Positive convexity

The sum w = JT a Zi is a linear function in Z and hence convex. The function ew is convex and so is the function ew with w being a linear function in z. Note that this expression is convex since it is a sum of positive convex functions (i.e., exponentials). 

Putable bonds exhibit a positive convexity, although lower than a conventional bond, above aU with rising interest rates. Figure 11.2 shows the changes of prices according to the interest rate. If the interest rates decrease, option free and putable bonds have the same convexity. If the interest rates rise, putable bonds become more valuable. 

EXHIBIT 4.18 Negative and Positive Convexity Exhibited by a Callable Bond... 

The relationship is reduced as maturity is increased because of the low liquidity of futures markets beyond two to five years, increased futures execution risk, and increased interest rate volatility. The convexity problem when using futures contracts (nonconvex instruments) to hedge interest rate swap positions (convex instruments) is more pronounced for long-term transactions, resulting in reduced hedge efficiency. The convexity issue is addressed in detail later in this chapter. 

A long position in FRAs or swaps and a short position in futures has net positive convexity. The short futures position has a positive payoff when interest rates rise and lower losses when interest rates fall, as they can be refinanced at a lower rate. This mark to market positive effect of futures contracts creates a bias in favor of short sellers of futures con-... 

Convexity comes into play when yield curve changes are moderate to large and serves to increase the value of the bond irrespective of whether the yield rises or falls. In other words, if yields rise, then bonds with positive convexity fall less than expected from duration alone, and when yields fall, bond prices rise more than expected. To put it bluntly, convexity is good for a bond portfolio, but it is exceptionally hard to actively manage a credit portfolio and maximise convexity at the same time. 

Bond traders wishing to hedge the interest rate risk of their bond positions have several tools to choose from, including other bonds, bond futures, and bond options, as well as swaps. Swaps, however, are particularly efficient hedging instruments, because they display positive convexity. As explained in chapter 2, this means that they increase in value when interest rates fall more than they lose when rates rise by a similar amount—just as plain vanilla bonds do. 

As explained in chapter 1, the curve representing a plain vanilla bond s price-yield relationship is essentially convex. The price-yield curve for a bond with an embedded option changes shape as the bond s price approaches par, at which point the bond is said to exhibit negative convexity. This means that its price will rise by a smaller amount for a decline in yield than it will fall for a rise in yield of the same magnitude. FIGURE 11.13 summarizes the price-yield relationships for both negatively and positively convex bonds. 

However, a fundamental difference with plasticity intervenes here. In viscoplasticity, a continuously differentiable dissipation potential, definite positive, convex and contains the origin, can be defined ... 

FIGURE 11.13 Price-Yield Relationships Associated with Negative and Positive Convexity ... 

PERCENT PRICE CHANCE FOR CHANGE IN YIELD POSITIVE CONVEXITY NEGATIVE CONVEXITY... 

Let K cV he a. convex closed subset of a reflexive Banach space V, I he a duality mapping, and P be a projection operator of V onto K. We are in a position to give a definition of a penalty operator. An operator (5 V V is called a penalty operator connected with the set K if the following conditions are fulfilled. Firstly, / is a monotonous bounded semicontinuous operator. Secondly, a kernel of / coincides with K, i.e. 

Here i —> i is the convex and continuous function describing a plasticity yield condition, the dot denotes a derivative with respect to t, n = (ni,ri2) is the unit normal vector to the boundary F. The function v describes a vertical velocity of the plate, rriij are bending moments, (5.175) is the equilibrium equation, and equations (5.176) give a decomposition of the curvature velocities —Vij as a sum of elastic and plastic parts aijkiirikiy Vijy respectively. Let aijki x) = ajiki x) = akuj x), i,j,k,l = 1,2, and there exist two positive constants ci,C2 such that for all m = rriij ... 

In transforming bis-ketone 45 to keto-epoxide 46, the elevated stereoselectivity was believed to be a consequence of tbe molecular shape — tbe sulfur ylide attacked preferentially from tbe convex face of the strongly puckered molecule of 45. Moreover, the pronounced chemoselectivity was attributed to tbe increased electropbilicity of the furanone versus the pyranone carbonyl, as a result of an inductive effect generated by tbe pair of spiroacetal oxygen substituents at tbe furanone a-position. ... 

The relationship between 20 and reserpine (1) is close like reserpine, intermediate 20 possesses the linear chain of all five rings and all six stereocenters. With the exception of the 3,4,5-tri-methoxybenzoate grouping, 20 differs from reserpine (1) in one very important respect the orientation of the ring C methine hydrogen at C-3 in 20 with respect to the molecular plane is opposite to that found in reserpine. Intermediate 20 is a reserpate stereoisomer, epimeric at position 3, and its identity was secured by comparison of its infrared spectrum with that of a sample of (-)-methyl-O-acetyl-isoreserpate, a derivative of reserpine itself.9 Intermediate 20 is produced by the addition of hydride to the more accessible convex face of 19, and it rests comfortably in a conformation that allows all of the large groups attached to the D/E ring skeleton to be equatorially disposed. 

One simple but useful way to demonstrate the convexity upward (downward) of a function is to show that it is the sum of convex upward (downward) functions. The proof of this property follows immediately from the definition of convexity. For functions of one variable convexity upward (downward) can also be demonstrated by showing that the second derivative is negative (positive) or zero over the interval of interest. Much of the usefulness of convex functions, for our purposes, stems from the following theorem ... 

Since 2, A, = 1, and since the coefficients of both logarithms are positive, we can use the convexity upward of the logarithm to give... 

If we imagine a line drawn on the primitive surface dividing all parts of the surface which are convex downwards in all directions from those which are concave downwards in one or both directions of principal curvature, this curve will have the equation (26), and is known as the spinodal carve. It divides the surface into two parts, which represent respectively states of stable and unstable equilibrium. For on one side A is positive, and on the other it is negative. If we assume that the tie-line of corresponding points on the connodal curve is ultimately tangent to that the direction of equations ... 

A partial pressure curve which is concave to the concentration axis, i.e. a positive curve, indicates the dissociation of a polymerised component, whilst a curve which is convex to the same axis, i.e., a negative curve, indicates the formation of a chemical compound of the two components. In the first case the concentration of the constituent passing into the vapour would be increased, in the second case reduced, by the assumed change. As examples, Dolezalek quotes ... 

The mixed and pure states of an A-particle fermion system can be described by positive and normalized operators,, which form a convex set... 

N-particle states,, has values in the convex set,, of positive functions in Ii ( Y) that integrate to the value N... 

While the functionin (B.l) is convex for all x, they( ) in (C.l) is concave from X = 0 to the inflection point x, = l/ /2 a and convex from Xi to oo. This means that the discretization error is negative for intervals between 0 and Xi and positive between Xj and oo, such that a partial cancellation of the error is possible. 

Unlike solid electrodes, the shape of the ITIES can be varied by application of an external pressure to the pipette. The shape of the meniscus formed at the pipette tip was studied in situ by video microscopy under controlled pressure [19]. When a negative pressure was applied, the ITIES shape was concave. As expected from the theory [25a], the diffusion current to a recessed ITIES was lower than in absence of negative external pressure. When a positive pressure was applied to the pipette, the solution meniscus became convex, and the diffusion current increased. The diffusion-limiting current increased with increasing height of the spherical segment (up to the complete sphere), as the theory predicts [25b]. Importantly, with no external pressure applied to the pipette, the micro-ITIES was found to be essentially flat. This observation was corroborated by numerous experiments performed with different concentrations of dissolved species and different pipette radii [19]. The measured diffusion current to such an interface agrees quantitatively with Eq. (6) if the outer pipette wall is silanized (see next section). The effective radius of a pipette can be calculated from Eq. (6) and compared to the value found microscopically [19]. 

Formation of products and intermediate species, as well as disappearance of reactants during the photocatalytic reactions can be discerned by the evolution of positive (i.e., concave shape) bands and negative (i.e., convex shape) bands, respectively. 

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