In Treasury-Bond markets, several options are traded where the underlying are illiquid instruments, usually referred to as Off-the-run bonds (OffTR in short). This is opposed to the reference or benchmark instruments, equally known as On-the-Run bonds (OnTR in short). In general, mainly due to a liquidity premium, OffTR instruments tend to trade at a spread above the OnTR bonds and display a different historical volatility when compared to the benchmark set.
When trying to price a derivative where the underlying is an OffTR bond, we have the issue of pricing the option in a way to avoid possible arbitrage with the swap market. This also stems from that implied volatilities are not available for OffTR products. To achieve this goal, we introduce two alternative solutions that both use information gathered from money markets. Although a swaption in a frictionless market may be seen as a portfolio of 0-coupon bonds, things in reality are far more complicated. In real markets Libor curves and Bond Yield curves differ substantially due to liquidity effects, REPOs and collateralisation features, just to mention a few. In addition, only forward swap rates volatilities are quoted, with the implicit assumption of lognormality under the swap measure (the one where the PVBP asset is the numeraire). Therefore, a direct equivalent of the swap-rate volatility surface in the bond option market does not exist.
We place ourselves in a market driven by a generic stochastic volatility model for the forward swap rates under the swap measure. The main idea consists in finding a suitable swaption (or a swaption portfolio) to replicate the payoff of the bond option. The hypothesis we introduce and justify assumes that Yield curves can be recovered from swap curves through a deterministic (although maturity-dependent) zero-coupon spread. Alternative possibilities are also discussed. In a first attempt to solve the problem, we determine an arbitrage relationship between OffTR bond options and swaptions that only involves a position in ATM options. We then generalize this approach to include the possibility of statically replicate the bond option payoff through a portfolio of swaptions at different strikes. In the second scenario, we are then able to price an option on a OffTR bond consistently with the swaption smile.