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Convexity is a measure of the curvature or 2nd derivative of how the price of a bond varies with interest rate, i.e. how the duration of a bond changes as the interest rate changes. [3] Specifically, one assumes that the interest rate is constant across the life of the bond and that changes in interest rates occur evenly.
Because interest rate caps/floors are equivalent to bond puts and calls respectively, the above analysis shows that caps and floors can be priced analytically in the Hull–White model. Jamshidian's trick applies to Hull–White (as today's value of a swaption in the Hull–White model is a monotonic function of today's short rate).
This difference in convexity can also be used to explain the price differential from an MBS to a Treasury bond. However, the OAS figure is usually preferred. The discussion of the "negative convexity" and "option cost" of a bond is essentially a discussion of a single MBS feature (rate-dependent cash flows) measured in different ways.
Callable bonds are a type of bond that the issuer can “call” or redeem before the maturity date. The specifics vary from bond to bond, but callable bonds always have one thing in common ...
Similar to the above, in these cases, it may be more correct to calculate an effective convexity. Mortgage-backed securities (pass-through mortgage principal prepayments) with US-style 15- or 30-year fixed-rate mortgages as collateral are examples of callable bonds.
In practice the most significant of these is bond convexity, the second derivative of bond price with respect to interest rates. As the second derivative is the first non-linear term, and thus often the most significant, "convexity" is also used loosely to refer to non-linearities generally, including higher-order terms.
By issuing numerous callable bonds, they have a natural hedge, as they can then call their own issues and refinance at a lower rate. The price behaviour of a callable bond is the opposite of that of puttable bond. Since call option and put option are not mutually exclusive, a bond may have both options embedded. [3]
For a bond with an embedded option, the standard yield to maturity based calculations of duration and convexity do not consider how changes in interest rates will alter the cash flows due to option exercise. To address this, effective duration and -convexity are introduced.