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### different approaches for valuing bonds

Bond valuation is the determination of the fair price of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the value of a bond is obtained by discounting the bond's expected cash flows to the present using the appropriate discount rate. In practice this discount rate is often determined by reference to other similar instruments, provided that such instruments exist.
If the bond includes embedded options, the valuation is more difficult and combines option pricing with discounting. Depending on the type of option, the option price as calculated is either added to or subtracted from the price of the "straight" portion. This total is then the value of the bond; the various yields can then be calculated for the total price

### Present value approach

Below is the formula for calculating a bond's price, which uses the basic present value (PV) formula for a given discount rate: (This formula assumes that a coupon payment has just been made; see below for adjustments on other dates.)
F = face value
iF = contractual interest rate
C = F * iF = coupon payment (periodic interest payment)
N = number of payments
i = market interest rate, or required yield, or yield to maturity (see below)
M = value at maturity, usually equals face value
P = market price of bond
If the market price of bond is less than its face value (par value), the bond is selling at a discount. Conversely, if the market price of bond is greater than its face value, the bond is selling at a premium.

### Relative price approach

Under this approach, the bond will be priced relative to a benchmark, usually a government security; see Relative valuation. Here, the yield to maturity on the bond is determined based on the bond's Credit rating relative to a government security with similar maturity or duration; see Credit spread (bond). The better the quality of the bond, the smaller the spread between its required return and the YTM of the benchmark. This required return, i in the formula, is then used to discount the bond cash flows as above to obtain the price.

### Arbitrage-free pricing approach

Under this approach, the bond price will reflect its arbitrage-free price. Here, each cash flow (coupon or face) is separately discounted at the same rate as a zero-coupon bond corresponding to the coupon date, and of equivalent credit worthiness (if possible, from the same issuer as the bond being valued, or if not, with the appropriate credit spread). Here, in general, we apply the rational pricing logic relating to "Assets with identical cash flows". In detail: (1) the bond's coupon dates and coupon amounts are known with certainty. Therefore (2) some multiple (or fraction) of zero-coupon bonds, each corresponding to the bond's coupon dates, can be specified so as to produce identical cash flows to the bond. Thus (3) the bond price today must be equal to the sum of each of its cash flows discounted at the discount rate implied by the value of the corresponding ZCB. Were this not the case, (4) the abitrageur could finance his purchase of whichever of the bond or the sum of the various ZCBs was cheaper, by short selling the other, and meeting his cash flow commitments using the coupons or maturing zeroes as appropriate. Then (5) his "risk free", arbitrage profit would be the difference between the two values.

### Stochastic calculus approach

The following is a partial differential equation (PDE) in stochastic calculus which is satisfied by any zero-coupon bond. This methodology recognises that since future interest rates are uncertain, the discount rate referred to above is not adequately represented by a single fixed number.
The solution to the PDE is given in 
where is the expectation with respect to risk-neutral probabilities, and R(t,T) is a random variable representing the discount rate; see also Martingale pricing.
Practically, to determine the bond price, specific short rate models are employed here. However, when using these models, it is often the case that no closed form solution exists, and a lattice- or simulation-based implementation of the model in question is employed. The approaches commonly used are:
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