ARTWare® Exotic Options:

 with Maxima® and Mathematica®

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A Maxima®  or Mathematica® Notebook looks like a document, but its "live".  As with most "symbolic manipulators" (see here), equations can be entered, and the system "understands" the equation in symbolic terms.  The symbolic equations can be used for further symbolic analysis, or to obtain more traditional numerical results.

The image to the right shows the basic Exotic Option (XO) Mathematica® Notebook with the major Chapter headings.  The Notebook is composed of "cells".  These are not quite like the cells in a spreadsheet, but they do hold "executable" (or "evaluatable") expressions.  Here, the cells are "collapsed" and only show the heading lines.

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By double clicking the right "cell boundary" of the Black-Scholes Chapter, the cells for that Chapter are expanded.  Here, several Sections/Subsections are visible, each with collapsed/nested cells of its own.

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Expanding the Black-Scholes Pricing Section shows that it has cells with text, and cells with formulas. The cell marked "In[42]=" is a cell which sets the values of the variables p, k, r, d, v, and t to ".".  This simply "clears" their contents.  Next, the formula BSOpt[p,k,v,r,d,t] is evaluated (just put the cursor in the cell, and press Shift + Enter). Since its input variables have been "cleared", it returns the "core equation" that it represents in the cell marked "Out[46]=".  Importantly, it "really understands" this as a symbolic mathematical formula, rather than some collection of letters.  For example, one may now use Mathematica® to analytically differentiate this formula to obtain, for example, the option's Vega, Theta, etc.

Of course, if the variable are provided values, then the evaluation will return the "answer" of the calculation in numerical terms.  The image to the right shows the evaluation of 105 strike call option with the stated parameters: the option's premium is returned as 4.9823.

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Once it is possible to have symbolic representation, it is possible to evaluate the custom formulas in more complex ways.  One very useful feature is the graphing of securities and derivatives value and risk formulas directly to provide "scenario analysis".  The image to the right shows the call option formula from above evaluated for a range of underlying prices from inception to expiration.

A slightly more sophisticated result is possible by using differentiation to produce graphs for the key option risks (i.e. the "Greeks").  Each risk is generated in terms of important market parameters, as shown here for a short call option position.

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It is a straightforward matter to create formulas for many different types of options/contracts.  Here, several Greeks for a Knock-Out-Call option are graphed in terms of market parameters, as was done with the vanilla call option above.  The Barrier option's risk profiles are considerably more complex, and have some interesting properties that are immediately evident from this type of "macro risk/scenario analysis", but would be near impossible to see with strict numerical analysis.  For example, the Theta profile clearly shows that the position is "simultaneously long and short vol", depending on the underlying market.  Thus, under some conditions Theta will be a cost, while under other conditions it will be a revenue.  Another important "macro risk" observation is that that Vega has a complex structure, but for the most part it is only "material" for a narrow range of volatilities.  This occurs since low vol will have the vanilla option likely expiry worthless, while high vol will likely knock-out the call, and so again expires worthless.  However, "in-between" those vols Vega hedging will be very tricky as its sensitivity various greatly.

This type of easy formulaic analysis can also be used for model verification or even "model arbitrage" analysis.  The image to the right shows comparative valuation profiles for an Asian call option based on two different Asian option valuation formulas, and compared to the vanilla option case for completeness.

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Comparative analysis can also be used to test synthetic replication strategies, and their effectiveness.  The image to the right shows the pay-out profiles for a Chooser option and two vanilla options (a call and put forming a "static replicating strangle").  The graph suggests quite good agreement, and implies that the strangle is expected to be a reasonably good (profile match) hedge for this Chooser option.

Just as important as showing that a replication/hedge strategy works, is knowing when a hedge strategy does not work.  The image to the right shows a portfolio composed of long a spread option, and short the underlying call and put options.  Notice that this was created symbolically by "plotting the portfolio" (i.e. the plotting is for "-1 Call - 1  Put + 1 SpreadOption ]").  Notice that the hedge is excellent in 2 "quadrants" (i.e. the P&L if flat), but very poor in the other 2 quadrants (where it is loosing increasing amounts of money for varying market conditions).  This immediately shows the weakness of this strategy (namely that an option on a portfolio is not the same as a portfolio of options), and it also shows under which market conditions this strategy will be unacceptable.

This type of "symbolic" implementation of securities and derivatives valuation, risk, and position analysis is not meant to replace the standard  day-to-day "ticketing" or "position keeping" machinery.  Rather, it is a useful tool to provide fast analysis of models, structures, hedging strategies, and related matters.  It is also an excellent tool for creating presentations to clients, shareholders, regulators, or for training purposes.

Currently there are two Mathematica® Notebooks and a Maxima®  package available:

TG2 Exotic Options "Complementary" Edition. This is an abridged "complementary" version (also available as part of ARTSchool and TG2Books)

ARTWare Symbolic Analysis Edition. This is the "full" commercial version

ARTicles: Static Hedging & Structuring 101 - Using Maxima® : "live" notebook and calculator for basic vanilla & exotic options static replication and structuring, available Jun/06.

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**Pr/rO  is also included as part of the ARTPr/rO advisory service.