Term structure based option-pricing models

Term structure models of pricing contingent claims have followed one of two approaches. One approach followed by Cox, Ingersoll and Ross (1985) actually model the expected returns from movements in the term structure in order to price the contingent claims. In effect, the term structure becomes endogenous to the pricing of the contingent claim.

The second approach followed by Ho and Lee (1986), Heath, Jarrow and Morton (1989), Black, Derman and Toy (1990) and Hull and White (1990) utilizes the volatilities of the various sectors of the term structure to derive a probability distribution of an arbitrage-free binomial, trinomial or multinomial lattice of the term structure. From this lattice, contingent claims are priced. These models all have one thing in common: they allow for the whole-term structure to be stochastic instead of the price of a single underlying instrument or a single interest rate. The whole-term structure is represented at each node of the binomial, trinomial or even multinomiaf lattice.

This methodology allows both long-term and short-term instruments to be valued in a consistent manner. Thus the various debt instruments, forwards, futures and swaps, as well as options, can be valued with an internal consistency that is not possible if ad hoc models are applied to different types of derivative.

The differences between these models are mainly to do with the assumptions that each makes about the way the term structure changes over short periods of time.

The Ho and Lee model

The Ho and Lee (1986) model uses a discrete time binomial approach to modelling, not just the short rate or the long rate, but the whole-term structure. Thus, given the term structure as known today, in the next time period the whole-term structure can either move up by a multiplicative function, h, or move down by a multiplicative function, h*.

All the spot rates that comprise the term structure are calculated and converted into discount factors. These factors are the current values of zero coupon bonds that pay one unit of par value at a maturity coinciding with the date that the spot rate applies to. The array of discount factors pertaining at any one time is a discount function, and it is this that represents the term structure at each node of the binomial lattice. The model assumes that the returns on all the zero coupon bonds are perfectly correlated. Unfortunately, this is not supported by the empirical evidence.

Ho and Lee impose two restrictions on the binomial process of these discount functions. First, the process is considered to be path independent: thus an up movement followed by a down movement gives the same value as a down movement followed by an up movement. Secondly, there must be no arbitrage possibilities in the subsequently generated term structures.

The path-independent assumption implies that interest rates are normally distributed. The no-arbitrage restriction means that the spot rates and the implicit forward rates must be compatible. The binomial lattice is constructed from knowledge of the current term structure and the values of h and h*. The parameters h and h* are themselves influenced by two parameters, it and 6. The first parameter, 7C , is analogous to the pseudo-binomial probability introduced in the binomial model (r — d) / (u — d). The parameter 6 is inversely related to the volatility of the term structure, and the more volatile the term structure, the lower will be the value of 6. This parameter determines the spread between h and h*. Ho and Lee derive the values of these parameters by applying a nonlinear regression model to data relating to contingent claims traded in the market-place. The technique is analogous to calculating the implied standard deviation from traded options and using that value in the option-pricing model. As these variables refer to movements in the term structure and not any particular contingent claim, the values of these variables should be the same for all contingent claims.

Thus the factors that perturb the term structure (h, h*) are influenced by the market’s consensus of expected volatility as embodied in the implied volatilities of market-traded contingent claims. The term structure is therefore perturbed to an extent compatible with the implied volatility of the term structure.

The weakness of this approach is that it assumes that volatility is the same at each vertex of the term structure. Yet we have already noted that long-term spot rates are less volatile than short-term spot rates and far out forward rates are less volatile than near term forward rates.

In addition, the assumption that interest rates are normally distributed leads to the situation where negative interest rates can result, yet such rates are contrary to theory and observed practice. Nevertheless, this model has led to further developments in binomial or multinomial modelling of the term structure and term structure dependent contingent claims.

The Heath-Jarrow-Morton model

The Heath—Jarrow—Morton (1989) model also uses all the information in the term structure. Moreover, they apply the methodology of a multi-factor model of term structure risk, so that the model can handle multiple causes of term structure movement. Thus the returns to zero coupon bonds of various maturities are not assumed to be perfectly correlated as is assumed by Ho and Lee (1986).

Their model incorporates continuous trading such that volatility functions, which play a role analogous to that of the pseudo-probabilities in the Ho and Lee model, can be calculated directly from data of changes in the term structure and not by modelling the value in traded contingent claims.

The model uses forward rates instead of zero coupon bond prices because of the greater stability of the forward rate process. As the price of zero coupon bonds must converge to the redemption price at maturity, this imposes non-stationarity on the volatility of bond prices in those models, such as the one by Ho and Lee, that use discount factors as manifestations of the term structure. However, this convergence does not apply to forward interest rates, thus again leading to a more stable volatility function.

Their 1989 paper develops a two-factor model, where the factors are the changing level and the changing slope of the term structure. As a result of specifying two factors of the stochastic process of the term structure, the model does not have to assume that all bond prices — or equivalently all forward rates — are perfectly correlated. And as a result of using a multi-factor model of the term structure process, the Heath—Jarrow—Morton model employs a multinomial instead of a binomial model of the term structure movement.

The result is an option pricing model that has many intellectual similarities to the Black—Scholes model, in that it only requires a knowledge of the underlying term structure and the volatilities of that term structure. However, the model is also able to accommodate the multi-factor framework of modelling the term structure of interest rates. In particular, by incorporating multiple volatility functions it is able to accommodate the volatility of the term structure resulting from changes in the level, changes in the slope and changes in the curvature of the term structure.

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