### down and out call

```BARRIER OPTIONS
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Agela Baze
Erjona Mimini
BARRIER OPTIONS

Barrier options are the most popular exotic options(traded OTC in the late ‘60). They are an
extension of standard options.

The pay-off of the option depends on the whole path followed by the underlying.

The contract fixes a specified level of the underlying price called barrier. The option pay-off
is conditional on the underlying price process hitting the barrier.

Barrier options can be classified according to:
1.
Pay-off type:

Call & Put
2.
Knocking type:

Knock-IN: The option comes INTO life and can be excercised only if the underlying’s
price touches the barrier within expiration.

Knock-OUT: The option comes OUT of life and cannot be excercised further if the
underlying’s price touches the barrier within expiration.
3.
Barrier type:

UP: The barrier is set above the underlying spot price and is an upper bound.

DOWN: The barrier is set below the underlying spot price and is a lower bound.

Relation between standard options and barrier options:
V standard option(t,S)=V(t,S)knock-out+V(t,S)knock-in IN-OUT PARITY
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BASIC BARRIER OPTIONS
Based on these classifications 8 different
basic types of barrier options can be
obtained starting from the plain vanilla Call
and Put.

Call:
 Up-and-in
 Up-and-out
 Down-and-in
 Down-and-out

Put:
 Up-and-in
 Up-and-out
 Down-and-in
 Down-and-out
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BARRIER OPTIONS: DOWN-AND-OUT CALL



It is an european call option which stops to exist if the
(is knocked out) if the underlying touches the barrier
Hmin before expiration. For the option to have an initial
value Hmin < S0.
Pay-off of the Down-and-out Call
 (ST-K)+ , if St>Hmin for all t<=T;
 0
, if St<=Hmin for at least one t<=T.
When St does not hit the barrier we are in the situation
of a Plain Vanilla Call.
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BARRIER OPTIONS: DOWN-AND-OUT CALL
 Consider
the case in which Hmin<K:
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BARRIER OPTIONS: DOWN-AND-OUT CALL
PRICING


First analytical formula was proposed in the 1970s.
There are several methods to price a Barrier Option:
 Static replicating portfolio of vanilla options (priced
with B&S)
 Law of the maximum (or minimum) of the underlying
Closed form solution;



PDE approach;
Monte Carlo simulation pricing;
Lattice evolution of the underlying approach.
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BARRIER OPTIONS: DOWN-AND-OUT CALL
PDE PRICING

All kinds of barrier options must satisfy the same Black&Scholes
equation:

Barrier Option pricing is a terminal boundary problem of the Black&Scholes
which compared to the vanilla pricing has one more boundary (here SB
denotes the barrier) and a corrisponding boundary condition

The pricing PDE of the down-and-out call option is :

After some manipulations we obtain the following relation:
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WHY BUYING A BARRIER OPTION?

i.
ii.
Buying a barrier option because:
They may more closely match your beliefs about
the market behavior
Lower cost with respect to the standard options
(call or put)
Knock out + knock in = vanilla option. -- >cheaper
iii.
You can hedge better with barrier options in some
cases.
They help traders who place directional bets enhance their
leverage and investors who accept to keep some residual
risk on their books reduce their hedging costs.
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1) Suppose an investor whose expectations about the

•
•

behavior of the market are that it will increase / highly
increase.
He can buy a down and out call:
If the market increases he can have unlimited profits, as in
the case of a standard call option.
Worthless when the price goes down more than H
He could also sell a down and in call with spot price S that
activates only if the market price falls below H. In this case,
he gains premiums from an unlikely market scenario
expectation i.e a deep fall of the market.
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2) The other two scenarios would be:

Sells an up and in call who expects that the market will decrease
or that will remain stable (and so will do the underlying stock).
The investor that is interested in buying this type of option expects
that the market will increase.

Buys an up and out call the investor that expects a small increase
or a rather stable market. Let’s suppose that the spot price is 100
and the barrier is H= 150. The option knockes out only if the price
goes higher than 150
Same reasoning in the case of a Barrier put option
Put
Down and out
Down and in
Up and out
Up and in
Where active
Above H
Below H
Below H
Above H
Crossed
worthless
Standard put
worthless
Standard put
Not crossed
Standard put
worthless
Standard put
worthless
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LOWER COST WITH RESPECT TO THE
STANDARD OPTIONS

You pay only for scenarios you think are more
probable to happen.
You avoid to pay for the cases in which the price may have
huge changes as a deep fall and then increasing again.
Once it is out, it will remain so. So reduces the premium.
Example: Knockout options won’t pay off if the stock
crosses the knockout barrier.
Cheaper than an otherwise identical option without the knockout feature. If you
think the chance of knockout is small, you can take advantage of the lower
premium and get the same benefits. Or, you can even pay more premium to
receive a cash rebate if the option is knocked out
Knock out = vanilla option - knock in
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IN SOME CASES YOU CAN HEDGE BETTER
WITH BARRIER OPTIONS
Let’s suppose an investor has a stock in his portfolio and his
expectations are that the market will decrease. Thus he could sell a
call option with strike K, let’s say = 80 and he would not loose from
the market fall. But it could happen also the opposite situation in
which the market goes up and he might have huge losses.
If he sells an up and out call with barrier H= 110, he could get
protected also from an increase in the market price.
If the spot price of the stock gets higher than 110, the option sold is
worthless, without a rebate, and the investor has gained the premium
+ the stock with a higher market value!
On the other side, he would limit his losses because of the lower cost of
the barrier call (he must buy it before selling) and because of a
market increase.


Example: A global bank might determine that the spread
between borrowing and lending rates of a foreign operation is
contingent on the foreign currency remaining within pre
specified barriers. In this case, a barrier option written on the
currency can be used to protect the profitability of these
operations in the event that the currency exits these bands.
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HEDGING BARRIER OPTIONS
Hedging barrier options is much more complicated than standard plain
vanilla because they combine characteristics of the plain vanilla
and of the barriers.
One can distinguish mainly two types of hedging:

Dynamic hedging - constant rebalancing of the portfolio (eg. BlackScholes extension of the European options)
If the barrier is hit before maturity, the barrier option value at that time is known to be
zero, or the price of the corresponding European option otherwise. Thus it is
sufficient to hedge the option before hitting the barrier. Prices of barrier options
are smooth functions of the underlying asset price in the active region so Ito
lemma can be applied up to the stopping time: Hold ∂f (t, St)/ ∂S shares of the
underlying asset for t< TH.

Static hedging - “buy and hold” strategy (eg. proof of the put-call parity)
1.Buy an option portfolio with K and T which partially replicates the payoﬀ at the
expiry.
2. Make the value of your portfolio equal to 0 at the barrier value by buying and
selling call options with strike equal to the barrier H, so to set the value of the
portfolio equal to 0 at the boundary.
Is not hit: Same payoff with the vanilla option
The barrier
Is hit: Option value = 0, thus portfolio value must be 0 for 13
St = H
SENSITIVITY- DELTA OF A 1 YEAR
DOWN AND OUT CALL
1.Delta of barrier
options can differ
significatively w.r.t
standard options
(∆ < 0 or ∆ >1)
2. A barrier option
values can actually
decrease with
increasing volatility
(you are long vol w.r.t
strike, but short vol w.r.t
barrier)
Price of the plain vanilla and down and out
option
Delta of the plain vanilla and down and out
option
* For high values of the spot, the delta behaves as if it was a standard
option.
** With a St > H just above the barrier, delta increases rapidly
*** It declines as the price declines (but remains positive) and becomes
zero when St = H because the option is worthless.
****The down-and-out call is always worth less than the corresponding
standard call, but approaches the same value at very high stock prices
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SENSITIVITY – GAMMA OF A 1 YEAR
DOWN AND OUT CALL
1 year down and out call option
Gamma is the sensitivity of delta to changes in stock
price. An option with greater gamma must be
rehedged more frequently; it takes more premium to
pay for greater rebalancing costs
For St <H the Gamma behaves the same as that of
the standard call option.
For St near to H the one-year call has a negative
gamma, reflecting the fact that the option holder is
really "short volatility," or short an option struck at the
barrier.
Finally, for St high the gamma has a similar behavior
of the plain vanilla call.
For stock prices near the barrier, an increase in
volatility would actually make knockout for the
down and out option more likely, and so decrease
overall option value.
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BARRIER OPTIONS: PERSPECTIVE

Tailor made options which at present come with
different variations from the standard ones:







Rebate (in case the option is knocked out);
Double Barrier(Double-no-touch,China 2011);
Discrete Barrier(Monitored at discrete intervals);
Window Barriers(Time ranges to monitor the price);
Parisian Barrier (Price can spend a certain period
out);
Fluffy Barriers(Depth of barrier crossing);
Useful in foreign exchange exposure managing.
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