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.
The points at which the straight-line slope intersects the intrinsic value of the option and the horizontal axis indicate the upper and lower breakeven futures prices for a “neutral hedge” in which the hedge ratio given by the LLP curve is used to hedge the underlying futures price.
In the table “LLP options price model” (see link at end of article) the input values and resulting predicted figures are illustrated by September 2007 put options on April 18, 2006. The strike prices, September 2007 futures price and market prices for corresponding put options are the values used to compute the LLP price curve.
Natural logarithms of S/E and W/E are used in the regression analysis to solve the following equations for each strike price:
Ln (W/E) = A + (B x Ln (S/E))
+ (C x Ln (S/E) ^ 2)
Option Price = (2.718282)^Ln (W/E) E
For example, when the strike price, E, is 95.50 and the futures price, S, is 94.875, the predicted option price is calculated as follows:
Ln (W/E) = -5.5995659
Ln (W/E) = - 4.85455
Option Price = (2.718282)^( - 4.85455) x 95.50 = 0.7442
The intrinsic value of each put option is the strike price less futures price (see “LLP options price model”). Intrinsic values for September 2007 puts on April 18, 2006 continue down to -1.875 when the futures price is 94.875 and the strike price is 93. Intrinsic values for these put options become negative between the strike prices that straddle the futures price. For puts, the hedge ratios are negative because the price curve slopes down to the right as the futures/strike price ratio increases. However, the meaning of the hedge ratio is the same as it is for call options.
Additional uses for the LLP pricing model include adding new data over a series of days or other periods to form a moving predictive option price curve analogous to a moving average, and forecasting option prices along the curve for futures prices where a strike price does not currently exist. In the case of futures or other assets with only a few option prices listed, data from several time periods may be combined to provide a basis for price predictions.
The time premium shown in “LLP options price model” compares the September 2007 puts and other options with the same expiration date. For example, market opinion on the prospects for the price variability of the Eurodollar, gold and soybean futures could be analyzed in terms of the time premiums of their option price curves.
In general, the longer the time to expiration, the higher the time premium. This measure tends to decline more rapidly with the approach of expiration. In comparing options on different assets, volatility of the underlying futures is a major factor. The more volatile, the more valuable the option; and this is reflected in time premiums.
Because the slope of the LLP curve is a straight line and the predicted option prices are on the curve, a hedge based on the slope should perform slightly better than would be expected based on the slope ratio. An example using put options on April 18 would be to use the slope indicated for strike price 95 and futures price of 94.875, a ratio of 0.5483. If the futures price changes by one-basis point, the option price, currently at 0.4225, should change by slightly more than half-of-a-basis point. A hedge would imply buying two put options to offset the movement of one long futures in the area of the curve close to the current price level and close to time to expiration.
The September 2007 futures price on April 19 was 94.855, -2-basis points, while the September 2007 put price for the 95 strike price was 0.4325, a gain of one-basis point. Holding two puts would have covered the price decline of one long futures contract.
To see the hedging capabilities over a longer period, it is possible to look backward to see the action along the price curve. On April 12, September 2007 Eurodollar futures were 94.80, lower than the price on April 18 by 7.5 basis points. The put contract for strike price of 95 at that time was 0.4700, an increase of 4.75-basis points. A 2:1 hedge ratio would have covered the decline in a futures. The additional coverage may have been related to the price curve increasing faster and losing more slowly than the straight-line slope. Because the slope is computed at just one point on the price curve, an improved hedging result should be expected in either direction.
The LLP option price model is a natural extension of the usual theoretical valuation methods that determine prices in the market for exchange-traded options. By concentrating on current price relationships, LLP permits traders to forecast option prices based on underlying assets. As an analytical instrument, LLP is easy to use, but works best when combined with knowledge of the capabilities and limitations of both theoretical and empirical pricing models.
Paul D. Cretien is a retired professor of finance at Baylor University and a chartered financial analyst. He is the author of The Basics of Bank Investments (Graduate School of Banking at LSU, 2004).