Bonds, the
underlyers in this case, exhibit what is known as
pull-to-par: as the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its
volatility. On the other hand, the
Black–Scholes model, which assumes constant volatility, does not reflect this
process, and cannot therefore be applied here; see . Addressing this, bond options are usually valued using the
Black model or with a
lattice-based short-rate model such as
Black-Derman-Toy,
Ho-Lee or
Hull–White. [http://pages.stern.nyu.edu/~eelton/debt_inst_class/option%20valuation.pdf The latter approach is theoretically more correct, although in practice the Black Model is more widely used for reasons of simplicity and speed. For
American- and
Bermudan- styled options, where exercise is permitted prior to maturity, only the lattice-based approach is applicable. • Using the Black model, the
spot price in the formula is not simply the market price of the
underlying bond, rather it is the
forward bond price. This forward price is calculated by first subtracting the present value of the coupons between the valuation date (i.e. today) and the exercise date from today's
dirty price, and then
forward valuing this amount to the exercise date. (These calculations are performed using today's
yield curve, as opposed to the bond's
YTM.) The reason that the Black Model may be applied in this way is that the
numeraire is then $1 at the time of delivery (whereas under
Black–Scholes, the numeraire is $1 today). This allows us to assume that (a) the bond price is a random variable at a future date, but also (b) that the risk-free rate between now and then is constant (since using the
forward measure moves the discounting outside of the expectation term [https://quant.stackexchange.com/questions/61151/martingale-pricing-with-time-dependent-risk-free-rate/61159#61159). Thus the valuation takes place in a
risk-neutral "forward world" where the expected future spot rate is the forward rate, and its
standard deviation is the same as in the "physical world"; see
Girsanov's theorem. The volatility used, is typically "read-off" an
Implied volatility surface. • The lattice-based model entails a tree of short rates – a zeroeth step – consistent with today's
yield curve and short rate (often
caplet) volatility, and where the final time step of the tree corresponds to the date of the underlying bond's maturity. Using this tree (1) the bond is valued at each node by "stepping backwards" through the tree: at the final nodes, bond value is simply
face value (or $1), plus coupon (in cents) if relevant; at each earlier node, it is the
discounted expected value of the up- and down-nodes in the later time step, plus coupon payments during the current time step. Then (2), the option is valued similar to the
approach for equity options: at nodes in the time-step corresponding to option maturity, value is based on
moneyness; at earlier nodes, it is the discounted expected value of the option at the up- and down-nodes in the later time step, and, depending on
option style (and other specifications – see
below), of the bond value at the node. [https://books.google.com/books?id=wF8yVzLI6EYC&q=Valuation+of+fixed+income+securities+and+derivatives For both steps, the discounting is at the short rate for the tree-node in question. (Note that the Hull-White tree is usually
Trinomial: the logic is as described, although there are then three nodes in question at each point.) See . ==Embedded options==