Option Pricing Model

Beta: a measure of how closely the movement of an individual stock tracks the movement of the entire stock market.

Gamma: Sensitivity of Delta to unit change in the underlying. Gamma indicates an absolute change in delta. For example, a Gamma change of 0.150 indicates the delta will increase by 0.150 if the underlying price increases or decreases by 1.0. Results may not be exact due to rounding.

Lambda: A measure of leverage. The expected percent change in the value of an option for a 1 percent change in the value of the underlying product. Lambda/Leverage.

Rho: Sensitivity of option value to change in interest rate. Rho indicates the absolute change in option value for a one percent change in the interest rate. For example, a Rho of .060 indicates the option's theoretical value will increase by .060 if the interest rate is decreased by 1.0. Results may not be exact due to rounding. Rho/Rate.

Theta: Sensitivity of option value to change in time. Theta indicates an absolute change in the option value for a 'one unit' reduction in time to expiration. The Option Calculator assumes 'one

unit' of time is 7 days. For example, a theta of -250 indicates the option's theoretical value will change by -.250 if the days to expiration is reduced by 7. Results may not be exact due to rounding. NOTE: 7-day Theta changes to 1 day Theta if days to expiration is 7 or less (see time decay). Theta/Time .

Vega (kappa, omega, tau): Sensitivity of option value to change in volatility. Vega indicates an absolute change in option value for a one percent change in volatility. For example, a Vega of .090 indicates an absolute change in the option's theoretical value will increase by .090 if the volatility percentage is increased by 1.0 or decreased by .090 if the volatility percentage is decreased by 1.0. Results may not be exact due to rounding. Vega/Volatility.

Because BS OPM isolates the effects of each variable’s effect  on pricing, it is said that these isolated, independent effects measure the sensitivity of the options value to changes in the underlying variables.

Volatility

·       Important factor in deciding what type of options to buy or sell. ·       Shows the range that a stock’s price has fluctuated in a certain period. ·       Volatility is denoted as the annualized standard deviation of a stock’s daily price change.

Volatility Measures

·       Statistical Volatility - a measure of actual asset price changes over a specific period of time ( a look - back) ·       Implied Volatility - a measure of how much the "market place" expects asset price to move, for an option price. That is, the volatility that the market itself is implying ( a look- ahead).

Implied Volatilities

·       The implied volatility calculated from a call option should be the same as that calculated from a put option when both have the same strike price and maturity.

More on Delta

·       Delta (D) describes how sensitive the option value is to changes in the underlying stock price.

Change in option price = Delta Change in stock price More on Gamma

·       Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset.

·       For example, a Gamma change of 0.150 indicates the delta will increase by 0.150 if the underlying price increases or decreases by 1.0.

Change in Delta          = Gamma Change in stock price

·       Gamma can be either positive or negative ·       Gamma is the only Greek that does not measure the sensitivity of an option to one of the underlying assets. – it measures changes to its Greek brother – Delta, as a result of changes to the stock price.

More on Theta

·       Theta (Q) of a derivative  is the rate of change of the value with respect to the passage of time. ·       Or sensitivity of option value to change in time

Change in Option Price          = THETA Change in time to Expiration

·       If time is measured in years and value in dollars, then a theta value of –10 means that as time to option expiration declines by .1 years, option value falls by $1. ·       AKA Time decay: o  A term used to describe how the theoretical value of an option "erodes" or reduces with the passage of time.

More on Vega

·       Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility ·       For example: o  a Vega of .090 indicates an absolute change in the option's theoretical value will increase by .090 if the volatility percentage is increased by 1.0 or decreased by .090 if the volatility percentage is decreased by 1.0.

Change in Option Price = Vega Change in volatility

·       Vega proves to us that the more volatile  the underlying stock, the more volatile the option price. ·       Vega is always a positive number.

More on Rho:

·       Rho is the rate of change of the value of a derivative with respect to the interest rate ·       For example:

a Rho of .060 indicates the option's theoretical value will increase by .060 if the interest rate is decreased by 1.0.

Change in option price = RHO Change in interest rate

·       Rho for calls is always positive ·       Rho for puts is always negative ·       A Rho of 25 means that a 1%  increase in the interest rate would: o  Increase the value of a call by $.25 o  Decrease the value of a put by $.25

business results.

Let's consider an example of foreign-currency risk with ACME Corporation, a hypothetical U.S.- based company that sells widgets in Germany. During the year, ACME Corp sells 100 widgets, each priced at 10 euros. Therefore, our constant assumption is that ACME sells 1,000 euros worth of widgets:

When the dollar-per-euro exchange rate increases from $1.33 to $1.50 to $1.75, it takes more dollars to buy one euro, or one euro translates into more dollars, meaning the dollar is depreciating or weakening. As the dollar depreciates, the same number of widgets sold translates into greater sales in dollar terms. This demonstrates how a weakening dollar is not all bad: it can boost export sales of U.S. companies. (Alternatively, ACME could reduce its prices abroad, which, because of the depreciating dollar, would not hurt dollar sales; this is another approach available to a U.S. exporter when the dollar is depreciating.)

The above example illustrates the "good news" event that can occur when the dollar depreciates, but a "bad news" event happens if the dollar appreciates and export sales end up being less. In the above example, we made a couple of very important simplifying assumptions that affect whether the dollar depreciation is a good or bad event:

(1) We assumed that ACME Corp manufactures its product in the U.S. and therefore incurs its inventory or production costs in dollars. If instead ACME manufactured its German widgets in Germany, production costs would be incurred in euros. So even if dollar sales increase due to depreciation in the dollar, production costs would go up too! This effect on both sales and costs is called a natural hedge: the economics of the business provide their own hedge mechanism. In such a case, the higher export sales (resulting when the euro is translated into dollars) are likely to be mitigated by higher production costs.

(2) We also assumed that all other things are equal, and often they are not. For example, we ignored

any secondary effects of inflation and whether ACME can adjust its prices.

Even after natural hedges and secondary effects, most multinational corporations are exposed to some form of foreign-currency risk.

Here is a different example of a perfect interest-rate hedge used by Johnson Controls, as noted in its 2004 annual report:

Fair Value Hedges - The Company [JCI] had two interest rate swaps outstanding at September 30,

Now let's look at the impact of the swap, illustrated below. The swap requires JCI to pay a fixed rate of interest while receiving floating-rate payments. The received floating-rate payments (shown in the upper half of the chart below) are used to pay the pre-existing floating-rate debt.

JCI is then left only with the floating-rate debt, and has therefore managed to convert a variable-rate obligation into a fixed-rate obligation with the addition of a derivative. And again, note the annual report implies JCI has a "perfect hedge": The variable-rate coupons that JCI received exactly compensates for the company's variable-rate obligations.

Commodity or Product Input Hedge Companies that depend heavily on raw-material inputs or commodities are sensitive, sometimes significantly, to the price change of the inputs. Airlines, for example, consume lots of jet fuel. Historically, most airlines have given a great deal of consideration to hedging against crude-oil price increases - although at the start of 2004 one major airline mistakenly settled (eliminating) all of its crude-oil hedges: a costly decision ahead of the surge in oil prices.

Monsanto (ticker: MON) produces agricultural products, herbicides and biotech-related products. It uses futures contracts to hedge against the price increase of soybean and corn inventory:

Conclusion We have reviewed three of the most popular types of corporate hedging with derivatives. There are many other derivative uses, and new types are being invented. For example, companies can hedge their weather risk to compensate them for extra cost of an unexpectedly hot or cold season. The derivatives we have reviewed are not generally speculative for the company. They help to protect the company from unanticipated events: adverse foreign-exchange or interest-rate movements and unexpected increases in input costs. The investor on the other side of the derivative transaction is the speculator. However, in no case are these derivatives free. Even if, for example, the company is surprised with a good-news event like a favorable interest-rate move, the company (because it had to pay for the derivatives) receives less on a net basis than it would have without the hedge.

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