John Hull mentioned in his book using Cash Price(Dirty Price) instead of Quoted Price(Clean Price) in pricing a bond option using Black-Scholes. It confuses me as it seems more natural to assume the return of bond price stripped off accrued interest follows a Geometric Brownian Motion. Accrued interest itself does not have any stochastic components.
Does anybody have a good explanation on this distinction?
Bond options are traded over the counter. Whatever the parties want to agree to, they can write down on their term sheet. Bond options are also not very common these days, but I've seen a few.
The strike is almost always a price of the bond. Usually, if the bond is quoted clean (without accrued, usually the case in developed markets), then the strike is also a clean price. Conversely, if the bond is quoted dirty (on proceeds, with acctrued), then the strike is also a dirty price. Much less often, the strike is a yield or an asset swap spread.
The dirty price of a (performing) bond looks like a saw: it grows every day, then drops when a coupon goes ex. Nothing stops the parties, e.g. from trading an option, whose strike is a clean price, even though the market convention of the underlying bond is a dirty price, or vice versa, if they prefer.
