Advances in Risk Management

88 AN ESSAY ON STOCHA ST IC VOLATILITY AND T HE YIELD CURVE

5.2 VARIATIONS ON STOCHASTIC VOLATILITY AND

CONDITIONAL VOLATILITY^4

It is usual to model stochastic volatility, like conditional volatility, by using
a product process. LetPbe the price of a financial instrument and let us
assume the following differential equation for the logarithm ofP:

d(log(P))=

dP

P

=μdt+σ(t)dz 1 t (5.1)

whereμis the expected yield of the financial instrument;σ, the volatility of
the yield; dt, a small time increment; and dz, a standard Wiener process.
Its discrete time approximation is the following product process:

xt=μ+σtUt (5.2)

withxt = log(Pt) andUt, a standard variable so thatE(Ut)=0 and
V(Ut)=1.
The conditional variance ofxtis equal to:

V(xt|σt)=V(μ+σtUt)=σ^2 t

σtis consequently the conditional standard deviation ofxt, or the conditional
volatility ofxt.
According to Mills (1999), a lognormal distribution is appropriate for this
conditional volatility, so:

ht=log(σt^2 )=γ 0 +γ 1 ht− 1 +ξt (5.3)

withξt∼N( 0 ,σξ^2 ). Equation (5.2) may be rewritten as:

xt=μ+Ute

ht

(^2) (5.4)
Mills (1999) assumes thatμis nil in equation (5.4) because daily and intra-
daily stocks and currencies returns have a mean equal to 0. To linearize
equation (5.4), we squarextand we take logarithms:
x^2 t=U^2 teht
log(x^2 t)=log(U^2 t)+ht (5.5)
We assume thatUt∼N(0,1), and we consequently know the distribution
of log(U^2 t). It is a logarithmicχ^2 distribution, with mean−1.27 and vari-
ance equal to 0.5π^2 , or approximately 4.93. Let us note that the distribution
of log(Ut^2 ) is very similar to the distribution of the payoffs of a short put
position. Consequently, this distribution is very appropriate for taking into
account left tail risk, a kind of risk associated with rare events like stock