I have gathered the following list of several empirical models, just to briefly touch on the methods of pricing options. The most commonly quantified measures are : rate trend; volatility to explain divergence from the expected trend; and a time series (which after the fact analysis will reveal contains insignificant random fluctuations) ;

The five most significant original models (described in over-simplified terms) are:

Black, Scholes, and Merton (1973) ; (see below listing of factors)

Vasicek (1977); applies constant volatility; mortgage backed securities. (see CIR below)

Cox, Ingersoll, and Ross (1985) ; traditional; dynamics of interest rates can depend on multiple sources of uncertainty; requires a complex function of time so as to be reduced to numerically tractable. Interest rates are shown as a time series continuum.

Ho and Lee (1986); the "first" second generation model; accepts givens in initial term structures of interest rates and forward rate volatilities. Variables known as "state variables" (uncertainties of the economy) are very difficult to quantify. (see HJM below)

Heath, Jarrow, and Morton (1992) builds upon Ho and Lee; looks at cross-section of bond prices (bonds and interest at a given point of time) HJM allows a higher volatility parameter than the CIR model.

Hellwig's (1980) model was mentioned, but particulars are scarce as of today. Duffie & Kan (1996) offer another traditional, one-factor "affine" model, in the vein of Vasicek (1977) and CIR (1985).

M. Scholes & F. Black (Merton added his ideas later) created the 1973 Black-Scholes option-pricing model :

1) strike price - price offered

2) time to expiration -

3) current price - watch kurtosis (fat tail distribution)more likely to get a highly improbable result - which changes the value of the option.

4) volatility - may be variable - called conditional heteroskedasiticity

5) risk-free interest rate.

(This model likes lognormal distribution, but studies indicate that it has application in asymmetrical distribution curves)

Black and Scholes's (1973) implied volatilities tend to be systematically related to the option's exercise price and time to expiration.

Deterministic Volatility Function (DVF) Option Valuation Model of Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) says that the asset price and time to expiration are dependent upon the volatility of the underlying asset's return.

Using a sample of Standard and Poors index of 500 companies (S&P 500) options during the period June 1988 through December 1993, we evaluate the economic significance of the implied deterministic volatility function by examining the predictive and hedging performance of the DVF option valuation model. " (Dumas, Fleming & Whaley 1996:44)

Derman and Kani (1994) also approach pricing of "barrier options" , which they determine to be both more complex and cheaper than standard options.

The barrier options are capped European style calls, floored European style puts, binary up & in calls, and binary down & in puts.

GARCH, a type of ARCH model, is a method to estimate volatility using discrete-time. There is one unifying effective lattice algorithm to price American and European options under discrete-time GARCH processes. We may use the GARCH option pricing model of Duan (1991), (who applied equilibrium-type arguments) to determine the option prices. (Ritchken & Trevor 1999)

Insurance companies today use actuarial tables to determine probability from known past events. We may use probabilistic and actuarial considerations for pricing options to determine what premium is needed to secure the expected loss. Balance those against the VAR (value at risk), which is the defacto industry standard. VAR represents what will be exceeded as a loss in 1 of 100 events over a time period.

Options have a delta, ∂, a mathematical measure of an option's expected sensitivity to changes in the underlying currency. This is the traditional method of pricing. It is called "delta hedging".

As a minor side issue, the put + call pairing (swaps) will make a high kurtosis option worth more because on one side or the other of the distribution curve, the highly improbable will become more likely - thus making the option more valuable. IN and OUT are significant descriptive terms for swaps. Out-of-the-money means you bet some unlikely event, for example, that 40 rises to 50 (low probability); in-the-money means higher probability, for example, 40 rises to 43. The out-of-the-money puts and in-the-money-calls (paired in a swap) are frequently underpriced. A future drop of value is always more probable than a rise in value.

The VIX index (Chicago Board Options Exchange Market Volatility Index), reached 60.63 before dropping back to 43.11, tracking price swings of the S&P 100 index options. (WSJ 14 Oct. 98) This is the highest volatility index since the 1987 market crash. Because options are dependent on underlying value, measurement of factors which effect underlying values are significant to all the option valuation models.