Convexity - Some Facts

Convexity- Assets


Financial instruments that gains one unit and loses one unit for every step up and down respectively are called linear instruments. These instruments will gain or lose same amount in magnitude in rising and falling market. An example of this kind of instrument is Futures contract. There are instruments that increase or decrease in value at different magnitudes for different level of changes in markets. This class of instruments is termed as convex products. In this article I will discuss, some ways to synthesize convexity in 3m Libor interest rates, Swap rates and also Volatility.

Convexity in 3M Libor interest rates can be traded using Futures vs Forwards and 3m Libor interest rate swaps vs 3m Libor arrear swaps products. These two products provide a way to buy and sell convexity implied by the markets. Normally, in interest rate markets we see forward rates implied by 3M euro dollar interest rate future contracts are adjusted for convexity. This practice arises due to differences in settlement conventions for futures and forward contracts. More intuitively, future contracts settle daily and forwards settle once on the rate resetting date. This nuance looks very trivial but in some instances this can translate into huge amounts. Due to daily settlement feature in futures, investor in long futures contract has to maintain maintenance margin that needs to replenish as market falls and withdraw excess funds over this margin when market rises. Whereas, the investor in the over the counter forwards market has to wait till settlement date to act. In nutshell, futures investor can realize the market changes daily and forwards investor has to wait till settlement date to realize the market changes. In financial markets, present valuing of the cash flows is a way to compare different products. In this way, futures cash flows are discounted daily and forwards are discounted at term rates. So when rates rise by 1 bp, futures contract gains $ 25 and rates rise by 10 bp, $ 250 change in value is realized. Similarly, the position will lose, $ 25 and $250 dollars respectively. In forwards market, gain and loses are different in magnitude. This is because, when rates rise, we discount the cash flows with higher rate so less value and in a falling rates market, we discount cash flows with lower rates and hence higher value. This is called convexity. This is mainly dependent on correlation between discount rate and 3m Libor rate, and their volatilities to measure. A pair contract, future and forward locks current market implied convexity and this can be bought and sold based on current Libor cap market volatilities. Now moving from future vs Forward convexity to Libor swaps that exhibit convexity.

Standard 3m Libor Interest rate swaps with reset advance and pay in arrears contract features allows to trade the level of interest rate yield curve. At the same time with a small tweak to reset feature from advance to arrears will create an arrear swap. This swap will present an investor to trade level of the swap rate along with a view on the speed at which forward rates change in current yield curve. These two products are convex products when measured against movement of yield curves. But Libor in arrears swaps since resets and pays in arrears has additional convexity that is absent in plain vanilla swap. Why convexity is coming into picture here? What is the benefit of this convexity? In today’s yield curve there is a forward curve structure. This structure once locked will define the value of the swap. In arrear swap since rates are reset in future on the same date of payment, it creates small mismatch between the term of the rate (3m Libor) observed in arrears and the payment time or date. In a regular swap, rate is observed in advance and allowed to accrue till payment date which happens to be its term. Where as in arrear swap this accrual doesn’t happen and this creates an opportunity to its holder. If rates rise, the payer has to higher rate but discounted with a higher rate (so lower value) and vice versa. Also, if investor thinks yield curve will realize embedded forward curve at slower pace wants to receive fixed rate (convexity adjusted). This gives additional yield pickup. This convexity is measured using interest rate cap volatilities.

Now let us move to CMS convexity products. This class of products provides investors to take view on the CMS rates where investor pays or receives CMS rate vs 3M Libor + spread. This spread is normally quoted in the broker market as spread over Libor. Now convexity is lurking behind CMS rates in this product also. Answer to this would be, due to mismatch between CMS rate being observed and accrued. For instance, CMS 10y rate is observed for 3m and paid at the end of period. This mismatch between accrual period and observed rates gives rise to convexity. Now how do I measure this? This is measured by looking into swaption volatility market. CMS swap can be replicated by series out of the money puts and calls. In normal put call parity frame work, Cap-floor is equal to swap. So if we can price a series of these caps and floors accurately we can measure the convexity effectively.