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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)
### Delivery
The program is a downloadable executable for MS Windows systems. Included is
a "Tutorial and Help" file, which also includes the License agreement
(Adobe Acrobat reader needed).
### License Pricing (OptionCity Calculator, Version 1.1)
- *
Please inquire about source code licenses.
Copyright © 2006, OptionCity.net |