Option-adjusted effective convexity (OAC)

Everyone knows that convexity measures changes in duration, but does it have a financial meaning on its own? Yes, it does. OAC is responsible for “diffusion return,” i.e. a systematic component of total return that is caused by interest rate volatility. Hence, convexity contributes both to return and, through duration, to risk.

Speaking formally, assets with higher (“better”) convexity are expected to have better returns —given everything even. However, this last pre-requisite almost never holds true in real life. For example, a 10-yr zero-coupon bond has a convexity of +1 whereas a 1-yr bill has a convexity of +0.01. Risk-neutral return expectations are identical for these instruments because the dynamics of interest-rates should preclude arbitrage. In usual arbitrage-free models, future rates evolve progressively above the forward rates, thereby compensating for the convexity difference.

Convexity and Vega

Vega measures price elasticity to change in volatility. Every option has a positive Vega; hence, embedded option bonds should have a negative Vega, but not necessarily a negative convexity. Recall that option-free bonds have positive convexity due to the convex nature of discounting. The presence of an option that belongs to the issuer should reduce the convexity of the investor’s position, possibly without reaching a negative level. Corporate bonds with remote or deep out-of-the-money call schedules may still have positive convexity; the same applies to high-premium MBS.

It is known that at the money (ATM) options have both the highest convexity (Gamma) and the highest Vega. However, short-dated options usually have small Vega, whereas long-dated options have small Gamma regardless of the strike. These facts are often used to conceptualize and structure a hedge strategy. With option straddles (put + call struck at-the-money) one can control Gamma (short-dated straddle) and Vega (long-dated straddle) separately without interfering with Delta (an ATM straddle has zero Delta).

 

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