Fixed income 2

Why embedded-option bonds need a tree

Option-free bonds can be valued with a simple spot rate curve, because their cash flows are fixed regardless of what rates do. For bonds with embedded options, changes in future rates affect the probability of the option being exercised and therefore the underlying future cash flows themselves. Valuing them requires a model that allows both rates and the cash flows to vary, which is what the binomial interest rate tree framework provides.

Structure of the binomial interest rate tree

Each nodal period holds a set of possible forward rates, and the relationship among them is a function of the interest rate volatility assumed to generate the tree. Adjacent forward rates within the same nodal period are two standard deviations apart, so for the first period the two rates are related by i(1,U) = i(1,L) × e^(2σ). Beyond the first nodal period, non-adjacent forward rates are a multiple of those two standard deviations depending on how separated they are — for example i(2,UU) = i(2,LL) × e^(4σ). The volatility estimate itself can be based on historical data or can be implied volatility derived from interest rate derivatives.

Pathwise valuation

Pathwise valuation is mathematically identical to the backward induction method in a binomial tree, but works forward along each possible route through the tree. For a binomial interest rate tree with n periods there are 2^(n−1) unique paths, so a three-period tree has 2² = 4 paths, each comprising one known spot rate and varying combinations of two unknown forward rates. Labelling the one-period spot rate S and the upper and lower forward rate outcomes U and L, those four paths are SUU, SUL, SLU and SLL. You value the bond along each path and the bond’s value is the average of the path values.

Path dependency

An important assumption of the binomial valuation process is that the value of cash flows at a given point in time is independent of the path interest rates followed to get there — cash flows are not path dependent. Mortgage-backed securities violate this. Prepayment risk resembles call risk, but unlike call risk it is affected not only by the level of interest rates at a point in time but also by the path rates took to reach it. Consider a pool formed at 6% where rates fall to 4%, rise to 6%, then fall to 4% again: many homeowners refinanced at the first dip, so on the second occurrence of 4% most of those who could refinance already have, producing lower prepayments than if 4% had never occurred before. Because MBS cash flows are path dependent, the binomial backward induction process cannot value them, and Monte Carlo simulation is used instead.

Monte Carlo forward-rate simulation

A Monte Carlo forward-rate simulation randomly generates a large number of interest rate paths using a model that incorporates a volatility assumption and an assumed probability distribution. Its key feature, and the reason it is used for mortgage-backed securities, is that the underlying cash flows can be path dependent. As with pathwise valuation, the value of the bond is the average of the values obtained from the various paths.

Calibration and drift adjustment

Simulated paths should be calibrated so that benchmark interest rate paths value benchmark securities at their market price — that is, so the model is arbitrage-free. The calibration process entails adding a constant to all rates when the value obtained from the simulated paths is too high relative to market prices, and subtracting a constant when the value is too low. A model calibrated this way is described as drift adjusted.

Mean reversion bounds in Monte Carlo

A Monte Carlo simulation may impose upper and lower bounds on interest rates as part of the model generating the simulated paths. These bounds rest on the notion of mean reversion: rates tend to rise when they are too low and to fall when they are too high.

Term structure models

Term structure models attempt to capture the statistical properties of interest rate movements and provide quantitatively precise descriptions of how interest rates will change. They split into equilibrium models and arbitrage-free models.

Arbitrage-free models

Arbitrage-free models of the term structure begin with the assumption that bonds trading in the market are correctly priced, and the model is calibrated to value those bonds consistently with their market price — hence the label. These models do not try to justify the current yield curve; they take the curve as given. The ability to calibrate to match current market prices is their central advantage over equilibrium models.

Ho-Lee model

The Ho-Lee model takes the form dr(t) = θ(t)dt + σdz(t), where θ(t) is a time-dependent drift term. It is derived using the relative pricing concepts of the Black-Scholes model and assumes that changes in the yield curve are consistent with a no-arbitrage condition. It is calibrated by using market prices to find the time-dependent drift term θ(t) that generates the current term structure, after which it can price zero-coupon bonds and determine the spot curve. The model assumes constant volatility and produces a symmetrical, normal distribution of future rates.

Kalotay-Williams-Fabozzi (KWF) model

The KWF model takes the form dln(r(t)) = θ(t)dt + σdz. It does not assume mean reversion and, like the Ho-Lee model, assumes constant volatility. The right-hand side of its equation is identical to Ho-Lee’s — the only difference is that KWF assumes the short-term rate is lognormally distributed, whereas Ho-Lee assumes a normal distribution.

Gauss+ model

Gauss+ is a multifactor model incorporating short-, medium- and long-term rates. The long-term rate is designed to be mean reverting and depends on macroeconomic variables. Medium-term rates revert to the long-term rate. The short-term rate is devoid of a random component, consistent with the role of the central bank in controlling it. The result is a hump-shaped volatility curve across tenors, with the volatility of medium-term rates being the highest.