S1
Stock
S2
How might this process lead to
meltdown?
Call
Call
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
∂ close to 1
∂ declines
Less in stock
More in bonds
B (X,t)
B(X,t)
Stock
S2
Stock
S1
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Practice (Problem 4)
•  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 want to maintain a protection window that
maintains a constant 90-day time horizon
•  Volatility of the index is 17.5%
•  Continuously compounded T-Bill rate is 5%
•  Dividend yield on the index now is 3% per year
with continuous compounding
Practice (Problem 4)
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
•  We can be somewhat more precise in finding the
new proportions to hold of equity and bonds in
order to mimic equity plus synthetic puts. First,
let’s calculate the new N:
Nnew = Nold (S/(S + Put)
Nnew = Nold
S
∂1S + (1– ∂2)B(X,t)
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Practice (Problem 4)
New N =
Nnew =
Nold S
∂1S + (1– ∂2)B(X,t)
21000 * 1224.36
(.5933 * 1224.36) + (1– .5592)*1204.05
Nnew = 20,453.71
Practice (Problem 4)
Number of Shares = ∂1 * 20453.71
Number of Shares = .5933 * 20453.71
Number of Shares = 12,134.58
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Practice (Problem 4)
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
Number of Bonds = Nnew * (1– ∂2)
Number of Bonds = 20453.71*(1– .5592)
Number of Bonds = 9014.98
Basic Premise of Futures
Hedge
•  We want to create a portfolio of stock and futures that
responds the same way to changes in stock price as a
portfolio of stock & put
•  From put-call parity, we know that
C(S,X,t) + B(X,t) = S + P(S,X,t)
•  We assume that for a very short time
C(S,X,t) = ∂1S – ∂2B(X,t)
•  Then for a short time
S + P(S,X,t) = B(X,t) + ∂C(S,X,t)
1S – ∂2B(X,t)
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
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Calculating the Futures Hedge
•  Goal:
∂V/ ∂S = Delta for insured portfolio = Nnew ∂C/ ∂S
•  ∂V/ ∂S = Nold + Nf (∂f / ∂S)
•  We know that ∂f / ∂S = e(r-d)t
This is because f0(t) = S0 e(r-d)t
•  So, Nold + Nf e(r-d)t = Nnew ∂C/ ∂S
S + P(S,X,t) = ∂1S + (1 – ∂2 ) B(X,t)
Calculating the Futures Hedge
•  Nold + Nf e(r-d)t = Nnew ∂C/ ∂S
•  Nf = Nold {[(Nnew / Nold)∂C/ ∂S]–1} e–(r–d)t
•  We remember that Nnew = Nold [S/(S+P)]
•  So:
Nf = Nold {[∂C/ ∂S S/(S+P)]–1} e–(r–d)t
•  Number of contracts = Nf /multiplier
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Practice (Problems 5&6)
•  Nf = Nold {[∂C/ ∂S S/(S+P)]–1} e–(r–d)t
•  Problem 5:
∂C/ ∂S S/(S+P) = .5933 * 1224.36/(1224.36+32.70) = .5778
•  Then in problem 6:
Nf = 21000 * [ (.5778 – 1) e–(.05–.03)(90/365)]
Nf = –8822.58
•  Number of contracts = –8822.58 /250 = –35.29
Break Forward Contract
•  This is a variation on an Equity Forward
•  Value at origination is zero
•  It is like a pay-later call, in which the buyer pays nothing at
time of purchase, but pays the future value of the call
premium at expiration
Ø  If the call expires in-the-money, the deferred premium is deducted
from the proceeds of exercise (this may still be negative if the
underlying falls below the breakeven)
Ø  If the call expires out-of-the-money, the buyer must pay the entire
deferred premium
•  The exercise price is the equilibrium forward price
Ø  X = S0e(r-d)t
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Break Forward Contract
•  Payoff at expiration depends on the value of the
underlying at expiration
Ø If ST > f0(T) the payoff is ST – f0(T) – C(S,X,t)ert
Ø If ST ≤ f0(T) the payment is –C(S,X,t)ert
Ø Payoff is positive if ST ≥ f0(T) + C(S,X,t)ert
•  Break forwards could be substituted for calls in a
call-bond portfolio insurance strategy
X = strike price of the call = f0(T)
f0(T) is the equilibrium forward price
f0(T) = S0e(r-d)t
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Break Forward Contract
•  The present value of the exercise price is S0, so the call starts out
exactly at the money
•  Put-call parity shows another analogy that is helpful in
understanding a break forward contract
C(S,X,T) +S = S + P(S,X,T)
C(S,X,T) – P(S,X,T) = 0
But, there is no short put in the break forward, so you must pay for
the option
•  The “loan” implicit in the break forward is the price of the call, for
which payment is deferred until expiration
X = strike price of the call = f0(T)
f0(T) is the equilibrium forward price
f0(T) = S0e(r-d)t
Further Practice (problems 8-10)
•  Strike price = $43.75 e(.0375*(270/365)) = $44.98
•  The puts and calls are each worth $2.70 per share
•  Face value of the “loan” would be $2.70 e(.0375*(270/365)) =
$2.7759 per share
•  Stock price rises to $46.50 at expiration
Ø  Payoff per share = $46.50 – $44.98 – $2.78 = –$1.26
•  Stock price falls to $42.50 at expiration
Ø  Option expires out of the money.
Ø  Payoff per share = – $2.78
•  Breakeven point is $44.98 + $2.78 = $47.76
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Exotic Options
•  Digital Options
•  Chooser Options
•  Path-Dependent Options
Ø Asian Options
Ø Lookback Options
Ø Barrier Options
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Digital Options Break a Call in
Two
•  Digital options, sometimes called binary
options, are of two types:
Ø Asset-or-nothing options
•  pay the holder the asset if the option expires in
the money and nothing otherwise.
Ø Cash-or-nothing options
•  pay the holder a fixed amount of cash (usually
$1) if the option expires in the money and
nothing otherwise
Decomposition of a European
Call
•  A balanced portfolio of long asset-ornothing options and short cash-ornothing options is equivalent to a
European Call
Ø Oaon = S0 N(d1)
Ø Ocon = e–rt N(d2)
•  So, C(S,X,t) = Oaon – XOcon
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Practice
•  For practice, see problems 11 through 13
on Set 7
Chooser Options
•  Also called as-you-like-it options
•  Enable investor to decide whether the
option will be a call or a put
Ø at a specific time after purchasing the option
Ø but before expiration
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Chooser Options
•  The chooser option is identical to
Ø an ordinary call expiring at T with exercise
price X plus
Ø an ordinary put expiring at t with exercise
price Xe-r(T-t)
•  How does this compare with a straddle?
Ø Different time to expiry
Ø Different exercise price
Decision must be made at time t < T
Practice
•  For practice, see problem 14 on Set 7
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Contingent-pay Option
•  Like a pay-later option
Ø Premium paid at expiration
•  Except:
Ø Premium paid only if the option expires inthe-money
Contingent-pay Option
•  It is a combination of a standard option and Ccp
cash-or-nothing calls
•  The value must be zero today so:
Ce(S,X,t) – Ccpe–rt N(d2) = 0
Ccp = Ce(S,X,t) / e–rt N(d2)
Ccp = Calle / Ocon
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Practice
•  For practice, see problem 15 on Set 7
Path-Dependent Options
•  Payoff is determined by
Ø sequence of prices followed by the asset
Ø not just by the asset’s price at expiration
•  1st example: Asian Option
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Path-Dependent Options
•  Lookback Option
Ø Also called a no-regrets option
Ø Permits purchase of the asset at its lowest
price during the option’s life or
Ø Sale of the asset at its highest price during
the option’s life
Path-Dependent Options
•  Four different types of Lookback Options
Ø lookback call: exercise price = minimum price
during option’s life
Ø lookback put: exercise price = maximum price
during option’s life
Ø fixed-strike lookback call: payoff based on
maximum price during option’s life (instead of final
price)
•  compared to fixed strike
Ø fixed-strike lookback put: payoff based on
minimum price during option’s life (instead of final
price)
•  compared to fixed strike
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Path-Dependent Options
•  Barrier Options:
Ø Terminate early if the asset price hits a certain level,
called the barrier
•  Called knock-out options (or simply out-options)
Ø Activate only if the asset price hits the barrier
•  called knock-in options (or simply in-options)
•  If the barrier is above the current price
Ø called an up-option
•  If the barrier is below the current price
Ø called a down-option
•  Normally cheaper than ordinary options
because fewer outcomes provide pay offs
Compound Option:
An Option on an Option
•  Four basic types
Ø  Call on a call
Ø  Put on a call
Ø  Call on a put
Ø  Put on a put
•  Two strikes; two expirations
•  Two Premia
Ø  1st paid up front
Ø  2nd paid if exercised
•  Often used in mkts where there is doubt about
volatility of ultimate underlying (such as currencies)
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Other Exotic Options
•
•
•
•
•
•
•
exchange options
multi-asset options
min-max options (rainbow options)
outperformance options
forward-start and tandem options
deferred strike options
shout, cliquet and lock-in options
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