How to Price Eurodollar Options: The LLP Model, a Unique Option-Pricing Model, Now Can Be Used to Price Eurodollar Futures. It Has Advantages in Generating Useful Pricing Formulas for Trading and Hedging Purposes
Cretien, Paul D., Modern Trader
The large number of strike prices covered by options on Eurodollar futures makes them an excellent source of data to use in building analytical pricing models. A typical set of option prices is shown in "Call option price curve" (below). This curve is computed by a system that depends on a log-log parabolic regression equation, the LLP option-pricing model.
LLP price curves permit forecasting option prices based on the relationship between underlying asset prices and strike prices. For example, LLP computed prices for September 2007 Eurodollar call options on Jan. 20, March 30, and April 19, 2006, which are shown on "LLP price curves" (right). Predictive formulas for option prices would have enabled traders to forecast movements in September 2007 calls based on changes in the futures price throughout several days or weeks following each date.
Eurodollar options, like other exchange-traded options, are valued continuously by computer trading systems based on theoretical models such as Black-Scholes. In contrast, the LLP system assumes all of the variables necessary to calculate theoretical option values are included in market prices. The resulting price curves reflect that both the LLP and theoretical models use logarithms in their calculations and that LLP parabolic price curves are closely related to those generated by theoretical models. Although the LLP method is useful for analyzing any options, the emphasis here is on its value in creating Eurodollar price curves.
EURODOLLAR OPTIONS FORECAST
Eurodollar futures prices are equal to 100 less the 90-day interest rate for a given forward month. For example, on April 19, 2006, the forward rate on September 2007 futures contracts was 5.145% and the price was 94.855. Eurodollar futures option prices are stated as proportions of 1%. Because the rate is applied to a $1 million 90-day deposit, each basis point is valued at $25, or (0.01 x $1,000,000 x (90/360) x (1/100)). On April 19, the September 2007 call option with a strike price of 94.75 was market-priced at 0.4025, or $1,006.25.
The intrinsic value of the September 2007 call option was the futures price less the strike price (for a put option the intrinsic value is the strike price minus the futures price), or 94.855 less 94.75, which is 0.105 ($262.50). The spread between the call option and intrinsic value on April 19 was $743.75.
When the futures price equals the strike price there is no intrinsic value and the option price reflects only time premium. As the time to expiration approaches the time premium shrinks.
"LLP price curves" shows the March and April prices are close together and both predictive curves have the same time premium of 0.38%, while the time premium in January was 0.43%.
The time premium generated by LLP price equations assist option traders in seeing the speed at which option values are declining and also helps assess market sentiment relating to potential changes in futures prices.
The April 12 LLP price equation forecasts option prices close to market prices six days later on April 18 (see "LLP prices: Six days in April," below). Only two prices are different by more than one-basis point, and those differences are less than 1.5-basis points.
HOW THE MODEL WORKS
Calculation of an LLP price curve requires regression analysis in which the natural logarithm of the option price divided by the strike price, Ln(W/E), is related to the natural logarithm of the futures price divided by the strike price, Ln(S/E). The predicted curve is a parabola in natural logarithms, as shown on "Put options log chart" (page 44).
One feature of the LLP price curve is that the slope of the curve is available for each strike price. The slope is a hedge ratio that shows how much the option price will change for a one-unit change in the futures price and indicates how many options should be bought or sold to hedge changes in one futures contract. …