Market practice with fixed-income derivatives is often to treat them as if there is an underlying asset that is lognormal. This is the methodology proposed by Black (1976).

Bond options

A simple example of Black '76 would be a European option on a bond, as long as the maturity of the bond is significantly greater than the expiration of the option. The relevant formulae are, for a call option

Here F is the forward price of the underlying bond at the option maturity date T. The volatility of this forward price is o. The interest rate r is the rate applicable to the option's expiration and K is the strike.

Caps and floors

A cap is made up of a string of caplets with a regular time interval between them. The payoff for the ith caplet is max(r,- — K,0) at time 7/+1 where ri is the interest rate applicable from ti to ti+1 and K is the strike.

Each caplet is valued under Black '76 as

where r is the continuously compounded interest rate applicable from t to T+i, F is the forward rate from time T to time T+1, K the strike and

where a is the volatility of the forward rate.

The floor let can be thought of in a similar way in terms of a put on the forward rate and so its formula is

Swaptions

A payer swaption, which is the right to pay fixed and receive floating, can be modelled as a call on the forward rate of the underlying swap. Its formula is then

where r is the continuously compounded interest rate applicable from t to T, the expiration, F is the forward swap rate, K the strike and

where a is the volatility of the forward swap rate. is the tenor of the swap and m the number of payments per year in the swap.

The receiver swaption is then

Spot rate models

The above method for pricing derivatives is not entirely internally consistent. For that reason there have been developed other interest rate models that are internally consistent.

In all of the spot rate models below we have

as the real process for the spot interest rate. The risk-neutral process which governs the value of fixed-income instruments is

where X is the market price of interest rate risk. In each case the stochastic differential equation we describe is for the risk-neutral spot rate process, not the real.

The differential equation governing the value of non-path-dependent contracts is

The value of fixed-income derivatives can also be interpreted as

where the expectation is with respect to the risk-neutral process.

Vasicek

In this model the risk-neutral process is

with a, b and c being constant. It is possible for r to go negative in this model.

There is a solution for bonds of the form exp(A(t; T) — B(t; T)r).

Cox, Injersoll & Ross

In this model the risk-neutral process is

with a, b and c being constant. As long as a is sufficiently large this process cannot go negative.

There is a solution for bonds of the form exp(A(t; T) —

B(t; T)r).

Ho & Lee

In this model the risk-neutral process is

with c being constant. It is possible for r to go negative in this model.

There is a solution for bonds of the form exp(A(t; T) — B(t; T)r).

The time-dependent parameter a(t) is chosen so that the theoretical yield curve matches the market yield curve initially. This is calibration.

Hull & White

There are Hull & White versions of the above models. They take the form

or

The functions of time allow various market data to be matched or calibrated.

There are solutions for bonds of the form exp(A(t; T) —

B(t; T)r).

Black & Karasinski

In this model the risk-neutral spot-rate process is

There are no closed-form solutions for simple bonds.

Two-factor models

In the two-factor models there are two sources of randomness, allowing a much richer structure of theoretical yield curves than can be achieved by single-factor models. Often, but not always, one of the factors is still the spot rate.

Brennan & Schwartz

In the Brennan & Schwartz model the risk-neutral spot rate process is

and the long rate satisfies

Fong & Vasicek

Fong & Vasicek consider the following model for risk-neutral variables

Thus they model the spot rate, and § the square root of the volatility of the spot rate.

Longstaff & Schwartz

Longstaff & Schwartz consider the following model for risk-neutral variables

where the spot interest rate is given by

Hull & White

The risk-neutral model,

is a two-factor version of the one-factor Hull & White. The function n(t) is used for fitting the initial yield curve.

All of the above, except for the Brennan & Schwartz model, have closed-form solutions for simple bonds in terms of the exponential of a linear function of the two variables.

The market price of risk as a random factor

Suppose that we have the two real random walks

where X is the market price of r risk. The zero-coupon bond pricing equation is then

Since the market price of risk is related to the slope of the yield curve as the short end, there is only one unobservable in this equation, X>L.

SABR

The SABR (stochastic, a, jj, p) model by Hagan, Kumar, Lesniewski & Woodward (2002) is a model for a forward rate, F, and its volatility, a, both of which are stochastic:

There are three parameters, jj, v and a correlation p. The model is designed for the special case where the volatility a and volatility of volatility, v, are both small. In this case there are relatively simple closed-form approximations (asymptotic solutions). The model is therefore most relevant for markets such as fixed income, rather than equity. Equity markets typically have large volatility making the model unsuitable.

The model calibrates well to simple fixed-income instruments of specified maturity, and if the parameters are allowed to be time dependent then a term structure can also be fitted.

Heath, Jarrow & Morton

In the Heath, Jarrow & Morton (HJM) model the evolution of the entire forward curve is modelled. The risk-neutral forward curve evolves according to

Zero-coupon bonds then have value given by

the principal at maturity is here scaled to $1. A hedging argument shows that the drift of the risk-neutral process for F cannot be specified independently of its volatility and so

This is equivalent to saying that the bonds, which are traded, grow at the risk-free spot rate on average in the risk-neutral world.

A multi-factor version of this results in the following risk-neutral process for the forward rate curve

In this the dXi are uncorrelated with each other.

Brace, Gatarek & Musiela

The Brace, Gatarek & Musiela (BGM) model is a discrete version of HJM where only traded bonds are modelled rather than the unrealistic entire continuous yield curve.

If Zi(t) = Z(t; T) is the value of a zero-coupon bond, maturing at T,at time t, then the forward rate applicable between Ti and Ti+1 is given by

where t = T+i — Ti. Assuming equal time periods between all maturities we have the risk-neutral process for the forward rates, given by

Modelling interest rates is then a question of the functional forms for the volatilities of the forward rates a, and the correlations between them, p,y.

Prices as expectations

For all of the above models the value of fixed-income derivatives can be interpreted as

Ef- [Present value of cash flows], where the expectation is with respect to the risk-neutral pro-cess(es). The 'present value' here is calculated path wise. If performing a simulation for valuation purposes you must discount cash flows for each path using the relevant discount factor for that path.

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