6.3 Option valuation
We now examine the equilibrium pricing of a European call option where the underlying is the risk-free zero coupon bond described by (6.7). Without loss of generality, the following assumptions can be made:
bond matures as time 1,
option expiry is at time where ‰ 1,
option exercise price is k ‰ 1.
Further we assume:
Markets are frictionless. That is:
no transaction costs, no taxes,
unlimited lending and borrowing ,
unlimited short sales.
The dynamics of the underlying risk-free zero coupon bond are described by (6.7), which is a non-standard Brownian Bridge process.
There exists a risk-free discount bond maturing at time , also described by (6.7). Assume that:
where is the instantaneous correlation between the unexpected returns on bonds maturing at time 1 and respectively. An initial upward sloping yield curve is assumed, that is P (0, ) > P (0, 1), which implies that initial forward rates are positive.
Investors are assumed to be rational and to have consensus views on the instantaneous standard deviations of bond returns and their distributions. They need not have the same views of the term structure or expected returns on bonds of various maturities.
By forming a hedge portfolio in bonds and the call, the equilibrium value of a European call may be calculated. The time t price of a European call on a discount bond with maturity 1, expiry and strike k is:
and N ( ·) is the cumulative normal distribution. Here ½ is the volatility, at time of the price of the bond with maturity time 1, i.e. it is the forward bond volatility. Consider the forward bond price P ( , 1) = P ( t, 1)/ P ( t , ). Its volatility may be calculated as:
where ƒ 1 and ƒ are the current (time t ) volatilities of the bonds with maturity time 1 and respectively.
In this model, BT assume that default-free discount bond prices follow a nonstandardised Brownian Bridge process. This assumption satisfies the pull-to-par characteristic of bond prices, but ignores the dependence of bond prices on the underlying interest rate term structure. No restrictions are placed on the dynamics of the bond price to ensure a model that precludes profitable arbitrage. This essential factor limits the usefulness and applicability of this model.
Chapter 7: The Hull and White Model
The Vasicek [ 50 ] and CIR [ 18 ] models, studied in Chapters 1 and 2 respectively, allow all interest rate contingent claims to be valued in a consistent manner, but involve unobservable parameters and do not provide a perfect fit for the current interest rate term structure.
However, the process describing the evolution of the short-term interest rate may be deduced from the observed term structure of interest rates and interest rate volatilities. Hence the Vasicek and CIR models may be extended so as to be consistent with the current term structure of interest rates and the current spot interest rate volatilities or current forward rate volatilities.
7.1 General model formulation
The Vasicek and CIR models are special cases of a general mean reverting process of the form:
where ² = 0 for the Vasicek model and ² = ½ for the CIR model.
Since market expectations of interest rate movements can be time-dependent, the drift and volatility parameters should be functions of time:
where ( t ) is the drift rate imposed on the interest rate which otherwise reverts to a constant level b . Rewriting (7.2) in the form:
gives a mean reverting model where the reversion level is a function of time.
Hull and White (HW) [ 28 ] make assumptions about the market price of interest rate risk and fit the Vasicek and CIR special cases of the above mean reverting model to the current term structure of interest rates and spot (or forward) interest rate volatilities.