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1. Binary options
Giampaolo Gabbi

2. Definition
• In finance, a binary option is a type of option where the payoff is
either some fixed amount of some asset or nothing at all. The two
main types of binary options are the cash-or-nothing binary option
and the asset-or-nothing binary option. The cash-or-nothing binary
option pays some fixed amount of cash if the option expires in-the-
money while the asset-or-nothing pays the value of the underlying
security.
• Thus, the options are binary in nature because there are only two
possible outcomes. They are also called all-or-nothing options,
digital options (more common in forex/interest rate markets), and
Fixed Return Options (FROs) (on the American Stock Exchange).
Binary options are usually European-style options.
• For example, a purchase is made of a binary cash-or-nothing call
option on XYZ Corp's stock struck at $100 with a binary payoff of
$1000. Then, if at the future maturity date, the stock is trading at or
above $100, $1000 is received. If its stock is trading below $100,
nothing is received.

3. Definition
Cash-or-nothing call
• This pays out one unit of cash if the spot is above the
strike at maturity
Cash-or-nothing put
• This pays out one unit of cash if the spot is below the
strike at maturity
Asset-or-nothing call
• This pays out one unit of asset if the spot is above the
strike at maturity
Asset-or-nothing put
• This pays out one unit of asset if the spot is below the
strike at maturity

4. Definition
• A key difference compared to vanilla
options for the option writer is that the
maximum possible downside is known in
advance.
• This makes selling binary options a much
more risk controlled and less negatively
skewed strategy than the typical short
volatility position.

5. Definition

6. Expressing a Range Trading View
• A binary option spread, such as one set
up by purchasing a binary call at a given
strike versus selling a binary call at a
higher strike, is the cleanest way of
implementing the view that the underlying
remains with a defined range.
• Since a binary option is similar to a call
spread, a binary call spread offers a risk
reward similar to a condor.

7. Expressing a Range Trading View
• For instance, with the SPX Aug12 1250
binary trading at 0.5 and the Aug12 1300
binary at 0.3, the 1250-1300 binary call
spread costs €0.2 and pays out €1 if the
index ends between 1250 and 1300 as of
the August expiration.
• In comparison, the condor wins in a similar
range but the boundaries are not as
clearly defined

8. Definition

9. Market
• Options are generally traded either OTC or on a national securities
exchange registered with the Securities and Exchange Commission
("SEC") or on a contract market designated by the Commodity
Futures Trading Commission ("CFTC"). A registered national
securities exchange or designated contract market are hereinafter
referred to collectively as "organized exchange“. An instrument is
described as trading OTC if it trades in some context other than on
or through an organized exchange. OTC derivatives are understood
to be specifically tailored to the needs and requirements of the end-
user, and therefore, lack the standardization and transparency
found on organized exchanges.
• The majority of derivative products are traded OTC. In such a
market, large financial institutions serve as derivatives dealers,
customizing products for the needs of particular clients. Contract
terms are negotiated between the parties, and typically each party
has only their contra-party to look to for performance of the contract.

10. Market
• Binary options have been traded for some time in an OTC
environment between institutional traders but not on a national
securities exchange. Contract markets have offered "binary options“
based on catastrophic events as well as on certain economic
indexes such as the Consumer Price Index (CPI). In France,
Germany and Austria, binary options have been traded OTC in a
one-sided market between investors and an institution. The
institution in these cases is the issuer of the contract and
establishes, if applicable, the market for the binary option.
• OTC binary options have several drawbacks and disadvantages.
One disadvantage is that OTC binary options are typically offered by
an institution on a non-fungible basis so that a customer can
purchase the option only from the institution, and cannot easily
resell to a third party because they are not standardized or traded
on an exchange. As a result, OTC binary options, as compared to
standardized exchange- traded options, lack important attributes of
a trading market such as transparency and liquidity.

11. Non exchange-traded binary
options
• Binary option contracts have long been available Over-
the-counter (OTC), i.e. sold directly by the issuer to the
buyer. They were generally considered "exotic"
instruments and there was no liquid market for trading
these instruments between their issuance and expiration.
They were often seen embedded in more complex option
contracts.
• Since mid-2008 binary options web-sites called binary
option trading platforms have been offering a simplified
version of exchange-traded binary options. It is
estimated that around 30 such platforms (including white
label products) have been in operation as of January
2011

12. Exchange-traded binary options
• In 2007, the Options Clearing Corporation proposed a rule change to allow binary
options, and the Securities and Exchange Commission approved listing cash-or-
nothing binary options in 2008.
• In May 2008, the American Stock Exchange (Amex) launched exchange-traded
European cash-or-nothing binary options, and the Chicago Board Options Exchange
(CBOE) followed in June 2008. The standardization of binary options allows them to
be exchange-traded with continuous quotations.
• Amex offers binary options on some ETFs and a few highly liquid equities such as
Citigroup and Google.
• Amex calls binary options "Fixed Return Options"; calls are named "Finish High" and
puts are named "Finish Low". To reduce the threat of market manipulation of single
stocks, Amex FROs use a "settlement index" defined as a volume-weighted average
of trades on the expiration day.
• CBOE offers binary options on the S&P 500 (SPX) and the CBOE Volatility Index
(VIX).
• The tickers for these are BSZ and BVZ, respectively. CBOE only offers calls, as
binary put options are trivial to create synthetically from binary call options. BSZ
strikes are at 5-point intervals and BVZ strikes are at 1-point intervals. The actual
underlying to BSZ and BVZ are based on the opening prices of index basket
members.
• Both Amex and CBOE listed options have values between $0 and $1, with a multiplier
of 100, and tick size of $0.01, and are cash settled

13. Example of a Binary Options Trade
• A trader who thinks that the EUR/USD strike
price will close at or above 1.2500 at 3:00 p.m.
can buy a call option on that outcome. A trader
who thinks that the EUR/USD strike price will
close at or below 1.2500 at 3:00 p.m. can buy a
put option or sell the contract.
• At 2:00 p.m. the EUR/USD spot price is 1.2490.
the trader believes this will increase, so he buys
10 call options for EUR/USD at or above 1.2500
at 3:00 p.m. at a cost of $40 each.

14. Example of a Binary Options Trade
• The risk involved in this trade is known. The trader’s
gross profit/loss follows the ‘all or nothing’ principle. He
can lose all the money he invested, which in this case is
$40 x 10 = $400, or make a gross profit of $100 x 10 =
$1000. If the EUR/USD strike price will close at or above
1.2500 at 3:00 p.m. the trader's net profit will be the
payoff at expiry minus the cost of the option: $1000 -
$400 = $600.
• The trader can also choose to liquidate (buy or sell to
close) his position prior to expiration, at which point the
option value is not guaranteed to be $100. The larger the
gap between the spot price and the strike price, the
value of the option decreases, as the option is less likely
to expire in the money.
• In this example, at 3:00 p.m. the spot has risen to
1.2505. The option has expired in the money and the
gross payoff is $1000. The trader's net profit is $600

15. CBOE to list binary options on S&P
500, VIX

16. CBOE Binary Options
• CBOE Binary Options are contracts that have an "all-or-
nothing" payout depending on the settlement price of the
underlying broad-based index relative to the strike price
of the binary option.
• Binary Call Options pay either 1) a fixed cash settlement
amount, if the underlying index settles at or above the
strike price at expiration; or 2) nothing at all, if the
underlying index settles below the strike price at
expiration.
• Binary Put Options pay either 1) a fixed cash settlement
amount, if the underlying index settles below the strike
price at expiration; or 2) nothing at all, if the underlying
index settles at or above the strike price at expiration.

17. CBOE Binary Options

18. CBOE Binary Options

19. CBOE Binary Options

20. CBOE Binary Options

21. Creating Contingent Premium
Options
• Contingent premium options are those in which
the buyer pays a premium only if the option
finishes in the money, and are commonly used
as a cheapening mechanism.
• A plain vanilla option in combination with a
binary option whose payoff at expiration would
equal the premium of the vanilla creates a
structure similar to a contingent premium option.
• Such a structure loses close to the strike in
compensation for the lower premium but does
not involve a premium payment if one’s
directional view happens to be incorrect.

22. Creating Contingent Premium
Options

23. Binary Option value
• For binary options with European exercise, pricing is
relatively straightforward as an analytical expression is
available in the Black Scholes world.
• For the European binary call that pays off 1 at expiration
T if the underlying S is over the strike K, the expiration
payoff can be summarized as

24. Relationship to vanilla options'
Greeks
• Since a binary call is a mathematical
derivative of a vanilla call with respect to
strike, the price of a binary call has the
same shape as the delta of a vanilla call,
• The delta of a binary call has the same
shape as the gamma of a vanilla call.

25. Interpretation of prices
• In a prediction market, binary options are used to find out a
population's best estimate of an event occurring - for example, a
price of 0.65 on a binary option triggered by the Democratic
candidate winning the next US Presidential election can be
interpreted as an estimate of 65% likelihood of him winning.
• In financial markets, expected returns on a stock or other instrument
are already priced into the stock. However, a binary options market
provides other information. Just as the regular options market
reveals the market's estimate of variance (volatility), i.e. the second
moment, a binary options market reveals the market's estimate of
skew, i.e. the third moment.
• A portfolio of binary options can also be used to synthetically
recreate (or valuate) any other option (analogous to integration).

26. Interpretation of prices (Intrade)

27. Interpretation of prices

28. Replication with Option Spreads
• The simplest listed instrument available that reasonably
mimics the payoff of a binary call option is a call spread.
The ideal hedge would be a spread of infinitesimal width,
but even with the strike intervals available in listed
options, replication of a binary is reasonably accurate
except very close to expiration.
• The imperfectness of such a hedge is a consequence of
the non-zero probability of the underlying finishing
between the two strikes. Since the 5-point interval
between strikes for near term SPX options is much
tighter than the 1-point interval between VIX option
strikes, this theoretical hedge implies a narrower range
for the bid-offer spread on SPX binaries.

29. Replication with Option Spreads
• To illustrate the construction of a replicating call
spread, we consider the 1300 strike binary call
option on the SPX expiring in Aug11. A potential
hedge for this is the 1295-1300 call spread on
SPX with the same expiration.
• Since the binary pays off $1 (corresponding to a
payoff of $100 per contract) if SPX closes at or
above 1300 at expiration, compared to a $5
payoff for the spread, we need 0.2 units of the
call spread for each binary option.

30. Replication with Option Spreads
• Next figure illustrates how the price of a binary
option varies as a function of the underlying as
expiration approaches.
• With several months to go in the life of the
option, the mark to market of the option is not
unlike that of the underlying itself.
• With about a week to expiration, it resembles a
call spread with a similar time to expiration.
• However, very close to expiration, the binary
becomes very convex just below the strike and
is highly negatively convex at levels slightly
above the strike.

31. Replication with Option Spreads