Theta

An options theta indicates how its value changes with its time to maturity. This is expressed formally in (9.6). 

Using the relationships embodied in the B-S model, the theta of a call option can be expressed mathematically as (9.7). 

A ( 4) = the probability that the option will be exercised o = the volatility of the underlying assets price 

Some option strategies exploit theta. Writing a short-term option and simultaneously purchasing a longer-term one with the same strike price, for instance, represents a play on the options theta if the time value of the longer option decays at a slower rate than that of the short-dated option, the trade will be profitable. 

An option s vega—also known as its epsilon (e), eta rj), or kappa (k)— indicates how much its value changes with changes in the price volatility of the underlying asset. For instance, an option with a vega of 12.75 will increase in price by 0.1275 for a 1 percent increase in volatility. Vega is expressed formally as (9.8). 

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Fig. XI-5. Adsorption isotherm from Ref. 61 for polystyrene on chrome in cyclohexane at the polymer theta condition. The polymer molecular weights x 10 are (-0) 11, (O) 67, (( )) 242, (( )) 762, and (O) 1340. Note that all the isotherms have a high-affinity form except for the two lowest molecular weights.

Lindek St, Cremer Chr and Stelzer E H K 1996 Confocal theta fluorescence microscopy using two-photon absorption and annular apertures Optik 02 131-4... 

Grassberger P 1997 Pruned-enriched Rosenbluth method simulations of theta polymers of chain length up to 1,000,000 Phys. Rev. E 56 3682... 

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

Table 8.3 Theta Temperatures for a Few Polymer-Solvent Systems...

Polymers in Solution. Polyacrylamide is soluble in water at all concentrations, temperatures, and pH values. An extrapolated theta temperature in water is approximately —40° C (17). Insoluble gel fractions are sometimes obtained owing to cross-link formation between chains or to the formation of imide groups along the polymer chains (18). In very dilute solution, polyacrylamide exists as unassociated coils which can have an eUipsoidal or beanlike stmcture (19). Large aggregates of polymer chains have been observed in hydrolyzed polyacrylamides (20) and in copolymers containing a small amount of hydrophobic groups (21). 

Activation of bayerite [20257-20-9] and nordstrandite [13840-05-6] quaUtatively follows the pattern shown in Figure 4, but the transition sequence is eta-theta-alpha. The stmctures of these transition phases are somewhat different from those obtained from gibbsite, reflecting the differences in crystal stmcture of the hydroxides (16). 

Support-phase changes or loss of surface area are, of course, irreversible, and replacement of the catalyst may be appropriate. Catalyst damage may take the form of phase changes to the alumina support from gamma to theta or alpha phase. The last is catalyticaky inert because of insignificant surface area. Theta alumina has a low surface area (< 100 /g) relative to gamma alumina (180 m /g) and has poor halogen retention. 

Heat-transfer rate kW Btu/b 0 Holland theta factor ... 

Name =Pendulumll Type =mamdani Numlnputs=4 InLabels= theta tdot... 

The measures of solid state reactivity to be described include experiments on solid-gas, solid-liquid, and solid-solid chemical reaction, solid-solid structural transitions, and hot pressing-sintering in the solid state. These conditions are achieved in catalytic activity measurements of rutile and zinc oxide, in studies of the dissolution of silicon nitride and rutile, the reaction of lead oxide and zirconia to form lead zirconate, the monoclinic to tetragonal transformation in zirconia, the theta-to-alpha transformation in alumina, and the hot pressing of aluminum nitride and aluminum oxide. 

Fig. 7.13. The conversion of theta- to alpha-phase alumina was found to be strongly affected by shock modification in work of Beauchamp and co-workers [90B01]. Whereas the unshocked powder showed evidence for an incubation period of 60 min, the shock-modified materials show immediate conversion typical of the presence of shock-formed nuclei.

Thermal treatment of shock-modified theta-phase alumina, which initially contained about 30% alpha phase, showed a dramatic change in the rate of transformation to the alpha phase [90B01]. As shown in Fig. 7.13, the shocked sample showed no evidence for an incubation period and displayed a rapid conversion to the alpha phase, in sharp contrast to the unshocked sample. Such behavior clearly indicates that the shock process resulted in formation of larger concentrations of alpha-phase nuclei. 

Although the properties of specific polymer/wall systems are no longer accessible, the various phase transitions of polymers in confined geometries can be treated (Fig. 1). For semi-infinite systems two distinct phase transitions occur for volume fraction 0 = 0 and chain length N oo, namely collapse in the bulk (at the theta-temperature 6 [26,27]) and adsorp-... 

A. Milchev, W. Paul, K. Binder. Off-lattice Monte Carlo simulation of dilute and concentrated polymer solutions under theta conditions. J Chem Phys 99 4786-4798, 1993. 

FIGURE 19.1 Theta condition effects on pMMA elution. Upper trace acetonitrile 65°C Lower trace THF 35°C... 

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