On-Line Options Pricing & Probability
Calculators
| 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), elasticity, 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.
| 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:
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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:
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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.
| 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:
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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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