Investors in assets with embedded options such as mortgages, callable or sinkable bonds expect to get some yield compensation for being short an option. They realize that options are exercised against the asset's value making its profile negatively convex. The more negatively convex the asset is, the more compensation is required.
Convexity cost is often identified with option cost and is assessed via rigorous option-adjusted valuation (OAV) computer systems. While this Learning Curve cannot replace mathematical sharpness and accuracy of full-scale OAV procedures (black boxes), it can help to understand and quantify the role of convexity and, in principle, even to trade convexity. The underlying OAV principle can be formulated (see center).
Static Return Spread and Static Convexity are measured along some average market-rate scenario, and with respect to a chosen interest-rate model factor. Relationships (1) and (2) are more rigorous that they look. In fact, they were obtained from a strict partial differential equation written for the expected instantaneous return on asset less that of the same-cashflow portfolio of Treasuries (see A. Levin and D. Love, The Concept of Instantaneous Return, in Yield Curve Dynamics, 1997). Convexity cost is expressed as a return component, namely the diffusion term in this equation, which is caused by a non-linear (convex or concave) relation between a random factor and the price.
Under some simplifying assumptions about the static return spread behavior along the chosen (average) interest rate path, one can replace it by the static discount spread. In this case, the right-hand side of formula (1) is called expected instantaneous return spread (EIRS), a surrogate for option-adjusted spread (OAS) obtained under some simplifying assumptions about the static spread behavior along the chosen (average) interest rate path. Note that one cannot immediately compute OAS from (1) as effective convexity depends on prices for the adjacent scenarios (factor-up, factor-down) where the static spreads are not yet known. Rather (1), (2) is a system of equations to be solved on a grid of scenarios complemented by some boundary conditions such as a zero convexity cost for the extreme scenarios.
In fact, the author has developed an efficient scheme that solves (1), (2) computing OAS with accuracy of 2-5 basis points and concurrently repricing the asset on the grid and thus delivering valuable information for asset/liability and risk managers. This scheme works much faster than the Monte-Carlo technique even if the latter is set up to achieve the same accuracy.
In addition to solving (1), (2) as a part of the actual OAV, these formulae can be also used for rough estimates and are helpful for checking broker's results, or just for a better comprehension of OAV and more confident trading position.
A closer look at convexity cost reveals some practical though not immediately obvious hints. First, convexity cost is proportional to the difference between effective convexity and static convexity. The latteXr is positive being a convexity for a long position in an option-free (static) asset, while the former is the resultant convexity. Although this difference is normally negative reducing the investor's return, effective convexity itself can be positive as observed for some mortgage-backs with relatively stable prepayment speeds or corporate bonds with remote call schedules.
Second, convexity cost is proportional to the volatility of the chosen interest rate factor, squared (this is a proxy for the actual dependency; if volatility changes, the entire OAV is affected including static discount spreads and the average paths themselves).
This sharp dependency requires a very accurate volatility assumption, and opens a room for intensive volatility arbitrage. Let us assume, for example that a trader considers purchasing an asset from a broker. A broker's system reports, among other results, 30 basis points of the option cost under a 15% volatility assumption. The trader has no access to the broker's model, but believes the broker understates volatility, and 18% should be used instead of 15%. Using the key formula (2) he immediately estimates (18/15)2*30 = 43 basis points of the actual convexity cost, a 13 basis point loss in expected return.
To continue our illustrations of the immediate practical worth of the convexity cost concept, let us consider a traditional trading approach usually involving static measures such as yield to maturity or a static return spread to the curve. Assume that all market participants agree on an 18% volatility assumption, the short rate is 5%, thus the absolute volatility level is 0.8%. One can establish a helpful index, convexity cost per unit of convexity,
Convexity Cost per unit of Convexity = 50*Volatility2 = 50 * 0.0082 = 0.0032.
Thus, each unit of convexity (measured with respect to the short rate) costs 32 basis points of OAS, and the "static" market should trade two securities approximately 32 basis points apart if their effective convexities (less static convexities) differ by 1 unit.
In a subsequent Learning Curve, we extend the convexity cost concept into the multi-factor analysis field and reveal hidden return components caused by factors not traded by the market.
This week's Learning Curve was written by Alexander Levin, a senior quantitative developer at The Dime Savings Bank of New York.
