Derman and Kani (1997) provide an analogy of DVF and
IVT with models of the Term Structure of interest rates. Heath, Jarrow, and Morton (1990, henceforth
HJM) develop a model for interest rates where the inputs are forward interest
rates, *f*_{t,T} as
follows

_{} (3)

This is the differential form of
the forward rate curve given an initial forward rate f_{0,T} for each
maturity T. m_{t,T} and s_{t,T}
are the mean and volatility of the forward curve and can be functions of the
brownian motion, dz, or past forward levels, f_{i,T} for i<T. Given that an initial forward curve, f_{0,T},
is established with all traded bond prices and with the assumption that the forward
curve is static as time passes. With the
term structure models, the term structure (bond prices) is specified and the
forward rates extracted. The forward
rates are risk adjusted expectations of future instantaneous short rate defined
as _{}. Duffie (1996) and
Derman and Kani (1997) show that the price of zero coupon bonds are described
by the deterministic backward equation, given the HJM framework as

_{},

where *f*_{t,T} is the for rate for each maturity as T evolves, *B*_{t,T} is the price of the zero
coupon bond defined as ** _{}. **Or conversely,
the forward equation

_{}.

The future evolutions of all bonds across all maturities are dependent upon this initial forward curve. The assumptions about the volatility process provide solutions to the differential equation presented above for the interest rate models of Ho and Lee (1986), Cox, Ingersol and Ross (1985), Vasicek (1977) and others. Baxter and Rennie (1996), Duffie (1996), and Ritchken (1996) provide detail reviews of the relationships and derivations of these models. This framework for interest rates can be extended to volatility and is discussed in the following section.

The effective theory of volatility has the local volatility surface being a deterministic function of a spectrum of option prices and futures prices. This deterministic function remains fixed (static) as time and the underlying asset change (Derman, Kani, and Chriss, 1996 and Derman and Kani, 1997). The effective theory of volatility based on deterministic volatility functions will then incorporate a time varying or modified stochastic process and allow for inferences about the terminal distribution of the underlying asset (Derman and Kani, 94). Based on Derman and Kani (1994 and 1997), the relationship between implied volatility and deterministic volatility functions for a futures contract can be shown. Assume that the futures contract stochastic process is

_{},

where dF_{t} is the change
in price of the futures contract, m is the mean of the process, s_{F}
is the volatility, and dz is a random Brownian motion. The solution to this equation is the Black-76
under the assumption of preference free or no arbitrage reasoning, geometric
brownian motion (Lognormal terminal distribution), constant volatility, and
constant risk free rate of interest, r. The call valuation formula is

_{},

where *F* is the futures price, *K*
is the strike price, r is the risk-free rate of interest, Dt is
the time to maturity in years, and *N( * )*
is the standard normal cumulative probability function. The variables d_{1} and d_{2}
are defined as

_{}.

.

This pricing model is the solution to the following stochastic differential equation or Kolmogorov backward equation

_{};

and the forward equation becomes

_{}.

This relationship between the underlying asset and the volatility of this asset is easily seen from the Cox, Ross, and Rubinstein (1979) binomial approximation as shown in figure ?. The equal spacing between the nodes of the tree at each time step exhibit the assumption of constant volatility. The violation of the constant volatility requires adjustment of the underlying process to account for the time variation. A modified stochastic process is required to capture the skewness and kurtosis that exists in the underlying asset.

To account for a violation of constant volatility, a simple stochastic process is

_{},

where s_{F,t} is the time
varying volatility conditioned on the underlying futures price and time. This is similar to defining a stochastic
volatility model, but here the volatility process is not defined as a random
process but behaves deterministically.
As stated, this is analogous to a static forward rate curve that defines
an effective theory of interest rates shown by Heath, Jarrow, and Morton (1990)
modeling of the term structure of interest rates (Derman and Kani, 1997) and is a single
factor continuous time model.

The option pricing model with time varying volatility is the solution to the stochastic differential equation

_{},

with the forward equation being

_{}**. ** (1)

Equation (1) can be solved for the volatility parameter for a given strike, K as

_{}. (2)

This is a local variance that is an
unbiased, risk adjusted expectation of future realized variance at time T
(Derman and Kani, 1997 and Fleming, 1998)[1]. The volatility s_{F,t} is a
function of the underlying futures price, F_{t}, and time, t, and s_{K,t}
is a function of the strike and time.
Fleming (1998), using Feinstein's (1989) derivation shows a similar
result for near expiration at-the-money options as

_{}

These equations allow for static or non-random local volatility surface to define an effective theory of volatility that is similar to the effective interest rate theory, previously discussed, used in modeling bond price dynamics (Derman and Kani, 1997). A relationship between strike and time to maturity is established (Dupire, 1994) if the volatility is known.

The deterministic volatility function, s_{F,t}
or s_{K,t},
will allow for the skewness that is observed.
In the CIR binomial frame work, a skewed tree results with larger
separation at lower index levels than at higher index levels as depicted in the
following figure. The local volatility
is easily viewed as the instantaneous volatility at a particular node. In a binomial volatility tree this is the
volatility over the next transition period and is a measure of distance between
the two nodes for the up or down paths in the implied tree. The constant volatility assumption of
Black-Scholes and Black-76 has equal spacing between nodes in a binomial tree,
whereas a local volatility provides for the skewness in the volatility smile
that is shown in figure ?.

Equation (2)** **shows
the volatility process is equivalent to averaging out all sources of stochastic
volatility. The underlying futures price
uncertainty is the only source of uncertainty that remains with the
deterministic structure being a function of the underlying futures price and
time. Figure 1 shows a representation of
this type of process assuming an out-of-the-money call option exists at strike
K that expires at time T with current commodity level F[2]. The volatility is assume to be a linearly
related to the underlying commodity and time specified as

_{} (3)

The implied volatility represents the expected volatility and is the average volatility over the time to expiration resented by a double integration over the possible price paths of the shaded area in figure ? and defined as

_{}

By substituting equation (3) and integrating, the expected volatility is

_{}.

This specification shows the implied volatility is a linear function of moneyness and time with volatility increasing at one-half the commodity level and decreasing at one-half the time to expiration.

Specification of deterministic functions of volatility should have theoretical base as Ball and Roma (1994) and Taylor and Xu (1994) show. As shown, the implied volatility is a function of time and commodity level. Again, research in term structure models provides some insight as to the volatility processes in general. The HJM model in equation (3) has a general specification for the volatility term as

_{}.

This specification has the forward volatility as
function of the spot volatility, s_{0}, the spot interest rate, and a mean
reversion parameter, k. The mean
reversion model is an exponentially dampened volatility structure (Ritchken,
1996). This general volatility structure
contains several common models of forward rates based on a single factor short
rate[3]. Restricting g=0, the Vasicek (1977)
model volatility structure is obtained.
With k=0 and g=1/2,
the Cox, Ingersol, and Ross (1986) square root volatility process is found, and
if g=1,
the Black Derman, and Toy (1990) model is obtained. Another model nested within this general
structure is the Ho and Lee (1986) model that requires g=0 and k=0 and this
reduces to a constant term structure volatility similar to the constant
volatility Black and Black-Scholes.
Forward volatility is not limited to this general specification, Amin
and Ng (1997) study the linear proportional model with the volatility process
described as [s_{0}+s_{1}(T-t)]f_{t,T}.

Each specification of volatility in the HJM framework is a dependent function of time and forward level. A deterministic volatility function should have the same structure. The deterministic function developed in equation (2) remains fixed as time and the underlying asset change, resulting in lower and higher implied volatilities resulting from the price evolution (Derman and Kani, 1997 and 1999). This type framework does not rule out a multi-factor process for volatility. The assumption here is that the process for volatility is unknown or not understood, so a specification is not being imposed as in the stochastic volatility models (Chriss, 1997). The process is being derived from options prices assuming that the market is efficient.

[1]
For proof, see Derman and Kani (1997) Appendix** **A pages 39-42.

[2] This example is similar to the one in Derman, Kani, and Zou (1996).

[3] Baxter and Rennie (1996) and Ritchken (1996) provide detail discussion on term structure models and their relationships, and Amin and Ng (1997) study of time series models of implied volatility of the HJM interest model in Eurodollar futures where additional discussion of the volatility relationships is found.