arbitrage profit would be the difference between the two values. Stochastic calculus approach When modelling a bond option, or other interest rate derivative (IRD), it is important to recognize that future interest rates are uncertain, and therefore, the discount rate(s) referred to above, under all three cases—i.e. whether for all coupons or for each individual coupon—is not adequately represented by a fixed (deterministic) number. In such cases, stochastic calculus is employed. The following is a partial differential equation (PDE) in stochastic calculus, which, by arbitrage arguments, is satisfied by any zero-coupon bond , over (instantaneous) time , for corresponding changes in , the short rate. The solution to the PDE (i.e. the corresponding formula for bond value) — given in Cox et al. — is: where is the expectation with respect to risk-neutral probabilities, and is a random variable representing the discount rate; see also Martingale pricing. To actually determine the bond price, the analyst must choose the specific short-rate model to be employed. The approaches commonly used are: the CIR model the Black–Derman–Toy model the Hull–White model the HJM framework the Chen model. Note that depending on the model selected, a closed-form (“Black like”) solution may not be available, and a lattice- or simulation-based implementation of the model in question is then employed. See also . Clean and dirty price When the bond is not valued precisely on a coupon date, the calculated price, using the methods above, will incorporate accrued interest: i.e. any interest due to the owner of the bond over the "stub period" since the previous coupon date (see day count convention). The price of a bond which includes this accrued interest is known as the "dirty price" (or "full price" or "all in price" or "Cash price"). The "clean price" is the price excluding any interest that has accrued. Clean prices are generally more stable over time than dirty prices. This is because the dirty price will drop suddenly when the bond goes "ex interest" and the purchaser is no longer entitled to receive the next coupon payment. In many markets, it is market practice to quote bonds on a clean-price basis. When a purchase is settled, the accrued interest is added to the quoted clean price to arrive at the actual amount to be paid. Yield and price relationships Once the price or value has been calculated, various yields relating the price of the