On-Line Options Pricing & Probability Calculators
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Option Pricing
Calculators |
 Employee stock
option (ESO) valuation: Standard Black-Scholes and lattice pricing
models cannot be
used to value ESOs due to
vesting requirements, the impact of staff turnover rates, and other ESO-specific factors
which are not a part of standard option pricing.
For tools which specifically handle
IFRS 2 and FASB 123R-compliant ESO valuation see
ESO valuation.
 Black-Scholes pricing
analysis -- Ignoring dividends: Lets you examine
graphically how changes in stock price, volatility, time to expiration
and interest rate affect the option price, time value, the derived
"Greeks" (delta, gamma, theta, vega, rho) and the
probability of the option closing in the money. For
simplicity, dividends are ignored so you just specify the time to
expiration in days rather than entering specific dates. See section below for
more
information, or Use
it now.
 Black-Scholes
pricing analysis -- Including dividends: Black Scholes
pricing and analysis of "Greeks" where a dividend is paid during the
life of the option. One dividend (an amount and an ex-dividend
date) can be specified. (The Excel add-in
available from this site will handle an unlimited number of dividends.) Use
it now.
 Binomial
tree graphical option calculator: Lets you calculate option prices
and view the binomial tree structure used in the calculation.
Either the original Cox, Ross & Rubinstein binomial tree can be
selected, or the equal probabilities tree. Both types of trees
normally produce very similar results. However the equal probabilities
tree has the advantage
over the C-R-R model of working correctly when the volatility is very
low and the interest rate very high. Both European and
American Exercise can be specified; dividends can be discrete or
a continuous yield. For American options the nodes
in the tree at which early exercise is assumed are highlighted. Use
the Cox, Ross & Rubinstein or Equal Probabilities
calculator now.
 Trinomial
tree graphical option calculator: Calculates option prices
using a trinomial tree and displays the tree used in the
calculation. Like the binomial model European and
American Exercise can be specified; dividends can be discrete or
a continuous yield and early exercise points are highlighted.
When a small number of tree steps is used the trinomial
model tends to give more accurate results than the binomial model.
As the number of steps increases the results from the binomial and
trinomial models (for vanilla options) rapidly converge. The
trinomial model (or adaptations of the trinomial model) is sometimes more stable and accurate than the binomial model for exotic
options (eg barrier options). Use it now.
 Barrier
option calculator using trinomial lattice: Calculates barrier
option prices, and hedge parameters, using a trinomial lattice, and
displays the tree structure used in the calculation. Key features
include American & European option pricing, dividends as continuous yield or
discrete payment, continuous or discrete monitoring of barrier, and
two methods of computation enhancement. Analytic prices, where
analytic formulas exist, are displayed for comparison.
See below for more information, or Use
it now.
 Black-Scholes/Binomial
convergence analysis: Display graphically the way in which options
priced under the binomial model converge with options priced under
Black-Scholes model as
the number of binomial steps increases. Whether the option is
out of the money, at the money, or in the money at the time of pricing
also has a significant impact on the way the two pricing models
converge and the calculator lets you examine how the strike price/spot
price relationship, as well as volatility, time to expiration and
interest rate affect the rate and 'shape' of convergence. Use it now.
 Dividend impact
analysis & American & European
option pricing comparison: Shows graphically how dividends paid during the
life of an option impact the price, and in particular the
sensitivity of the option price to different ex-dividend
dates. It also lets you compare American and European
pricing on the one graph for different option durations and
ex-dividend dates. See section below for more
information, or Use it now.
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Stock Price
Distribution & Probability Calculators |
 Stock price
lognormal distribution
analysis calculator: Lets you plot the lognormal distribution
curve of stock prices. You can change your assumptions for
starting price, volatility, number of days and expected growth rate in
the underlying stock and see how these changes affect the shape of the
distribution curve . The calculator will also indicate the
probabilities of prices being below or above a range of stock prices,
and will produce other statistics such as the expected
continuously
compounded annual return and standard deviations of prices and
expected returns. Use it now.
 Stock price
probability calculator: Computes the probability of a stock
price exceeding, or falling between, upper and lower boundary prices.
The results show both closing probabilities (ie at end of period) and
the probabilities of the boundary prices ever being exceeded (ie
the probability that the boundary prices will be exceeded at any time during the period).
It also calculates the probabilities of either of the boundary prices
being touched and of both boundaries being touched at any time. The calculator allows for both continuous monitoring of stock prices,
and for discrete monitoring (eg once per trading day). The number of
hours per trading day and number of trading days per year can be
varied by the user. The Hoadley
Finance Add-in for Excel lets you calculate these probabilities from
Excel spreadsheets. See section below for more
information, or Use it now.
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More about the Option Pricing
Calculators |
| Black-Scholes calculators |
You can use the on-line options
pricing analysis calculators to see, in tabular form and graphically, how
changing each of the Black-Scholes variables impacts the option price, time value
and the derived "Greeks".
You can also examine how changes in the Black-Scholes
variables affect the probability of the option being in the money (ITM) at
expiration. In calculating the probability of closing ITM a lognormal
distribution of stock prices with the stock earning a rate of return equal to the risk
free interest rate is assumed.
For simplicity, dividends are ignored in the first
pricing calculator and the calculator applies European-style
pricing. This provides an easy-to-use way to see how the Black-Scholes
pricing model works.
Dividends, however, are important when evaluating option
strategies. The second pricing calculator which
takes dividends into account.
Note that the pricing in both calculators is for European style options so don't use the model
to try to accurately price equity options, particularly those with an ex-dividend date very
close to expiration. If you want accurate prices for American options then
either use the on-line binomial or trinomial
pricing calculators, the Options
Strategy Evaluation Todel,
or the Finance Add-in for Excel.
| Barrier
option calculator using trinomial lattice |
This calculator calculates the
value of standard barrier options (up and out, down and out, up and in,
down and in) using a trinomial lattice. Whilst analytic formulas
exist for some European options (eg with continuous dividends) most
barrier options --eg American barrier options with discrete
dividends -- have no analytic
solution and thus numerical methods, such as the trinomial lattice, must
be used.
Where an analytic formula exists
the option value calculated using the analytic formula is displayed along with the trinomial result so you can see how close
the trinomial price is to the analytic price.
As well as calculating option
values and hedge parameters the calculator displays the first six levels
of the trinomial tree structure. The Nodes which fall just inside
the barrier are highlighted. Note that these barrier nodes will
normally be horizontal, except when there is a discrete dividend payment
during the life of the option, in which case there may be a step effect as
the tree is adjusted to take account of the changed underlying asset
price following the ex-dividend date.
The following notes explain some
of the inputs to and outputs from the calculator:
 | Dividend & days to
ex-dividend date: Dividends can be specified as either a
continuous annual yield (eg. 6.5%pa) or as one discrete payment.
For a continuous yield, days to ex-dividend should be left blank (or
zero). For a discrete payment the days to ex-dividend date must be
specified.
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| Discrete monitoring:
This specifies how often the underlying asset price is observed for
the purpose of deciding whether or not the barrier has been touched.
It can be specified as continuous, hourly per trading day, once per
trading day, weekly (every 7 days) or once per calendar month. The
discrete monitoring adjustment is by Broadie, Glasserman and Kou
(1997).
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| Hours per trading day, and
trading days per annum: Used in conjunction with the discrete
monitoring adjustment.
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| Calculation enhancement:
A major problem with using binomial or trinomial trees to price
barrier options is the fact that a very large number of tree steps is
required to achieve an accurate result due to the fact the tree nodes
will only rarely be aligned with the true barrier. Even small misalignments
produce large pricing errors. Unlike plain vanilla options,
convergence of option prices to 'true value' using trees is very
slow and prices exhibit a highly irregular saw-toothed pattern.
The number of tree steps required for reasonable accuracy and
stability can be five thousand or more.
A number of solutions exist to greatly improve accuracy without having to
use huge trees. Two of these are implemented in this calculator:
Align nodes with barrier (Boyle and Lau (1994)): an algorithm which adjusts the number of time steps (taking into account
time to expiration, volatility, underlying asset price and the
barrier) so that the nodes of the tree line up on or very close to the
barrier. The Boyle and Lau-based solution works well in many
situations. Unfortunately however, it also has its limitations. For
example if the barrier is very close to the underlying asset price the
number of steps required to align the nodes becomes prohibitively
large; if
there are discrete dividend payments which cause the nodes in the tree
to move then this adjustment is not effective.
Interpolate between nodes on barrier boundaries (Derman, Kani,
Ergener, and Bardhan (1995)): this approach basically calculates
prices for nodes which straddle the "specified barrier"
(using a "modified barrier" and an "effective
barrier") and interpolates between
the two prices taking into account how far each node is from the
"specified" barrier to arrive at a corrected price. This approach,
while generally slightly less accurate than the Boyle and Lau
adjustment when conditions are ideal, is more accurate when the
barrier is not "horizontal", when there are discrete dividends, and
when a volatility smile is taken into account. It also handles
double barriers. Like the Boyle and Lau method, interpolation does not
work well when the barrier is very close to the
underlying asset price.
The calculator lets you select: no correction (which is only there for
comparison as it generally produces useless results), node alignment,
interpolation or both alignment and interpolation (which
often, but by no means always, produces the best result). To see how well the correction
works you can compare the option value computed using the trinomial
lattice with the value computed using the analytic formula which is
displayed when available. The tree structure displayed is also useful
for seeing how the nodes line up with the barrier.
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| Number of tree (time)
steps: This has been arbitrarily limited to 100 steps. The
Hoadley Finance Add-in for Excel
allows an unlimited number of steps for greater accuracy. If the node alignment
correction is selected then the actual number of steps used will usually
be greater than the number specified. If the application of the node
alignment correction would cause the number of tree steps to exceed
250 then the node alignment correction will be disabled.
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| Hedge parameters
("Greeks"): Calculated using the trinomial tree, using
the specified number of steps and calculation enhancement method.
The "Greeks" are accurate unless the barrier is close to the
underlying asset price.
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Single barrier and double barrier
functions are also available in the
Hoadley Finance Add-in for Excel available from
this site.
| Dividend impact analysis for American &
European options |
This set of calculators lets you examine how a
discrete dividend paid
during the life of American options, and the timing of the dividend
payment,
impacts option prices by using the dividend impact analysis calculator.
This calculator will also let you compare the pricing
for American and European options on the one graph.
Three analyses are available. They are easier to use
than to explain, but basically they enable you to:
| see how option prices change as the value date gets
closer to expiration, for a range of ex-dividend dates, value dates
and expiration dates.
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| compare the impact of
different ex-dividend dates on American option prices with their impact on
European option prices.
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For European options, the timing of the ex-dividend date
has relatively little effect on the price of the option -- whether
the ex-dividend date is close to today's date, or close to the
expiration date generally has only a slight effect.
With American options, however, the timing of the
ex-dividend date, particularly for calls, can have a major effect on the option price.
You therefore need
to know this when looking at the attractiveness or otherwise of specific
deals.
How much difference does it make to an option
price if the ex-dividend date of the underlying stock is closer to the
expiration date of the option compared with an ex-dividend date closer to
today's date? The dividend impact analysis
calculator will graph put or call prices by time to expiration for
multiple ex-dividend dates on the one chart thereby proving answers to
this and other similarly important questions.
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More About Stock
Price Distribution & Probability Calculators |
Both the stock
price lognormal distribution analysis calculator, and the stock
price probability calculator are based on a rigorous implementation of
the mathematics underlying the Black-Scholes model: that stock prices
follow a stochastic process described by geometric brownian motion.
The calculators have been
thoroughly tested in two ways:
| Internal consistency.
For example, the standard deviation of the long-term continuously compounded annual
return of stock prices should equal the annual volatility if
the period is exactly one year -- and it does.
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| Monte Carlo simulation.
all probability numbers are
very close to those derived by simulation, and the larger the number
of simulation iterations the closer they become. You can check this out for
yourself by using the Finance Add-in
for Excel
which contains a Monte Carlo simulation component for calculating
probabilities.
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Testing the calculators against
other similar calculators is not so easy as there are very few
good ones around. And of the few that exist, most only provide simple "end of period"
probabilities (the calculators on this site also provide the more useful but more difficult
to calculate "at any time during the period" probabilities),
some others don't allow you to
specify key inputs (like expected return or frequency of observation of
underlying asset prices) and others which ostensibly look
okay produce garbage. And some of these you have to pay for...
Note
on expected return: A key input to the stock price distribution
and probability calculators is the expected return of the asset. This is the return that the asset is
expected to earn, on average, in one year -- capital growth, and
dividends. It's sometimes referred to as the arithmetic return. It may be regarded as being the
long term government bond rate
plus an equity risk premium, representing the return investors expect from their investment
at any point in time.
An example might be: 5% bond rate plus 4.5% equity risk premium
giving an expected return of 9.5% pa.
The rate is expressed as a
continuously compounded return. Thus, for example, if the expected return
on a stock is 10% with annual compounding then the
equivalent rate with continuous compounding would be approximately 9.53% (ie
ln(1+10%), where ln is the natural logarithm function).
Continuously compounded rates
are used to be consistent with the option pricing calculators which all
assume continuously compounded rates. From a practical point of view it
makes very little difference whether the rate is annually compounded,
continuously compounded, or something in between.
Note that the expected average
annual return is not the same as the geometric return. The expected return is required
as an input to the calculators whereas the geometrically
compounded return over the period concerned is one of the outputs. The
geometric return will always be
less than the arithmetic average of the annual returns (unless
volatility is zero). And
the higher the volatility the smaller geometric return will be, and the smaller the median asset price at the end of the
period will be. The mean, or average, asset price at the end of the period
on the other hand is simply a function of the expected average return and
is unaffected by volatility. You can see the
interrelationship between these variables by using the two calculators.
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