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Calculator with a Smile chart expanded. For a full screenshot, click here.

Beyond the Black-Scholes' model. If you're in the options market, you need a good calculator. The OptionCity Calculator uses two advances from modern option pricing theory: stochastic volatility and stock price jumps. These two ingredients make for a good calculator. We'll explain why.

Volatility is not constant. If you've traded options, you know that volatility is ever-changing. (Click on the thumbnail image to see a long-run chart of the SPX volatility). For example, you've probably seen this effect: when the stock market falls, index option volatility tends to move up and vice-versa. It's sometimes called the "leverage effect" in academic circles. Yet, for option prices to follow the celebrated Black-Scholes' 1973 model, volatility must be a constant -- the leverage effect shouldn't exist. In a nutshell, there's a contradiction. That contradiction is a key reason why the Black-Scholes' model is often at odds with the marketplace.

Smiles and skews. In fact, not only does actual volatility jump around over time, but the Black-Scholes' implied volatility differs from strike to strike. This is the smile or skew effect. If option prices were determined by the Black-Scholes' model, there would be no smile or skew (It would be flat). In reality, you know that as many index options approach expiration, their smile charts become very steep. In other words, out-of-the-money puts trade at very high implied volatilities relative to the at-the-moneys.

For example, for the S&P500-type data in the screen shot above, with a current volatility of 20% per annum, the Black-Scholes price for a put striking at 1050 (with the index at 1200) and about 1 month to go is only \$0.16. Yet the calculator model price, which uses plausible assumptions, is over 8 times higher: \$1.29 with an implied volatility of about 27%. Most important, the calculator price is much closer to the market price. By comparing a similar market price with only the Black-Scholes price, a trader might assume that these puts were grossly over-priced in the market and engage in a systematic selling program. As the calculator shows, that might be a mistake, even though it could indeed prove to be a profitable trading strategy for quite a while. Why? Primarily, the occasional negative jumps (about once every 4 years in the screenshot), with the accompanying volatility surge, take back all your profits. At a minimum, the expected earnings over time are probably much lower than the hypothetical trader expected. This shows how the calculator can save money by reducing mistakes.

How the Calculator models work. The option pricing models built into the OptionCity Calculator handle these effects in a realistic way. First, they allow volatility to change; today's volatility is expected to be different tomorrow. Volatility is described by a random process with realistic features. That's the stochastic volatility part of the Calculator's models.

The correlation parameter. Our stochastic volatility models generate the leverage effect by using a "correlation" parameter. You can see the input slot at the end of the third input row in the screenshot. (The screenshot shows an earlier release). You enter a number between -1 and +1: in the screenshot, the entry is -0.65. The correlation describes the relationship between stock prices changes and volatility changes. A negative entry like this means that volatility tends to move up when stock price moves down and vice-versa -- paralleling what you usually see in the equity marketplace! The closer the entry is to -1.0, the stronger the effect. For some options, like currency options, there isn't much of a skew (the smile is fairly symmetrical). To model this in the calculator, you enter a 0 or very small correlation.

Very steep skews. What about the very steep skews (i.e., the relatively expensive prices) associated with the expiring, out-of-the money, stock index puts? (Like the S&P500 options). These skews are often explained by the demand of portfolio hedgers for protection against losses. That's valid as far as it goes, but do the hedgers pay too much? too little?

In the OptionCity Calculator, these skews are generated by two effects that work together. First, there's the correlation (leverage) effect that we just mentioned. The more negative the correlation, the steeper the skew. This makes sense: if you're a put buyer, you will pay more if you know that, if the market falls, then volatility will probably rise. After all, your puts will become valuable in two different ways. (1: they go in-the-money and 2: volatility is higher). To compensate, sellers have to charge more.

How the market moves. But, if you're a portfolio hedger, you will also think about how a market fall might happen. Hypothetically, if the fall was guaranteed to be fairly steady (the academics call this a 'continuous sample path'), then you can hedge by selling some stock on the way down. In reality, you know that the fall might instead be a sharp jump (the market opens down 10% some morning) that cannot be hedged by trading shares. Because of this possibility, you may pay even more for those puts. Especially if a 10% loss hurts more than a 10% gain helps. Again, the sellers must compensate for the possible jumps, too.

Good matches to the market. By including the possibility of (negative) stock price jumps, the skew charts generated by our calculator become even steeper and very good matches to the market can be achieved. For example, the screenshot above shows an OptionCity Calculator generated  smile that is very similar to the actual SPX smile at the time of this writing. (The at-the-money point is about 1200 in the chart and the options have about one month to go). So, back to the question: do the put buyers pay too much?  With the OptionCity Calculator, you can translate your own assumptions about jumps and volatility into fair prices for options. Like any calculator, it's a general purpose tool. You provide the input. We do the heavy arithmetic.

Try it. The models we use are fairly sophisticated, but no model is perfect. We hope you're convinced that ours are closer to reality than some older theories.  Interested? Try the free download below. The program comes with a Tutorial with more information.

### Key Benefits of the OptionCity Calculator

•  Flexible models with stochastic volatility and stock price jumps
•  Option prices with Greeks (sensitivity to parameters)
•  Realistic Smile charts
•  Fast evaluations
•  Self-validating results. (You validate calculations by selecting a different numerical method: Lattice, Series, or Monte Carlo)

How to get it