关于convexity

问题如下：

A portfolio contains a long position in an option contract on a US Treasury bond. The option exhibits positive convexity across the entire range of potential returns for the underlying bond. This positive convexity:

选项：

A.Implies that the option’s value increases at a decreasing rate as the option goes further into the money.

B.Makes a long option position a superior investment compared to a long bond position of equivalent duration.

C.Can be effectively hedged by the sale of a negatively convex financial instrument.

D.Implies that the option increases in value as market volatility increases.

解释：

The relationship between convexity and volatility for a security can be seen most clearly through the second-order Taylor approximation of the change in price given a small change in yield. The resulting change in price can be estimated as:

where D is equal to the duration, C is the convexity and y is the change in the interest rate. Since Δ𝒚² is always positive, positive convexity will lead to an increase in return as long as interest rates move, with larger interest moves in either direction leading to a greater return benefit from the positive convexity. Therefore, a position in a security with positive convexity can be considered a long position in volatility.

This relationship can also be explained graphically. The price curve of a security with positive convexity will lie above and tangentially to the price curve of the underlying. If volatility of the underlying increases, then so will the volatility of either a long call or a long put, but the deviation from the price of the underlying will be positive when there is positive convexity, and negative with negative convexity. Therefore, the expected terminal value over the in-the-money region will increase while the expected terminal value over the out-of-the-money region will remain zero, an aggregate effect of increasing the total expected value of the option.