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Pricing analysis
ignoring dividends

Dividend impact
analysis

### On-Line Options Pricing & Probability Calculators

 Option Pricing Calculators

 Employee stock options 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. On-line option pricing 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. 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.

 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: 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. 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). Hours per trading day, and trading days per annum: Used in conjunction with the discrete monitoring adjustment. 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. 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. 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.

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. compare the impact of different ex-dividend dates on American option prices with their impact on European option prices.

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.

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.