Exotic Options Chapter 19

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Exotic OptionsChapter 19
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EXOTIC OPTIONS So far we studied and analyzed
options strategies that included a variety of
calls puts and short or long positions in the
underlying asset. All the options that we studied
were standard European or standard American
style options. We now turn to study another class
of options that are non standard options. They
are also labeled exotic options.
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EXOTIC OPTIONS These options are non standard in
the sense that one or several of the usually
standardized options contractual stipulations are
replaced with conditions that are tailored to
suit the buyer and seller specific needs.
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EXOTIC OPTIONS EXAMPLES Bermudan options
American options with a predetermined set of
possible exercise dates. Asian options
Options whose position at expiration is
determined by an average of the underlying asset
price during a pre specified period. Barrier
options Options that come to existence or cease
to exist if the underlying asset price reaches a
predetermined threshold level.
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Collars(19.1) Often, investors buy the underlying
asset and purchase protective European puts at
some exercise price K1 K1lt S. In order to
finance the purchase of the protective puts, the
investor may short European calls with K2 K1lt S
lt K2 for the same expiration, T. At times, the
investor chooses the exercise prices such that
the call premium is equal to the put
premium c(S, T-t, K1) p(S, T-t, K2).
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Collars AT EXPIRATION Strategy ICF STlt K1
K1ltSTlt K2 STgt K2 Buy stock -S ST
ST ST Buy put(K1) -p K1 ST
0 0 Sell call(K2) c 0
0 K2 ST TOTAL -S K1 ST
K2 P/L K1 S ST - S K2
S
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Collars The self financing Collar guarantees that
the asset, which was purchased for S, will sell
for K1 or better, up to K2. Given that the
probability of ST to exceed K2 is very low, the
possibility of losing the upper side of the
assets price distribution is close to zero. This
strategy guarantees a specific price range for
the assets selling price at T. ? a Range
forward contract.
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Collars Several variations of collars are
possible with or without holding the asset and
with the put and call prices not equal. These
strategies depend on whether the investor wishes
to guarantee a selling price or a purchasing
price for the asset at T and whether the investor
wishes to open a self financing strategy or
not. In all of the above situations the valuation
of the strategy is based on the Black and Scholes
valuation of the call and the put.
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Forward start options(19.3) These are
at-the-money options that will begin on a
specified future date, T1 , say and will expire
at T2 . The value of such an option at its
writing time, say 0, is the NPV of the options
value at its initial date of existence, T1, with
expiration at T2. It can be easily shown that it
has the same value of an at-the-money option
with T T2 - T1.
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Forward start options(19.3) In general, however,
the option may begin with the exercise price set
at X ?S. If ? 1 then the option is
at-the-money. Otherwise, it will be out or in the
money, depending on ? being greater or less than
1 and the type of the option call or put.
Moreover, one may face a sequence of Forward
start options where the i1st option begins at
the expiration of the i-th option and its
exercise price is set at ? times the asset price
at the expiration of the i-th option.
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Forward start options Suppose that we face n such
options, the value of the entire strategy is
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Forward start options The assumption here is that
? remains the same throughout the n periods. An
Example Consider an employee that, as part of
his/her compensation package, receives a call
with forward start three months from now. The
options parameters are S 60 ?1.1 r .08 q
.04 ? .30 T1 .25 and T2 1. Substituting
these parameters into the formula with only one
option we obtain the call value C 4.4064.
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Compound options(19.4) or Options on Options The
underlying asset of a compound option is an
option. Thus, upon exercise of a compound option,
the holder will either receive or deliver another
option. The holder of the compound option pays a
premium on an option with K1 that expires on T1.
If exercised, the holder will buy or sell for K1
an option with K2 that expires T2 time periods
from today.
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• Compound options Four possibilities
• The payoff
• A call on a call max0, c(S, K2 ,T2) K1)
• A put on a call max0, K1 - c(S, K2 ,T2)
• A call on a put max0, p(S, K2 ,T2) K1)
• A put on a put max0, K1 p(S, K2 ,T2).
• c and p are the Black and Scholes values with
  exercise price K2 and time to expiration T2.
• K1 is the exercise price of the option on the
  underlying option, with T1 time to expiration.

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Compound options Example 1 A put on a call. The
underlying call is the following call on the
index c(S500 K2520 T2.5) The put option on
this call is pc(S500K2520T2.5), K150
T1.25 The payoff on this compound option
is Max0, K1 c(S, K2,T2) Max0, 50 -
c(500, 520, .5).
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Compound options The value of this put on call
when r .08 q .03 and underlying stock
index volatility ? .35 is 21.1965.
CONCLUSION You pay 21.20 for a put on a call
on the index. If the put is exercised, you will
receive 50 for selling a call on the index with
exercise price of 520 and a time to expiration of
.25 yrs from then.
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Compound options Example 2 A call on a put. The
following is a very common situation for foreign
multinational firms A foreign firm submits a
bid for selling equipment in the U.S.A. for a
fixed amount, M, of foreign currency. At time T1
the firm will find out if it won or lost the bid.
If it did win the bid, it will sell the equipment
and receive the USD equivalent of M on date T2.
The firm is clearly exposed to exchange rate risk
and may wish to hedge this risk.
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Compound options Time line 0. T1 .
T2 BID ACCEPTED PAYMENT and or
REJECTED DELIVERY If the foreign currency
depreciates against The USD, the USD amount
equivalent to M will be smaller.
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Compound options The firm could purchase a
protective put on the foreign currency for T2 and
pay the full premium, ignoring the fact that it
may not win the bid. INSTEAD
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Compound options The firm could buy a call for
T1, which will give the firm the right to
purchase the FORX put in case it won the bid and
the call is in the money at T1. In this way, the
firm will have two payments. The call premium,
will typically be smaller than the premium on the
outright put. The put premium payment will occur
if the firm wins the bid and the call is in the
money. The sum of these two payments may or may
not exceed the outright put premium.
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Compound options EXAMPLE 2 a call on a put A
Canadian firm submits a bid to sell equipment in
the U.S.A. for CD10M. The firm will find whether
it won the bid or not in 25 days. If the bid was
won, it will deliver the equipment and be paid in
full 24 days later. The payment will be in USD.
The current exchange rate is USD.6303/CD.
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Compound options Had the deal been done today
the firm would have received USD6.303M. However,
if the firm wins the bid and the CD depreciates
against the USD, the firm will realize a smaller
amount. The firm may buy protective puts.
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Compound options If the firm decides to purchase
a protective put on the foreign currency for T2
49 days and pay the full premium, ignoring the
fact that it may not win the bid, we use S
USD.6303/CD K1 USD.6303/CD. p p(.6303
.6303 49/365 r.0404 ?.028) USD.2669/CD 0r,
a total of p USD.2669/CDCD10M USD26,690.
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Compound options INSTEAD A call on a put. The
underlying put option is a put on the CD with
the following parameters p(S.6303 K2.6303
T249/365) The call option on this put
is cp(S.6303K2.6303T249/365), K1?
T125/365?.028
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Compound options The payoff on this compound
option depends on K1. In order to decide on the
exercise value of the compound call, the firm
calculates the value of an at-the-money put with
24 days to maturity p USD.185/CD and in order
to compare the compound option strategy with the
outright protective put strategy, the firm
calculates the compound option value for a range
of striking prices .16, .18, .20, .22 and .24
cents per CD.
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Compound options Note The put value will
increase when the CD depreciates against the USD,
thus, increasing the compound option value.
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Compound options Again. The underlying put option
is a put on the CD with the following
parameters p(S.6303 K2.6303 T249/365) The
call option on this put is cp(S.6303K2.6303T2
49/365), K1? T125/365?.028 Calculation of
the compound option shows an increasing compound
option value with a decreasing exercise price K1
cp(S, K2,T2) K1? T125/365?.028 .24 USD
.1096/CD ?USD10,960 .22 USD.1184/CD ?USD11,840 .
20 USD.1277/CD ?USD12,770 .18
USD.1377/CD ?USD13,770 .16 USD.1485/CD ?USD14,
850
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Compound options A call on a put.
CONCLUSION The Canadian firm will pay ICF for
the compound option ( call on a put) today. If it
wins the bid in 25 days and the call ends up
in-the-money, it will pay the exercise price of
the call in order to purchase the put. This will
lead to the following possibilities WIN BID
and K1 ICF p gt .6303 TOTAL .24 USD10,960 USD2
4,000 USD34,960 .22 USD11,840 USD22,000 USD33,840
.20 USD12,770 USD20,000 USD32,770 .18
USD13,770 USD18,000 USD31,770 .16 USD14,850 USD1
6,000 USD30,850
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Compound options EXAMPLE 3 A call on a put. An
American firm submits a bid for a project in
Germany for EUR100M. The firm will find whether
it won the bid or not in three months. If the bid
was won, the project will begin in 91 days
(immediately upon winning the bid) and will be
completed and paid for in six months. The payment
will be in EURs that will be exchanged into USDs
and deposited in USDs immediately. The current
exchange rate is USD.9/EUR. Had the deal been
done today the firm would have received USD90M.
However, if the firm wins the bid and the USD
depreciates against the EUR, for example to
USD.8/EUR the firm will realize a smaller
amount. Lets analyze the two hedging alternatives
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Compound options Example 3 continued A
protective put If the firm decides to purchase a
protective put on the foreign currency for T2
.5yrs and pay the full premium, ignoring the fact
that it may not win the bid, we use S
USD.9/EUR K1 USD.9/EUR This put will cost p
p(.9 .9 .5 rusd.06 reur.03 ?.01)
USD.0188/EUR 0r, a total of p
USD.0188/EUREUR100M p USD1,880,000. AGAINThe
outright purchase of the six months protective
put ignores the possibility that the American
firm will lose its bid.
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Compound options Example 3 continued The
underlying put option is a put on the EUR with
the following parameters p(S.9 K2.9
T2.5) The call option on this put
is cp(S.9K2.9T2.5), K1? T1.25?.01
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Compound options The payoff on this compound
option depends on K1. In order to decide on the
exercise value of the compound call, the firm
calculates the value of an at-the-money put with
3 months to maturity p USD.0146/EUR. Thus the
American firm decides to set K1USD.0146/EUR and
calculates the compound option value. Whatever
this value is, the result is that three months
from now, the firm will know whether it won or
lost the bid. If it lost the bid then the cost is
limited to the compound option value, which is
considerably less than the cost of the outright
six-month put.
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Compound options Example 3 continued If, on the
other hand the call ends up in-the- money, the
put is then purchased for USD.0146/EUREUR100M
USD1,460,000. If the call is out-of-the money,
it is not exercised and the put premium is saved.
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The value of unprotected American calls an
application of compound options When a stock
pays out cash dividends, the stock price falls by
the dividend amount. This price fall causes the
premiums of calls on this stock to to decrease.
The exchanges do not compensate call holders for
the lost value caused by cash dividend payments.
Hence the title unprotected American calls. One
may argue that investors, being aware of the
expected cash dividend payments, take this into
account and that market prices adjust
accordingly. While this is true, nonetheless, we
still face the following problem Suppose that
the stock will pay a known cash dividend, D, at a
known future date. What is the call (fair) market
value?
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Unprotected American calls It follows that on xd,
the call holder faces the MaxSxd - K D
c(Sxd, K, T-xd). On that day, the put-call
parity implies that c(Sxd, K,T-xd) p(Sxd, K,
T-xd) Sxd Ke- r(T xd). Substitute the
put-call parity into the option value to
obtain Sxd D K Max0, p(Sxd, K, T-xd)
D - K(1-e- r(T xd)).
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Unprotected American calls The current value of
this cash flow is S - Ke- r(xd - t) the
compound option value call on a put. The Call is
for expiration at xd and with exercise price D -
K(1-e- r(T xd)) If exercised, the call holder
will buy a put that expires on T and with
exercise price K.
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Unprotected American calls What is the meaning of
exercising the compound option? It means that you
pay the call exercise price D - K(1-e- r(T
xd)) and receive the put, a total cash flow
of Sxd D Kp(Sxd, K, T-xd)DK(1-e- r(T
xd)). But Upon substitution of p(Sxd, K,T-xd)
c(Sxd, K,T-xd) - Sxd Ke- r(T xd).
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Unprotected American calls The value received
upon exercising the compound option is c(Sxd,
K,T-xd)?? not to exercise the American call
Moreover, if the compound option is not
exercised, the value in the investors hand
is Sxd D K??exercise the American call.
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Compound options Finally, another application of
compound options Consider a firm with equity and
debt. For simplicity, assume that the entire debt
issue is a pure discount bond maturing T time
periods hence. At T, stock holders must pay bond
holder the face value of the debt, F, or else,
bond holders take over the firm. Assume that
stock holders wish to wait until T, pay back the
debt and liquidate the firm. The firm value at T
is ST and therefore, the cash flow to the stock
holders at T can be summarized as follows CF
max0, ST F.
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Compound options CF max0, ST F, is the
cash flow of a call on the value of the firm,
given by the bond holders to the stock holders.
Here comes the surprising conclusion An option
on the firms stock is a compound option. Upon
its exercise, the holder buys or sells the firms
stock I.e., buys or sells the right to buy the
firm back from the bond holders at time T.
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Chooser options(19.5) The option is traded now,
at time 0, determining a future time T1 at which
the option holder must decide whether the option
will be a call or a put. Let c and p denote the
options underlying the Chooser option, then, at
T1 the value of the option is Maxc,p. Suppose
that an investor expects that the market will
make a strong swing but is not sure whether it
will be a down or an up swing. The standard
strategy is a straddle buy a call and a put in
order to capture the expected volatility of the
underlying asset.
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Chooser options The chooser option is an
alternative to a straddle, an alternative whose
premium is lower than the straddle premium. The
straddle premium includes the put premium and the
call premium. The chooser premium is
Maxc,p. In general c c(S1, T2, K2) and p
p(S1, T3, K3). Case I. Both options are for the
same expiration date T and the same exercise
price, K. In this relatively simple case,
substitute the put call parity for European
options p(S1,T,K ) c(S1,T,K) - S1e- q(T- T1)
Ke- r(T- T1) Into Maxc(S1,T,K), p(S1,T,K )
Maxc, c - S1e- q(T- T1) Ke- r(T- T1),
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Chooser options which can be rewritten c
Max0, Ke- r(T- T1) - S1e- q(T- T1) or c e-
q(T- T1)Max0, Ke- (r-q)(T- T1) - S1. From the
last expression we see that the Chooser option
is a Call, expiring at T with exercise price K,
plus e- q(T- T1) puts, expiring at T1 with
exercise price Ke- (r-q)(T- T1) .
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Chooser options Example Stock XYZ is trading for
125.9375/share. A Straddle with K 125, T 35
days, r .0446 and ? .83 will cost c
13.21 p 12.09 25.30. The Chooser
option with T1 20 days will be worth c 13.21
plus the put value with T 15 days .0411yrs
p 7.80. And the Chooser option costs 21.01.
It costs less than the Straddle because there is
a possibility that the payoff at expiration will
be zero.
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Chooser options Example Stock XYZ is trading for
50/share. A chooser option with a decision time
at T1 .25yrs and expiration T .5yrs is with
K 50, r .08, q 0, and ? .25. The
Chooser option premium is 6.11/share.
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Chooser options Case II The call and the put are
for different exercise prices and times to
expiration. c c(S1, T2, K2) and p p(S1, T3,
K3). To evaluate this option, let S be the
underlying asset price at which todays premium
of a call with K2 that expires at T2 is equal to
the T1 value of a put with K2 and T2. c0(S, K2,
T2) p(S, K3,T3). By definition, chose the
call for S gt S Chose the put for S lt S. The
payoff to the Chooser option, given S, can be
written as
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Chooser options
This payoff is equivalent to the payoff of two
compound options A call on a call with zero
strike price Plus A call on a put with zero
strike price.
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Chooser options
Example At time T1 .25yrs, the option holder
must chose between a call, c(S 50, K 55, T
6 months) and a put, p(S 50, K 48, T 7
months). r .1, ? .35 and q .05. The
Chooser options premium is 6.05/share.
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Chooser options
Example An multinational American firms
division in the UK receives and pays cash flows
in British sterling, , on a regular basis. The
UK division, is required by the parent firm to
exchange the into USD upon it receipt. Thus,
the firm is exposed to exchange rate risk. For
instance, during the next 100 days the firm will
receive payment 68 days hence and will pay
100 days hence. Clearly the exposure could be
hedged against by purchasing a sterling put for
68 days and buying a sterling put for 100 days.
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Chooser options
Suppose that the firm expects an announcement by
the UK central bank in 30 days. This announcement
is believed to be making some long term impact on
the FORX market. In this case, the firm may chose
to buy Chooser option for 30 days on a call,
c(1.58/, T70days, K 1,60/) and a
put, P(1.58/,T 38days, K 1.56) . The cost
of this Chooser option .0548/.
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Barrier options(19.6) Barrier options are a
modified form of standard option. The strike
price determines the payoff at expiration. But if
the underlying assets market price crosses or
does not cross a predetermined BARRIER price the
option may or may not exist. Knock in option
the option becomes a standard option if the
barrier was crossed some time before expiration.
It will then pay if it ends up in-the-money. Knock
out option the option is a standard option as
long as the barrier is not crossed. It ceases to
exist once the barrier is crossed. The barrier
may be crossed from below or from above and
therefore, we can categorize barrier options as
follows
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Barrier Barrier
effect on payoff Option Type Location Crossed
Not crossed Call Down-and-out BltS
Worthless Standard Down-and-in BltS
Standard Worthless Up-and-out BgtS
Worthless Standard Up-and-in BgtS
Standard Worthless Put Down-and-out BltS
Worthless Standard Down-and-in BltS
Standard Worthless Up-and-out BgtS
Worthless Standard Up-and-in BgtS
Standard Worthless In principle, barrier
options may pay a rebate to the option holder
when the barrier is crossed. In practice only put
option pay a rebate as a consolation prize when
the option is knocked out.
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• Barrier options
• Barrier options are traded for several reasons
• As a buyer you eliminate paying for scenarios you
  think are unlikely. As a seller, you may enhance
  your income by shorting a barrier option that
  pays off on scenarios that you think are
  improbable.
• Example You expect the stock price to rise next
  year by some 105 of spot. However, even though
  you believe that the market will rise, if it will
  not rise by a support level of 95 it will then
  decline.
• Buy a down-and-out call with K (1.05)S and a
  barrier at (.95)Market, or
• Sell a down-and-in call with K (1.05)S and gets
  knocked in only if the market fall below 95.

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• Barrier options
• Barrier options may match hedging needs more
  closely than similar standard options.
• Example
• You wish to protect the value of a stock that you
  just bought. You intend to sell the stock if its
  price rises by 10 or more. By purchasing a
  protective put you pay a premium that protects
  you even when the price rise 10, 15 etc.
• Instead, you may buy an up-and-out put with the
  same exercise price but with a barrier set at
  (1.1)S. This way you have the same protection on
  the down side and no protection if S increases by
  more than 10 and you sell the stock.

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• Barrier options
• Barrier options premiums are generally lower than
  those of standard options. Since you pay for the
  option only if it is knocked in and you do not
  pay for the option if it is knocked out, the
  premium is paid for specific scenarios
• Down-and-out call its value is close to the
  standard call value, because it gets knocked out
  for low stock values where the standard call has
  little value
• Down-and-out put worth much less than a
  standard put, because it gets knocked out at low
  stock values where the standard put is deep in
  the money.

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Barrier options Example S 100 K 100 B
90 T 1 q 5 r 10 ?
15. Standard European Down-and-out
Call 7.84 7.22 Put 3.75 0.28
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Barrier options Down-and-in call worth much
less than the standard call, because it gets
knocked in only when the stock price has made a
large and unlikely down move. Down-and-in put
its value is close to the standard put value,
because it gets knocked in for low stock values
where the put is deep in the money. Example S
100 K 100 B 90 T 1 q 5 r
10 ? 15. Standard European
Down-and-in Call 7.84 0.62 Put 3.75
3.46
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Barrier options Up-and-out call worth only a
fraction of the standard call value, because it
gets knocked out for high stock values where the
standard call is deep in the money. Up-and-out
put worth almost the same as much as the
standard put, because it gets knocked out at high
stock values where the standard put is out of the
money. Example S 100 K 100 B 120 T
1 q 5 r 10 ? 25. Standard European
Up-and-out Call 11.43 0.66 Put 7.34
6.70
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Barrier options Up-and-in call worth almost the
same as the standard call, because it gets
knocked in only when the stock price is up where
the standard call also gets most of its
value. Up-and-in put its value is much less
than the standard put value, because it gets
knocked in for high stock values where the put is
deep out of the money and thus, has very low
value. Example S 100 K 100 B 120 T
1 q 5 r 10 ? 25. Standard
European Up-and-in Call 11.43 10.78 Pu
t 7.34 0.64
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Barrier options Example Down-and-out call In 51
days a firm From Denmark will make a payment of
200M on the principal of a Euro Danish bond. S
USD.18213/ If the payment were to be today the
payment would have been USD36,426,000. If, the
Danish currency appreciates against the dollar,
however, the firm will have to pay more in USD. A
standard call option with S .18213 T 51
days K .1950 ? 14.4 rUSA 4.18 rDEN
5.24 will cost 0.0462 per Danish or
USD92,400. This call guarantees that the firm
will buy the dollars for no more than USD.195/
. A barrier option a down-and-out call with
the barrier set at B USD.1750 will cost .0453
or USD90,600. New risk The exchange rate may go
down, knock the call out and then rise again.
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Barrier options Example Up-and-out put. In 40
days an American firm will receive JY100M. The
current spot exchange rate is USD.011009/JY. The
payment now would be USD1,100,900. If the JY
depreciates against the dollar, however, the
payment will be less. The firm buys an
up-and-out put with a barrier at USD.011047/JY.
If the barrier is crossed the put is out and the
firm feels that its risk exposure to the JY
depreciating is negligible. If the barrier is not
crossed, the firm holds a protective put.
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Barrier options Example Up-and-in call. In 40
days an American firm will pay EUR50M on a
Eurobond that matures in 100 days. If the EURO
appreciates against the dollar, the firm payment
in USD increases. The firm expects the EURO to
stay the same or even depreciate over the next
100 days, but it still wishes to protect itself
against an appreciating EURO. The firm decides
to purchase an Up-and-in call. This way, if the
firms expectations materialize, the EURO will
depreciate against the dollar and the call will
not come into existence. If, however, the firms
prediction turns out to be incorrect, the EURO
will appreciate against the dollar, the barrier
will be crossed and the call will come into
existence, protecting the firm payment in USD.
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Barrier options There are many other types of
barrier options. Some are Barrier option for a
pre specified time only, others are barriers on
BINARY options to be discussed later. One type
of barrier options that is very interesting is a
Two-asset barrier option Consider a Norwegian
oil producer. As oil is typically sold for
USD/bbl, the producer income in EUROs depends on
the USD oil price and on the USD-EURO exchange
rate. A standard currency option on the EURO
will hedge the FORX risk exposure. Instead, A
currency option that is knocked out if the oil
price increases beyond a particular level, will
give the producer some flexibility between the
oil price level and the FORX rate. It will be
cheaper than the outright option on the EURO.
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Binary Options (page 441)

• Cash-or-nothing pays Q if S gt K at time T,
  otherwise pays 0. Value erT Q N(d2)
• Asset-or-nothing pays S if S gt K at time T,
  otherwise pays 0. Value S0 N(d1)

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Decomposition of a Call Option

• Long Asset-or-Nothing option
• Short Cash-or-Nothing option where payoff is K
• Value S0 N(d1) erT KN(d2)

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Lookback Options (page 441)

• Lookback call pays ST Smin at time T
• Allows buyer to buy stock at lowest observed
  price in some interval of time
• Lookback put pays Smax ST at time T
• Allows buyer to sell stock at highest observed
  price in some interval of time

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Shout Options (page 443)

• Buyer can shout once during option life
• Final payoff is either
• Usual option payoff, max(ST K, 0), or
• Intrinsic value at time of shout, St K
• Payoff max(ST St , 0) St K
• Similar to lookback option but cheaper
• How can a binomial tree be used to value a shout
  option?

69
Asian Options (page 443)

• Payoff related to average stock price
• Average Price options pay
• max(Save K, 0) (call), or
• max(K Save , 0) (put)
• Average Strike options pay
• max(ST Save , 0) (call), or
• max(Save ST , 0) (put)

70
Asian Options

• No analytic solution
• Can be valued by assuming (as an approximation)
  that the average stock price is lognormally
  distributed

71
Exchange Options (page 445)

• Option to exchange one asset for another
• For example, an option to exchange U for V
• Payoff is max(VT UT, 0)

72
Basket Options (page 446)

• A basket option is an option to buy or sell a
  portfolio of assets
• This can be valued by calculating the first two
  moments of the value of the basket and then
  assuming it is lognormal