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.
|
frakjo
week_9_lecture_notes.htm
0
