Option Pricing Models Lecture Notes: This week’s assignment is quite complex. Keep in mind that the theory behind these pricing models is the important thing to remember for this week’s assignment. If you feel the need to understand the Black Scholes (BSOPM) model in greater detail, I direct you to   and http://en.wikipedia.org/wiki/Black_Scholes.  The models we discuss this week can be used via MS Excel templates, which you will find uploaded to the course content section of our classroom under this week’s folder.  There is also an alternative calculator, courtesy of 888options.com located at the Binomial & Black Scholes Calculator link.  I strongly encourage you to try these out to get a feel for how the different variables play into the final determination of pricing. 1.   Binomial options pricing model In finance, the binomial options pricing model provides a generalisable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein (1979). Essentially, the model uses a "discrete-time" model of the varying price over time of the underlying financial instrument. Option valuation is then via application of therisk neutrality assumption over the life of the option, as the price of the underlying instrument evolves. Use of the model The Binomial options pricing model approach is widely used as it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM models the underlying instrument over time - as opposed to at a particular point. For example, the model is used to value American options which can be exercised at any point and Bermudan options which can be exercised at various points.  The model is also relatively simple, mathematically, and can therefore be readily implemented in a software (or even spreadsheet) environment. Although slower than the Black-Scholes model, it is considered more accurate, particularly for longer-dated options, and options on securities with dividend payments. For these reasons, various versions of the binomial model are widely used by practitioners in the options markets. For options with several sources of uncertainty (e.g. real options), or for options with complicated features (e.g. Asian options), lattice methods face several difficulties and are not practical. Monte Carlo option models are generally used in these cases. Monte Carlo simulation is, however, time-consuming in terms of computation, and is not used when the Lattice approach (or a formula) will suffice. See Monte Carlo methods in finance. Methodology The binomial pricing model uses a "discrete-time framework" to trace the evolution of the option's key underlying variable via a binomial lattice (tree), for a given number of time steps between valuation date and option expiration. Each node in the lattice represents a possible price of the underlying, at a particular point in time. This price evolution forms the basis for t.

1. Option Pricing Models Lecture Notes:
This week’s assignment is quite complex. Keep in mind that
the theory behind these pricing models is the important thing to
remember for this week’s assignment.
If you feel the need to understand the Black Scholes (BSOPM)
model in greater detail, I direct you to
and http://en.wikipedia.org/wiki/Black_Scholes.
The models we discuss this week can be used via MS Excel
templates, which you will find uploaded to the course content
section of our classroom under this week’s folder. There is
also an alternative calculator, courtesy
of 888options.com located at the Binomial & Black Scholes
Calculator link. I strongly encourage you to try these out to get
a feel for how the different variables play into the final
determination of pricing.
1. Binomial options pricing model
In finance, the binomial options pricing model provides a
generalisable numerical method for the valuation of options.
The binomial model was first proposed by Cox, Ross and
Rubinstein (1979). Essentially, the model uses a "discrete-time"
model of the varying price over time of the underlying financial
instrument. Option valuation is then via application of therisk
neutrality assumption over the life of the option, as the price of
the underlying instrument evolves.
Use of the model
The Binomial options pricing model approach is widely used as
it is able to handle a variety of conditions for which other
models cannot easily be applied. This is largely because the
BOPM models the underlying instrument over time - as opposed
to at a particular point. For example, the model is used to
value American options which can be exercised at any point
and Bermudan options which can be exercised at various
points.

2. The model is also relatively simple, mathematically, and can
therefore be readily implemented in a software (or
even spreadsheet) environment. Although slower than
the Black-Scholes model, it is considered more accurate,
particularly for longer-dated options, and options on securities
with dividend payments. For these reasons, various versions of
the binomial model are widely used by practitioners in the
options markets.
For options with several sources of uncertainty (e.g. real
options), or for options with complicated features (e.g. Asian
options), lattice methods face several difficulties and are not
practical. Monte Carlo option models are generally used in
these cases. Monte Carlo simulation is, however, time-
consuming in terms of computation, and is not used when the
Lattice approach (or a formula) will suffice. See Monte Carlo
methods in finance.
Methodology
The binomial pricing model uses a "discrete-time framework" to
trace the evolution of the option's key underlying variable via a
binomial lattice (tree), for a given number of time steps between
valuation date and option expiration.
Each node in the lattice represents a possible price of the
underlying, at a particular point in time. This price evolution
forms the basis for the option valuation.
The valuation process is iterative, starting at each final node,
and then working backwards through the tree to the first node
(valuation date), where the calculated result is the value of the
option.
Option valuation using this method is, as described, a three step
process:
1. price tree generation
2. calculation of option value at each final node
3. progressive calculation of option value at each earlier node;
the value at the first node is the value of the option.
For a more detailed explanation of the BOPM see:
· Cox JC, Ross SA and Rubinstein M. 1979. Options pricing: a

3. simplified approach, Journal of Financial Economics, 7:229-
263.1
2. Black-Scholes Model
Probably the most famous tool associated with option pricing.
Black and Scholes developed a simple model that can be
programmed in a spreadsheet or on a hand calculator to price
options the Black Scholes valuation is often called a risk neutral
valuation.
The Black-Scholes formula was the first widely used model for
option pricing. 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. While
the Black-Scholes model does not perfectly describe real-world
options markets, it is still often used in the valuation and
trading of options.
I. The variables of the Black Scholes formula are:
· Stock Price
· Strike Price
· Time remaining until expiration expressed as a percent of a
year
· Current risk-free interest rate
· Volatility measured by annual standard deviation.
II. Why Is Black-Scholes So Attractive?
· It is Easy
· Four of the five necessary parameters are observable
· Investor's risk aversion does not affect value; Formula can be
used by anyone, regardless of willingness to bear risk
· It does not depend on the expected return of the stock
· Investors with different assessments of the stock's expected
return will nevertheless agree on the call price.
3. The Greeks
The Greeks are a collection of statistical values (expressed as
percentages) that give the investor a better overall view of how
a stock has been performing. These statistical values can be
helpful in deciding what options strategies are best to use. The

4. investor should remember that statistics show trends based on
past performance. It is not guaranteed that the future
performance of the stock will behave according to the historical
numbers. These trends can change drastically based on new
stock performance.
The Greeks are vital tools in risk management. Each Greek
(with the exception of theta) represents a specific measure
of risk in owning an option, and option portfolios can be
adjusted accordingly ("hedged") to achieve a desired exposure;
see for example Delta hedging.
As a result, a desirable property of a model of a financial
market is that it allows for easycomputation of the Greeks. The
Greeks in the Black-Scholes model are very easy to calculate
and this is one reason for the model's continued popularity in
the market(downloaded
from http://en.wikipedia.org/wiki/The_Greeks.).
Beta: a measure of how closely the movement of an individual
stock tracks the movement of the entire stock market.
Delta: The Delta is a measure of the relationship between an
option price and the underlying stock price. For a call option, a
Delta of .50 means a half-point rise in premium for every dollar
that the stock goes up. For a put option contract, the premium
rises as stock prices fall. As options near expiration, in the
money contracts approach a Delta of 1.
Gamma: Sensitivity of Delta to unit change in the underlying.
Gamma indicates an absolute change in delta. For example, a
Gamma change of 0.150 indicates the delta will increase by
0.150 if the underlying price increases or decreases by 1.0.
Results may not be exact due to rounding.
Lambda: A measure of leverage. The expected percent change in
the value of an option for a 1 percent change in the value of the

5. underlying product. Lambda/Leverage.
Rho: Sensitivity of option value to change in interest rate. Rho
indicates the absolute change in option value for a one percent
change in the interest rate. For example, a Rho of .060 indicates
the option's theoretical value will increase by .060 if the
interest rate is decreased by 1.0. Results may not be exact due
to rounding. Rho/Rate.
Theta: Sensitivity of option value to change in time. Theta
indicates an absolute change in the option value for a 'one unit'
reduction in time to expiration. The Option Calculator assumes
'one unit' of time is 7 days. For example, a theta of -250
indicates the option's theoretical value will change by -.250 if
the days to expiration is reduced by 7. Results may not be exact
due to rounding. NOTE: 7-day Theta changes to 1 day Theta if
days to expiration is 7 or less (see time decay). Theta/Time .
Vega (kappa, omega, tau): Sensitivity of option value to change
in volatility. Vega indicates an absolute change in option value
for a one percent change in volatility. For example, a Vega of
.090 indicates an absolute change in the option's theoretical
value will increase by .090 if the volatility percentage is
increased by 1.0 or decreased by .090 if the volatility
percentage is decreased by 1.0. Results may not be exact due to
rounding. Vega/Volatility.
WEEK TEN
Lecture Notes: THE GREEKS & AMERICAN OPTION
PRICING
This week’s lesson gives you a bit more time to study the
“Greeks” and how they are used. The lesson continues to detail
the material from last week’s lesson in terms of the various
pricing models and elements thereof.

6. Before moving any further into the material, please spend
plenty of time on “The Greeks” (aka option sensitivities). Here
are my lecture notes on this important topic, along with some
info on hedging:
The Greeks1
The Greeks are a collection of statistical values (expressed as
percentages) that give the
investor a better overall view of how a stock has been
performing. These statistical values can be helpful in deciding
what options strategies are best to use. The investor should
remember that statistics show trends based on past performance.
It is not guaranteed that the future performance of the stock will
behave according to the historical numbers. These trends can
change drastically based on new stock performance.
The Greeks are vital tools in risk management. Each Greek
(with the exception of theta) represents a specific measure
of riskin owning an option, and option portfolios can be
adjusted accordingly ("hedged") to achieve a desired exposure;
see for example Delta hedging.
As a result, a desirable property of a modelof a financial
marketis that it allows for easy computationof the Greeks.
The Greeks in the Black-Scholes modelare very easy to
calculate and this is one reason for the model's continued
popularity in the market.
Beta: a measure of how closely the movement of an individual
stock tracks the movement of the entire stock market.
Gamma: Sensitivity of Delta to unit change in the underlying.
Gamma indicates an absolute change in delta. For example, a
Gamma change of 0.150 indicates the delta will increase by
0.150 if the underlying price increases or decreases by 1.0.
Results may not be exact due to rounding.
Lambda: A measure of leverage. The expected percent change in
the value of an option for a 1 percent change in the value of the
underlying product. Lambda/Leverage.
Rho: Sensitivity of option value to change in interest rate. Rho
indicates the absolute change in option value for a one percent

7. change in the interest rate. For example, a Rho of .060 indicates
the option's theoretical value will increase by .060 if the
interest rate is decreased by 1.0. Results may not be exact due
to rounding. Rho/Rate.
Theta: Sensitivity of option value to change in time. Theta
indicates an absolute change in the option value for a 'one unit'
reduction in time to expiration. The Option Calculator assumes
'one
unit' of time is 7 days. For example, a theta of -250 indicates
the option's theoretical value will change by -.250 if the days to
expiration is reduced by 7. Results may not be exact due to
rounding. NOTE: 7-day Theta changes to 1 day Theta if days to
expiration is 7 or less (see time decay). Theta/Time .
Vega (kappa, omega, tau): Sensitivity of option value to change
in volatility. Vega indicates an absolute change in option value
for a one percent change in volatility. For example, a Vega of
.090 indicates an absolute change in the option's theoretical
value will increase by .090 if the volatility percentage is
increased by 1.0 or decreased by .090 if the volatility
percentage is decreased by 1.0. Results may not be exact due to
rounding. Vega/Volatility.
Because BS OPM isolates the effects of each variable’s effect
on pricing, it is said that these isolated, independent effects
measure the sensitivity of the options value to changes in the
underlying variables.
Volatility
· Important factor in deciding what type of options to buy
or sell.
· Shows the range that a stock’s price has fluctuated in a
certain period.
· Volatility is denoted as the annualized standard deviation
of a stock’s daily price change.
Volatility Measures
· Statistical Volatility - a measure of actual asset price
changes over a specific period of time ( a look - back)

8. · Implied Volatility - a measure of how much the "market
place" expects asset price to move, for an option price. That is,
the volatility that the market itself is implying ( a look- ahead).
Implied Volatilities
· The implied volatility calculated from a call option should
be the same as that calculated from a put option when both have
the same strike price and maturity.
More on Delta
· Delta (D) describes how sensitive the option value is to
changes in the underlying stock price.
Change in option price = Delta
Change in stock price
More on Gamma
· Gamma (G) is the rate of change of delta (D) with respect
to the price of the underlying asset.
· For example, a Gamma change of 0.150 indicates the delta
will increase by 0.150 if the underlying price increases or
decreases by 1.0.
Change in Delta = Gamma
Change in stock price
· Gamma can be either positive or negative
· Gamma is the only Greek that does not measure the
sensitivity of an option to one of the underlying assets. – it
measures changes to its Greek brother – Delta, as a result of
changes to the stock price.
More on Theta
· Theta (Q) of a derivative is the rate of change of the
value with respect to the passage of time.
· Or sensitivity of option value to change in time

9. Change in Option Price = THETA Change in time to
Expiration
· If time is measured in years and value in dollars, then
a theta value of –10 means that as time to option expiration
declines by .1 years, option value falls by $1.
· AKA Time decay:
o A term used to describe how the theoretical value of an
option "erodes" or reduces with the passage of time.
More on Vega
· Vega (n) is the rate of change of the value of a derivatives
portfolio with respect to volatility
· For example:
o a Vega of .090 indicates an absolute change in the option's
theoretical value will increase by .090 if the volatility
percentage is increased by 1.0 or decreased by
.090 if the volatility percentage is decreased by 1.0.
Change in Option Price = Vega
Change in volatility
· Vega proves to us that the more volatile the underlying
stock, the more volatile the option price.
· Vega is always a positive number.
More on Rho:
· Rho is the rate of change of the value of a derivative with
respect to the interest rate
· For example:
a Rho of .060 indicates the option's theoretical value will
increase by .060 if the interest rate is decreased by 1.0.
Change in option price = RHO Change in interest rate
· Rho for calls is always positive
· Rho for puts is always negative
· A Rho of 25 means that a 1% increase in the interest rate
would:

10. o Increase the value of a call by $.25
o Decrease the value of a put by $.25
Corporate Use Of Derivatives For Hedging
January 4, 2005 | By David Harper, (Contributing Editor -
Investopedia Advisor)
If you are considering a stock investment and you read that the
company uses derivatives to hedge some risk, should you be
concerned or reassured? Warren Buffett's stand is famous: he
has attacked all derivatives, saying he and his company "view
them as time bombs, both for the parties that deal in them and
the economic system" (2003 Berkshire Hathaway Annual
Report). On the other hand, the trading volume of derivatives
has escalated rapidly, and non-financial companies continue to
purchase and trade them in ever-greater numbers. Consider
the Chicago Mercantile Exchange,which is the largest exchange
for futures contractsin the United States. As of November
2004, the average daily volume of futures contracts reached 3.2
million, up a stunning 40% from the previous year. In the same
month, foreign-exchange futures set a new record for single-day
volume, reaching more than half-a-million contracts, with a
notional value of over $72 billion.
To help you evaluate a company's use of derivatives for
hedgingrisk, we'll look at the three most common ways to
use derivativesfor hedging.
Foreign-Exchange Risks
One of the more common corporate uses of derivatives is for
hedging foreign-currency risk, or
foreign-exchange risk, which is the risk that a change in
currency exchange rates adversely impacts
business results.
Let's consider an example of foreign-currency risk with ACME
Corporation, a hypothetical U.S.- based company that sells
widgets in Germany. During the year, ACME Corp sells 100

11. widgets, each priced at 10 euros. Therefore, our constant
assumption is that ACME sells 1,000 euros worth of widgets:
When the dollar-per-euro exchange rate increases from $1.33 to
$1.50 to $1.75, it takes more dollars to buy one euro, or one
euro translates into more dollars, meaning the dollar is
depreciating or weakening. As the dollar depreciates, the same
number of widgets sold translates into greater sales in dollar
terms. This demonstrates how a weakening dollar is not all bad:
it can boost export sales of U.S. companies. (Alternatively,
ACME could reduce its prices abroad, which, because of the
depreciating dollar, would not hurt dollar sales; this is another
approach available to a U.S. exporter when the dollar is
depreciating.)
The above example illustrates the "good news" event that can
occur when the dollar depreciates, but a "bad news" event
happens if the dollar appreciates and export sales end up being
less. In the above example, we made a couple of very important
simplifying assumptions that affect whether the dollar
depreciation is a good or bad event:
(1) We assumed that ACME Corp manufactures its product in
the U.S. and therefore incurs its inventory or production costs
in dollars. If instead ACME manufactured its German widgets in
Germany, production costs would be incurred in euros. So even
if dollar sales increase due to depreciation in the dollar,
production costs would go up too! This effect on both sales and
costs is called a natural hedge: the economics of the business
provide their own hedge mechanism. In such a case, the higher
export sales (resulting when the euro is translated into dollars)
are likely to be mitigated by higher production costs.
(2) We also assumed that all other things are equal, and often
they are not. For example, we ignored
any secondary effects of inflation and whether ACME can
adjust its prices.
Even after natural hedges and secondary effects, most

12. multinational corporations are exposed to some form of foreign-
currency risk.
Now let's illustrate a simple hedge that a company like ACME
might use. To minimize the effects of any USD/EUR exchange
rates, ACME purchases 800 foreign-exchange futures contracts
against the USD/EUR exchange rate. The value of the futures
contracts will not, in practice, correspond exactly on a 1:1 basis
with a change in the current exchange rate (that is, the futures
rate won't change exactly with the spot rate), but we will
assume it does anyway. Each futures contract has a value equal
to the "gain" above the $1.33 USD/EUR rate. (Only because
ACME took this side of the futures position, somebody - the
counter-party - will take the opposite position):
In this example, the futures contract is a separate transaction;
but it is designed to have an inverse relationship with the
currency exchange impact, so it is a decent hedge. Of course,
it's not a free lunch: if the dollar were to weaken instead, then
the increased export sales are mitigated (partially offset) by
losses on the futures contracts.
Hedging Interest-Rate Risk
Companies can hedge interest-rate riskin various ways.
Consider a company that expects to sell a
division in one year and at that time to receive a cash windfall
that it wants to "park" in a good risk- free investment. If the
company strongly believes that interest rates will drop between
now and then,
it could purchase (or 'take a long position on') a
Treasuryfutures contract. The company is effectively locking in
the future interest rate.
Here is a different example of a perfect interest-rate hedge used
by Johnson Controls, as noted in its
2004 annual report:
Fair Value Hedges - The Company [JCI] had two interest rate
swaps outstanding at September 30,

13. 2004 designated as a hedge of the fair value of a portion of
fixed-rate bonds…The change in fair value of the swaps exactly
offsets the change in fair value of the hedged debt, with no net
impact on earnings. (JCI 10K, 11/30/04 Notes to Financial
Statements)
Source: www.10kwizard.com.
Johnson Controls is using an interest rate swap.Before it
entered into the swap, it was paying a variable interest rateon
some of its bonds. (For example, a common arrangement would
be to pay LIBORplus something and to reset the rate every six
months). We can illustrate these variable rate payments with a
down-bar chart:
Now let's look at the impact of the swap, illustrated below. The
swap requires JCI to pay a fixed rate of interest while receiving
floating-rate payments. The received floating-rate payments
(shown in the upper half of the chart below) are used to pay the
pre-existing floating-rate debt.
JCI is then left only with the floating-rate debt, and has
therefore managed to convert a variable-rate obligation into a
fixed-rate obligation with the addition of a derivative. And
again, note the annual report implies JCI has a "perfect hedge":
The variable-rate coupons that JCI received exactly
compensates for the company's variable-rate obligations.
Commodity or Product Input Hedge
Companies that depend heavily on raw-material inputs or
commodities are sensitive, sometimes
significantly, to the price change of the inputs. Airlines, for
example, consume lots of jet fuel. Historically, most airlines
have given a great deal of consideration to hedging against
crude-oil price increases - although at the start of 2004 one
major airline mistakenly settled (eliminating) all of its crude-oil
hedges: a costly decision ahead of the surge in oil prices.

14. Monsanto (ticker: MON) produces agricultural products,
herbicides and biotech-related products. It uses futures
contracts to hedge against the price increase of soybean and
corn inventory:
Changes in Commodity Prices: Monsanto uses futures contracts
to protect itself against commodity price increases… these
contracts hedge the committed or future purchases of, and the
carrying value of payables to growers for soybean and corn
inventories. A 10 percent decrease in the prices would have a
negative effect on the fair value of those futures of $10 million
for soybeans and $5 million for corn. We also use natural-gas
swaps to manage energy input costs. A 10 percent decrease in
price of gas would have a negative effect on the fair value of
the swaps of $1 million. (Monsanto 10K,11/04/04 Notes to
Financial Statements)
Source: www.10kwizard.com,
Conclusion
We have reviewed three of the most popular types of corporate
hedging with derivatives. There are
many other derivative uses, and new types are being invented.
For example, companies can hedge their weather risk to
compensate them for extra cost of an unexpectedly hot or cold
season. The derivatives we have reviewed are not generally
speculative for the company. They help to protect the company
from unanticipated events: adverse foreign-exchange or
interest-rate movements and unexpected increases in input
costs. The investor on the other side of the derivative
transaction is the speculator. However, in no case are these
derivatives free. Even if, for example, the company is surprised
with a good-news event like a favorable interest-rate move, the
company (because it had to pay for the derivatives) receives
less on a net basis than it would have without the hedge.
By David Harper, (Contributing Editor - Investopedia Advisor)
In addition to being a writer for Investopedia, David Harper,
CFA, FRM, is the founder of The Bionic Turtle, a set of study

15. aids designed to help finance professionals prepare for
certification exams. He is a contributing editor to the
Investopedia Advisorand Principal of investor alternatives, a
firm that conducts quantitative research, consulting (e.g.,
derivatives valuation), litigation support and financial
education.
1Downloaded on 01/01/207
from http://en.wikipedia.org/wiki/Binomial_options_pricing_mo
del.