Download The first widely-used model for option pricing is the Black Scholes

In finance, an option is a financial instrument that gives its owner the right,
but not the obligation, to engage in some specific transaction on an asset.
Options are derivative instruments, as their fair price derives from the value
of the other asset, called the underlying. The underlying is commonly a
stock, a bond, or a futures contract, though many other types of options
exist, and options can in principle be created for any type of valuable asset.
An option to buy something is called a call; an option to sell is called a put.
The price specified at which the underlying may be traded is called the strike
price. The process of activating an option and thereby trading the underlying
at the agreed-upon price is referred to as exercising it. Most options have an
expiration date. If the option is not exercised by the expiration date, it
becomes void and worthless.
In return for granting the option, called writing the option, the originator of
the option collects a payment, the premium, from the buyer. The writer of an
option must make good on delivering (or receiving) the underlying asset or
its cash equivalent, if the option is exercised.
An option can usually be sold by its original buyer to another party. Many
options are created in standardized form and traded on an anonymous
options exchange among the general public, while other over-the-counter
options are customized to the desires of the buyer on an ad hoc basis, usually
by an investment bank.
Option valuation
The theoretical value of an option is evaluated according to any of several
mathematical models. These models, which are developed by quantitative
analysts, attempt to predict how the value of an option changes in response
to changing conditions. Hence, the risks associated with granting, owning, or
trading options may be quantified and managed with a greater degree of
precision, perhaps, than with some other investments. Exchange-traded
options form an important class of options which have standardized contract
features and trade on public exchanges, facilitating trading among
independent parties. Over-the-counter options are traded between private
parties, often well-capitalized institutions that have negotiated separate
trading and clearing arrangements with each other.
[edit] Contract specifications
Every financial option is a contract between the two counterparties with the
terms of the option specified in a term sheet. Option contracts may be quite
complicated; however, at minimum, they usually contain the following
specifications:[3]
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whether the option holder has the right to buy (a call option) or the
right to sell (a put option)
the quantity and class of the underlying asset(s) (e.g. 100 shares of
XYZ Co. B stock)
the strike price, also known as the exercise price, which is the price at
which the underlying transaction will occur upon exercise
the expiration date, or expiry, which is the last date the option can be
exercised
the settlement terms, for instance whether the writer must deliver the
actual asset on exercise, or may simply tender the equivalent cash
amount
the terms by which the option is quoted in the market to convert the
quoted price into the actual premium-–the total amount paid by the
holder to the writer of the option.
[edit] Types of options
The primary types of financial options are:
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Exchange traded options (also called "listed options") are a class of
exchange traded derivatives. Exchange traded options have
standardized contracts, and are settled through a clearing house with
fulfillment guaranteed by the credit of the exchange. Since the
contracts are standardized, accurate pricing models are often
available. Exchange traded options include:[4][5]
o stock options,
o commodity options,
o bond options and other interest rate options
o stock market index options or, simply, index options and
o options on futures contracts
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Over-the-counter options (OTC options, also called "dealer
options") are traded between two private parties, and are not listed on
an exchange. The terms of an OTC option are unrestricted and may be
individually tailored to meet any business need. In general, at least
one of the counterparties to an OTC option is a well-capitalized
institution. Option types commonly traded over the counter include:
1. interest rate options
2. currency cross rate options, and
3. options on swaps or swaptions.
[edit] Other option types
Another important class of options, particularly in the U.S., are employee
stock options, which are awarded by a company to their employees as a form
of incentive compensation. Other types of options exist in many financial
contracts, for example real estate options are often used to assemble large
parcels of land, and prepayment options are usually included in mortgage
loans. However, many of the valuation and risk management principles
apply across all financial options.
[edit] Option styles
Main article: Option style
Naming conventions are used to help identify properties common to many
different types of options. These include:
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European option - an option that may only be exercised on
expiration.
American option - an option that may be exercised on any trading day
on or before expiry.
Bermudan option - an option that may be exercised only on specified
dates on or before expiration.
Barrier option - any option with the general characteristic that the
underlying security's price must pass a certain level or "barrier" before
it can be exercised
Exotic option - any of a broad category of options that may include
complex financial structures.[6]
Vanilla option - by definition, any option that is not exotic.
[edit] Valuation models
Main article: Valuation of options
The value of an option can be estimated using a variety of quantitative
techniques based on the concept of risk neutral pricing and using stochastic
calculus. The most basic model is the Black-Scholes model. More
sophisticated models are used to model the volatility smile. These models
are implemented using a variety of numerical techniques.[7] In general,
standard option valuation models depend on the following factors:
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The current market price of the underlying security,
the strike price of the option, particularly in relation to the current
market price of the underlier (in the money vs. out of the money),
the cost of holding a position in the underlying security, including
interest and dividends,
the time to expiration together with any restrictions on when exercise
may occur, and
an estimate of the future volatility of the underlying security's price
over the life of the option.
More advanced models can require additional factors, such as an estimate of
how volatility changes over time and for various underlying price levels, or
the dynamics of stochastic interest rates.
The following are some of the principal valuation techniques used in
practice to evaluate option contracts.
[edit] Black-Scholes
Main article: Black–Scholes
In the early 1970s, Fischer Black and Myron Scholes made a major
breakthrough by deriving a differential equation that must be satisfied by the
price of any derivative dependent on a non-dividend-paying stock. By
employing the technique of constructing a risk neutral portfolio that
replicates the returns of holding an option, Black and Scholes produced a
closed-form solution for a European option's theoretical price.[8] At the same
time, the model generates hedge parameters necessary for effective risk
management of option holdings. While the ideas behind the Black-Scholes
model were ground-breaking and eventually led to Scholes and Merton
receiving the Swedish Central Bank's associated Prize for Achievement in
Economics (often mistakenly referred to as the Nobel Prize),[9] the
application of the model in actual options trading is clumsy because of the
assumptions of continuous (or no) dividend payment, constant volatility, and
a constant interest rate. Nevertheless, the Black-Scholes model is still one of
the most important methods and foundations for the existing financial
market in which the result is within the reasonable range.[10]
[edit] Stochastic volatility models
Main article: Heston model
Since the market crash of 1987, it has been observed that market implied
volatility for options of lower strike prices are typically higher than for
higher strike prices, suggesting that volatility is stochastic, varying both for
time and for the price level of the underlying security. Stochastic volatility
models have been developed including one developed by S.L. Heston.[11]
One principal advantage of the Heston model is that it can be solved in
closed-form, while other stochastic volatility models require complex
numerical methods.[11]
[edit] Model implementation
Further information: Valuation of options
Once a valuation model has been chosen, there are a number of different
techniques used to take the mathematical models to implement the models.
[edit] Analytic techniques
In some cases, one can take the mathematical model and using analytical
methods develop closed form solutions such as Black-Scholes and the Black
model. The resulting solutions are useful because they are rapid to calculate,
and their "Greeks" are easily obtained.
[edit] Binomial tree pricing model
Main article: Binomial options pricing model
Closely following the derivation of Black and Scholes, John Cox, Stephen
Ross and Mark Rubinstein developed the original version of the binomial
options pricing model.[12] [13] It models the dynamics of the option's
theoretical value for discrete time intervals over the option's duration. The
model starts with a binomial tree of discrete future possible underlying stock
prices. By constructing a riskless portfolio of an option and stock (as in the
Black-Scholes model) a simple formula can be used to find the option price
at each node in the tree. This value can approximate the theoretical value
produced by Black Scholes, to the desired degree of precision. However, the
binomial model is considered more accurate than Black-Scholes because it is
more flexible, e.g. discrete future dividend payments can be modeled
correctly at the proper forward time steps, and American options can be
modeled as well as European ones. Binomial models are widely used by
professional option traders.
[edit] Monte Carlo models
Main article: Monte Carlo methods for option pricing
For many classes of options, traditional valuation techniques are intractable
due to the complexity of the instrument. In these cases, a Monte Carlo
approach may often be useful. Rather than attempt to solve the differential
equations of motion that describe the option's value in relation to the
underlying security's price, a Monte Carlo model uses simulation to generate
random price paths of the underlying asset, each of which results in a payoff
for the option. The average of these payoffs can be discounted to yield an
expectation value for the option.[14]
[edit] Finite difference models
Main article: Finite difference methods for option pricing
The equations used to model the option are often expressed as partial
differential equations (see for example Black–Scholes PDE). Once
expressed in this form, a finite difference model can be derived, and the
valuation obtained. This approach is useful for extensions of the option
pricing model, such as changing assumptions as to dividends, that are not
able to be represented with a closed form analytic solution. A number of
implementations of finite difference methods exist for option valuation,
including: explicit finite difference, implicit finite difference and the CrankNicholson method. A trinomial tree option pricing model can be shown to be
a simplified application of the explicit finite difference method.
[edit] Other models
Other numerical implementations which have been used to value options
include finite element methods. Additionally, various short rate models have
been developed for the valuation of interest rate derivatives, bond options
and swaptions. These, similarly, allow for closed-form, lattice-based, and
simulation-based modelling, with corresponding advantages and
considerations.
[edit] Risks
As with all securities, trading options entails the risk of the option's value
changing over time. However, unlike traditional securities, the return from
holding an option varies non-linearly with the value of the underlier and
other factors. Therefore, the risks associated with holding options are more
complicated to understand and predict.
In general, the change in the value of an option can be derived from Ito's
lemma as:
where the Greeks Δ, Γ, κ and θ are the standard hedge parameters calculated
from an option valuation model, such as Black-Scholes, and dS, dσ and dt
are unit changes in the underlier price, the underlier volatility and time,
respectively.
Thus, at any point in time, one can estimate the risk inherent in holding an
option by calculating its hedge parameters and then estimating the expected
change in the model inputs, dS, dσ and dt, provided the changes in these
values are small. This technique can be used effectively to understand and
manage the risks associated with standard options. For instance, by
offsetting a holding in an option with the quantity − Δ of shares in the
underlier, a trader can form a delta neutral portfolio that is hedged from loss
for small changes in the underlier price. The corresponding price sensitivity
formula for this portfolio Π is:
[edit] Example
A call option expiring in 99 days on 100 shares of XYZ stock is struck at
$50, with XYZ currently trading at $48. With future realized volatility over
the life of the option estimated at 25%, the theoretical value of the option is
$1.89. The hedge parameters Δ, Γ, κ, θ are (0.439, 0.0631, 9.6, and -0.022),
respectively. Assume that on the following day, XYZ stock rises to $48.5
and volatility falls to 23.5%. We can calculate the estimated value of the call
option by applying the hedge parameters to the new model inputs as:
Under this scenario, the value of the option increases by $0.0614 to $1.9514,
realizing a profit of $6.14. Note that for a delta neutral portfolio, where by
the trader had also sold 44 shares of XYZ stock as a hedge, the net loss
under the same scenario would be ($15.86).
[edit] Pin risk
Main article: Pin risk
A special situation called pin risk can arise when the underlier closes at or
very close to the option's strike value on the last day the option is traded
prior to expiration. The option writer (seller) may not know with certainty
whether or not the option will actually be exercised or be allowed to expire
worthless. Therefore, the option writer may end up with a large, unwanted
residual position in the underlier when the markets open on the next trading
day after expiration, regardless of their best efforts to avoid such a residual.
[edit] Counterparty risk
A further, often ignored, risk in derivatives such as options is counterparty
risk. In an option contract this risk is that the seller won't sell or buy the
underlying asset as agreed. The risk can be minimized by using a financially
strong intermediary able to make good on the trade, but in a major panic or
crash the number of defaults can overwhelm even the strongest
intermediaries.
[edit] Trading
The most common way to trade options is via standardized options contracts
that are listed by various futures and options exchanges. [15] Listings and
prices are tracked and can be looked up by ticker symbol. By publishing
continuous, live markets for option prices, an exchange enables independent
parties to engage in price discovery and execute transactions. As an
intermediary to both sides of the transaction, the benefits the exchange
provides to the transaction include:
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fulfillment of the contract is backed by the credit of the exchange,
which typically has the highest rating (AAA),
counterparties remain anonymous,
enforcement of market regulation to ensure fairness and transparency,
and
maintenance of orderly markets, especially during fast trading
conditions.
Over-the-counter options contracts are not traded on exchanges, but instead
between two independent parties. Ordinarily, at least one of the
counterparties is a well-capitalized institution. By avoiding an exchange,
users of OTC options can narrowly tailor the terms of the option contract to
suit individual business requirements. In addition, OTC option transactions
generally do not need to be advertised to the market and face little or no
regulatory requirements. However, OTC counterparties must establish credit
lines with each other, and conform to each others clearing and settlement
procedures.
With few exceptions,[16] there are no secondary markets for employee stock
options. These must either be exercised by the original grantee or allowed to
expire worthless.
Swap (finance)
In finance, a swap is a derivative in which counterparties exchange certain
benefits of one party's financial instrument for those of the other party's
financial instrument. The benefits in question depend on the type of financial
instruments involved. For example, in the case of a swap involving two
bonds, the benefits in question can be the periodic interest (or coupon)
payments associated with the bonds. Specifically, the two counterparties
agree to exchange one stream of cash flows against another stream. These
streams are called the legs of the swap. The swap agreement defines the
dates when the cash flows are to be paid and the way they are calculated.[1]
Usually at the time when the contract is initiated at least one of these series
of cash flows is determined by a random or uncertain variable such as an
interest rate, foreign exchange rate, equity price or commodity price.[1]
The cash flows are calculated over a notional principal amount, which is
usually not exchanged between counterparties. Consequently, swaps can be
used to in cash or collateral.
Swaps can be used to hedge certain risks such as interest rate risk, or to
speculate on changes in the expected direction of underlying prices.
The first swaps were negotiated in the early 1980s.[1] David Swensen, a Yale
Ph.D. at Salomon Brothers, engineered the first swap transaction according
to "When Genius Failed: The Rise and Fall of Long-Term Capital
Management" by Roger Lowenstein. Today, swaps are among the most
heavily traded financial contracts in the world: the total amount of interest
rates and currency swaps outstanding is more thаn $426.7 trillion in 2009,
according to International Swaps and Derivatives Association.
Swap market
Most swaps are traded over-the-counter (OTC), "tailor-made" for the
counterparties. Some types of swaps are also exchanged on futures markets
such as the Chicago Mercantile Exchange Holdings Inc., the largest U.S.
futures market, the Chicago Board Options Exchange,
IntercontinentalExchange and Frankfurt-based Eurex AG.
The Bank for International Settlements (BIS) publishes statistics on the
notional amounts outstanding in the OTC derivatives market. At the end of
2006, this was USD 415.2 trillion, more than 8.5 times the 2006 gross world
product. However, since the cash flow generated by a swap is equal to an
interest rate times that notional amount, the cash flow generated from swaps
is a substantial fraction of but much less than the gross world product—
which is also a cash-flow measure. The majority of this (USD 292.0 trillion)
was due to interest rate swaps. These split by currency as:
Types of swaps
The five generic types of swaps, in order of their quantitative importance,
are: interest rate swaps, currency swaps, credit swaps, commodity swaps and
equity swaps. There are also many other types.
Interest rate swaps
The most common type of swap is a “plain Vanilla” interest rate swap. It is
the exchange of a fixed rate loan to a floating rate loan. The life of the swap
can range from 2 years to over 15 years. The reason for this exchange is to
take benefit from comparative advantage. Some companies may have
comparative advantage in fixed rate markets while other companies have a
comparative advantage in floating rate markets. When companies want to
borrow they look for cheap borrowing i.e. from the market where they have
comparative advantage. However this may lead to a company borrowing
fixed when it wants floating or borrowing floating when it wants fixed. This
is where a swap comes in. A swap has the effect of transforming a fixed rate
loan into a floating rate loan or vice versa.
For example, party B makes periodic interest payments to party A based on
a variable interest rate of LIBOR +70 basis points. Party A in return makes
periodic interest payments based on a fixed rate of 8.65%. The payments are
calculated over the notional amount. The first rate is called variable,
because it is reset at the beginning of each interest calculation period to the
then current reference rate, such as LIBOR. In reality, the actual rate
received by A and B is slightly lower due to a bank taking a spread.
[edit] Currency swaps
Main article: Currency swap
A currency swap involves exchanging principal and fixed rate interest
payments on a loan in one currency for principal and fixed rate interest
payments on an equal loan in another currency. Just like interest rate swaps,
the currency swaps also are motivated by comparative advantage.
[edit] Commodity swaps
Main article: Commodity swap
A commodity swap is an agreement whereby a floating (or market or spot)
price is exchanged for a fixed price over a specified period. The vast
majority of commodity swaps involve crude oil.
[edit] Equity Swap
Main article: equity swap
An equity swap is a special type of total return swap, where the underlying
asset is a stock, a basket of stocks, or a stock index. Compared to actually
owning the stock, in this case you do not have to pay anything up front, but
you do not have any voting or other rights that stock holders do have.
[edit] Credit default swaps
Main article: Credit default swap
A credit default swap (CDS) is a swap contract in which the buyer of the
CDS makes a series of payments to the seller and, in exchange, receives a
payoff if a credit instrument - typically a bond or loan - goes into default
(fails to pay). Less commonly, the credit event that triggers the payoff can be
a company undergoing restructuring, bankruptcy or even just having its
credit rating downgraded. CDS contracts have been compared with
insurance, because the buyer pays a premium and, in return, receives a sum
of money if one of the events specified in the contract occur. Unlike an
actual insurance contract the buyer is allowed to profit from the contract and
may also cover an asset to which the buyer has no direct exposure.
[edit] Other variations
There are myriad different variations on the vanilla swap structure, which
are limited only by the imagination of financial engineers and the desire of
corporate treasurers and fund managers for exotic structures.[1]
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A total return swap is a swap in which party A pays the total return
of an asset, and party B makes periodic interest payments. The total
return is the capital gain or loss, plus any interest or dividend
payments. Note that if the total return is negative, then party A
receives this amount from party B. The parties have exposure to the
return of the underlying stock or index, without having to hold the
underlying assets. The profit or loss of party B is the same for him as
actually owning the underlying asset.
An option on a swap is called a swaption. These provide one party
with the right but not the obligation at a future time to enter into a
swap.
A variance swap is an over-the-counter instrument that allows one to
speculate on or hedge risks associated with the magnitude of
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movement, a CMS, is a swap that allows the purchaser to fix the
duration of received flows on a swap.
An Amortising swap is usually an interest rate swap in which the
notional principal for the interest payments declines during the life of
the swap, perhaps at a rate tied to the prepayment of a mortgage or to
an interest rate benchmark such as the LIBOR.
[edit] Valuation
Further information: Rational pricing#Swaps and Arbitrage
The value of a swap is the net present value (NPV) of all estimated future
cash flows. A swap is worth zero when it is first initiated, however after this
time its value may become positive or negative.[1] There are two ways to
value swaps: in terms of bond prices, or as a portfolio of forward contracts.[1]
[edit] Using bond prices
While principal payments are not exchanged in an interest rate swap,
assuming that these are received and paid at the end of the swap does not
change its value. Thus, from the point of view of the floating-rate payer, a
swap position in a fixed-rate bond (i.e. receiving fixed interest payments),
and a short position in a floating rate note (i.e. making floating interest
payments):
Vswap = Bfixed − Bfloating
From the point of view of the fixed-rate payer, the swap can be viewed as
having the opposite positions. That is,
Vswap = Bfloating − Bfixed
Similarly, currency swaps can be regarded as having positions in bonds
whose cash flows correspond to those in the swap. Thus, the home currency
value is:
Vswap = Bdomestic − S0Bforeign, where Bdomestic is the domestic cash flows of
the swap, Bforeign is the foreign cash flows of the LIBOR is the rate of
interest offered by banks on deposit from other banks in the
eurocurrency market. One-month LIBOR is the rate offered for 1month deposits, 3-month LIBOR for three months deposits, etc.
LIBOR rates are determined by trading between banks and change
continuously as economic conditions change. Just like the prime rate of
interest quoted in the domestic market, LIBOR is a reference rate of interest
in the International Market.
[edit] Arbitrage arguments
As mentioned, to be arbitrage free, the terms of a swap contract are such
that, initially, the NPV of these future cash flows is equal to zero. Where this
is not the case, arbitrage would be possible.
For example, consider a plain vanilla fixed-to-floating interest rate swap
where Party A pays a fixed rate, and Party B pays a floating rate. In such an
agreement the fixed rate would be such that the present value of future fixed
rate payments by Party A are equal to the present value of the expected
future floating rate payments (i.e. the NPV is zero). Where this is not the
case, an Arbitrageur, C, could:
1. assume the position with the lower present value of payments, and
borrow funds equal to this present value
2. meet the cash flow obligations on the position by using the borrowed
funds, and receive the corresponding payments - which have a higher
present value
3. use the received payments to repay the debt on the borrowed funds
4. pocket the difference - where the difference between the present value
of the loan and the present value of the inflows is the arbitrage profit.
Subsequently, once traded, the price of the Swap must equate to the price of
the various corresponding instruments as mentioned above. Where this is not
true, an arbitrageur could similarly short sell the overpriced instrument, and
use the proceeds to purchase the correctly priced instrument, pocket the
difference, and then use payments generated to service the instrument which
he is short.
What is an Option?
Options Basics - Part One
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Equity Options
Calls and Puts
Options Premium
Expiration Friday
Options Basics - Part Two
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Leverage & Risk
In-the-money, At-the-money, Out-of-the-money
Time Decay
Expiration
Long
Short
Open
Close
Options Basics - Part Three
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Exercise
Assignment
Net Price
Early Exercise
•
Volatility
Tools
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An option is a contract to buy or sell a specific financial product officially known as the
option's underlying instrument or underlying interest. For equity options, the underlying
instrument is a stock, exchange-traded fund (ETF), or similar product. The contract itself
is very precise. It establishes a specific price, called the strike price, at which the contract
may be exercised, or acted on. And it has an expiration date. When an option expires, it
no longer has value and no longer exists.
Options come in two varieties, calls and puts, and you can buy or sell either type. You
make those choices - whether to buy or sell and whether to choose a call or a put - based
on what you want to achieve as an options investor.
Buying and Selling
If you buy a call, you have the right to buy the underlying instrument at the strike price
on or before the expiration date. If you buy a put, you have the right to sell the underlying
instrument on or before expiration. In either case, as the option holder, you also have the
right to sell the option to another buyer during its term or to let it expire worthless.
The situation is different if you write, or "sell to open", an option. Selling to open a short
option position obligates you, the writer, to fulfill your side of the contract if the holder
wishes to exercise. When you sell a call as an opening transaction, you're obligated to sell
the underlying interest at the strike price, if you're assigned. When you sell a put as an
opening transaction, you're obligated to buy the underlying interest, if assigned. As a
writer, you have no control over whether or not a contract is exercised, and you need to
recognize that exercise is always possible at any time until the expiration date. But just as
the buyer can sell an option back into the market rather than exercising it, as a writer you
can purchase an offsetting contract, provided you have not been assigned, and end your
obligation to meet the terms of the contract. When offsetting a short option position, you
would enter a "buy to close" transaction.
At a Premium
When you buy an option, the purchase price is called the premium. If you sell, the
premium is the amount you receive. The premium isn't fixed and changes constantly - so
the premium you pay today is likely to be higher or lower than the premium yesterday or
tomorrow. What those changing prices reflect is the give and take between what buyers
are willing to pay and what sellers are willing to accept for the option. The point at which
there's agreement becomes the price for that transaction, and then the process begins
again.
If you buy options, you start out with what's known as a net debit. That means you've
spent money you might never recover if you don't sell your option at a profit or exercise
it. And if you do make money on a transaction, you must subtract the cost of the premium
from any income you realize to find your net profit.
As a seller, on the other hand, you begin with a net credit because you collect the
premium. If the option is never exercised, you keep the money. If the option is exercised,
you still get to keep the premium, but are obligated to buy or sell the underlying stock if
you're assigned.
The Value of Options
What a particular options contract is worth to a buyer or seller is measured by how likely
it is to meet their expectations. In the language of options, that's determined by whether
or not the option is, or is likely to be, in-the-money or out-of-the-money at expiration. A
call option is in-the-money if the current market value of the underlying stock is above
the exercise price of the option, and out-of-the-money if the stock is below the exercise
price. A put option is in-the-money if the current market value of the underlying stock is
below the exercise price and out-of-the-money if it is above it. If an option is not in-themoney at expiration, the option is assumed to be worthless.
An option's premium has two parts: an intrinsic value and a time value. Intrinsic value is
the amount by which the option is in-the-money. Time value is the difference between
whatever the intrinsic value is and what the premium is. The longer the amount of time
for market conditions to work to your benefit, the greater the time value.
Options Prices
Several factors, including supply and demand in the market where the option is traded,
affect the price of an option, as is the case with an individual stock. What's happening in
the overall investment markets and the economy at large are two of the broad influences.
The identity of the underlying instrument, how it traditionally behaves, and what it is
doing at the moment are more specific ones. Its volatility is also an important factor, as
investors attempt to gauge how likely it is that an option will move in-the-money.
OPTION PRICING MODEL
The first widely-used model for option pricing is the Black Scholes.
This formula can be used to calculate a theoretical value for an option
using current stock prices, expected dividends, the option's strike
price, expected interest rates, time to expiration and expected stock
volatility. ...
Option Pricing Models and the "Greeks"
Pricing Models Used
The Black-Scholes model and the Cox, Ross and Rubinstein binomial model
are the primary pricing models used by the software available from this site
(Finance Add-in for Excel, the Options Strategy Evaluation Tool, and the
on-line pricing calculators.)
Both models are based on the same theoretical foundations and assumptions
(such as the geometric Brownian motion theory of stock price behaviour and
risk-neutral valuation). However, there are also some some important
differences between the two models and these are highlighted below.
The Black-Scholes Model
The Black-Scholes model is used to calculate a theoretical call price
(ignoring dividends paid during the life of the option) using the five key
determinants of an option's price: stock price, strike price, volatility, time to
expiration, and short-term (risk free) interest rate.
The original formula for calculating the theoretical option price (OP) is as
follows:
Where:
The variables are:
S = stock price
X = strike price
t = time remaining until expiration, expressed as a percent of a year
r = current continuously compounded risk-free interest rate
v = annual volatility of stock price (the standard deviation of the short-term
returns over one year). See below for how to estimate volatility.
ln = natural logarithm
N(x) = standard normal cumulative distribution function
e = the exponential function
Lognormal distribution -- but how realistic?
The model is based on a normal distribution of underlying asset returns
which is the same thing as saying that the underlying asset prices themselves
are lognormally distributed. A lognormal distribution has a longer right tail
compared with a normal, or bell-shaped, distribution. The lognormal
distribution allows for a stock price distribution of between zero and infinity
(ie no negative prices) and has an upward bias (representing the fact that a
stock price can only drop 100% but can rise by more than 100%).
In practice underlying asset price distributions often depart significantly
from the lognormal. For example historical distributions of underlying asset
returns often have fatter left and right tails than a normal distribution
indicating that dramatic market moves occur with greater frequency than
would be predicted by a normal distribution of returns-- ie more very high
returns and more very low returns.
A corollary of this is the volatility smile -- the way in which at-the-money
options often have a lower volatility than deeply out-of- the-money options
or deeply in-the- money options.
The Excel add-in which can be downloaded from this site contains three sets
of tools for dealing with non-lognormally distributed asset prices and the
volatility smile:
Modified Black-Scholes and binomial pricing (using implied binomial
trees) for European and American option pricing with non-lognormal
distributions. These models can be used to see the impact on option prices
of non-lognormal price distributions (as measured by coefficents of
skewness (symmetry) and kurtosis (fatness of distribution tails and height
of peaks)), and to calculate and plot the volatility smile implied by these
distributions.
Measuring the degree to which historical asset price distributions diverge
from the lognormal (as measured by coefficents of skewness and kurtosis).
Plotting non-lognormal distribution curves for specified coefficients of
skewness and Kurtosis (as well as volatility etc) to see how they differ
from the lognormal.
In addition to using the add-in you can use the on-line stock price
distribution calculator to examine the sensitivity of the shape of the
lognormal stock price distribution curve to changes in time, volatility, and
expected growth rates. And you can also use the on-line stock price
probability calculator to look at the probabilities of stock prices exceeding
upper and lower boundary prices -- both at the end of a specified number of
days and at any time during the period.
Volatility -- implied or historical?
This is the most critical parameter for option pricing -- option prices are very
sensitive to changes in volatility. Volatility however cannot be directly
observed and must be estimated.
Whilst implied volatility -- the volatility of the option implied by current
market prices -- is commonly used the argument that this is the "best"
estimate is somewhat circular. Skilled options traders will not rely solely on
implied volatility but will look behind the estimates to see whether or not
they are higher or lower than you would expect from historical and current
volatilities, and hence whether options are more expensive or cheaper than
perhaps they should be.
It's a slight over simplification, but basically implied volatility will give you
the price of an option; historical volatility will give you an indication of its
value. It's important to understand both. For instance, if your forecast of
volatility based on historical prices is greater than current implied volatility
(options under valued) you might want to buy a straddle; if your historical
forecast is less than implied volatility you might want to sell a straddle.
This site contains one of the most comprehensive sets of tools available for
getting a handle on volatility. The tools include an Historic Volatility
Calculator (which automatically extracts historic prices from the web, and
calculates and graphs volatility), an Implied Volatility Calculator (which
retrieves and calculates implied volatility for an entire option chain), and an
Excel Add-in (for those who like to build their own Excel applications). The
volatility functions in the add-in include:
Implied volatility calculation (American and European options, with and
without dividends).
Equally weighted historical volatility estimation using historical prices:
one or more of high-low, closing, and open prices. All the price data
required are available at no charge at sites such as Yahoo.
Exponentially weighted historical volatility estimation using the EWMA
(exponentially weighted moving average) model or the GARCH model.
These models give greater emphasis to more recent prices.
Volatility forecasting using the GARCH model, which lets you see how
volatility is likely to move in the future. A common application of this is
to create volatility term structures for the weeks or months ahead to
answer questions like "what volatility should I use for pricing an option
with a life of three months?".
The implied volatility, historical volatility, and forecast volatility tools are
complementary. With volatility being such a critical factor a good options
trader will use all three sets of tools to help form a view about the volatility
to use in pricing options.
See the Finance Add-in for Excel and Volatility FAQs pages, Historic
Volatility Calculator page, Implied Volatility Calculator page, and the online demos for more information.
Risk-neutral valuation -- does the expected stock return matter?
Unlike volatility, which is all important for determining the fair value of an
option, views about the future direction of an underlying asset (ie whether
you think it will go up or down in the future and by how much) are
completely irrelevant.
Significantly, the expected rate of return of the stock (which would
incorporate risk preferences of investors as an equity risk premium) is not
one of the variables in the Black-Scholes model (or any other model for
option valuation). The important implication is that the value of an option is
completely independent of the expected growth of the underlying asset (and
is therefore risk neutral).
Thus, while any two investors may strongly disagree on the rate of return
they expect on a stock they will, given agreement to the assumptions of
volatility and the risk free rate, always agree on the fair value of the option
on that underlying asset.
The fact that a prediction of the future price of the underlying asset is not
necessary to value an option may appear to be counter intuitive, but it can
easily be shown to be correct. Dynamically hedging a call using underlying
asset prices generated from Monte Carlo simulation is a particularly
convincing way of demonstrating this. Irrespective of the assumptions
regarding stock price growth built into the Monte Carlo simulation the cost
of hedging a call (ie dynamically maintaining a delta neutral position by
buying & selling the underlying asset) will always be the same, and will be
very close to the Black-Scholes value. (The Excel add-in available from this
site contains a Monte Carlo simulation component which can be used for
this purpose.)
Putting it another way, whether the stock price rises or falls after, eg, writing
a call, it will always cost the same (providing volatility remains constant) to
dynamically hedge the call and this cost, when discounted back to present
value at the risk free rate, is very close to the Black-Scholes value.
Which is hardly surprising given that the Black-Scholes price is nothing
more than the amount an option writer would require as compensation for
writing a call and completely hedging the risk. The important point is that
the hedger's view about future stock prices is irrelevant.
This key concept underlying the valuation of all derivatives -- that fact that
the price of an option is independent of the risk preferences of investors -- is
called risk-neutral valuation. It means that all derivatives can be valued by
assuming that the return from their underlying assets is the risk free rate.
Advantages & Limitations
Advantage: The main advantage of the Black-Scholes model is speed -- it
lets you calculate a very large number of option prices in a very short time.
Limitation: The Black-Scholes model has one major limitation: it cannot
be used to accurately price options with an American-style exercise as it
only calculates the option price at one point in time -- at expiration. It does
not consider the steps along the way where there could be the possibility of
early exercise of an American option.
As all exchange traded equity options have American-style exercise (ie they
can be exercised at any time as opposed to European options which can only
be exercised at expiration) this is a significant limitation.
The exception to this is an American call on a non-dividend paying asset. In
this case the call is always worth the same as its European equivalent as
there is never any advantage in exercising early.
Various adjustments are sometimes made to the Black-Scholes price to
enable it to approximate American option prices (eg the Fischer Black
Pseudo-American method) but these only work well within certain limits
and they don't really work well for puts.
The Binomial Model
The binomial model breaks down the time to expiration into potentially a
very large number of time intervals, or steps. A tree of stock prices is
initially produced working forward from the present to expiration. At each
step it is assumed that the stock price will move up or down by an amount
calculated using volatility and time to expiration. This produces a binomial
distribution, or recombining tree, of underlying stock prices. The tree
represents all the possible paths that the stock price could take during the life
of the option.
At the end of the tree -- ie at expiration of the option -- all the terminal
option prices for each of the final possible stock prices are known as they
simply equal their intrinsic values.
Next the option prices at each step of the tree are calculated working back
from expiration to the present. The option prices at each step are used to
derive the option prices at the next step of the tree using risk neutral
valuation based on the probabilities of the stock prices moving up or down,
the risk free rate and the time interval of each step. Any adjustments to
stock prices (at an ex-dividend date) or option prices (as a result of early
exercise of American options) are worked into the calculations at the
required point in time. At the top of the tree you are left with one option
price.
To get a feel for how the binomial model works you can use the on-line
binomial tree calculators: either using the original Cox, Ross, & Rubinstein
tree or the equal probabilities tree, which produces equally accurate results
while overcoming some of the limitations of the C-R-R model. The
calculators let you calculate European or American option prices and display
graphically the tree structure used in the calculation. Dividends can be
specified as being discrete or as an annual yield, and points at which early
exercise is assumed for American options are highlighted.
Advantages & Limitations
Advantage: The big advantage the binomial model has over the BlackScholes model is that it can be used to accurately price American options.
This is because with the binomial model it's possible to check at every point
in an option's life (ie at every step of the binomial tree) for the possibility of
early exercise (eg where, due to eg a dividend, or a put being deeply in the
money the option price at that point is less than its intrinsic value).
Where an early exercise point is found it is assumed that the option holder
would elect to exercise, and the option price can be adjusted to equal the
intrinsic value at that point. This then flows into the calculations higher up
the tree and so on.
The on-line binomial tree graphical option calculator highlights those points
in the tree structure where early exercise would have have caused an
American price to differ from a European price.
The binomial model basically solves the same equation, using a
computational procedure that the Black-Scholes model solves using an
analytic approach and in doing so provides opportunities along the way to
check for early exercise for American options.
Limitation: The main limitation of the binomial model is its relatively slow
speed. It's great for half a dozen calculations at a time but even with today's
fastest PCs it's not a practical solution for the calculation of thousands of
prices in a few seconds.
Relationship to the Black-Scholes model
The same underlying assumptions regarding stock prices underpin both the
binomial and Black-Scholes models: that stock prices follow a stochastic
process described by geometric brownian motion. As a result, for European
options, the binomial model converges on the Black-Scholes formula as the
number of binomial calculation steps increases. In fact the Black-Scholes
model for European options is really a special case of the binomial model
where the number of binomial steps is infinite. In other words, the binomial
model provides discrete approximations to the continuous process
underlying the Black-Scholes model.
Whilst the Cox, Ross & Rubinstein binomial model and the Black-Scholes
model ultimately converge as the number of time steps gets infinitely large
and the length of each step gets infinitesimally small this convergence,
except for at-the-money options, is anything but smooth or uniform. To
examine the way in which the two models converge see the on-line BlackScholes/Binomial convergence analysis calculator. This lets you examine
graphically how convergence changes as the number of steps in the binomial
calculation increases as well as the impact on convergence of changes to the
strike price, stock price, time to expiration, volatility and risk free interest
rate.
Other Models used by the Software for American Options
For rapid calculation of a large number of prices, analytic models, like
Black-Scholes, are the only practical option on even the fastest PCs.
However, the pricing of American options (other than calls on non-dividend
paying assets) using analytic models is more difficult than for European
options.
To handle American option pricing in an efficient manner other models have
been developed. Three of the most widely used models which are used
where appropriate in the the software available from this site include:
Roll, Geske and Whaley analytic solution: The RGW formula can be
used for pricing an American call on a stock paying discrete dividends.
Because it is an analytic solution it is relatively fast.
Black's approximation for American calls: Although the RGW formula
is an analytic solution it involves solving equations iteratively and thus it
is slower than Black-Scholes. Black's approximation basically involves
using the Black-Scholes model after making adjustments to the stock price
and expiration date to take account of early exercise.
Barone-Adesi and Whaley quadratic approximation: An analytic
solution for American puts and calls paying a continuous dividend. Like
the RGW formula it involves solving equations iteratively so whilst it is
much faster than the binomial model it is still much slower than BlackScholes.
The Delta
A by-product of the Black-Scholes model is the calculation of the delta: the
degree to which an option price will move given a small change in the
underlying stock price. For example, an option with a delta of 0.5 will move
half a cent for every full cent movement in the underlying stock.
A deeply out-of-the-money call will have a delta very close to zero; a
deeply in-the-money call will have a delta very close to 1.
The formula for a the delta of a European call on a non-dividend paying
stock is:
Delta = N(d1) (see Black-Scholes formula above for d1)
Call deltas are positive; put deltas are negative, reflecting the fact that the
put option price and the underlying stock price are inversely related. The
put delta equals the call delta - 1.
The delta is often called the hedge ratio: If you have a portfolio short n
options (eg you have written n calls) then n multiplied by the delta gives you
the number of shares (ie units of the underlying) you would need to create a
riskless position - ie a portfolio which would be worth the same whether the
stock price rose by a very small amount or fell by a very small amount. In
such a "delta neutral" portfolio any gain in the value of the shares held due
to a rise in the share price would be exactly offset by a loss on the value of
the calls written, and vice versa.
Note that as the delta changes with the stock price and time to expiration the
number of shares would need to be continually adjusted to maintain the
hedge. How quickly the delta changes with the stock price is given by
gamma (see "Greeks" below).
The Options Strategy Evaluation Tool, which can be downloaded from this
site, calculates and displays the delta for each individual option trade entered
into the tool. If you set up a covered call in the Options Strategy Evaluation
Tool using Black-Scholes European pricing (ie sell n calls and buy n
underlying shares) then change the number of shares bought to be equal to
the number of options multiplied by the delta you will have an example of a
hedged position. Notice how the time line (ie the curved line showing the
profit at the number of days to expiration) on the payoff diagram just
touches (but doesn't pass through) the horizontal axis at one point only: the
point equal to the current share price. Moving a short distance in either
direction on this line will have the same impact on profit. ie you are delta
hedged.
The Options Strategy Evaluation Tool also calculates the position delta for a
range of stock prices and days to expiration -- that is, the delta of the entire
strategy consisting of multiple option trades and trades in the underlying
stock. The position delta, sometimes called the Equivalent Stock Position
(ESP) lets you see, for example, how a dollar rise in the underlying stock
prices will affect the overall profitability of the entire strategy. For example,
if the ESP of a portfolio, or strategy, is -2,300 it means that the market
exposure of the portfolio is equivalent to a portfolio short 2,300 shares. Thus
a one dollar rise in the stock price will cause the profitability of the entire
position to fall by $2,300.
The other position "Greeks" are also calculated by the model as well -- see
below.
You can also see how the delta changes with stock price, volatility, time to
expiration and interest rate by using the on-line options calculator.