Introduction

We discovered equations that disprove the Gaussian model of price movements.

Millions of Trades Daily Disprove the Old Model

Millions of options trade on the Chicago Board of Options Exchange (CBOE) every business day. [1]  Investors, traders, and institutions offer bid-ask prices for options on thousands of stocks and funds at dozens of strike prices for several expiration dates.  Those investors, traders, and institutions buy and sell options using a very different model.  We show that our equations implement that model in the accompanying spreadsheets.

Apple Inc. Example

On 16-May, 2011, near the close, Apple shares traded at $333.70. (Symbol:  AAPL)

The table shows the bid prices for Apple July calls, the strike prices, and the bid prices for Apple July puts which expire at the market’s close on 15-Jul, 2011.

Intrinsic Value

An option’s intrinsic value equals its value if exercised this instant.

 Apple Inc. traded at $333.70 on 16-May, 2011. The intrinsic value of a $300 call equaled $33.70. Bids of $37.55 for a $300 call may or may not offer a seller a nice profit – more on this later.

Time Value, or Time Premium

The premium of an option over its intrinsic value results from the time remaining until expiration. The time premium decreases as the days-until-expiration decrease, proportional to the square-root of the days remaining. It may also result from the optimism of buyers “taking a chance” by buying way-out-of-the-money call options.

The time premium, hereinafter, the “time value,” also depends directly on the volatility of the underlying asset.

Volatility

A measure of the amount by which an underlying [equity] is expected to fluctuate in a given period of time. Volatility is a primary determinant in the valuation of options premiums and time value. There are two basic kinds of volatility, implied and historical (statistical). Implied volatility is calculated by using an option pricing model…[2]

Each equity has a unique implied volatility which represents The Market’s assessment of the prospects for that equity.  (Hereinafter, the term, volatility, means implied volatility.)  The Market takes everything into account, competition, technology, market saturation, profit margins, book value, extraordinary expenses and earnings, productivity, union relations, market dominance, pending legislation, management competence, workforce, threats of war and terrorist attacks, natural disasters, etc., and rolls it all up into the one measure which we call volatility.

Volatility can and does change due to changes in the market, consumer demand, competition, legislation, etc.  In addition, if The Market itself seems more uncertain, then we expect higher volatilities across the board and vice versa.  Volatility equates to uncertainty about the equity’s future worth.

Value Equations for American-Style Options

We discovered closed form equations which calculate the fair market value of American-style options.  Our equations use the option’s type, the option’s Strike Price, the option’s time-until-expiration, T, the underlying equity’s Spot Price, and the underlying equity’s volatility, V, to determine the option’s value.  Because of put-call symmetry, the equation for puts differs from that for calls only by a sign change.

The equations require one variable, Volatility, to determine the option’s time value.

Closed-Form

The closed form of our equations make implementation a breeze.  The attached Excel^© spreadsheets calculated the time values in a jiffy.

Known Volatility

If we know the volatility of the underlying equity, then we may use the Option Value Equations to compute the option’s value from the option type, Spot Price, Spot Date, Strike Price and Expiration Date.

No Volatility “Smile”

If we do not know the implied volatility of the underlying equity, then we use the Option Value Equations to discover it. Because our probability distribution function (PDF) doesn’t have “skinny tails” we have a single, best value for implied volatility over a wide range of strike prices.

No “Greeks”

Our equations, one for puts and one for calls, use the spot and strike prices, the days until expiration, and the volatility to determine the options’ values. We need no other “constants” to determine the behavior of the options.

The equations’ first and second derivatives also have closed forms.

Undetermined Volatility – Multiple Strike Prices

We find the volatility which best approximates the bid price for each strike price using the method of least squares:  We determined a best value of 10.36 for Apple Inc.’s Implied Volatility on 16-May, 2011.

The above graph uses an implied volatility equal to 0.01.

The above graph uses an implied volatility equal to 2.0.

The above graph uses an implied volatility equal to 4.0.

The above graph uses an implied volatility equal to 6.0.

The above graph uses an implied volatility equal to 8.0.

The above graph uses an implied volatility equal to 10.0.

The above graphic shows the curves with the least sum of squared errors between the time values as bid and those calculated by our formula, by using an implied volatility of 10.36.

Opportunities?

The graphic above suggests over-bidding for deep-in-the-money calls, at strikes from $270 through $300. It also suggests over-bidding for way-out-of-the-money calls, at strikes from $380 through $405.

As the days-until-expiration decrease, we expect those differences to diminish markedly.

Normalized Volatility

The figure below shows the normalized put and call volatilities on 9/28/2010 for Vanguard Total Index fund (VTI), IBM, General Electric (GE), British Petroleum (BP), Caterpillar (CAT), Pan American Silver (PAAS), Apple Inc., Ford (F), US Steel (X), Amylin Pharmaceuticals (AMLN), Wynn Resorts (WYNN), UAL Corporation (UAUA), DryShips Inc. (DRYS), Avis Budget Group, Inc. (CAR), InterOil Corporation (IOC), and Dendreon Corporation (DNDN).

The figure shows that normalized volatility differed noticeably among different equities on 9/28/2010.  It also shows put-call symmetry:  the normalized volatility for calls approximately equals that for puts.

A Prediction?

Apple’s normalized volatility on 16-May, 2011, equaled 3.11 percent. On 28-Sep, 2010, Apple had an implied volatility near 4 percent. Did that difference presage the increase in Apple’s spot price, from $288 to $337?

Second-Order Effects

Interest rates and dividends affect our equations very little, in large part because our PDF doesn’t have “skinny” tails. Further, even though all equity prices have a lower bound of zero, we see little effect on the values our equations deliver for options on lower-priced equities.

Summary

Past mathematical models and equations assumed that equity prices walk away from their Spot Prices randomly, in “random walks,” which in turn led to pricing models based on Gaussian, log-normal, or similar probability distribution functions.  In the real world of buying and selling equities and options those models don’t compute.

We discovered equations that calculate the value of American-style puts and calls in simple, closed form.  Our new equations compute fair market option values from the equity’s spot price and volatility, the option’s type, its time-until-expiration, and its strike price.  If the underlying equity has an undetermined volatility, then we use our equations to determine its volatility from the prices of its options.

Our new equations materially advance the state of the art:

·         Simple to understand

·         Easy to use

·         Accurate results

·         Widespread applicability

Because our option value equations accurately portray the real world, institutions and investors can use it for:

·         Equity allocation

·         Volatility hedging

·         Risk assessment

·         Accurately forecasting probabilities of profit and losses for option strategies such as bear and bull spreads, calendar spreads, collars, and straddles.

·         Arbitrage

·         Portfolio rebalancing

Real-World Validation

Investors, traders, and institutions buy and sell options on thousands of equities at a dozens of strike prices for several expiration dates.  Millions of times each trading day, those transactions validate our option value equations.

Purchasing/Licensing

Direct further, serious inquiries to burkecaltech@cox.net.