Posted: June 20th, 2022

please se Option procing assigm

Please list your answer to each question, by question number.  I grade on quality not quantity.  If you use outside sources, you MUST properly reference them.  If you are not familiar with standardized referencing, please go to www.citationmachine.net for assistance.  My preference in style is APA.  Please pay attention to your grammar, format and spelling – points will be deducted for errors in these areas besides errors in content. 

 

Your assignment is to value two (2) call options, using two (2) different option-pricing calculators, and/ or pricing programs.  Therefore, your answer will include 4 valuations (2 for each expiration).  Some places to look for option pricing models are Google, Yahoo! Finance,

You must properly reference the sites or sources where you found your pricing models.

 

The two calls you are to value are:

1.    The August 2014 $50 Whole Foods call option ($50 is the strike price)

2.    The Jan 2015 $50 Whole Foods call option ($50 is the strike price)

 

Use a standard deviation of 20 and a risk free rate of 2%.

   

  1. Copy and paste your quantitative results into a word document and explain your results.  If there is a difference between the results that your two models arrive at, please discuss why you think this happened.  You MUST provide a properly referenced citation for where you obtained your model in the form of a footnote to your answer to this question.

 

  1. Compare the Aug 2014 call Greeks and the Jan 2015 call Greeks and explain why there are differences for EACH Greek (include Delta, Gamma, Theta, Vega & Rho).  No credit will be given if you just provide the definitions of the Greeks & I would prefer if you omitted basic definitions and references to the puts; you must apply the definitions to the situation and be specific. 

 

  1. Go to Yahoo! Finance and look up the actual market prices (premiums) for these two calls (please indicate the date and time of the quotes in your answer).  Explain why you believe the calculated price may not be the same as the actual market price. 

 

  1. Based upon your research, take a position in Whole Foods– either bullish or bearish and explain how you would act on your opinion in the market, using what you have leaweek_11_lecture_notes_new.htmrned in this class – what would you buy or sell and why; what might your potential profit and loss be (specific numbers are required); why is your choice the best choice for capitalizing on your opinion.  You can use actual share transactions and/or naked or covered derivatives of any month or strike; you are not limited to the calls that you are analyzing in this assignment.  The assumption is that you have no existing positions in Whole Foods stock or Whole Foods derivatives.   Doing nothing is not an acceptable answer. 

 

Please keep in mind that this is more of a critical thinking exercise than a quantitative exercise.  I will assess your answer to #1 & # 4 (the calculations) in a quantitative manor, but the rest of your assignment will be assessed on how well you present and explain your results.  If you use any outside sources, you MUST reference them properly – see www.citationmachine.net for referencing assistance. 

    
ATTACHED FILE(S)

 

 
 

 
 

Lecture noTES

FINANCIAL RISK MANAGEMENT

FIN 4486

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.
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 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 easy computationof 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.
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 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)
·       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
 
 
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:
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 contracts
in
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  hedging
risk, we’ll look at the three most common ways to use derivatives
for 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
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 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 risk
in 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  Treasury
futures 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,

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 rate
on some of its bonds. (For example, a common arrangement would be to pay LIBOR
plus
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.
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 aids designed to help finance professionals prepare for
certification exams. He is a contributing editor to the  Investopedia Advisor
and 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_model
.
** This article and more are available at Investopedia.com – Your Source for Investing
Education **

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Copyright © 1998 –
2014. All rights reserved worldwide.
 

 
 

 
 

Lecture noTES

FINANCIAL RISK MANAGEMENT

FIN 4486

wEEK Nine

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 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 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
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 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. 

 

Florida International University Online
Copyright © 1998 –
2014. All rights reserved worldwide.

Option Pricing Models:
The Black-Scholes-Merton Model aka Black – Scholes Option Pricing Model (BSOPM)

*

Important Concepts
The Black-Scholes-Merton option pricing model
The relationship of the model’s inputs to the option price
How to adjust the model to accommodate dividends and put options
The concepts of historical and implied volatility
Hedging an option position

*

The Black-Scholes-Merton Formula
Brownian motion and the works of Einstein, Bachelier, Wiener, Itô
Black, Scholes, Merton and the 1997 Nobel Prize
Recall the binomial model and the notion of a dynamic risk-free hedge in which no arbitrage opportunities are available.
The binomial model is in discrete time. As you decrease the length of each time step, it converges to continuous time.

*

Some Assumptions of the Model
Stock prices behave randomly and evolve according to a lognormal distribution.
The risk-free rate and volatility of the log return on the stock are constant throughout the option’s life
There are no taxes or transaction costs
The stock pays no dividends
The options are European

*

Background
Put and call prices are affected by
Price of underlying asset
Option’s exercise price
Length of time until expiration of option
Volatility of underlying asset
Risk-free interest rate
Cash flows such as dividends
Premiums can be derived from the above factors

*

Option Valuation
The value of an option is the present value of its intrinsic value at expiration. Unfortunately, there is no way to know this intrinsic value in advance.
Black & Scholes developed a formula to price call options
This most famous option pricing model is the often referred to as “Black-Scholes OPM”.

*
Note: There are many other OPMs in existence. These are mostly variations on the Black-Scholes model, and the Black-Scholes model is the most used.

The Concepts Underlying Black-Scholes
The option price and the stock price depend on the same underlying source of uncertainty
We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty
The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate

*

Option Valuation Variables
There are five variables in the Black-Scholes OPM (in order of importance):
Price of underlying security
Strike price
Annual volatility (standard deviation)
Time to expiration
Risk-free interest rate

*

Option Valuation Variables: Underlying Price
The current price of the underlying security is the most important variable.
For a call option, the higher the price of the underlying security, the higher the value of the call.
For a put option, the lower the price of the underlying security, the higher the value of the put.

*

Option Valuation Variables: Strike Price
The strike (exercise) price is fixed for the life of the option, but every underlying security has several strikes for each expiration month
For a call, the higher the strike price, the lower the value of the call.
For a put, the higher the strike price, the higher the value of the put.

*

Option Valuation Variables: Volatility
Volatility is measured as the annualized standard deviation of the returns on the underlying security.
All options increase in value as volatility increases.
This is due to the fact that options with higher volatility have a greater chance of expiring in-the-money.

*

Option Valuation Variables: Time to Expiration
The time to expiration is measured as the fraction of a year.
As with volatility, longer times to expiration increase the value of all options.
This is because there is a greater chance that the option will expire in-the-money with a longer time to expiration.

*

Option Valuation Variables: Risk-free Rate
The risk-free rate of interest is the least important of the variables.
It is used to discount the strike price
The risk-free rate, when it increases, effectively decreases the strike price. Therefore, when interest rates rise, call options increase in value and put options decrease in value.

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Implied Volatility
The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price
The is a one-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices

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Nature of Volatility
Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed
For this reason time is usually measured in “trading days” not calendar days when options are valued

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A Nobel Formula
The Black-Scholes-Merton model gives the correct formula for a European call under these assumptions.
The model is derived with complex mathematics but is easily understandable. The formula is

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Screen Shot of the Excel template for the BSOPM

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OPM & The Measurement of Portfolio Risk Exposure
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.

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The Greeks
Greeks are derivatives of the option price function :
Delta
Gamma
Theta
Vega
Rho
The Greeks are also called hedge parameters as they are often used in hedging operations by big financial institutions

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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

A
B
Slope = D
Stock price

Option
price

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Delta Application
Suppose that the delta of a call is .8944.
What does this mean????

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Delta Neutral
In other words, we want the delta to be zero.
Example:
Current stock price is $100
Call price (per opm) is $11.84
Delta = .8944
We must buy .8944 shares of stock for each option sole to produce a delta –neutral portfolio

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Delta Neutrality
Exists when small changes in the price of the stock does not affect the value of the portfolio.
However, this “neutrality” is dynamic, as the value of delta itself changes as the stock price changes.
This idea of neutrality can be extended to the other sensitivity measures.

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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

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Gamma Application
Can be either positive or negative
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.

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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
Change in Option Price = THETA
Change in time to Expiration

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Theta Application
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:
A term used to describe how the theoretical value of an option “erodes” or reduces with the passage of time.

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Vega
Vega (n) is the rate of change of the value of a derivatives portfolio with respect to 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.
Change in Option Price = Vega
Change in volatility

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Vega Application
Proves to us that the more volatile the underlying stock, the more volatile the option price.
Vega is always a positive number.

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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

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Rho Application
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:
Increase the value of a call by $.25
Decrease the value of a put by $.25

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